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.

Informally, a morphism between two objects in a class is any map which respects all the structures of that class. For the three categories we are interested in, the formal definition is as follows.

Definition 1. (Morphisms)

  1. A morphism \phi: (X,T) \to (Y,S) between two dynamical systems is a map \phi: X \to Y which intertwines T and S in the sense that S \circ \phi = \phi \circ T.
  2. A morphism \phi: (X, {\mathcal F}, T) \to (Y, {\mathcal G}, S) between two topological dynamical systems is a morphism \phi: (X,T) \to (Y,S) of dynamical systems which is also continuous, thus \phi^{-1}(U) \in {\mathcal F} for all U \in {\mathcal G}.
  3. A morphism \phi: (X, {\mathcal X},\mu, T) \to (Y, {\mathcal Y}, \nu, S) between two measure-preserving systems is a morphism \phi: (X,T) \to (Y,S) of dynamical systems which is also 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}). Equivalently, \nu = \phi_*(\mu) is the push-forward of \mu by \phi.

When it is clear what category we are working in, and what the shifts are, we shall often refer to a system by its underlying space, thus for instance a morphism \phi: (X,{\mathcal X},\mu,T) \to (Y,{\mathcal Y},\nu,S) might be abbreviated as \phi: X \to Y.

If a morphism \phi: X \to Y has an inverse \phi^{-1}: Y \to X which is also a morphism, we say that \phi is an isomorphism, and that X and Y are isomorphic or conjugate.

It is easy to see that morphisms obey the axioms of a (concrete) category, thus the identity map \hbox{id}_X: X \to X on a system is always a morphism, and the composition \psi \circ \phi: X \to Z of two morphisms \phi: X \to Y and \psi: Y \to Z is again a morphism.

Let’s give some simple examples of morphisms.

  • Example 1. If (X,T) is a dynamical system, a topological dynamical system, or a measure-preserving dynamical system, then T^n: X \to X is an isomorphism for any integer n. (Indeed, one can view the map X \mapsto T^n as a covariant functor natural transformation from the identity functor on the category of dynamical systems (or topological dynamical systems, etc.) to itself, although we will not take this perspective here.)
  • Example 2 (Subsystems). Let (X,T) be a dynamical system, and let E be a subset of X which is T-invariant in the sense that T^n E = E for all n. Then the restriction of (E, T\downharpoonright_E) of (X,T) to E is itself a dynamical system, and the inclusion map \iota: E \to X is a morphism. In the category of topological dynamical systems (X,{\mathcal F},T), we have the same assertion so long as E is closed (hence compact, since X is compact). In the category of measure-preserving systems (X,{\mathcal X},\mu,T), we have the same assertion so long as E has full measure (thus E \in {\mathcal X} and \mu(E)=1). We thus see that subsystems are not very common in measure-preserving systems and will in fact play very little role there; however, subsystems (and specifically, minimal subsystems) will play a fundamental role in topological dynamics.
  • Example 3 (Skew shift). Let \alpha \in {\Bbb R} be a fixed real number. Let (X,T) be the dynamical system X := ({\Bbb R}/{\Bbb Z})^2, T: (x_1,x_2) \mapsto (x_1+\alpha,x_2+x_1), let (Y,S) be the dynamical system Y := {\Bbb R}/{\Bbb Z}, S: y \mapsto y+\alpha, and let \pi: X \to Y be the projection map \pi: (x_1,x_2) \to x_1. Then \pi is a morphism. If one converts X and Y into either a topological dynamical system or a measure-preserving system in the obvious manner, then \pi remains a morphism. Observe that \pi foliates the big space X “upstairs” into “vertical” fibres \pi^{-1}(\{y\}), y \in Y indexed by the small “horizontal” space “downstairs”; the shift S on the factor space Y downstairs determines how the fibres move (the shift T upstairs sends each vertical fibre \pi^{-1}(\{y\}) to another vertical fibre \pi^{-1}(\{S y\}), but does not govern the dynamics within each fibre. More generally, any factor map (i.e. a surjective morphism) exhibits this type of behaviour. (Another example of a factor map is the map \pi: {\Bbb Z}/N{\Bbb Z} \to {\Bbb Z}/M{\Bbb Z} between two cyclic groups (with the standard shift x \mapsto x+1) given by \pi: x \mapsto x \hbox{ mod } M. This is a well-defined factor map when M is a factor of N, which may help explain the terminology. If we wanted to adhere strictly to the category theoretic philosophy, we should use epimorphisms rather than surjections, but we will not require this subtle distinction here.)
  • Example 4 (Universal pointed dynamical system). Let {\Bbb Z} = ({\Bbb Z}, +1) be the dynamical system given by the integers with the standard shift n \mapsto n+1. Then given any other dynamical system (X,T) with a distinguished point x \in X, the orbit map \phi: n \mapsto T^n x is a morphism from {\Bbb Z} to X. This allows us to lift most questions about dynamical systems (with a distinguished point x) to those for a single “universal” dynamical system, namely the integers (with distinguished point 0). One cannot pull off the same trick directly with topological dynamical systems or measure-preserving systems, because {\Bbb Z} is non-compact and does not admit a shift-invariant probability measure. As we shall see later, the former difficulty can be resolved by passing to a universal compactification of the integers, namely the Stone–Čech compactification {\beta} {\Bbb Z} (or equivalently, the space of ultrafilters on the integers), though with the important caveat that this compactification is not metrisable. To resolve the second difficulty (with the assistance of a distinguished set rather than a distinguished point), see the next example.
  • Example 5 (Universal dynamical system with distinguished set). Recall the boolean Bernoulli system (2^{\Bbb Z},U) (Example 6 from the previous lecture). Given any other dynamical system (X,T) with a distinguished set A \subset X, the recurrence map \phi: X \to 2^{\Bbb Z} defined by \phi(x) := \{ n \in {\Bbb Z}: T^n x \in A \} is a morphism. Observe that A = \phi^{-1}(B), where B is the cylinder set B := \{ E \in 2^{\Bbb Z}: 0 \in E \}. Thus we can push forward an arbitrary dynamical system (X,T,A) with distinguished set to a universal dynamical system (2^{\Bbb Z},U,B). Actually one can restrict (2^{\Bbb Z},U,B) to the subsystem (\phi(X), U\downharpoonright_{\phi(X)}, B \cap \phi(X)), which is easily seen to be shift-invariant. In the category of topological dynamical systems, the above assertions still hold (giving 2^{\Bbb Z} the product topology), so long as A is clopen. In the category of measure-preserving systems (X,{\mathcal X},\mu,T), the above assertions hold as long as A is measurable, 2^{\Bbb Z} is given the product \sigma-algebra, and the push-forward measure \phi_*(\mu).

