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Note: this post is of a particularly technical nature, in particular presuming familiarity with nilsequences, nilsystems, characteristic factors, etc., and is primarily intended for experts.

As mentioned in the previous post, Ben Green, Tamar Ziegler, and myself proved the following inverse theorem for the Gowers norms:

Theorem 1 (Inverse theorem for Gowers norms) Let {N \geq 1} and {s \geq 1} be integers, and let {\delta > 0}. Suppose that {f: {\bf Z} \rightarrow [-1,1]} is a function supported on {[N] := \{1,\dots,N\}} such that

\displaystyle \frac{1}{N^{s+2}} \sum_{n,h_1,\dots,h_{s+1}} \prod_{\omega \in \{0,1\}^{s+1}} f(n+\omega_1 h_1 + \dots + \omega_{s+1} h_{s+1}) \geq \delta.

Then there exists a filtered nilmanifold {G/\Gamma} of degree {\leq s} and complexity {O_{s,\delta}(1)}, a polynomial sequence {g: {\bf Z} \rightarrow G}, and a Lipschitz function {F: G/\Gamma \rightarrow {\bf R}} of Lipschitz constant {O_{s,\delta}(1)} such that

\displaystyle \frac{1}{N} \sum_n f(n) F(g(n) \Gamma) \gg_{s,\delta} 1.

This result was conjectured earlier by Ben Green and myself; this conjecture was strongly motivated by an analogous inverse theorem in ergodic theory by Host and Kra, which we formulate here in a form designed to resemble Theorem 1 as closely as possible:

Theorem 2 (Inverse theorem for Gowers-Host-Kra seminorms) Let {s \geq 1} be an integer, and let {(X, T)} be an ergodic, countably generated measure-preserving system. Suppose that one has

\displaystyle \lim_{N \rightarrow \infty} \frac{1}{N^{s+1}} \sum_{h_1,\dots,h_{s+1} \in [N]} \int_X \prod_{\omega \in \{0,1\}^{s+1}} f(T^{\omega_1 h_1 + \dots + \omega_{s+1} h_{s+1}}x)\ d\mu(x)

\displaystyle > 0

for all non-zero {f \in L^\infty(X)} (all {L^p} spaces are real-valued in this post). Then {(X,T)} is an inverse limit (in the category of measure-preserving systems, up to almost everywhere equivalence) of ergodic degree {\leq s} nilsystems, that is to say systems of the form {(G/\Gamma, x \mapsto gx)} for some degree {\leq s} filtered nilmanifold {G/\Gamma} and a group element {g \in G} that acts ergodically on {G/\Gamma}.

It is a natural question to ask if there is any logical relationship between the two theorems. In the finite field category, one can deduce the combinatorial inverse theorem from the ergodic inverse theorem by a variant of the Furstenberg correspondence principle, as worked out by Tamar Ziegler and myself, however in the current context of {{\bf Z}}-actions, the connection is less clear.

One can split Theorem 2 into two components:

Theorem 3 (Weak inverse theorem for Gowers-Host-Kra seminorms) Let {s \geq 1} be an integer, and let {(X, T)} be an ergodic, countably generated measure-preserving system. Suppose that one has

\displaystyle \lim_{N \rightarrow \infty} \frac{1}{N^{s+1}} \sum_{h_1,\dots,h_{s+1} \in [N]} \int_X \prod_{\omega \in \{0,1\}^{s+1}} T^{\omega_1 h_1 + \dots + \omega_{s+1} h_{s+1}} f\ d\mu

\displaystyle > 0

for all non-zero {f \in L^\infty(X)}, where {T^h f := f \circ T^h}. Then {(X,T)} is a factor of an inverse limit of ergodic degree {\leq s} nilsystems.

Theorem 4 (Pro-nilsystems closed under factors) Let {s \geq 1} be an integer. Then any factor of an inverse limit of ergodic degree {\leq s} nilsystems, is again an inverse limit of ergodic degree {\leq s} nilsystems.

Indeed, it is clear that Theorem 2 implies both Theorem 3 and Theorem 4, and conversely that the two latter theorems jointly imply the former. Theorem 4 is, in principle, purely a fact about nilsystems, and should have an independent proof, but this is not known; the only known proofs go through the full machinery needed to prove Theorem 2 (or the closely related theorem of Ziegler). (However, the fact that a factor of a nilsystem is again a nilsystem was established previously by Parry.)

The purpose of this post is to record a partial implication in reverse direction to the correspondence principle:

Proposition 5 Theorem 1 implies Theorem 3.

As mentioned at the start of the post, a fair amount of familiarity with the area is presumed here, and some routine steps will be presented with only a fairly brief explanation.

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A few years ago, Ben Green, Tamar Ziegler, and myself proved the following (rather technical-looking) inverse theorem for the Gowers norms:

Theorem 1 (Discrete inverse theorem for Gowers norms) Let {N \geq 1} and {s \geq 1} be integers, and let {\delta > 0}. Suppose that {f: {\bf Z} \rightarrow [-1,1]} is a function supported on {[N] := \{1,\dots,N\}} such that

\displaystyle  \frac{1}{N^{s+2}} \sum_{n,h_1,\dots,h_{s+1}} \prod_{\omega \in \{0,1\}^{s+1}} f(n+\omega_1 h_1 + \dots + \omega_{s+1} h_{s+1}) \geq \delta.

Then there exists a filtered nilmanifold {G/\Gamma} of degree {\leq s} and complexity {O_{s,\delta}(1)}, a polynomial sequence {g: {\bf Z} \rightarrow G}, and a Lipschitz function {F: G/\Gamma \rightarrow {\bf R}} of Lipschitz constant {O_{s,\delta}(1)} such that

\displaystyle  \frac{1}{N} \sum_n f(n) F(g(n) \Gamma) \gg_{s,\delta} 1.

For the definitions of “filtered nilmanifold”, “degree”, “complexity”, and “polynomial sequence”, see the paper of Ben, Tammy, and myself. (I should caution the reader that this blog post will presume a fair amount of familiarity with this subfield of additive combinatorics.) This result has a number of applications, for instance to establishing asymptotics for linear equations in the primes, but this will not be the focus of discussion here.

The purpose of this post is to record the observation that this “discrete” inverse theorem, together with an equidistribution theorem for nilsequences that Ben and I worked out in a separate paper, implies a continuous version:

Theorem 2 (Continuous inverse theorem for Gowers norms) Let {s \geq 1} be an integer, and let {\delta>0}. Suppose that {f: {\bf R} \rightarrow [-1,1]} is a measurable function supported on {[0,1]} such that

\displaystyle  \int_{{\bf R}^{s+1}} \prod_{\omega \in \{0,1\}^{s+1}} f(t+\omega_1 h_1 + \dots + \omega_{s+1} h_{s+1})\ dt dh_1 \dots dh_{s+1} \geq \delta. \ \ \ \ \ (1)

Then there exists a filtered nilmanifold {G/\Gamma} of degree {\leq s} and complexity {O_{s,\delta}(1)}, a (smooth) polynomial sequence {g: {\bf R} \rightarrow G}, and a Lipschitz function {F: G/\Gamma \rightarrow {\bf R}} of Lipschitz constant {O_{s,\delta}(1)} such that

\displaystyle  \int_{\bf R} f(t) F(g(t) \Gamma)\ dt \gg_{s,\delta} 1.

The interval {[0,1]} can be easily replaced with any other fixed interval by a change of variables. A key point here is that the bounds are completely uniform in the choice of {f}. Note though that the coefficients of {g} can be arbitrarily large (and this is necessary, as can be seen just by considering functions of the form {f(t) = \cos( \xi t)} for some arbitrarily large frequency {\xi}).

It is likely that one could prove Theorem 2 by carefully going through the proof of Theorem 1 and replacing all instances of {{\bf Z}} with {{\bf R}} (and making appropriate modifications to the argument to accommodate this). However, the proof of Theorem 1 is quite lengthy. Here, we shall proceed by the usual limiting process of viewing the continuous interval {[0,1]} as a limit of the discrete interval {\frac{1}{N} \cdot [N]} as {N \rightarrow \infty}. However there will be some problems taking the limit due to a failure of compactness, and specifically with regards to the coefficients of the polynomial sequence {g: {\bf N} \rightarrow G} produced by Theorem 1, after normalising these coefficients by {N}. Fortunately, a factorisation theorem from a paper of Ben Green and myself resolves this problem by splitting {g} into a “smooth” part which does enjoy good compactness properties, as well as “totally equidistributed” and “periodic” parts which can be eliminated using the measurability (and thus, approximate smoothness), of {f}.

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Szemerédi’s theorem asserts that any subset of the integers of positive upper density contains arbitrarily large arithmetic progressions. Here is an equivalent quantitative form of this theorem:

Theorem 1 (Szemerédi’s theorem) Let {N} be a positive integer, and let {f: {\bf Z}/N{\bf Z} \rightarrow [0,1]} be a function with {{\bf E}_{x \in {\bf Z}/N{\bf Z}} f(x) \geq \delta} for some {\delta>0}, where we use the averaging notation {{\bf E}_{x \in A} f(x) := \frac{1}{|A|} \sum_{x \in A} f(x)}, {{\bf E}_{x,r \in A} f(x) := \frac{1}{|A|^2} \sum_{x, r \in A} f(x)}, etc.. Then for {k \geq 3} we have

\displaystyle  {\bf E}_{x,r \in {\bf Z}/N{\bf Z}} f(x) f(x+r) \dots f(x+(k-1)r) \geq c(k,\delta)

for some {c(k,\delta)>0} depending only on {k,\delta}.

The equivalence is basically thanks to an averaging argument of Varnavides; see for instance Chapter 11 of my book with Van Vu or this previous blog post for a discussion. We have removed the cases {k=1,2} as they are trivial and somewhat degenerate.

There are now many proofs of this theorem. Some time ago, I took an ergodic-theoretic proof of Furstenberg and converted it to a purely finitary proof of the theorem. The argument used some simplifying innovations that had been developed since the original work of Furstenberg (in particular, deployment of the Gowers uniformity norms, as well as a “dual” norm that I called the uniformly almost periodic norm, and an emphasis on van der Waerden’s theorem for handling the “compact extension” component of the argument). But the proof was still quite messy. However, as discussed in this previous blog post, messy finitary proofs can often be cleaned up using nonstandard analysis. Thus, there should be a nonstandard version of the Furstenberg ergodic theory argument that is relatively clean. I decided (after some encouragement from Ben Green and Isaac Goldbring) to write down most of the details of this argument in this blog post, though for sake of brevity I will skim rather quickly over arguments that were already discussed at length in other blog posts. In particular, I will presume familiarity with nonstandard analysis (in particular, the notion of a standard part of a bounded real number, and the Loeb measure construction), see for instance this previous blog post for a discussion.

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I’ve just uploaded to the arXiv my paper “Failure of the {L^1} pointwise and maximal ergodic theorems for the free group“, submitted to Forum of Mathematics, Sigma. This paper concerns a variant of the pointwise ergodic theorem of Birkhoff, which asserts that if one has a measure-preserving shift map {T: X \rightarrow X} on a probability space {X = (X,\mu)}, then for any {f \in L^1(X)}, the averages {\frac{1}{N} \sum_{n=1}^N f \circ T^{-n}} converge pointwise almost everywhere. (In the important case when the shift map {T} is ergodic, the pointwise limit is simply the mean {\int_X f\ d\mu} of the original function {f}.)

The pointwise ergodic theorem can be extended to measure-preserving actions of other amenable groups, if one uses a suitably “tempered” Folner sequence of averages; see this paper of Lindenstrauss for more details. (I also wrote up some notes on that paper here, back in 2006 before I had started this blog.) But the arguments used to handle the amenable case break down completely for non-amenable groups, and in particular for the free non-abelian group {F_2} on two generators.

Nevo and Stein studied this problem and obtained a number of pointwise ergodic theorems for {F_2}-actions {(T_g)_{g \in F_2}} on probability spaces {(X,\mu)}. For instance, for the spherical averaging operators

\displaystyle  {\mathcal A}_n f := \frac{1}{4 \times 3^{n-1}} \sum_{g \in F_2: |g| = n} f \circ T_g^{-1}

(where {|g|} denotes the length of the reduced word that forms {g}), they showed that {{\mathcal A}_{2n} f} converged pointwise almost everywhere provided that {f} was in {L^p(X)} for some {p>1}. (The need to restrict to spheres of even radius can be seen by considering the action of {F_2} on the two-element set {\{0,1\}} in which both generators of {F_2} act by interchanging the elements, in which case {{\mathcal A}_n} is determined by the parity of {n}.) This result was reproven with a different and simpler proof by Bufetov, who also managed to relax the condition {f \in L^p(X)} to the weaker condition {f \in L \log L(X)}.

