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One of the basic objects of study in combinatorics are finite strings {(a_n)_{n=0}^N} or infinite strings {(a_n)_{n=0}^\infty} of symbols {a_n} from some given alphabet {{\mathcal A}}, which could be either finite or infinite (but which we shall usually take to be compact). For instance, a set {A} of natural numbers can be identified with the infinite string {(1_A(n))_{n=0}^\infty} of {0}s and {1}s formed by the indicator of {A}, e.g. the even numbers can be identified with the string {1010101\ldots} from the alphabet {\{0,1\}}, the multiples of three can be identified with the string {100100100\ldots}, and so forth. One can also consider doubly infinite strings {(a_n)_{n \in {\bf Z}}}, which among other things can be used to describe arbitrary subsets of integers.

On the other hand, the basic object of study in dynamics (and in related fields, such as ergodic theory) is that of a dynamical system {(X,T)}, that is to say a space {X} together with a shift map {T: X \rightarrow X} (which is often assumed to be invertible, although one can certainly study non-invertible dynamical systems as well). One often adds additional structure to this dynamical system, such as topological structure (giving rise topological dynamics), measure-theoretic structure (giving rise to ergodic theory), complex structure (giving rise to complex dynamics), and so forth. A dynamical system gives rise to an action of the natural numbers {{\bf N}} on the space {X} by using the iterates {T^n: X \rightarrow X} of {T} for {n=0,1,2,\ldots}; if {T} is invertible, we can extend this action to an action of the integers {{\bf Z}} on the same space. One can certainly also consider dynamical systems whose underlying group (or semi-group) is something other than {{\bf N}} or {{\bf Z}} (e.g. one can consider continuous dynamical systems in which the evolution group is {{\bf R}}), but we will restrict attention to the classical situation of {{\bf N}} or {{\bf Z}} actions here.

There is a fundamental correspondence principle connecting the study of strings (or subsets of natural numbers or integers) with the study of dynamical systems. In one direction, given a dynamical system {(X,T)}, an observable {c: X \rightarrow {\mathcal A}} taking values in some alphabet {{\mathcal A}}, and some initial datum {x_0 \in X}, we can first form the forward orbit {(T^n x_0)_{n=0}^\infty} of {x_0}, and then observe this orbit using {c} to obtain an infinite string {(c(T^n x_0))_{n=0}^\infty}. If the shift {T} in this system is invertible, one can extend this infinite string into a doubly infinite string {(c(T^n x_0))_{n \in {\bf Z}}}. Thus we see that every quadruplet {(X,T,c,x_0)} consisting of a dynamical system {(X,T)}, an observable {c}, and an initial datum {x_0} creates an infinite string.

Example 1 If {X} is the three-element set {X = {\bf Z}/3{\bf Z}} with the shift map {Tx := x+1}, {c: {\bf Z}/3{\bf Z} \rightarrow \{0,1\}} is the observable that takes the value {1} at the residue class {0 \hbox{ mod } 3} and zero at the other two classes, and one starts with the initial datum {x_0 = 0 \hbox{ mod } 3}, then the observed string {(c(T^n x_0))_{n=0}^\infty} becomes the indicator {100100100\ldots} of the multiples of three.

In the converse direction, every infinite string {(a_n)_{n=0}^\infty} in some alphabet {{\mathcal A}} arises (in a decidedly non-unique fashion) from a quadruple {(X,T,c,x_0)} in the above fashion. This can be easily seen by the following “universal” construction: take {X} to be the set {X:= {\mathcal A}^{\bf N}} of infinite strings {(b_i)_{n=0}^\infty} in the alphabet {{\mathcal A}}, let {T: X \rightarrow X} be the shift map

\displaystyle  T(b_i)_{n=0}^\infty := (b_{i+1})_{n=0}^\infty,

let {c: X \rightarrow {\mathcal A}} be the observable

\displaystyle  c((b_i)_{n=0}^\infty) := b_0,

and let {x_0 \in X} be the initial point

\displaystyle  x_0 := (a_i)_{n=0}^\infty.

Then one easily sees that the observed string {(c(T^n x_0))_{n=0}^\infty} is nothing more than the original string {(a_n)_{n=0}^\infty}. Note also that this construction can easily be adapted to doubly infinite strings by using {{\mathcal A}^{\bf Z}} instead of {{\mathcal A}^{\bf N}}, at which point the shift map {T} now becomes invertible. An important variant of this construction also attaches an invariant probability measure to {X} that is associated to the limiting density of various sets associated to the string {(a_i)_{n=0}^\infty}, and leads to the Furstenberg correspondence principle, discussed for instance in these previous blog posts. Such principles allow one to rigorously pass back and forth between the combinatorics of strings and the dynamics of systems; for instance, Furstenberg famously used his correspondence principle to demonstrate the equivalence of Szemerédi’s theorem on arithmetic progressions with what is now known as the Furstenberg multiple recurrence theorem in ergodic theory.