Now we begin our analysis of dynamical systems. When studying other mathematical objects (e.g. groups or representations), often one of the first steps in the theory is to decompose general objects into “irreducible” ones, and then hope to classify the latter. Let’s see how this works for dynamical systems (X,T) and topological dynamical systems (X,{\mathcal F},T). (For measure-preserving systems, the analogous decomposition will be the ergodic decomposition, which we will discuss later in this course.)

Define a minimal dynamical system to be a system (X,T) which has no proper subsystems (Y,S). Similarly define a minimal topological dynamical system to be a system (X,{\mathcal F},T) with no proper subsystems (Y, {\mathcal G},S). [One could make the same definition for measure-preserving systems, but it tends to be a bit vacuous – given any measure preserving system that contains points of measure zero, one can make it trivially smaller by removing the orbit T^{\Bbb Z} x := \{ T^n x: n \in {\Bbb Z}\} of any point x of measure zero. One could place a topology on the space X and demand that it be compact, in which case minimality just means that the probability measure \mu has full support.]

For a dynamical system, it is not hard to see that for any x \in X, the orbit Y = T^{\Bbb Z} x = \{ T^n x: n \in {\Bbb Z}\} is a minimal system, and conversely that all minimal systems arise in this manner; in particular, every point is contained in a minimal orbit. It is also easy to see that any two minimal systems (i.e. orbits) are either disjoint or coincident. Thus every dynamical system can be uniquely decomposed into the disjoint union of minimal systems. Also, every orbit T^{\Bbb Z} x is isomorphic to {\Bbb Z}/\hbox{Stab}(x), where \hbox{Stab}(x) := \{ n \in {\Bbb Z}: T^n x = x \} is the stabiliser group of x. Since we know what all the subgroups of {\Bbb Z}, we conclude that every minimal system is either equivalent to a cyclic group shift ({\Bbb Z}/N{\Bbb Z}, x \mapsto x+1) for some N \geq 1, or to the integer shift ({\Bbb Z}, x \mapsto x+1). Thus we have completely classified all dynamical systems up to isomorphism as the arbitrary union of these minimal examples. [In the case of finite dynamical systems, the integer shift does not appear, and we have recovered the classical fact that every permutation is uniquely decomposable as the product of disjoint cycles.]