The question remained open as to whether the pointwise ergodic theorem for {F_2}-actions held if one only assumed that {f} was in {L^1(X)}. Nevo and Stein were able to establish this for the Cesáro averages {\frac{1}{N} \sum_{n=1}^N {\mathcal A}_n}, but not for {{\mathcal A}_n} itself. About six years ago, Assaf Naor and I tried our hand at this problem, and was able to show an associated maximal inequality on {\ell^1(F_2)}, but due to the non-amenability of {F_2}, this inequality did not transfer to {L^1(X)} and did not have any direct impact on this question, despite a fair amount of effort on our part to attack it.

Inspired by some recent conversations with Lewis Bowen, I returned to this problem. This time around, I tried to construct a counterexample to the {L^1} pointwise ergodic theorem – something Assaf and I had not seriously attempted to do (perhaps due to being a bit too enamoured of our {\ell^1(F_2)} maximal inequality). I knew of an existing counterexample of Ornstein regarding a failure of an {L^1} ergodic theorem for iterates {P^n} of a self-adjoint Markov operator – in fact, I had written some notes on this example back in 2007. Upon revisiting my notes, I soon discovered that the Ornstein construction was adaptable to the {F_2} setting, thus settling the problem in the negative:

Theorem 1 (Failure of {L^1} pointwise ergodic theorem) There exists a measure-preserving {F_2}-action on a probability space {X} and a non-negative function {f \in L^1(X)} such that {\sup_n {\mathcal A}_{2n} f(x) = +\infty} for almost every {x}.

To describe the proof of this theorem, let me first briefly sketch the main ideas of Ornstein’s construction, which gave an example of a self-adjoint Markov operator {P} on a probability space {X} and a non-negative {f \in L^1(X)} such that {\sup_n P^n f(x) = +\infty} for almost every {x}. By some standard manipulations, it suffices to show that for any given {\alpha > 0} and {\varepsilon>0}, there exists a self-adjoint Markov operator {P} on a probability space {X} and a non-negative {f \in L^1(X)} with {\|f\|_{L^1(X)} \leq \alpha}, such that {\sup_n P^n f \geq 1-\varepsilon} on a set of measure at least {1-\varepsilon}. Actually, it will be convenient to replace the Markov chain {(P^n f)_{n \geq 0}} with an ancient Markov chain {(f_n)_{n \in {\bf Z}}} – that is to say, a sequence of non-negative functions {f_n} for both positive and negative {f}, such that {f_{n+1} = P f_n} for all {n \in {\bf Z}}. The purpose of requiring the Markov chain to be ancient (that is, to extend infinitely far back in time) is to allow for the Markov chain to be shifted arbitrarily in time, which is key to Ornstein’s construction. (Technically, Ornstein’s original argument only uses functions that go back to a large negative time, rather than being infinitely ancient, but I will gloss over this point for sake of discussion, as it turns out that the {F_2} version of the argument can be run using infinitely ancient chains.)

For any {\alpha>0}, let {P(\alpha)} denote the claim that for any {\varepsilon>0}, there exists an ancient Markov chain {(f_n)_{n \in {\bf Z}}} with {\|f_n\|_{L^1(X)} = \alpha} such that {\sup_{n \in {\bf Z}} f_n \geq 1-\varepsilon} on a set of measure at least {1-\varepsilon}. Clearly {P(1)} holds since we can just take {f_n=1} for all {n}. Our objective is to show that {P(\alpha)} holds for arbitrarily small {\alpha}. The heart of Ornstein’s argument is then the implication

\displaystyle  P(\alpha) \implies P( \alpha (1 - \frac{\alpha}{4}) ) \ \ \ \ \ (1)

for any {0 < \alpha \leq 1}, which upon iteration quickly gives the desired claim.

Let’s see informally how (1) works. By hypothesis, and ignoring epsilons, we can find an ancient Markov chain {(f_n)_{n \in {\bf Z}}} on some probability space {X} of total mass {\|f_n\|_{L^1(X)} = \alpha}, such that {\sup_n f_n} attains the value of {1} or greater almost everywhere. Assuming that the Markov process is irreducible, the {f_n} will eventually converge as {n \rightarrow \infty} to the constant value of {\|f_n\|_{L^1(X)}}, in particular its final state will essentially stay above {\alpha} (up to small errors).

Now suppose we duplicate the Markov process by replacing {X} with a double copy {X \times \{1,2\}} (giving {\{1,2\}} the uniform probability measure), and using the disjoint sum of the Markov operators on {X \times \{1\}} and {X \times \{2\}} as the propagator, so that there is no interaction between the two components of this new system. Then the functions {f'_n(x,i) := f_n(x) 1_{i=1}} form an ancient Markov chain of mass at most {\alpha/2} that lives solely in the first half {X \times \{1\}} of this copy, and {\sup_n f'_n} attains the value of {1} or greater on almost all of the first half {X \times \{1\}}, but is zero on the second half. The final state of {f'_n} will be to stay above {\alpha} in the first half {X \times \{1\}}, but be zero on the second half.

Now we modify the above example by allowing an infinitesimal amount of interaction between the two halves {X \times \{1\}}, {X \times \{2\}} of the system (I mentally think of {X \times \{1\}} and {X \times \{2\}} as two identical boxes that a particle can bounce around in, and now we wish to connect the boxes by a tiny tube). The precise way in which this interaction is inserted is not terribly important so long as the new Markov process is irreducible. Once one does so, then the ancient Markov chain {(f'_n)_{n \in {\bf Z}}} in the previous example gets replaced by a slightly different ancient Markov chain {(f''_n)_{n \in {\bf Z}}} which is more or less identical with {f'_n} for negative times {n}, or for bounded positive times {n}, but for very large values of {n} the final state is now constant across the entire state space {X \times \{1,2\}}, and will stay above {\alpha/2} on this space.

Finally, we consider an ancient Markov chain {F_n} which is basically of the form

\displaystyle  F_n(x,i) \approx f''_n(x,i) + (1 - \frac{\alpha}{2}) f_{n-M}(x) 1_{i=2}

for some large parameter {M} and for all {n \leq M} (the approximation becomes increasingly inaccurate for {n} much larger than {M}, but never mind this for now). This is basically two copies of the original Markov process in separate, barely interacting state spaces {X \times \{1\}, X \times \{2\}}, but with the second copy delayed by a large time delay {M}, and also attenuated in amplitude by a factor of {1-\frac{\alpha}{2}}. The total mass of this process is now {\frac{\alpha}{2} + \frac{\alpha}{2} (1 -\frac{\alpha}{2}) = \alpha (1 - \alpha/4)}. Because of the {f''_n} component of {F_n}, we see that {\sup_n F_n} basically attains the value of {1} or greater on the first half {X \times \{1\}}. On the second half {X \times \{2\}}, we work with times {n} close to {M}. If {M} is large enough, {f''_n} would have averaged out to about {\alpha/2} at such times, but the {(1 - \frac{\alpha}{2}) f_{n-M}(x)} component can get as large as {1-\alpha/2} here. Summing (and continuing to ignore various epsilon losses), we see that {\sup_n F_n} can get as large as {1} on almost all of the second half of {X \times \{2\}}. This concludes the rough sketch of how one establishes the implication (1).

It was observed by Bufetov that the spherical averages {{\mathcal A}_n} for a free group action can be lifted up to become powers {P^n} of a Markov operator, basically by randomly assigning a “velocity vector” {s \in \{a,b,a^{-1},b^{-1}\}} to one’s base point {x} and then applying the Markov process that moves {x} along that velocity vector (and then randomly changing the velocity vector at each time step to the “reduced word” condition that the velocity never flips from {s} to {s^{-1}}). Thus the spherical average problem has a Markov operator interpretation, which opens the door to adapting the Ornstein construction to the setting of {F_2} systems. This turns out to be doable after a certain amount of technical artifice; the main thing is to work with {F_2}-measure preserving systems that admit ancient Markov chains that are initially supported in a very small region in the “interior” of the state space, so that one can couple such systems to each other “at the boundary” in the fashion needed to establish the analogue of (1) without disrupting the ancient dynamics of such chains. The initial such system (used to establish the base case {P(1)}) comes from basically considering the action of {F_2} on a (suitably renormalised) “infinitely large ball” in the Cayley graph, after suitably gluing together the boundary of this ball to complete the action. The ancient Markov chain associated to this system starts at the centre of this infinitely large ball at infinite negative time {n=-\infty}, and only reaches the boundary of this ball at the time {n=0}.

An extremely large portion of mathematics is concerned with locating solutions to equations such as

\displaystyle  f(x) = 0

or

\displaystyle  \Phi(x) = x \ \ \ \ \ (1)

for {x} in some suitable domain space (either finite-dimensional or infinite-dimensional), and various maps {f} or {\Phi}. To solve the fixed point iteration equation (1), the simplest general method available is the fixed point iteration method: one starts with an initial approximate solution {x_0} to (1), so that {\Phi(x_0) \approx x_0}, and then recursively constructs the sequence {x_1, x_2, x_3, \dots} by {x_n := \Phi(x_{n-1})}. If {\Phi} behaves enough like a “contraction”, and the domain is complete, then one can expect the {x_n} to converge to a limit {x}, which should then be a solution to (1). For instance, if {\Phi: X \rightarrow X} is a map from a metric space {X = (X,d)} to itself, which is a contraction in the sense that

\displaystyle  d( \Phi(x), \Phi(y) ) \leq (1-\eta) d(x,y)

for all {x,y \in X} and some {\eta>0}, then with {x_n} as above we have

\displaystyle  d( x_{n+1}, x_n ) \leq (1-\eta) d(x_n, x_{n-1} )

for any {n}, and so the distances {d(x_n, x_{n-1} )} between successive elements of the sequence decay at at least a geometric rate. This leads to the contraction mapping theorem, which has many important consequences, such as the inverse function theorem and the Picard existence theorem.

A slightly more complicated instance of this strategy arises when trying to linearise a complex map {f: U \rightarrow {\bf C}} defined in a neighbourhood {U} of a fixed point. For simplicity we normalise the fixed point to be the origin, thus {0 \in U} and {f(0)=0}. When studying the complex dynamics {f^2 = f \circ f}, {f^3 = f \circ f \circ f}, {\dots} of such a map, it can be useful to try to conjugate {f} to another function {g = \psi^{-1} \circ f \circ \psi}, where {\psi} is a holomorphic function defined and invertible near {0} with {\psi(0)=0}, since the dynamics of {g} will be conjguate to that of {f}. Note that if {f(0)=0} and {f'(0)=\lambda}, then from the chain rule any conjugate {g} of {f} will also have {g(0)=0} and {g'(0)=\lambda}. Thus, the “simplest” function one can hope to conjugate {f} to is the linear function {z \mapsto \lambda z}. Let us say that {f} is linearisable (around {0}) if it is conjugate to {z \mapsto \lambda z} in some neighbourhood of {0}. Equivalently, {f} is linearisable if there is a solution to the Schröder equation

\displaystyle  f( \psi(z) ) = \psi(\lambda z) \ \ \ \ \ (2)

for some {\psi: U' \rightarrow {\bf C}} defined and invertible in a neighbourhood {U'} of {0} with {\psi(0)=0}, and all {z} sufficiently close to {0}. (The Schröder equation is normalised somewhat differently in the literature, but this form is equivalent to the usual form, at least when {\lambda} is non-zero.) Note that if {\psi} solves the above equation, then so does {z \mapsto \psi(cz)} for any non-zero {c}, so we may normalise {\psi'(0)=1} in addition to {\psi(0)=0}, which also ensures local invertibility from the inverse function theorem. (Note from winding number considerations that {\psi} cannot be invertible near zero if {\psi'(0)} vanishes.)

We have the following basic result of Koenigs:

Theorem 1 (Koenig’s linearisation theorem) Let {f: U \rightarrow {\bf C}} be a holomorphic function defined near {0} with {f(0)=0} and {f'(0)=\lambda}. If {0 < |\lambda| < 1} (attracting case) or {1 < |\lambda| < \infty} (repelling case), then {f} is linearisable near zero.

Proof: Observe that if {f, \psi, \lambda} solve (2), then {f^{-1}, \psi^{-1}, \lambda^{-1}} solve (2) also (in a sufficiently small neighbourhood of zero). Thus we may reduce to the attractive case {0 < |\lambda| < 1}.

Let {r>0} be a sufficiently small radius, and let {X} denote the space of holomorphic functions {\psi: B(0,r) \rightarrow {\bf C}} on the complex disk {B(0,r) := \{z \in {\bf C}: |z| < r \}} with {\psi(0)=0} and {\psi'(0)=1}. We can view the Schröder equation (2) as a fixed point equation

\displaystyle  \psi = \Phi(\psi)

where {\Phi: X' \rightarrow X} is the partially defined function on {X} that maps a function {\psi: B(0,r) \rightarrow {\bf C}} to the function {\Phi(\psi): B(0,r) \rightarrow {\bf C}} defined by

\displaystyle  \Phi(\psi)(z) := f^{-1}( \psi( \lambda z ) ),

assuming that {f^{-1}} is well-defined on the range of {\psi(B(0,\lambda r))} (this is why {\Phi} is only partially defined).

