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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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The last two lectures of this course will be on Ratner’s theorems on equidistribution of orbits on homogeneous spaces. Due to lack of time, I will not be able to cover all the material here that I had originally planned; in particular, for an introduction to this family of results, and its connections with number theory, I will have to refer readers to my previous blog post on these theorems. In this course, I will discuss two special cases of Ratner-type theorems. In this lecture, I will talk about Ratner-type theorems for discrete actions (of the integers {\Bbb Z}) on nilmanifolds; this case is much simpler than the general case, because there is a simple criterion in the nilmanifold case to test whether any given orbit is equidistributed or not. Ben Green and I had need recently to develop quantitative versions of such theorems for a number-theoretic application. In the next and final lecture of this course, I will discuss Ratner-type theorems for actions of SL_2({\Bbb R}), which is simpler in a different way (due to the semisimplicity of SL_2({\Bbb R}), and lack of compact factors).

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We continue our study of basic ergodic theorems, establishing the maximal and pointwise ergodic theorems of Birkhoff. Using these theorems, we can then give several equivalent notions of the fundamental concept of ergodicity, which (roughly speaking) plays the role in measure-preserving dynamics that minimality plays in topological dynamics. A general measure-preserving system is not necessarily ergodic, but we shall introduce the ergodic decomposition, which allows one to express any non-ergodic measure as an average of ergodic measures (generalising the decomposition of a permutation into disjoint cycles).

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