For topological dynamical systems, it is still true that any two minimal systems are either disjoint or coincident (why?), but the situation nevertheless is more complicated. First of all, orbits need not be closed (consider for instance the circle shift ({\Bbb R}/{\Bbb Z}, x \mapsto x+\alpha) with \alpha irrational). If one considers the orbit closure \overline{T^{\Bbb Z} x} of a point x, then this is now a subsystem (why?) , and every minimal system is the orbit closure of any of its elements (why?), but in the converse direction, not all orbit closures are minimal. Consider for instance the boolean Bernoulli system (2^{\Bbb Z}, A \mapsto A-1) with x = {\Bbb N} := \{ 0,1,2,\ldots\} \in 2^{\Bbb Z} being the natural numbers. Then the orbit T^{\Bbb Z} x of x consists of all the half-lines \{ a,a+1,\ldots,\} \in 2^{\Bbb Z} for a \in {\Bbb Z}, but it is not closed; it has the point {\Bbb Z} \in 2^{\Bbb Z} and the point \emptyset \in 2^{\Bbb Z} as limit points (recall that 2^{\Bbb Z} is given the product (i.e. pointwise) topology). Each of these points is an invariant point of T and thus forms its own orbit closure, which is obviously minimal. [In particular, this shows that x itself is not contained in any minimal system – why?]

Thus we see that finite dynamical systems do not quite form a perfect model for topological dynamical systems. A slightly better (but still imperfect) model would be that of non-invertible finite dynamical systems (X,T), in which T: X \to X is now just a function rather than a permutation. Then we can still verify that all minimal orbits are given by disjoint cycles, but they no longer necessarily occupy all of X; it is quite possible for the orbit T^{\Bbb N} x = \{ T^n x: n \in {\Bbb N} \} of a point x to start outside of any of the minimal cycles, although it will eventually be absorbed in one of them.

In the above examples, the limit points of an orbit formed their own minimal orbits. In some cases, one has to pass to limits multiple times before one reaches a minimal orbit (cf. the “Glaeser refinements” in this lecture of Charlie Fefferman). For instance, consider the boolean Bernoulli system again, but now consider the point

y := \bigcup_{n=0}^\infty [4^n,2 \times 4^n] = [1,2] \cup [4,8] \cup [16,32] \cup \ldots \in 2^{\Bbb Z}

where we use the notation {}[N,M] := \{ n \in {\Bbb Z}: N \leq n \leq M \}. Observe that the point x defined earlier is not in the orbit T^{\Bbb N} y, but lies in the orbit closure, as it is the limit of T^{4^n} y. On the other hand, the orbit closure of x does not contain y. So the orbit closure of x is a subsystem of that of y, and then inside the former system one has the minimal systems \{ {\Bbb Z} \} and \{ \emptyset \}. It is not hard to iterate this type of example and see that we can have quite intricate hierarchies of systems.

Exercise 1. Construct a topological dynamical system (X,{\mathcal F},T) and a sequence of orbit closures \overline{T^{\Bbb Z} x_n} in X which form a proper nested sequence, thus

\overline{T^{\Bbb Z} x_1} \supsetneq \overline{T^{\Bbb Z} x_2} \supsetneq \overline{T^{\Bbb Z} x_3} \supsetneq \ldots

[Hint: Take a countable family of nested Bernoulli systems, and find a way to represent each one as a orbit closure.] \diamond

Despite this apparent complexity, we can always terminate such hierarchies of subsystems at a minimal system:

Lemma 1. Every topological dynamical system (X,{\mathcal F},T) contains a minimal dynamical system.

