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J. HawkinsErgodic DynamicsGraduate Texts in Mathematics289https://doi.org/10.1007/978-3-030-59242-4_2

2. Dynamical Properties of Measurable Transformations

Jane Hawkins1  
(1)
Department of Mathematics, University of North Carolina at Chapel Hill, Chapel Hill, NC, USA
 

In dynamical systems, both mathematical and physical, there is often a split in behavior between predictable behavior, as is seen in the presence of an attractor for example, and chaotic behavior. There is also the important notion of recurrence which refers to a subset of the domain of a dynamical system returning to itself, infinitely often.

In this chapter we present the basic definitions and properties of some dynamical systems of interest with a focus on conservativity, ergodicity, and recurrence.

2.1 The Basic Definitions

We start with a description of the spaces we work with throughout the book. A Polish space X is a separable, metrizable space, complete with respect to some choice of metric. We always use $$\mathcal {B}$$ to denote the σ-algebra of Borel sets, the smallest collection of sets closed under complements and countable unions containing the open sets of X (see Definition A.11). We assume throughout the book that we have a standard space $$(X, \mathcal B)$$ , (a Polish space with Borel structure) with a measure μ on $$\mathcal B$$ . We let $$\mathcal T$$ denote the topology on X, the collection of open sets (see Definition A.15). We do not assume that μ(X) < ∞ unless explicitly stated, but we always assume measures are σ-finite. We call a standard space X a probability space when μ(X) = 1. We refer to Appendix A for background details on this terminology. For a standard space $$(X,\mathcal B)$$ and a map f : X → X, if $$U \in \mathcal B$$ , we define
$$\displaystyle \begin{aligned} f^{-1}U = \{x \in X \ : \ f(x) \in U\}. \end{aligned} $$
(2.1)
In particular for a point y ∈ X, f −1 y = {x ∈ X  :  f(x) = y} is defined as a set whether or not the map f is invertible. A set U ⊂ X is (completely) invariant under f if U = f −1(U) = f(U) and forward invariant if U = f(U).
Definition 2.1
  1. 1.

    The map f : X → X is continuous if for every set $$U \in \mathcal T$$ , $$f^{-1}U \in \mathcal T$$ as well.

     
  2. 2.

    The map f is Borel measurable if for every Borel set $$B \in \mathcal B$$ , $$f^{-1} B \in \mathcal B$$ . Since X is a standard space, we just call f measurable.

     
  3. 3.

    If f is continuous, invertible, and f −1 is defined and continuous, we call f a homeomorphism .

     
  4. 4.

    If f is measurable and there exists a measurable set N ⊂ X with μ(N) = 0 such that f : X ∖ N → X ∖ N is invertible, and if f −1 is measurable as well, then we call f a measurable automorphism. Frequently f is called an automorphism in this case, but the word automorphism is used in many settings and means many different things for a dynamical system, so we include the adjective measurable when needed.

     
  5. 5.

    More generally, for a standard measure space $$(Y, \mathcal F)$$ , the map f : X → Y  is measurable if for every Borel measurable set $$C \in \mathcal F$$ , $$f^{-1} C \in \mathcal B$$ .

     
  6. 6.

    We call f a measurable transformation or map on X when Y = X.

     
  7. 7.

    The notation $$(X,\mathcal B,\mu , f)$$ denotes the dynamical system f, which consists of a transformation of a standard space X along with its measurable structure. We will specify when f is continuous, but f is always assumed to be measurable. We also assume that except possibly on a set of μ measure 0, {f −1(x)} is finite or countably infinite.

     

We define what it means for two dynamical systems to be “the same”; that is, conjugate, or isomorphic. As in every mathematical setting, these notions depend on the context.

Definition 2.2
  1. 1.
    Given two dynamical systems $$(X_1,\mathcal B_1,\mu _1,f_1)$$ and $$(X_2,\mathcal B_2,\mu _2,f_2)$$ , we say they are isomorphic (or measure theoretically isomorphic) if there exist sets $$Y_j \in \mathcal B_j$$ , such that μ j(X j ∖ Y j) = 0, for j = 1, 2, and a measurable invertible map ϕ : Y 1 → Y 2, with μ 1(ϕ −1 A) = μ 2(A) for all $$A \in \mathcal B_2$$ , and such that
    $$\displaystyle \begin{aligned}\phi \circ f_1 (x) = f_2 \circ \phi(x) \, \mbox{for every} \, x \in Y_1.\end{aligned} $$
    (2.2)
     
  2. 2.
    We say f 1 and f 2 are topologically conjugate if f j : X j → X j, j = 1, 2 are continuous and there exists a homeomorphism Φ : X 1 → X 2 such that for all x ∈ X 1
    $$\displaystyle \begin{aligned}\Phi \circ f_1 (x) = f_2 \circ \Phi(x).\end{aligned} $$
    (2.3)
     
  3. 3.