We can solve this equation by the fixed point iteration method, if {r} is small enough. Namely, we start with {\psi_0: B(0,r) \rightarrow {\bf C}} being the identity map, and set {\psi_1 := \Phi(\psi_0), \psi_2 := \Phi(\psi_1)}, etc. We equip {X} with the uniform metric {d( \psi, \tilde \psi ) := \sup_{z \in B(0,r)} |\psi(z) - \tilde \psi(z)|}. Observe that if {d( \psi, \psi_0 ), d(\tilde \psi, \psi_0) \leq r}, and {r} is small enough, then {\psi, \tilde \psi} takes values in {B(0,2r)}, and {\Phi(\psi), \Phi(\tilde \psi)} are well-defined and lie in {X}. Also, since {f^{-1}} is smooth and has derivative {\lambda^{-1}} at {0}, we have

\displaystyle  |f^{-1}(z) - f^{-1}(w)| \leq (1+\varepsilon) |\lambda|^{-1} |z-w|

if {z, w \in B(0,r)}, {\varepsilon>0} and {r} is sufficiently small depending on {\varepsilon}. This is not yet enough to establish the required contraction (thanks to Mario Bonk for pointing this out); but observe that the function {\frac{\psi(z)-\tilde \psi(z)}{z^2}} is holomorphic on {B(0,r)} and bounded by {d(\psi,\tilde \psi)/r^2} on the boundary of this ball (or slightly within this boundary), so by the maximum principle we see that

\displaystyle  |\frac{\psi(z)-\tilde \psi(z)}{z^2}| \leq \frac{1}{r^2} d(\psi,\tilde \psi)

on all of {B(0,r)}, and in particular

\displaystyle  |\psi(z)-\tilde \psi(z)| \leq |\lambda|^2 d(\psi,\tilde \psi)

on {B(0,\lambda r)}. Putting all this together, we see that

\displaystyle  d( \Phi(\psi), \Phi(\tilde \psi)) \leq (1+\varepsilon) |\lambda| d(\psi, \tilde \psi);

since {|\lambda|<1}, we thus obtain a contraction on the ball {\{ \psi \in X: d(\psi,\psi_0) \leq r \}} if {\varepsilon} is small enough (and {r} sufficiently small depending on {\varepsilon}). From this (and the completeness of {X}, which follows from Morera’s theorem) we see that the iteration {\psi_n} converges (exponentially fast) to a limit {\psi \in X} which is a fixed point of {\Phi}, and thus solves Schröder’s equation, as required. \Box

Koenig’s linearisation theorem leaves open the indifferent case when {|\lambda|=1}. In the rationally indifferent case when {\lambda^n=1} for some natural number {n}, there is an obvious obstruction to linearisability, namely that {f^n = 1} (in particular, linearisation is not possible in this case when {f} is a non-trivial rational function). An obstruction is also present in some irrationally indifferent cases (where {|\lambda|=1} but {\lambda^n \neq 1} for any natural number {n}), if {\lambda} is sufficiently close to various roots of unity; the first result of this form is due to Cremer, and the optimal result of this type for quadratic maps was established by Yoccoz. In the other direction, we have the following result of Siegel:

Theorem 2 (Siegel’s linearisation theorem) Let {f: U \rightarrow {\bf C}} be a holomorphic function defined near {0} with {f(0)=0} and {f'(0)=\lambda}. If {|\lambda|=1} and one has the Diophantine condition {\frac{1}{|\lambda^n-1|} \leq C n^C} for all natural numbers {n} and some constant {C>0}, then {f} is linearisable at {0}.

The Diophantine condition can be relaxed to a more general condition involving the rational exponents of the phase {\theta} of {\lambda = e^{2\pi i \theta}}; this was worked out by Brjuno, with the condition matching the one later obtained by Yoccoz. Amusingly, while the set of Diophantine numbers (and hence the set of linearisable {\lambda}) has full measure on the unit circle, the set of non-linearisable {\lambda} is generic (the complement of countably many nowhere dense sets) due to the above-mentioned work of Cremer, leading to a striking disparity between the measure-theoretic and category notions of “largeness”.

Siegel’s theorem does not seem to be provable using a fixed point iteration method. However, it can be established by modifying another basic method to solve equations, namely Newton’s method. Let us first review how this method works to solve the equation {f(x)=0} for some smooth function {f: I \rightarrow {\bf R}} defined on an interval {I}. We suppose we have some initial approximant {x_0 \in I} to this equation, with {f(x_0)} small but not necessarily zero. To make the analysis more quantitative, let us suppose that the interval {[x_0-r_0,x_0+r_0]} lies in {I} for some {r_0>0}, and we have the estimates

\displaystyle  |f(x_0)| \leq \delta_0 r_0

\displaystyle  |f'(x)| \geq \eta_0

\displaystyle  |f''(x)| \leq \frac{1}{\eta_0 r_0}

for some {\delta_0 > 0} and {0 < \eta_0 < 1/2} and all {x \in [x_0-r_0,x_0+r_0]} (the factors of {r_0} are present to make {\delta_0,\eta_0} “dimensionless”).

Lemma 3 Under the above hypotheses, we can find {x_1} with {|x_1 - x_0| \leq \eta_0 r_0} such that

\displaystyle  |f(x_1)| \ll \delta_0^2 \eta_0^{-O(1)} r_0.

In particular, setting {r_1 := (1-\eta_0) r_0}, {\eta_1 := \eta_0/2}, and {\delta_1 = O(\delta_0^2 \eta_0^{-O(1)})}, we have {[x_1-r_1,x_1+r_1] \subset [x_0-r_0,x_0+r_0] \subset I}, and

\displaystyle  |f(x_1)| \leq \delta_1 r_1

\displaystyle  |f'(x)| \geq \eta_1

\displaystyle  |f''(x)| \leq \frac{1}{\eta_1 r_1}

for all {x \in [x_1-r_1,x_1+r_1]}.

The crucial point here is that the new error {\delta_1} is roughly the square of the previous error {\delta_0}. This leads to extremely fast (double-exponential) improvement in the error upon iteration, which is more than enough to absorb the exponential losses coming from the {\eta_0^{-O(1)}} factor.

Proof: If {\delta_0 > c \eta_0^{C}} for some absolute constants {C,c>0} then we may simply take {x_0=x_1}, so we may assume that {\delta_0 \leq c \eta_0^{C}} for some small {c>0} and large {C>0}. Using the Newton approximation {f(x_0+h) \approx f(x_0) + h f'(x_0)} we are led to the choice

\displaystyle  x_1 := x_0 - \frac{f(x_0)}{f'(x_0)}

for {x_1}. From the hypotheses on {f} and the smallness hypothesis on {\delta} we certainly have {|x_1-x_0| \leq \eta_0 r_0}. From Taylor’s theorem with remainder we have

\displaystyle  f(x_1) = f(x_0) - \frac{f(x_0)}{f'(x_0)} f'(x_0) + O( \frac{1}{\eta_0 r_0} |\frac{f(x_0)}{f'(x_0)}|^2 )

\displaystyle  = O( \frac{1}{\eta_0 r_0} (\frac{\delta_0 r_0}{\eta_0})^2 )

and the claim follows. \Box

We can iterate this procedure; starting with {x_0,\eta_0,r_0,\delta_0} as above, we obtain a sequence of nested intervals {[x_n-r_n,x_n+r_n]} with {f(x_n)| \leq \delta_n}, and with {\eta_n,r_n,\delta_n,x_n} evolving by the recursive equations and estimates

\displaystyle  \eta_n = \eta_{n-1} / 2

\displaystyle  r_n = (1 - \eta_{n-1}) r_{n-1}

\displaystyle  \delta_n = O( \delta_{n-1}^2 \eta_{n-1}^{-O(1)} )

\displaystyle  |x_n - x_{n-1}| \leq \eta_{n-1} r_{n-1}.

If {\delta_0} is sufficiently small depending on {\eta_0}, we see that {\delta_n} converges rapidly to zero (indeed, we can inductively obtain a bound of the form {\delta_n \leq \eta_0^{C (2^n + n)}} for some large absolute constant {C} if {\delta_0} is small enough), and {x_n} converges to a limit {x \in I} which then solves the equation {f(x)=0} by the continuity of {f}.

As I recently learned from Zhiqiang Li, a similar scheme works to prove Siegel’s theorem, as can be found for instance in this text of Carleson and Gamelin. The key is the following analogue of Lemma 3.

Lemma 4 Let {\lambda} be a complex number with {|\lambda|=1} and {\frac{1}{|\lambda^n-1|} \ll n^{O(1)}} for all natural numbers {n}. Let {r_0>0}, and let {f_0: B(0,r_0) \rightarrow {\bf C}} be a holomorphic function with {f_0(0)=0}, {f'_0(0)=\lambda}, and

\displaystyle  |f_0(z) - \lambda z| \leq \delta_0 r_0 \ \ \ \ \ (3)

for all {z \in B(0,r_0)} and some {\delta_0>0}. Let {0 < \eta_0 \leq 1/2}, and set {r_1 := (1-\eta_0) r_0}. Then there exists an injective holomorphic function {\psi_0: B(0, r_1) \rightarrow B(0, r_0)} and a holomorphic function {f_1: B(0,r_1) \rightarrow {\bf C}} such that

\displaystyle  f_0( \psi_1(z) ) = \psi_1(f_1(z)) \ \ \ \ \ (4)

for all {z \in B(0,r_1)}, and such that

\displaystyle  |\psi_1(z) - z| \ll \delta_0 \eta_0^{-O(1)} r_1

and

\displaystyle  |f_1(z) - \lambda z| \leq \delta_1 r_1

for all {z \in B(0,r_1)} and some {\delta_1 = O(\delta_0^2 \eta_0^{-O(1)})}.

Proof: By scaling we may normalise {r_0=1}. If {\delta_0 > c \eta_0^C} for some constants {c,C>0}, then we can simply take {\psi_1} to be the identity and {f_1=f_0}, so we may assume that {\delta_0 \leq c \eta_0^C} for some small {c>0} and large {C>0}.

To motivate the choice of {\psi_1}, we write {f_0(z) = \lambda z + \hat f_0(z)} and {\psi_1(z) = z + \hat \psi(z)}, with {\hat f_0} and {\hat \psi_1} viewed as small. We would like to have {f_0(\psi_1(z)) \approx \psi_1(\lambda z)}, which expands as

\displaystyle  \lambda z + \lambda \hat \psi_1(z) + \hat f_0( z + \hat \psi_1(z) ) \approx \lambda z + \hat \psi_1(\lambda z).

As {\hat f_0} and {\hat \psi} are both small, we can heuristically approximate {\hat f_0(z + \hat \psi_1(z) ) \approx \hat f_0(z)} up to quadratic errors (compare with the Newton approximation {f(x_0+h) \approx f(x_0) + h f'(x_0)}), and arrive at the equation

\displaystyle  \hat \psi_1(\lambda z) - \lambda \hat \psi_1(z) = \hat f_0(z). \ \ \ \ \ (5)

This equation can be solved by Taylor series; the function {\hat f_0} vanishes to second order at the origin and thus has a Taylor expansion

\displaystyle  \hat f_0(z) = \sum_{n=2}^\infty a_n z^n

and then {\hat \psi_1} has a Taylor expansion

\displaystyle  \hat \psi_1(z) = \sum_{n=2}^\infty \frac{a_n}{\lambda^n - \lambda} z^n.

We take this as our definition of {\hat \psi_1}, define {\psi_1(z) := z + \hat \psi_1(z)}, and then define {f_1} implicitly via (4).

Let us now justify that this choice works. By (3) and the generalised Cauchy integral formula, we have {|a_n| \leq \delta_0} for all {n}; by the Diophantine assumption on {\lambda}, we thus have {|\frac{a_n}{\lambda^n - \lambda}| \ll \delta_0 n^{O(1)}}. In particular, {\hat \psi_1} converges on {B(0,1)}, and on the disk {B(0, (1-\eta_0/4))} (say) we have the bounds

\displaystyle  |\hat \psi_1(z)|, |\hat \psi'_1(z)| \ll \delta_0 \sum_{n=2}^\infty n^{O(1)} (1-\eta_0/4)^n \ll \eta_0^{-O(1)} \delta_0. \ \ \ \ \ (6)

In particular, as {\delta_0} is so small, we see that {\psi_1} maps {B(0, (1-\eta_0/4))} injectively to {B(0,1)} and {B(0,1-\eta_0)} to {B(0,1-3\eta_0/4)}, and the inverse {\psi_1^{-1}} maps {B(0, (1-\eta_0/2))} to {B(0, (1-\eta_0/4))}. From (3) we see that {f_0} maps {B(0,1-3\eta_0/4)} to {B(0,1-\eta_0/2)}, and so if we set {f_1: B(0,1-\eta_0) \rightarrow B(0,1-\eta_0/4)} to be the function {f_1 := \psi_1^{-1} \circ f_0 \circ \psi_1}, then {f_1} is a holomorphic function obeying (4). Expanding (4) in terms of {\hat f_0} and {\hat \psi_1} as before, and also writing {f_1(z) = \lambda z + \hat f_1(z)}, we have

\displaystyle  \lambda z + \lambda \hat \psi_1(z) + \hat f_0( z + \hat \psi_1(z) ) = \lambda z + \hat f_1(z) + \hat \psi_1(\lambda z + \hat f_1(z))

for {z \in B(0, 1-\eta_0)}, which by (5) simplifies to

\displaystyle  \hat f_1(z) = \hat f_0( z + \hat \psi_1(z) ) - \hat f_0(z) + \hat \psi_1(\lambda z) - \hat \psi_1(\lambda z + \hat f_1(z)).