Proof. Observe that the intersection of any chain of subsystems of X is again a subsystem (here we use the finite intersection property of compact sets to guarantee that the intersection is non-empty, and we also use the fact that the arbitrary intersection of closed or T-invariant sets is again closed or T-invariant). The claim then follows from Zorn’s lemma. [We will always assume the axiom of choice throughout this course.] \Box

Exercise 2. Every compact metrisable space is second countable and thus has a countable base. Suppose we are given an explicit enumeration V_1,V_2,\ldots of such a base. Then find a proof of Lemma 1 which avoids the axiom of choice. \diamond

It would be nice if we could use Lemma 1 to decompose topological dynamical systems into the union of minimal subsystems, as we did in the case of non-topological dynamical systems. Unfortunately this does not work so well; the problem is that the complement of a minimal system is an open set rather than a closed set, and so we cannot cleanly separate a minimal system from its complement. (In any case, the preceding examples already show that there can be some points in a system that are not contained in any minimal subsystem. Also, in contrast with non-invertible non-topological dynamical systems, our examples also show that a closed orbit can contain multiple minimal subsystems, so we cannot reduce to some sort of “nilpotent” system that has only one minimal system.)
We will study minimal dynamical systems in detail in the next few lectures. I’ll close now with some examples of minimal systems.

Example 6 (Cyclic group shift). The cyclic group shift ({\Bbb Z}/N{\Bbb Z}, x\mapsto x+1), where N is a positive integer, is a minimal system, and these are the only discrete minimal topological dynamical systems. More generally, if x is a periodic point of a topological dynamical system (thus T^N x = x for some N \geq 1), then the closed orbit of x is isomorphic to a cyclic group shift and is thus minimal.

Example 7 (Torus shift). Consider a torus shift ( ({\Bbb R}/{\Bbb Z})^d, x \mapsto x+\alpha), where \alpha \in {\Bbb R}^d is a fixed vector. It turns out that this system is minimal if and only if \alpha is totally irrational, which means that n \cdot \alpha is not an integer for any non-zero n \in {\Bbb Z}^d. (The “if” part is slightly non-trivial, requiring Weyl’s equidistribution theorem; but the “only if” part is easy, and is left as an exercise.)

Example 8 (Morse sequence): Let A = \{a,b\} be a two-letter alphabet, and consider the Bernoulli system (A^{\Bbb Z},T) formed from doubly infinite words

\ldots x_{-2} x_{-1}.x_0 x_1 x_2 \ldots

in A with the left-shift. Now define the sequence of finite words

w_1 := a.b, w_2 := abba.baab, w_3 := abbabaabbaababba.baababbaabbabaab, etc.

by the recursive formula

w_1 := a.b; \quad w_{i+1} := f(w_i)

where f(w) denotes the word formed from w by replacing each occurrence of a and b by abba and baab respectively. These words w_i converge pointwise to an infinite word

w = \ldots abbabaababbabaabbaababba.baababbaabbabaababbabaab\ldots.

Exercise 3. Show that w is not a periodic element of A^{\Bbb Z}, but that the orbit \overline{T^{\Bbb Z} w} is both closed and minimal. [Hint: find large subwords of w which appear syndetically, which means that the gaps between each appearance are bounded. In fact, all subwords of w appear syndetically. One can also work with a more explicit description of w involving the number of non-zero digits in the binary expansion of the index.] This set is an example of a substitution minimal set. \diamond

Exercise 4. Let (X, {\mathcal F}, T) and (Y, {\mathcal G}, S) be topological dynamical systems. Define the product of these systems to be (X \times Y, {\mathcal F} \times {\mathcal G}, T \times S), where X \times Y is the Cartesian product, {\mathcal F} \times {\mathcal G} is the product topology, and T \times S is the map (x,y) \mapsto (Tx,Sy). Note that there are obvious projection morphisms from this product system to the two original systems. Show that this product system is indeed a product in the sense of category theory. Do analogous claims hold in the categories of dynamical systems and measure-preserving systems? \diamond

Exercise 5. Let (X, {\mathcal F}, T) and (Y, {\mathcal G}, S) be topological dynamical systems. Define the disjoint union of these systems to be (X \uplus Y, {\mathcal F} \uplus {\mathcal G}, T \uplus S) where (X \uplus Y, {\mathcal F} \uplus {\mathcal G}) is the disjoint union of (X,{\mathcal F}) and (Y,{\mathcal G}), and T \uplus S is the map which agrees with T on X and agrees with S on Y. Note that there are obvious embedding morphisms from the original two systems into the disjoint union. Show that the disjoint union is a coproduct in the sense of category theory. Are analogous claims true for the categories of dynamical systems and measure-preserving systems? \diamond

[Update, Jan 11: several corrections.]

[Update, Jan 14: A required to be clopen in the topological version of Example 4.]

[Update, Jan 17: Slight change to Example 7.]

[Update, Jan 20: More exercises added.]