In the case when the alphabet {{\mathcal A}} is the binary alphabet {\{0,1\}}, and (for technical reasons related to the infamous non-injectivity {0.999\ldots = 1.00\ldots} of the decimal representation system) the string {(a_n)_{n=0}^\infty} does not end with an infinite string of {1}s, then one can reformulate the above universal construction by taking {X} to be the interval {[0,1)}, {T} to be the doubling map {Tx := 2x \hbox{ mod } 1}, {c: X \rightarrow \{0,1\}} to be the observable that takes the value {1} on {[1/2,1)} and {0} on {[0,1/2)} (that is, {c(x)} is the first binary digit of {x}), and {x_0} is the real number {x_0 := \sum_{n=0}^\infty a_n 2^{-n-1}} (that is, {x_0 = 0.a_0a_1\ldots} in binary).

The above universal construction is very easy to describe, and is well suited for “generic” strings {(a_n)_{n=0}^\infty} that have no further obvious structure to them, but it often leads to dynamical systems that are much larger and more complicated than is actually needed to produce the desired string {(a_n)_{n=0}^\infty}, and also often obscures some of the key dynamical features associated to that sequence. For instance, to generate the indicator {100100100\ldots} of the multiples of three that were mentioned previously, the above universal construction requires an uncountable space {X} and a dynamics which does not obviously reflect the key features of the sequence such as its periodicity. (Using the unit interval model, the dynamics arise from the orbit of {2/7} under the doubling map, which is a rather artificial way to describe the indicator function of the multiples of three.)

A related aesthetic objection to the universal construction is that of the four components {X,T,c,x_0} of the quadruplet {(X,T,c,x_0)} used to generate the sequence {(a_n)_{n=0}^\infty}, three of the components {X,T,c} are completely universal (in that they do not depend at all on the sequence {(a_n)_{n=0}^\infty}), leaving only the initial datum {x_0} to carry all the distinctive features of the original sequence. While there is nothing wrong with this mathematically, from a conceptual point of view it would make sense to make all four components of the quadruplet to be adapted to the sequence, in order to take advantage of the accumulated intuition about various special dynamical systems (and special observables), not just special initial data.

One step in this direction can be made by restricting {X} to the orbit {\{ T^n x_0: n \in {\bf N} \}} of the initial datum {x_0} (actually for technical reasons it is better to restrict to the topological closure {\overline{\{ T^n x_0: n \in {\bf N} \}}} of this orbit, in order to keep {X} compact). For instance, starting with the sequence {100100100\ldots}, the orbit now consists of just three points {100100100\ldots}, {010010010\ldots}, {001001001\ldots}, bringing the system more in line with the example in Example 1. Technically, this is the “optimal” representation of the sequence by a quadruplet {(X,T,c,x_0)}, because any other such representation {(X',T',c',x'_0)} is a factor of this representation (in the sense that there is a unique map {\pi: X \rightarrow X'} with {T' \circ \pi = \pi \circ T}, {c' \circ \pi = c}, and {x'_0 = \pi(x_0)}). However, from a conceptual point of view this representation is still somewhat unsatisfactory, given that the elements of the system {X} are interpreted as infinite strings rather than elements of a more geometrically or algebraically rich object (e.g. points in a circle, torus, or other homogeneous space).

For general sequences {(a_n)_{n=0}^\infty}, locating relevant geometric or algebraic structure in a dynamical system generating that sequence is an important but very difficult task (see e.g. this paper of Host and Kra, which is more or less devoted to precisely this task in the context of working out what component of a dynamical system controls the multiple recurrence behaviour of that system). However, for specific examples of sequences {(a_n)_{n=0}^\infty}, one can use an informal procedure of educated guesswork in order to produce a more natural-looking quadruple {(X,T,c,x_0)} that generates that sequence. This is not a particularly difficult or deep operation, but I found it very helpful in internalising the intuition behind the correspondence principle. Being non-rigorous, this procedure does not seem to be emphasised in most presentations of the correspondence principle, so I thought I would describe it here.

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Before we begin or study of dynamical systems, topological dynamical systems, and measure-preserving systems (as defined in the previous lecture), it is convenient to give these three classes the structure of a category. One of the basic insights of category theory is that a mathematical objects in a given class (such as dynamical systems) are best studied not in isolation, but in relation to each other, via morphisms. Furthermore, many other basic concepts pertaining to these objects (e.g. subobjects, factors, direct sums, irreducibility, etc.) can be defined in terms of these morphisms. One advantage of taking this perspective here is that it provides a unified way of defining these concepts for the three different categories of dynamical systems, topological dynamical systems, and measure-preserving systems that we will study in this course, thus sparing us the need to give any of our definitions (except for our first one below) in triplicate.

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