From (6), the fundamental theorem of calculus, and the smallness of {\delta_0} we have

\displaystyle  |\hat \psi_1(\lambda z) - \hat \psi_1(\lambda z + \hat f_1(z))| \leq \frac{1}{2} |\hat f_1(z)|

and thus

\displaystyle  |\hat f_1(z)| \leq 2 |\hat f_0( z + \hat \psi_1(z) ) - \hat f_0(z)|.

From (3) and the Cauchy integral formula we have {\hat f'_0(z) = O( \delta_0 \eta_0^{-O(1)})} on (say) {B(0,1-\eta_0/4)}, and so from (6) and the fundamental theorem of calculus we conclude that

\displaystyle  |\hat f_1(z)| \ll \delta_0^2 \eta_0^{-O(1)}

on {B(0,1-\eta_0)}, and the claim follows. \Box

If we set {\eta_0 := 1/2}, {f_0 := f}, and {\delta_0>0} to be sufficiently small, then (since {f(z)-\lambda z} vanishes to second order at the origin), the hypotheses of this lemma will be obeyed for some sufficiently small {r_0}. Iterating the lemma (and halving {\eta_0} repeatedly), we can then find sequences {\eta_n, \delta_n, r_n > 0}, injective holomorphic functions {\psi_n: B(0,r_n) \rightarrow B(0,r_{n-1})} and holomorphic functions {f_n: B(0,r_n) \rightarrow {\bf C}} such that one has the recursive identities and estimates

\displaystyle  \eta_n = \eta_{n-1} / 2

\displaystyle  r_n = (1 - \eta_{n-1}) r_{n-1}

\displaystyle  \delta_n = O( \delta_{n-1}^2 \eta_{n-1}^{-O(1)} )

\displaystyle  |\psi_n(z) - z| \ll \delta_{n-1} \eta_{n-1}^{-O(1)} r_n

\displaystyle  |f_n(z) - \lambda z| \leq \delta_n r_n

\displaystyle  f_{n-1}( \psi_n(z) ) = \psi_n(f_n(z))

for all {n \geq 1} and {z \in B(0,r_n)}. By construction, {r_n} decreases to a positive radius {r_\infty} that is a constant multiple of {r_0}, while (for {\delta_0} small enough) {\delta_n} converges double-exponentially to zero, so in particular {f_n(z)} converges uniformly to {\lambda z} on {B(0,r_\infty)}. Also, {\psi_n} is close enough to the identity, the compositions {\Psi_n := \psi_1 \circ \dots \circ \psi_n} are uniformly convergent on {B(0,r_\infty/2)} with {\Psi_n(0)=0} and {\Psi'_n(0)=1}. From this we have

\displaystyle  f( \Psi_n(z) ) = \Psi_n(f_n(z))

on {B(0,r_\infty/4)}, and on taking limits using Morera’s theorem we obtain a holomorphic function {\Psi} defined near {0} with {\Psi(0)=0}, {\Psi'(0)=1}, and

\displaystyle  f( \Psi(z) ) = \Psi(\lambda z),

obtaining the required linearisation.

Remark 5 The idea of using a Newton-type method to obtain error terms that decay double-exponentially, and can therefore absorb exponential losses in the iteration, also occurs in KAM theory and in Nash-Moser iteration, presumably due to Siegel’s influence on Moser. (I discuss Nash-Moser iteration in this note that I wrote back in 2006.)

The von Neumann ergodic theorem (the Hilbert space version of the mean ergodic theorem) asserts that if {U: H \rightarrow H} is a unitary operator on a Hilbert space {H}, and {v \in H} is a vector in that Hilbert space, then one has

\displaystyle \lim_{N \rightarrow \infty} \frac{1}{N} \sum_{n=1}^N U^n v = \pi_{H^U} v

in the strong topology, where {H^U := \{ w \in H: Uw = w \}} is the {U}-invariant subspace of {H}, and {\pi_{H^U}} is the orthogonal projection to {H^U}. (See e.g. these previous lecture notes for a proof.) The same proof extends to more general amenable groups: if {G} is a countable amenable group acting on a Hilbert space {H} by unitary transformations {T^g: H \rightarrow H} for {g \in G}, and {v \in H} is a vector in that Hilbert space, then one has

\displaystyle \lim_{N \rightarrow \infty} \mathop{\bf E}_{g \in \Phi_N} T^g v = \pi_{H^G} v \ \ \ \ \ (1)

 

for any Folner sequence {\Phi_N} of {G}, where {H^G := \{ w \in H: T^g w = w \hbox{ for all }g \in G \}} is the {G}-invariant subspace, and {\mathop{\bf E}_{a \in A} f(a) := \frac{1}{|A|} \sum_{a \in A} f(a)} is the average of {f} on {A}. Thus one can interpret {\pi_{H^G} v} as a certain average of elements of the orbit {Gv := \{ T^g v: g \in G \}} of {v}.

In a previous blog post, I noted a variant of this ergodic theorem (due to Alaoglu and Birkhoff) that holds even when the group {G} is not amenable (or not discrete), using a more abstract notion of averaging:

Theorem 1 (Abstract ergodic theorem) Let {G} be an arbitrary group acting unitarily on a Hilbert space {H}, and let {v} be a vector in {H}. Then {\pi_{H^G} v} is the element in the closed convex hull of {Gv := \{ T^g v: g \in G \}} of minimal norm, and is also the unique element of {H^G} in this closed convex hull.

I recently stumbled upon a different way to think about this theorem, in the additive case {G = (G,+)} when {G} is abelian, which has a closer resemblance to the classical mean ergodic theorem. Given an arbitrary additive group {G = (G,+)} (not necessarily discrete, or countable), let {{\mathcal F}} denote the collection of finite non-empty multisets in {G} – that is to say, unordered collections {\{a_1,\dots,a_n\}} of elements {a_1,\dots,a_n} of {G}, not necessarily distinct, for some positive integer {n}. Given two multisets {A = \{a_1,\dots,a_n\}}, {B = \{b_1,\dots,b_m\}} in {{\mathcal F}}, we can form the sum set {A + B := \{ a_i + b_j: 1 \leq i \leq n, 1 \leq j \leq m \}}. Note that the sum set {A+B} can contain multiplicity even when {A, B} do not; for instance, {\{ 1,2\} + \{1,2\} = \{2,3,3,4\}}. Given a multiset {A = \{a_1,\dots,a_n\}} in {{\mathcal F}}, and a function {f: G \rightarrow H} from {G} to a vector space {H}, we define the average {\mathop{\bf E}_{a \in A} f(a)} as

\displaystyle \mathop{\bf E}_{a \in A} f(a) = \frac{1}{n} \sum_{j=1}^n f(a_j).

Note that the multiplicity function of the set {A} affects the average; for instance, we have {\mathop{\bf E}_{a \in \{1,2\}} a = \frac{3}{2}}, but {\mathop{\bf E}_{a \in \{1,2,2\}} a = \frac{5}{3}}.

We can define a directed set on {{\mathcal F}} as follows: given two multisets {A,B \in {\mathcal F}}, we write {A \geq B} if we have {A = B+C} for some {C \in {\mathcal F}}. Thus for instance we have {\{ 1, 2, 2, 3\} \geq \{1,2\}}. It is easy to verify that this operation is transitive and reflexive, and is directed because any two elements {A,B} of {{\mathcal F}} have a common upper bound, namely {A+B}. (This is where we need {G} to be abelian.) The notion of convergence along a net, now allows us to define the notion of convergence along {{\mathcal F}}; given a family {x_A} of points in a topological space {X} indexed by elements {A} of {{\mathcal F}}, and a point {x} in {X}, we say that {x_A} converges to {x} along {{\mathcal F}} if, for every open neighbourhood {U} of {x} in {X}, one has {x_A \in U} for sufficiently large {A}, that is to say there exists {B \in {\mathcal F}} such that {x_A \in U} for all {A \geq B}. If the topological space {V} is Hausdorff, then the limit {x} is unique (if it exists), and we then write

\displaystyle x = \lim_{A \rightarrow G} x_A.

When {x_A} takes values in the reals, one can also define the limit superior or limit inferior along such nets in the obvious fashion.

We can then give an alternate formulation of the abstract ergodic theorem in the abelian case:

Theorem 2 (Abelian abstract ergodic theorem) Let {G = (G,+)} be an arbitrary additive group acting unitarily on a Hilbert space {H}, and let {v} be a vector in {H}. Then we have

\displaystyle \pi_{H^G} v = \lim_{A \rightarrow G} \mathop{\bf E}_{a \in A} T^a v

in the strong topology of {H}.

Proof: Suppose that {A \geq B}, so that {A=B+C} for some {C \in {\mathcal F}}, then

\displaystyle \mathop{\bf E}_{a \in A} T^a v = \mathop{\bf E}_{c \in C} T^c ( \mathop{\bf E}_{b \in B} T^b v )

so by unitarity and the triangle inequality we have

\displaystyle \| \mathop{\bf E}_{a \in A} T^a v \|_H \leq \| \mathop{\bf E}_{b \in B} T^b v \|_H,

thus {\| \mathop{\bf E}_{a \in A} T^a v \|_H^2} is monotone non-increasing in {A}. Since this quantity is bounded between {0} and {\|v\|_H}, we conclude that the limit {\lim_{A \rightarrow G} \| \mathop{\bf E}_{a \in A} T^a v \|_H^2} exists. Thus, for any {\varepsilon > 0}, we have for sufficiently large {A} that

\displaystyle \| \mathop{\bf E}_{b \in B} T^b v \|_H^2 \geq \| \mathop{\bf E}_{a \in A} T^a v \|_H^2 - \varepsilon

for all {B \geq A}. In particular, for any {g \in G}, we have

\displaystyle \| \mathop{\bf E}_{b \in A + \{0,g\}} T^b v \|_H^2 \geq \| \mathop{\bf E}_{a \in A} T^a v \|_H^2 - \varepsilon.

We can write

\displaystyle \mathop{\bf E}_{b \in A + \{0,g\}} T^b v = \frac{1}{2} \mathop{\bf E}_{a \in A} T^a v + \frac{1}{2} T^g \mathop{\bf E}_{a \in A} T^a v

and so from the parallelogram law and unitarity we have

\displaystyle \| \mathop{\bf E}_{a \in A} T^a v - T^g \mathop{\bf E}_{a \in A} T^a v \|_H^2 \leq 4 \varepsilon

for all {g \in G}, and hence by the triangle inequality (averaging {g} over a finite multiset {C})

\displaystyle \| \mathop{\bf E}_{a \in A} T^a v - \mathop{\bf E}_{b \in A+C} T^b v \|_H^2 \leq 4 \varepsilon

for any {C \in {\mathcal F}}. This shows that {\mathop{\bf E}_{a \in A} T^a v} is a Cauchy sequence in {H} (in the strong topology), and hence (by the completeness of {H}) tends to a limit. Shifting {A} by a group element {g}, we have

\displaystyle \lim_{A \rightarrow G} \mathop{\bf E}_{a \in A} T^a v = \lim_{A \rightarrow G} \mathop{\bf E}_{a \in A + \{g\}} T^a v = T^g \lim_{A \rightarrow G} \mathop{\bf E}_{a \in A} T^a v

and hence {\lim_{A \rightarrow G} \mathop{\bf E}_{a \in A} T^a v} is invariant under shifts, and thus lies in {H^G}. On the other hand, for any {w \in H^G} and {A \in {\mathcal F}}, we have

\displaystyle \langle \mathop{\bf E}_{a \in A} T^a v, w \rangle_H = \mathop{\bf E}_{a \in A} \langle v, T^{-a} w \rangle_H = \langle v, w \rangle_H

and thus on taking strong limits

\displaystyle \langle \lim_{A \rightarrow G} \mathop{\bf E}_{a \in A} T^a v, w \rangle_H = \langle v, w \rangle_H

and so {v - \lim_{A \rightarrow G} \mathop{\bf E}_{a \in A} T^a v} is orthogonal to {H^G}. Combining these two facts we see that {\lim_{A \rightarrow G} \mathop{\bf E}_{a \in A} T^a v} is equal to {\pi_{H^G} v} as claimed. \Box

To relate this result to the classical ergodic theorem, we observe

Lemma 3 Let {G} be a countable additive group, with a F{\o}lner sequence {\Phi_n}, and let {f_g} be a bounded sequence in a normed vector space indexed by {G}. If {\lim_{A \rightarrow G} \mathop{\bf E}_{a \in A} f_a} exists, then {\lim_{n \rightarrow \infty} \mathop{\bf E}_{a \in \Phi_n} f_a} exists, and the two limits are equal.

Proof: From the F{\o}lner property, we see that for any {A} and any {\varepsilon>0}, the averages {\mathop{\bf E}_{a \in \Phi_n} f_a} and {\mathop{\bf E}_{a \in A+\Phi_n} f_a} differ by at most {\varepsilon} in norm if {n} is sufficiently large depending on {A}, {\varepsilon} (and the {f_a}). On the other hand, by the existence of the limit {\lim_{A \rightarrow G} \mathop{\bf E}_{a \in A} f_a}, the averages {\mathop{\bf E}_{a \in A} f_a} and {\mathop{\bf E}_{a \in A + \Phi_n} f_a} differ by at most {\varepsilon} in norm if {A} is sufficiently large depending on {\varepsilon} (regardless of how large {n} is). The claim follows. \Box

It turns out that this approach can also be used as an alternate way to construct the GowersHost-Kra seminorms in ergodic theory, which has the feature that it does not explicitly require any amenability on the group {G} (or separability on the underlying measure space), though, as pointed out to me in comments, even uncountable abelian groups are amenable in the sense of possessing an invariant mean, even if they do not have a F{\o}lner sequence.

Given an arbitrary additive group {G}, define a {G}-system {({\mathrm X}, T)} to be a probability space {{\mathrm X} = (X, {\mathcal X}, \mu)} (not necessarily separable or standard Borel), together with a collection {T^g: X \rightarrow X} of invertible, measure-preserving maps, such that {T^0} is the identity and {T^g T^h = T^{g+h}} (modulo null sets) for all {g,h \in G}. This then gives isomorphisms {T^g: L^p({\mathrm X}) \rightarrow L^p({\mathrm X})} for {1 \leq p \leq \infty} by setting {T^g f(x) := f(T^{-g} x)}. From the above abstract ergodic theorem, we see that

\displaystyle {\mathbf E}( f | {\mathcal X}^G ) = \lim_{A \rightarrow G} \mathop{\bf E}_{a \in A} T^g f

in the strong topology of {L^2({\mathrm X})} for any {f \in L^2({\mathrm X})}, where {{\mathcal X}^G} is the collection of measurable sets {E} that are essentially {G}-invariant in the sense that {T^g E = E} modulo null sets for all {g \in G}, and {{\mathbf E}(f|{\mathcal X}^G)} is the conditional expectation of {f} with respect to {{\mathcal X}^G}.

In a similar spirit, we have

Theorem 4 (Convergence of Gowers-Host-Kra seminorms) Let {({\mathrm X},T)} be a {G}-system for some additive group {G}. Let {d} be a natural number, and for every {\omega \in\{0,1\}^d}, let {f_\omega \in L^{2^d}({\mathrm X})}, which for simplicity we take to be real-valued. Then the expression

\displaystyle \langle (f_\omega)_{\omega \in \{0,1\}^d} \rangle_{U^d({\mathrm X})} := \lim_{A_1,\dots,A_d \rightarrow G}

\displaystyle \mathop{\bf E}_{h_1 \in A_1-A_1,\dots,h_d \in A_d-A_d} \int_X \prod_{\omega \in \{0,1\}^d} T^{\omega_1 h_1 + \dots + \omega_d h_d} f_\omega\ d\mu

converges, where we write {\omega = (\omega_1,\dots,\omega_d)}, and we are using the product direct set on {{\mathcal F}^d} to define the convergence {A_1,\dots,A_d \rightarrow G}. In particular, for {f \in L^{2^d}({\mathrm X})}, the limit

\displaystyle \| f \|_{U^d({\mathrm X})}^{2^d} = \lim_{A_1,\dots,A_d \rightarrow G}

\displaystyle \mathop{\bf E}_{h_1 \in A_1-A_1,\dots,h_d \in A_d-A_d} \int_X \prod_{\omega \in \{0,1\}^d} T^{\omega_1 h_1 + \dots + \omega_d h_d} f\ d\mu

converges.

We prove this theorem below the fold. It implies a number of other known descriptions of the Gowers-Host-Kra seminorms {\|f\|_{U^d({\mathrm X})}}, for instance that

\displaystyle \| f \|_{U^d({\mathrm X})}^{2^d} = \lim_{A \rightarrow G} \mathop{\bf E}_{h \in A-A} \| f T^h f \|_{U^{d-1}({\mathrm X})}^{2^{d-1}}

for {d > 1}, while from the ergodic theorem we have

\displaystyle \| f \|_{U^1({\mathrm X})} = \| {\mathbf E}( f | {\mathcal X}^G ) \|_{L^2({\mathrm X})}.

This definition also manifestly demonstrates the cube symmetries of the Host-Kra measures {\mu^{[d]}} on {X^{\{0,1\}^d}}, defined via duality by requiring that

\displaystyle \langle (f_\omega)_{\omega \in \{0,1\}^d} \rangle_{U^d({\mathrm X})} = \int_{X^{\{0,1\}^d}} \bigotimes_{\omega \in \{0,1\}^d} f_\omega\ d\mu^{[d]}.

In a subsequent blog post I hope to present a more detailed study of the {U^2} norm and its relationship with eigenfunctions and the Kronecker factor, without assuming any amenability on {G} or any separability or topological structure on {{\mathrm X}}.

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The 2014 Fields medallists have just been announced as (in alphabetical order of surname) Artur Avila, Manjul Bhargava, Martin Hairer, and Maryam Mirzakhani (see also these nice video profiles for the winners, which is a new initiative of the IMU and the Simons foundation). This time four years ago, I wrote a blog post discussing one result from each of the 2010 medallists; I thought I would try to repeat the exercise here, although the work of the medallists this time around is a little bit further away from my own direct area of expertise than last time, and so my discussion will unfortunately be a bit superficial (and possibly not completely accurate) in places. As before, I am picking these results based on my own idiosyncratic tastes, and they should not be viewed as necessarily being the “best” work of these medallists. (See also the press releases for Avila, Bhargava, Hairer, and Mirzakhani.)

Artur Avila works in dynamical systems and in the study of Schrödinger operators. The work of Avila that I am most familiar with is his solution with Svetlana Jitormiskaya of the ten martini problem of Kac, the solution to which (according to Barry Simon) he offered ten martinis for, hence the name. The problem involves perhaps the simplest example of a Schrödinger operator with non-trivial spectral properties, namely the almost Mathieu operator {H^{\lambda,\alpha}_\omega: \ell^2({\bf Z}) \rightarrow \ell^2({\bf Z})} defined for parameters {\alpha,\omega \in {\bf R}/{\bf Z}} and {\lambda>0} by a discrete one-dimensional Schrödinger operator with cosine potential:

\displaystyle (H^{\lambda,\alpha}_\omega u)_n := u_{n+1} + u_{n-1} + 2\lambda (\cos 2\pi(\theta+n\alpha)) u_n.

This is a bounded self-adjoint operator and thus has a spectrum {\sigma( H^{\lambda,\alpha}_\omega )} that is a compact subset of the real line; it arises in a number of physical contexts, most notably in the theory of the integer quantum Hall effect, though I will not discuss these applications here. Remarkably, the structure of this spectrum depends crucially on the Diophantine properties of the frequency {\alpha}. For instance, if {\alpha = p/q} is a rational number, then the operator is periodic with period {q}, and then basic (discrete) Floquet theory tells us that the spectrum is simply the union of {q} (possibly touching) intervals. But for irrational {\alpha} (in which case the spectrum is independent of the phase {\theta}), the situation is much more fractal in nature, for instance in the critical case {\lambda=1} the spectrum (as a function of {\alpha}) gives rise to the Hofstadter butterfly. The “ten martini problem” asserts that for every irrational {\alpha} and every choice of coupling constant {\lambda > 0}, the spectrum is homeomorphic to a Cantor set. Prior to the work of Avila and Jitormiskaya, there were a number of partial results on this problem, notably the result of Puig establishing Cantor spectrum for a full measure set of parameters {(\lambda,\alpha)}, as well as results requiring a perturbative hypothesis, such as {\lambda} being very small or very large. The result was also already known for {\alpha} being either very close to rational (i.e. a Liouville number) or very far from rational (a Diophantine number), although the analyses for these two cases failed to meet in the middle, leaving some cases untreated. The argument uses a wide variety of existing techniques, both perturbative and non-perturbative, to attack this problem, as well as an amusing argument by contradiction: they assume (in certain regimes) that the spectrum fails to be a Cantor set, and use this hypothesis to obtain additional Lipschitz control on the spectrum (as a function of the frequency {\alpha}), which they can then use (after much effort) to improve existing arguments and conclude that the spectrum was in fact Cantor after all!

Manjul Bhargava produces amazingly beautiful mathematics, though most of it is outside of my own area of expertise. One part of his work that touches on an area of my own interest (namely, random matrix theory) is his ongoing work with many co-authors on modeling (both conjecturally and rigorously) the statistics of various key number-theoretic features of elliptic curves (such as their rank, their Selmer group, or their Tate-Shafarevich groups). For instance, with Kane, Lenstra, Poonen, and Rains, Manjul has proposed a very general random matrix model that predicts all of these statistics (for instance, predicting that the {p}-component of the Tate-Shafarevich group is distributed like the cokernel of a certain random {p}-adic matrix, very much in the spirit of the Cohen-Lenstra heuristics discussed in this previous post). But what is even more impressive is that Manjul and his coauthors have been able to verify several non-trivial fragments of this model (e.g. showing that certain moments have the predicted asymptotics), giving for the first time non-trivial upper and lower bounds for various statistics, for instance obtaining lower bounds on how often an elliptic curve has rank {0} or rank {1}, leading most recently (in combination with existing work of Gross-Zagier and of Kolyvagin, among others) to his amazing result with Skinner and Zhang that at least {66\%} of all elliptic curves over {{\bf Q}} (ordered by height) obey the Birch and Swinnerton-Dyer conjecture. Previously it was not even known that a positive proportion of curves obeyed the conjecture. This is still a fair ways from resolving the conjecture fully (in particular, the situation with the presumably small number of curves of rank {2} and higher is still very poorly understood, and the theory of Gross-Zagier and Kolyvagin that this work relies on, which was initially only available for {{\bf Q}}, has only been extended to totally real number fields thus far, by the work of Zhang), but it certainly does provide hope that the conjecture could be within reach in a statistical sense at least.

Martin Hairer works in at the interface between probability and partial differential equations, and in particular in the theory of stochastic differential equations (SDEs). The result of his that is closest to my own interests is his remarkable demonstration with Jonathan Mattingly of unique invariant measure for the two-dimensional stochastically forced Navier-Stokes equation

\displaystyle \partial_t u + (u \cdot \nabla u) = \nu \Delta u - \nabla p + \xi

\displaystyle \nabla \cdot u = 0

on the two-torus {({\bf R}/{\bf Z})^2}, where {\xi} is a Gaussian field that forces a fixed set of frequencies. It is expected that for any reasonable choice of initial data, the solution to this equation should asymptotically be distributed according to Kolmogorov’s power law, as discussed in this previous post. This is still far from established rigorously (although there are some results in this direction for dyadic models, see e.g. this paper of Cheskidov, Shvydkoy, and Friedlander). However, Hairer and Mattingly were able to show that there was a unique probability distribution to almost every initial data would converge to asymptotically; by the ergodic theorem, this is equivalent to demonstrating the existence and uniqueness of an invariant measure for the flow. Existence can be established using standard methods, but uniqueness is much more difficult. One of the standard routes to uniqueness is to establish a “strong Feller property” that enforces some continuity on the transition operators; among other things, this would mean that two ergodic probability measures with intersecting supports would in fact have a non-trivial common component, contradicting the ergodic theorem (which forces different ergodic measures to be mutually singular). Since all ergodic measures for Navier-Stokes can be seen to contain the origin in their support, this would give uniqueness. Unfortunately, the strong Feller property is unlikely to hold in the infinite-dimensional phase space for Navier-Stokes; but Hairer and Mattingly develop a clean abstract substitute for this property, which they call the asymptotic strong Feller property, which is again a regularity property on the transition operator; this in turn is then demonstrated by a careful application of Malliavin calculus.

Maryam Mirzakhani has mostly focused on the geometry and dynamics of Teichmuller-type moduli spaces, such as the moduli space of Riemann surfaces with a fixed genus and a fixed number of cusps (or with a fixed number of boundaries that are geodesics of a prescribed length). These spaces have an incredibly rich structure, ranging from geometric structure (such as the Kahler geometry given by the Weil-Petersson metric), to dynamical structure (through the action of the mapping class group on this and related spaces), to algebraic structure (viewing these spaces as algebraic varieties), and are thus connected to many other objects of interest in geometry and dynamics. For instance, by developing a new recursive formula for the Weil-Petersson volume of this space, Mirzakhani was able to asymptotically count the number of simple prime geodesics of length up to some threshold {L} in a hyperbolic surface (or more precisely, she obtained asymptotics for the number of such geodesics in a given orbit of the mapping class group); the answer turns out to be polynomial in {L}, in contrast to the much larger class of non-simple prime geodesics, whose asymptotics are exponential in {L} (the “prime number theorem for geodesics”, developed in a classic series of works by Delsart, Huber, Selberg, and Margulis); she also used this formula to establish a new proof of a conjecture of Witten on intersection numbers that was first proven by Kontsevich. More recently, in two lengthy papers with Eskin and with Eskin-Mohammadi, Mirzakhani established rigidity theorems for the action of {SL_2({\bf R})} on such moduli spaces that are close analogues of Ratner’s celebrated rigidity theorems for unipotently generated groups (discussed in this previous blog post). Ratner’s theorems are already notoriously difficult to prove, and rely very much on the polynomial stability properties of unipotent flows; in this even more complicated setting, the unipotent flows are no longer tractable, and Mirzakhani instead uses a recent “exponential drift” method of Benoist and Quint with as a substitute. Ratner’s theorems are incredibly useful for all sorts of problems connected to homogeneous dynamics, and the analogous theorems established by Mirzakhani, Eskin, and Mohammadi have a similarly broad range of applications, for instance in counting periodic billiard trajectories in rational polygons.

As laid out in the foundational work of Kolmogorov, a classical probability space (or probability space for short) is a triplet {(X, {\mathcal X}, \mu)}, where {X} is a set, {{\mathcal X}} is a {\sigma}-algebra of subsets of {X}, and {\mu: {\mathcal X} \rightarrow [0,1]} is a countably additive probability measure on {{\mathcal X}}. Given such a space, one can form a number of interesting function spaces, including

  • the (real) Hilbert space {L^2(X, {\mathcal X}, \mu)} of square-integrable functions {f: X \rightarrow {\bf R}}, modulo {\mu}-almost everywhere equivalence, and with the positive definite inner product {\langle f, g\rangle_{L^2(X, {\mathcal X}, \mu)} := \int_X f g\ d\mu}; and
  • the unital commutative Banach algebra {L^\infty(X, {\mathcal X}, \mu)} of essentially bounded functions {f: X \rightarrow {\bf R}}, modulo {\mu}-almost everywhere equivalence, with {\|f\|_{L^\infty(X, {\mathcal X}, \mu)}} defined as the essential supremum of {|f|}.

There is also a trace {\tau = \tau_\mu: L^\infty(X, {\mathcal X}, \mu) \rightarrow {\bf C}} on {L^\infty} defined by integration: {\tau(f) := \int_X f\ d\mu}.

One can form the category {\mathbf{Prb}} of classical probability spaces, by defining a morphism {\phi: (X, {\mathcal X}, \mu) \rightarrow (Y, {\mathcal Y}, \nu)} between probability spaces to be a function {\phi: X \rightarrow Y} which is measurable (thus {\phi^{-1}(E) \in {\mathcal X}} for all {E \in {\mathcal Y}}) and measure-preserving (thus {\mu(\phi^{-1}(E)) = \nu(E)} for all {E \in {\mathcal Y}}).

Let us now abstract the algebraic features of these spaces as follows; for want of a better name, I will refer to this abstraction as an algebraic probability space, and is very similar to the non-commutative probability spaces studied in this previous post, except that these spaces are now commutative (and real).

Definition 1 An algebraic probability space is a pair {({\mathcal A}, \tau)} where

  • {{\mathcal A}} is a unital commutative real algebra;
  • {\tau: {\mathcal A} \rightarrow {\bf R}} is a homomorphism such that {\tau(1)=1} and {\tau( f^2 ) \geq 0} for all {f \in {\mathcal A}};
  • Every element {f} of {{\mathcal A}} is bounded in the sense that {\sup_{k \geq 1} \tau( f^{2k} )^{1/2k} < \infty}. (Technically, this isn’t an algebraic property, but I need it for technical reasons.)

A morphism {\phi: ({\mathcal A}_1, \tau_1) \rightarrow ({\mathcal A}_2, \tau_2)} is a homomorphism {\phi^*: {\mathcal A}_2 \rightarrow {\mathcal A}_1} which is trace-preserving, in the sense that {\tau_1(\phi^*(f)) = \tau_2(f)} for all {f \in {\mathcal A}_2}.

For want of a better name, I’ll denote the category of algebraic probability spaces as {\mathbf{AlgPrb}}. One can view this category as the opposite category to that of (a subcategory of) the category of tracial commutative real algebras. One could emphasise this opposite nature by denoting the algebraic probability space as {({\mathcal A}, \tau)^{op}} rather than {({\mathcal A},\tau)}; another suggestive (but slightly inaccurate) notation, inspired by the language of schemes, would be {\hbox{Spec}({\mathcal A},\tau)} rather than {({\mathcal A},\tau)}. However, we will not adopt these conventions here, and refer to algebraic probability spaces just by the pair {({\mathcal A},\tau)}.

By the previous discussion, we have a covariant functor {F: \textbf{Prb} \rightarrow \textbf{AlgPrb}} that takes a classical probability space {(X, {\mathcal X}, \mu)} to its algebraic counterpart {(L^\infty(X, {\mathcal X},\mu), \tau_\mu)}, with a morphism {\phi: (X, {\mathcal X}, \mu) \rightarrow (Y, {\mathcal Y}, \nu)} of classical probability spaces mapping to a morphism {F(\phi): (L^\infty(X, {\mathcal X},\mu), \tau_\mu) \rightarrow (L^\infty(Y, {\mathcal Y},\nu), \tau_\nu)} of the corresponding algebraic probability spaces by the formula

\displaystyle  F(\phi)^* f := f \circ \phi

for {f \in L^\infty(Y, {\mathcal Y}, \nu)}. One easily verifies that this is a functor.

In this post I would like to describe a functor {G: \textbf{AlgPrb} \rightarrow \textbf{Prb}} which partially inverts {F} (up to natural isomorphism), that is to say a recipe for starting with an algebraic probability space {({\mathcal A}, \tau)} and producing a classical probability space {(X, {\mathcal X}, \mu)}. This recipe is not new – it is basically the (commutative) Gelfand-Naimark-Segal construction (discussed in this previous post) combined with the Loomis-Sikorski theorem (discussed in this previous post). However, I wanted to put the construction in a single location for sake of reference. I also wanted to make the point that {F} and {G} are not complete inverses; there is a bit of information in the algebraic probability space (e.g. topological information) which is lost when passing back to the classical probability space. In some future posts, I would like to develop some ergodic theory using the algebraic foundations of probability theory rather than the classical foundations; this turns out to be convenient in the ergodic theory arising from nonstandard analysis (such as that described in this previous post), in which the groups involved are uncountable and the underlying spaces are not standard Borel spaces.

Let us describe how to construct the functor {G}, with details postponed to below the fold.

  1. Starting with an algebraic probability space {({\mathcal A}, \tau)}, form an inner product on {{\mathcal A}} by the formula {\langle f, g \rangle := \tau(fg)}, and also form the spectral radius {\rho(f) :=\lim_{k \rightarrow \infty} \tau(f^{2^k})^{1/2^k}}.
  2. The inner product is clearly positive semi-definite. Quotienting out the null vectors and taking completions, we arrive at a real Hilbert space {L^2 = L^2({\mathcal A},\tau)}, to which the trace {\tau} may be extended.
  3. Somewhat less obviously, the spectral radius is well-defined and gives a norm on {{\mathcal A}}. Taking {L^2} limits of sequences in {{\mathcal A}} of bounded spectral radius gives us a subspace {L^\infty = L^\infty({\mathcal A},\tau)} of {L^2} that has the structure of a real commutative Banach algebra.
  4. The idempotents {1_E} of the Banach algebra {L^\infty} may be indexed by elements {E} of an abstract {\sigma}-algebra {{\mathcal B}}.
  5. The Boolean algebra homomorphisms {\delta_x: {\mathcal B} \rightarrow \{0,1\}} (or equivalently, the real algebra homomorphisms {\iota_x: L^\infty \rightarrow {\bf R}}) may be indexed by elements {x} of a space {X}.
  6. Let {{\mathcal X}} denote the {\sigma}-algebra on {X} generated by the basic sets {\overline{E} := \{ x \in X: \delta_x(E) = 1 \}} for every {E \in {\mathcal B}}.
  7. Let {{\mathcal N}} be the {\sigma}-ideal of {{\mathcal X}} generated by the sets {\bigcap_n \overline{E_n}}, where {E_n \in {\mathcal B}} is a sequence with {\bigcap_n E_n = \emptyset}.
  8. One verifies that {{\mathcal B}} is isomorphic to {{\mathcal X}/{\mathcal N}}. Using this isomorphism, the trace {\tau} on {L^\infty} can be used to construct a countably additive measure {\mu} on {{\mathcal X}}. The classical probability space {(X, {\mathcal X}, \mu)} is then {G( {\mathcal A}, \tau )}, and the abstract spaces {L^2, L^\infty} may now be identified with their concrete counterparts {L^2(X, {\mathcal X}, \mu)}, {L^\infty(X, {\mathcal X}, \mu)}.
  9. Every algebraic probability space morphism {\phi: ({\mathcal A}_1,\tau_1) \rightarrow ({\mathcal A}_2,\tau_2)} generates a classical probability morphism {G(\phi): (X_1, {\mathcal X}_1, \mu_1) \rightarrow (X_2, {\mathcal X}_2, \mu_2)} via the formula

    \displaystyle  \delta_{G(\phi)(x_1)}( E_2 ) = \delta_{x_1}( \phi^*(E_2) )

    using a pullback operation {\phi^*} on the abstract {\sigma}-algebras {{\mathcal B}_1, {\mathcal B}_2} that can be defined by density.

Remark 1 The classical probability space {X} constructed by the functor {G} has some additional structure; namely {X} is a {\sigma}-Stone space (a Stone space with the property that the closure of any countable union of clopen sets is clopen), {{\mathcal X}} is the Baire {\sigma}-algebra (generated by the clopen sets), and the null sets are the meager sets. However, we will not use this additional structure here.

The partial inversion relationship between the functors {F: \textbf{Prb} \rightarrow \textbf{AlgPrb}} and {G: \textbf{AlgPrb} \rightarrow \textbf{Prb}} is given by the following assertion:

  1. There is a natural transformation from {F \circ G: \textbf{AlgPrb} \rightarrow \textbf{AlgPrb}} to the identity functor {I: \textbf{AlgPrb} \rightarrow \textbf{AlgPrb}}.

More informally: if one starts with an algebraic probability space {({\mathcal A},\tau)} and converts it back into a classical probability space {(X, {\mathcal X}, \mu)}, then there is a trace-preserving algebra homomorphism of {{\mathcal A}} to {L^\infty( X, {\mathcal X}, \mu )}, which respects morphisms of the algebraic probability space. While this relationship is far weaker than an equivalence of categories (which would require that {F \circ G} and {G \circ F} are both natural isomorphisms), it is still good enough to allow many ergodic theory problems formulated using classical probability spaces to be reformulated instead as an equivalent problem in algebraic probability spaces.

Remark 2 The opposite composition {G \circ F: \textbf{Prb} \rightarrow \textbf{Prb}} is a little odd: it takes an arbitrary probability space {(X, {\mathcal X}, \mu)} and returns a more complicated probability space {(X', {\mathcal X}', \mu')}, with {X'} being the space of homomorphisms {\iota_x: L^\infty(X, {\mathcal X}, \mu) \rightarrow {\bf R}}. while there is “morally” an embedding of {X} into {X'} using the evaluation map, this map does not exist in general because points in {X} may well have zero measure. However, if one takes a “pointless” approach and focuses just on the measure algebras {({\mathcal X}, \mu)}, {({\mathcal X}', \mu')}, then these algebras become naturally isomorphic after quotienting out by null sets.

Remark 3 An algebraic probability space captures a bit more structure than a classical probability space, because {{\mathcal A}} may be identified with a proper subset of {L^\infty} that describes the “regular” functions (or random variables) of the space. For instance, starting with the unit circle {{\bf R}/{\bf Z}} (with the usual Haar measure and the usual trace {\tau(f) = \int_{{\bf R}/{\bf Z}} f}), any unital subalgebra {{\mathcal A}} of {L^\infty({\bf R}/{\bf Z})} that is dense in {L^2({\bf R}/{\bf Z})} will generate the same classical probability space {G( {\mathcal A}, \tau )} on applying the functor {G}, namely one will get the space {({\bf R}/{\bf Z})'} of homomorphisms from {L^\infty({\bf R}/{\bf Z})} to {{\bf R}} (with the measure induced from {\tau}). Thus for instance {{\mathcal A}} could be the continuous functions {C( {\bf R}/{\bf Z} )}, the Wiener algebra {A({\bf R}/{\bf Z})} or the full space {L^\infty({\bf R}/{\bf Z})}, but the classical space {G( {\mathcal A}, \tau )} will be unable to distinguish these spaces from each other. In particular, the functor {F \circ G} loses information (roughly speaking, this functor takes an algebraic probability space and completes it to a von Neumann algebra, but then forgets exactly what algebra was initially used to create this completion). In ergodic theory, this sort of “extra structure” is traditionally encoded in topological terms, by assuming that the underlying probability space {X} has a nice topological structure (e.g. a standard Borel space); however, with the algebraic perspective one has the freedom to have non-topological notions of extra structure, by choosing {{\mathcal A}} to be something other than an algebra {C(X)} of continuous functions on a topological space. I hope to discuss one such example of extra structure (coming from the Gowers-Host-Kra theory of uniformity seminorms) in a later blog post (this generalises the example of the Wiener algebra given previously, which is encoding “Fourier structure”).

A small example of how one could use the functors {F, G} is as follows. Suppose one has a classical probability space {(X, {\mathcal X}, \mu)} with a measure-preserving action of an uncountable group {\Gamma}, which is only defined (and an action) up to almost everywhere equivalence; thus for instance for any set {E} and any {g, h \in \Gamma}, {T^{gh} E} and {T^g T^h E} might not be exactly equal, but only equal up to a null set. For similar reasons, an element {E} of the invariant factor {{\mathcal X}^\Gamma} might not be exactly invariant with respect to {\Gamma}, but instead one only has {T^g E} and {E} equal up to null sets for each {g \in \Gamma}. One might like to “clean up” the action of {\Gamma} to make it defined everywhere, and a genuine action everywhere, but this is not immediately achievable if {\Gamma} is uncountable, since the union of all the null sets where something bad occurs may cease to be a null set. However, by applying the functor {F}, each shift {T^g: X \rightarrow X} defines a morphism {T^g: L^\infty(X, {\mathcal X}, \mu) \rightarrow L^\infty(X, {\mathcal X}, \mu)} on the associated algebraic probability space (i.e. the Koopman operator), and then applying {G}, we obtain a shift {T^g: X' \rightarrow X'} on a new classical probability space {(X', {\mathcal X}', \mu')} which now gives a genuine measure-preserving action of {\Gamma}, and which is equivalent to the original action from a measure algebra standpoint. The invariant factor {{\mathcal X}^\Gamma} now consists of those sets in {{\mathcal X}'} which are genuinely {\Gamma}-invariant, not just up to null sets. (Basically, the classical probability space {(X', {\mathcal X}', \mu')} contains a Boolean algebra {\overline{\mathcal B}} with the property that every measurable set {A \in {\mathcal X}'} is equivalent up to null sets to precisely one set in {\overline{\mathcal B}}, allowing for a canonical “retraction” onto {\overline{\mathcal B}} that eliminates all null set issues.)

More indirectly, the functors {F, G} suggest that one should be able to develop a “pointless” form of ergodic theory, in which the underlying probability spaces are given algebraically rather than classically. I hope to give some more specific examples of this in later posts.

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There are a number of ways to construct the real numbers {{\bf R}}, for instance

  • as the metric completion of {{\bf Q}} (thus, {{\bf R}} is defined as the set of Cauchy sequences of rationals, modulo Cauchy equivalence);
  • as the space of Dedekind cuts on the rationals {{\bf Q}};
  • as the space of quasimorphisms {\phi: {\bf Z} \rightarrow {\bf Z}} on the integers, quotiented by bounded functions. (I believe this construction first appears in this paper of Street, who credits the idea to Schanuel, though the germ of this construction arguably goes all the way back to Eudoxus.)

There is also a fourth family of constructions that proceeds via nonstandard analysis, as a special case of what is known as the nonstandard hull construction. (Here I will assume some basic familiarity with nonstandard analysis and ultraproducts, as covered for instance in this previous blog post.) Given an unbounded nonstandard natural number {N \in {}^* {\bf N} \backslash {\bf N}}, one can define two external additive subgroups of the nonstandard integers {{}^* {\bf Z}}:

  • The group {O(N) := \{ n \in {}^* {\bf Z}: |n| \leq CN \hbox{ for some } C \in {\bf N} \}} of all nonstandard integers of magnitude less than or comparable to {N}; and
  • The group {o(N) := \{ n \in {}^* {\bf Z}: |n| \leq C^{-1} N \hbox{ for all } C \in {\bf N} \}} of nonstandard integers of magnitude infinitesimally smaller than {N}.

The group {o(N)} is a subgroup of {O(N)}, so we may form the quotient group {O(N)/o(N)}. This space is isomorphic to the reals {{\bf R}}, and can in fact be used to construct the reals:

Proposition 1 For any coset {n + o(N)} of {O(N)/o(N)}, there is a unique real number {\hbox{st} \frac{n}{N}} with the property that {\frac{n}{N} = \hbox{st} \frac{n}{N} + o(1)}. The map {n + o(N) \mapsto \hbox{st} \frac{n}{N}} is then an isomorphism between the additive groups {O(N)/o(N)} and {{\bf R}}.

Proof: Uniqueness is clear. For existence, observe that the set {\{ x \in {\bf R}: Nx \leq n + o(N) \}} is a Dedekind cut, and its supremum can be verified to have the required properties for {\hbox{st} \frac{n}{N}}. \Box

In a similar vein, we can view the unit interval {[0,1]} in the reals as the quotient

\displaystyle  [0,1] \equiv [N] / o(N) \ \ \ \ \ (1)

where {[N]} is the nonstandard (i.e. internal) set {\{ n \in {\bf N}: n \leq N \}}; of course, {[N]} is not a group, so one should interpret {[N]/o(N)} as the image of {[N]} under the quotient map {{}^* {\bf Z} \rightarrow {}^* {\bf Z} / o(N)} (or {O(N) \rightarrow O(N)/o(N)}, if one prefers). Or to put it another way, (1) asserts that {[0,1]} is the image of {[N]} with respect to the map {\pi: n \mapsto \hbox{st} \frac{n}{N}}.

In this post I would like to record a nice measure-theoretic version of the equivalence (1), which essentially appears already in standard texts on Loeb measure (see e.g. this text of Cutland). To describe the results, we must first quickly recall the construction of Loeb measure on {[N]}. Given an internal subset {A} of {[N]}, we may define the elementary measure {\mu_0(A)} of {A} by the formula

\displaystyle  \mu_0(A) := \hbox{st} \frac{|A|}{N}.

This is a finitely additive probability measure on the Boolean algebra of internal subsets of {[N]}. We can then construct the Loeb outer measure {\mu^*(A)} of any subset {A \subset [N]} in complete analogy with Lebesgue outer measure by the formula

\displaystyle  \mu^*(A) := \inf \sum_{n=1}^\infty \mu_0(A_n)

where {(A_n)_{n=1}^\infty} ranges over all sequences of internal subsets of {[N]} that cover {A}. We say that a subset {A} of {[N]} is Loeb measurable if, for any (standard) {\epsilon>0}, one can find an internal subset {B} of {[N]} which differs from {A} by a set of Loeb outer measure at most {\epsilon}, and in that case we define the Loeb measure {\mu(A)} of {A} to be {\mu^*(A)}. It is a routine matter to show (e.g. using the Carathéodory extension theorem) that the space {{\mathcal L}} of Loeb measurable sets is a {\sigma}-algebra, and that {\mu} is a countably additive probability measure on this space that extends the elementary measure {\mu_0}. Thus {[N]} now has the structure of a probability space {([N], {\mathcal L}, \mu)}.

Now, the group {o(N)} acts (Loeb-almost everywhere) on the probability space {[N]} by the addition map, thus {T^h n := n+h} for {n \in [N]} and {h \in o(N)} (excluding a set of Loeb measure zero where {n+h} exits {[N]}). This action is clearly seen to be measure-preserving. As such, we can form the invariant factor {Z^0_{o(N)}([N]) = ([N], {\mathcal L}^{o(N)}, \mu\downharpoonright_{{\mathcal L}^{o(N)}})}, defined by restricting attention to those Loeb measurable sets {A \subset [N]} with the property that {T^h A} is equal {\mu}-almost everywhere to {A} for each {h \in o(N)}.

The claim is then that this invariant factor is equivalent (up to almost everywhere equivalence) to the unit interval {[0,1]} with Lebesgue measure {m} (and the trivial action of {o(N)}), by the same factor map {\pi: n \mapsto \hbox{st} \frac{n}{N}} used in (1). More precisely:

Theorem 2 Given a set {A \in {\mathcal L}^{o(N)}}, there exists a Lebesgue measurable set {B \subset [0,1]}, unique up to {m}-a.e. equivalence, such that {A} is {\mu}-a.e. equivalent to the set {\pi^{-1}(B) := \{ n \in [N]: \hbox{st} \frac{n}{N} \in B \}}. Conversely, if {B \in [0,1]} is Lebesgue measurable, then {\pi^{-1}(B)} is in {{\mathcal L}^{o(N)}}, and {\mu( \pi^{-1}(B) ) = m( B )}.

More informally, we have the measure-theoretic version

\displaystyle  [0,1] \equiv Z^0_{o(N)}( [N] )

of (1).

Proof: We first prove the converse. It is clear that {\pi^{-1}(B)} is {o(N)}-invariant, so it suffices to show that {\pi^{-1}(B)} is Loeb measurable with Loeb measure {m(B)}. This is easily verified when {B} is an elementary set (a finite union of intervals). By countable subadditivity of outer measure, this implies that Loeb outer measure of {\pi^{-1}(E)} is bounded by the Lebesgue outer measure of {E} for any set {E \subset [0,1]}; since every Lebesgue measurable set differs from an elementary set by a set of arbitrarily small Lebesgue outer measure, the claim follows.

Now we establish the forward claim. Uniqueness is clear from the converse claim, so it suffices to show existence. Let {A \in {\mathcal L}^{o(N)}}. Let {\epsilon>0} be an arbitrary standard real number, then we can find an internal set {A_\epsilon \subset [N]} which differs from {A} by a set of Loeb measure at most {\epsilon}. As {A} is {o(N)}-invariant, we conclude that for every {h \in o(N)}, {A_\epsilon} and {T^h A_\epsilon} differ by a set of Loeb measure (and hence elementary measure) at most {2\epsilon}. By the (contrapositive of the) underspill principle, there must exist a standard {\delta>0} such that {A_\epsilon} and {T^h A_\epsilon} differ by a set of elementary measure at most {2\epsilon} for all {|h| \leq \delta N}. If we then define the nonstandard function {f_\epsilon: [N] \rightarrow {}^* {\bf R}} by the formula

\displaystyle  f(n) := \hbox{st} \frac{1}{\delta N} \sum_{m \in [N]: m \leq n \leq m+\delta N} 1_{A_\epsilon}(m),

then from the (nonstandard) triangle inequality we have

\displaystyle  \frac{1}{N} \sum_{n \in [N]} |f(n) - 1_{A_\epsilon}(n)| \leq 3\epsilon

(say). On the other hand, {f} has the Lipschitz continuity property

\displaystyle  |f(n)-f(m)| \leq \frac{2|n-m|}{\delta N}

and so in particular we see that

\displaystyle  \hbox{st} f(n) = \tilde f( \hbox{st} \frac{n}{N} )

for some Lipschitz continuous function {\tilde f: [0,1] \rightarrow [0,1]}. If we then let {E_\epsilon} be the set where {\tilde f \geq 1 - \sqrt{\epsilon}}, one can check that {A_\epsilon} differs from {\pi^{-1}(E_\epsilon)} by a set of Loeb outer measure {O(\sqrt{\epsilon})}, and hence {A} does so also. Sending {\epsilon} to zero, we see (from the converse claim) that {1_{E_\epsilon}} is a Cauchy sequence in {L^1} and thus converges in {L^1} for some Lebesgue measurable {E}. The sets {A_\epsilon} then converge in Loeb outer measure to {\pi^{-1}(E)}, giving the claim. \Box

Thanks to the Lebesgue differentiation theorem, the conditional expectation {{\bf E}( f | Z^0_{o(N)}([N]))} of a bounded Loeb-measurable function {f: [N] \rightarrow {\bf R}} can be expressed (as a function on {[0,1]}, defined {m}-a.e.) as

\displaystyle  {\bf E}( f | Z^0_{o(N)}([N]))(x) := \lim_{\epsilon \rightarrow 0} \frac{1}{2\epsilon} \int_{[x-\epsilon N,x+\epsilon N]} f\ d\mu.

By the abstract ergodic theorem from the previous post, one can also view this conditional expectation as the element in the closed convex hull of the shifts {T^h f}, {h = o(N)} of minimal {L^2} norm. In particular, we obtain a form of the von Neumann ergodic theorem in this context: the averages {\frac{1}{H} \sum_{h=1}^H T^h f} for {H=O(N)} converge (as a net, rather than a sequence) in {L^2} to {{\bf E}( f | Z^0_{o(N)}([N]))}.

If {f: [N] \rightarrow [-1,1]} is (the standard part of) an internal function, that is to say the ultralimit of a sequence {f_n: [N_n] \rightarrow [-1,1]} of finitary bounded functions, one can view the measurable function {F := {\bf E}( f | Z^0_{o(N)}([N]))} as a limit of the {f_n} that is analogous to the “graphons” that emerge as limits of graphs (see e.g. the recent text of Lovasz on graph limits). Indeed, the measurable function {F: [0,1] \rightarrow [-1,1]} is related to the discrete functions {f_n: [N_n] \rightarrow [-1,1]} by the formula

\displaystyle  \int_a^b F(x)\ dx = \hbox{st} \lim_{n \rightarrow p} \frac{1}{N_n} \sum_{a N_n \leq m \leq b N_n} f_n(m)

for all {0 \leq a < b \leq 1}, where {p} is the nonprincipal ultrafilter used to define the nonstandard universe. In particular, from the Arzela-Ascoli diagonalisation argument there is a subsequence {n_j} such that

\displaystyle  \int_a^b F(x)\ dx = \lim_{j \rightarrow \infty} \frac{1}{N_{n_j}} \sum_{a N_{n_j} \leq m \leq b N_{n_j}} f_n(m),

thus {F} is the asymptotic density function of the {f_n}. For instance, if {f_n} is the indicator function of a randomly chosen subset of {[N_n]}, then the asymptotic density function would equal {1/2} (almost everywhere, at least).

I’m continuing to look into understanding the ergodic theory of {o(N)} actions, as I believe this may allow one to apply ergodic theory methods to the “single-scale” or “non-asymptotic” setting (in which one averages only over scales comparable to a large parameter {N}, rather than the traditional asymptotic approach of letting the scale go to infinity). I’m planning some further posts in this direction, though this is still a work in progress.

The von Neumann ergodic theorem (the Hilbert space version of the mean ergodic theorem) asserts that if {U: H \rightarrow H} is a unitary operator on a Hilbert space {H}, and {v \in H} is a vector in that Hilbert space, then one has

\displaystyle  \lim_{N \rightarrow \infty} \frac{1}{N} \sum_{n=1}^N U^n v = \pi_{H^U} v

in the strong topology, where {H^U := \{ w \in H: Uw = w \}} is the {U}-invariant subspace of {H}, and {\pi_{H^U}} is the orthogonal projection to {H^U}. (See e.g. these previous lecture notes for a proof.) The same proof extends to more general amenable groups: if {G} is a countable amenable group acting on a Hilbert space {H} by unitary transformations {g: H \rightarrow H}, and {v \in H} is a vector in that Hilbert space, then one has

\displaystyle  \lim_{N \rightarrow \infty} \frac{1}{|\Phi_N|} \sum_{g \in \Phi_N} gv = \pi_{H^G} v \ \ \ \ \ (1)

for any Folner sequence {\Phi_N} of {G}, where {H^G := \{ w \in H: gw = w \hbox{ for all }g \in G \}} is the {G}-invariant subspace. Thus one can interpret {\pi_{H^G} v} as a certain average of elements of the orbit {Gv := \{ gv: g \in G \}} of {v}.

I recently discovered that there is a simple variant of this ergodic theorem that holds even when the group {G} is not amenable (or not discrete), using a more abstract notion of averaging:

Theorem 1 (Abstract ergodic theorem) Let {G} be an arbitrary group acting unitarily on a Hilbert space {H}, and let {v} be a vector in {H}. Then {\pi_{H^G} v} is the element in the closed convex hull of {Gv := \{ gv: g \in G \}} of minimal norm, and is also the unique element of {H^G} in this closed convex hull.

Proof: As the closed convex hull of {Gv} is closed, convex, and non-empty in a Hilbert space, it is a classical fact (see e.g. Proposition 1 of this previous post) that it has a unique element {F} of minimal norm. If {T_g F \neq F} for some {g}, then the midpoint of {T_g F} and {F} would be in the closed convex hull and be of smaller norm, a contradiction; thus {F} is {G}-invariant. To finish the first claim, it suffices to show that {v-F} is orthogonal to every element {h} of {H^G}. But if this were not the case for some such {h}, we would have {\langle T_g v - F, h \rangle = \langle v-F,h\rangle \neq 0} for all {g \in G}, and thus on taking convex hulls {\langle F-F,h\rangle = \langle f-F,f\rangle \neq 0}, a contradiction.

Finally, since {T_g v - F} is orthogonal to {H^G}, the same is true for {F'-F} for any {F'} in the closed convex hull of {Gv}, and this gives the second claim. \Box

This result is due to Alaoglu and Birkhoff. It implies the amenable ergodic theorem (1); indeed, given any {\epsilon>0}, Theorem 1 implies that there is a finite convex combination {v_\epsilon} of shifts {gv} of {v} which lies within {\epsilon} (in the {H} norm) to {\pi_{H^G} v}. By the triangle inequality, all the averages {\frac{1}{|\Phi_N|} \sum_{g \in \Phi_N} gv_\epsilon} also lie within {\epsilon} of {\pi_{H^G} v}, but by the Folner property this implies that the averages {\frac{1}{|\Phi_N|} \sum_{g \in \Phi_N} gv} are eventually within {2\epsilon} (say) of {\pi_{H^G} v}, giving the claim.

It turns out to be possible to use Theorem 1 as a substitute for the mean ergodic theorem in a number of contexts, thus removing the need for an amenability hypothesis. Here is a basic application:

Corollary 2 (Relative orthogonality) Let {G} be a group acting unitarily on a Hilbert space {H}, and let {V} be a {G}-invariant subspace of {H}. Then {V} and {H^G} are relatively orthogonal over their common subspace {V^G}, that is to say the restrictions of {V} and {H^G} to the orthogonal complement of {V^G} are orthogonal to each other.

Proof: By Theorem 1, we have {\pi_{H^G} v = \pi_{V^G} v} for all {v \in V}, and the claim follows. (Thanks to Gergely Harcos for this short argument.) \Box

Now we give a more advanced application of Theorem 1, to establish some “Mackey theory” over arbitrary groups {G}. Define a {G}-system {(X, {\mathcal X}, \mu, (T_g)_{g \in G})} to be a probability space {X = (X, {\mathcal X}, \mu)} together with a measure-preserving action {(T_g)_{g \in G}} of {G} on {X}; this gives an action of {G} on {L^2(X) = L^2(X,{\mathcal X},\mu)}, which by abuse of notation we also call {T_g}:

\displaystyle  T_g f := f \circ T_{g^{-1}}.

(In this post we follow the usual convention of defining the {L^p} spaces by quotienting out by almost everywhere equivalence.) We say that a {G}-system is ergodic if {L^2(X)^G} consists only of the constants.

(A technical point: the theory becomes slightly cleaner if we interpret our measure spaces abstractly (or “pointlessly“), removing the underlying space {X} and quotienting {{\mathcal X}} by the {\sigma}-ideal of null sets, and considering maps such as {T_g} only on this quotient {\sigma}-algebra (or on the associated von Neumann algebra {L^\infty(X)} or Hilbert space {L^2(X)}). However, we will stick with the more traditional setting of classical probability spaces here to keep the notation familiar, but with the understanding that many of the statements below should be understood modulo null sets.)

A factor {Y = (Y, {\mathcal Y}, \nu, (S_g)_{g \in G})} of a {G}-system {X = (X,{\mathcal X},\mu, (T_g)_{g \in G})} is another {G}-system together with a factor map {\pi: X \rightarrow Y} which commutes with the {G}-action (thus {T_g \pi = \pi S_g} for all {g \in G}) and respects the measure in the sense that {\mu(\pi^{-1}(E)) = \nu(E)} for all {E \in {\mathcal Y}}. For instance, the {G}-invariant factor {Z^0_G(X) := (X, {\mathcal X}^G, \mu\downharpoonright_{{\mathcal X}^G}, (T_g)_{g \in G})}, formed by restricting {X} to the invariant algebra {{\mathcal X}^G := \{ E \in {\mathcal X}: T_g E = E \hbox{ a.e. for all } g \in G \}}, is a factor of {X}. (This factor is the first factor in an important hierachy, the next element of which is the Kronecker factor {Z^1_G(X)}, but we will not discuss higher elements of this hierarchy further here.) If {Y} is a factor of {X}, we refer to {X} as an extension of {Y}.

From Corollary 2 we have

Corollary 3 (Relative independence) Let {X} be a {G}-system for a group {G}, and let {Y} be a factor of {X}. Then {Y} and {Z^0_G(X)} are relatively independent over their common factor {Z^0_G(Y)}, in the sense that the spaces {L^2(Y)} and {L^2(Z^0_G(X))} are relatively orthogonal over {L^2(Z^0_G(Y))} when all these spaces are embedded into {L^2(X)}.

This has a simple consequence regarding the product {X \times Y = (X \times Y, {\mathcal X} \times {\mathcal Y}, \mu \times \nu, (T_g \oplus S_g)_{g \in G})} of two {G}-systems {X = (X, {\mathcal X}, \mu, (T_g)_{g \in G})} and {Y = (Y, {\mathcal Y}, \nu, (S_g)_{g \in G})}, in the case when the {Y} action is trivial:

Lemma 4 If {X,Y} are two {G}-systems, with the action of {G} on {Y} trivial, then {Z^0_G(X \times Y)} is isomorphic to {Z^0_G(X) \times Y} in the obvious fashion.

This lemma is immediate for countable {G}, since for a {G}-invariant function {f}, one can ensure that {T_g f = f} holds simultaneously for all {g \in G} outside of a null set, but is a little trickier for uncountable {G}.

Proof: It is clear that {Z^0_G(X) \times Y} is a factor of {Z^0_G(X \times Y)}. To obtain the reverse inclusion, suppose that it fails, thus there is a non-zero {f \in L^2(Z^0_G(X \times Y))} which is orthogonal to {L^2(Z^0_G(X) \times Y)}. In particular, we have {fg} orthogonal to {L^2(Z^0_G(X))} for any {g \in L^\infty(Y)}. Since {fg} lies in {L^2(Z^0_G(X \times Y))}, we conclude from Corollary 3 (viewing {X} as a factor of {X \times Y}) that {fg} is also orthogonal to {L^2(X)}. Since {g} is an arbitrary element of {L^\infty(Y)}, we conclude that {f} is orthogonal to {L^2(X \times Y)} and in particular is orthogonal to itself, a contradiction. (Thanks to Gergely Harcos for this argument.) \Box

Now we discuss the notion of a group extension.

Definition 5 (Group extension) Let {G} be an arbitrary group, let {Y = (Y, {\mathcal Y}, \nu, (S_g)_{g \in G})} be a {G}-system, and let {K} be a compact metrisable group. A {K}-extension of {Y} is an extension {X = (X, {\mathcal X}, \mu, (T_g)_{g \in G})} whose underlying space is {X = Y \times K} (with {{\mathcal X}} the product of {{\mathcal Y}} and the Borel {\sigma}-algebra on {K}), the factor map is {\pi: (y,k) \mapsto y}, and the shift maps {T_g} are given by

\displaystyle  T_g ( y, k ) = (S_g y, \rho_g(y) k )

where for each {g \in G}, {\rho_g: Y \rightarrow K} is a measurable map (known as the cocycle associated to the {K}-extension {X}).

An important special case of a {K}-extension arises when the measure {\mu} is the product of {\nu} with the Haar measure {dk} on {K}. In this case, {X} also has a {K}-action {k': (y,k) \mapsto (y,k(k')^{-1})} that commutes with the {G}-action, making {X} a {G \times K}-system. More generally, {\mu} could be the product of {\nu} with the Haar measure {dh} of some closed subgroup {H} of {K}, with {\rho_g} taking values in {H}; then {X} is now a {G \times H} system. In this latter case we will call {X} {H}-uniform.

If {X} is a {K}-extension of {Y} and {U: Y \rightarrow K} is a measurable map, we can define the gauge transform {X_U} of {X} to be the {K}-extension of {Y} whose measure {\mu_U} is the pushforward of {\mu} under the map {(y,k) \mapsto (y, U(y) k)}, and whose cocycles {\rho_{g,U}: Y \rightarrow K} are given by the formula

\displaystyle  \rho_{g,U}(y) := U(gy) \rho_g(y) U(y)^{-1}.

It is easy to see that {X_U} is a {K}-extension that is isomorphic to {X} as a {K}-extension of {Y}; we will refer to {X_U} and {X} as equivalent systems, and {\rho_{g,U}} as cohomologous to {\rho_g}. We then have the following fundamental result of Mackey and of Zimmer:

Theorem 6 (Mackey-Zimmer theorem) Let {G} be an arbitrary group, let {Y} be an ergodic {G}-system, and let {K} be a compact metrisable group. Then every ergodic {K}-extension {X} of {Y} is equivalent to an {H}-uniform extension of {Y} for some closed subgroup {H} of {K}.

This theorem is usually stated for amenable groups {G}, but by using Theorem 1 (or more precisely, Corollary 3) the result is in fact also valid for arbitrary groups; we give the proof below the fold. (In the usual formulations of the theorem, {X} and {Y} are also required to be Lebesgue spaces, or at least standard Borel, but again with our abstract approach here, such hypotheses will be unnecessary.) Among other things, this theorem plays an important role in the Furstenberg-Zimmer structural theory of measure-preserving systems (as well as subsequent refinements of this theory by Host and Kra); see this previous blog post for some relevant discussion. One can obtain similar descriptions of non-ergodic extensions via the ergodic decomposition, but the result becomes more complicated to state, and we will not do so here.

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