    If the map ϕ in (2.2) is surjective onto Y 2 but not necessarily one-to-one, we call ϕ a factor map and say that f 2 is a measurable factor of f 1.

     
  4. 4.

    If the map Φ in (2.3) is surjective onto X 2 but not necessarily one-to-one, we call Φ a continuous factor map and say that f 2 is a continuous factor of f 1.

     
Given a dynamical system $$(X,\mathcal B,\mu , f)$$ it is often useful to think of the transformation f as an experiment to perform and the points in X as the possible states of the system on which it is performed. Then f transforms x ∈ X into f(x) ∈ X, and iteration of the map f represents the passage of time or repeated applications of the experiment. As introduced in Example 1.​2 and used throughout the book, we denote the n-fold composition of f by
$$\displaystyle \begin{aligned} f^n \equiv f \circ f \circ \cdots \circ f \quad (n \mbox{ times}), \end{aligned} $$
(2.4)
and refer to it as the nth iteration of f. We are interested in understanding limn→∞ f n(x), if and when it exists, for some or all x ∈ X which amounts to “predicting the future” of state x. We already referred to the orbit of a point several times; when the dynamical system is invertible the meaning is clear. For noninvertible systems, there are various types of orbits one can consider.
Definition 2.3
Assume $$(X,\mathcal B)$$ is a standard space and f : X → X is a map, not necessarily invertible. Given x ∈ X,

  • the forward orbit of x is $$ \mathcal {O}^+(x) = \{f^n(x) \}_{n \in \mathbb N \cup \{0\}}$$ ,

  • the backward orbit of x is the set $$ \mathcal {O}^-(x) = \cup _{n \in \mathbb N \cup \{0\}} \{f^{-n}(x) \}$$ .

  • The smallest set containing x, which is completely invariant under the map f is the grand orbit of x, and is defined by
    $$\displaystyle \begin{aligned}\mathcal{O}(x) = \bigcup_{n \geq 1}\bigcup_{m\geq 1}\{f^{-n}\left(f^m x \right)\}.\end{aligned}$$

  • When f is invertible, the two-sided orbit of x is the set $$ \mathcal {O}^{\pm }(x) =\cup _{n \in \mathbb Z } f^{-n}(x) $$ .

When f is noninvertible, $$\mathcal {O}^-(x)$$ is often shown as a tree-like structure as in the left side of Figure 2.1.
../images/491076_1_En_2_Chapter/491076_1_En_2_Fig1_HTML.png
Fig. 2.1

Using f(x) = 2x ( mod 1), the structure of f −3(.2) is shown on the left, and a diagram of the finite forward orbit $$\mathcal {O}^+(.2)$$ is shown on the right. The backward orbit $$\mathcal {O}^-(.2) = \{f^n(.2) \}_{n \in \mathbb Z}$$ is infinite; the points shown are: .2 (top row), f −1(.2) = {.1, .6}(second row), f −2(.2) = {.05, .55, .3, .8} (third row), and f −3(.2) = {.025, .525, .275, .775, .15, .65, .4, .9} (fourth row). The orbit on the right appears on the left with the same coloring (pastel shades).

Even if limn→∞ f n(x) does not exist, there are related quantities of interest in ergodic theory that carry a lot of information about the dynamical system, such as the limit of the average value of a measurable function along a generic orbit. This is studied in Chapter 4, and a good overview of the role of individual orbits in appears in [185].

The next result shows that every continuous transformation is measurable. The converse is false, which we can see if we consider the example of simple functions (see Definition B.2), a large class of measurable functions that are typically not continuous.

Proposition 2.4

If $$(X, \mathcal B, \mu ,f)$$ is a continuous dynamical system, then f is measurable.

Proof
If U ⊂ X is open, then $$U \in \mathcal B$$ , and the continuity of f implies that f −1 U is also open, and therefore $$f^{-1}U \in \mathcal B$$ . Since
$$\displaystyle \begin{aligned}f^{-1}(X \setminus U) = \{x \,:\, f(x) \in X \setminus U\}=X \setminus f^{-1}(U),\end{aligned}$$
then the complement to f −1 U is also in $$\mathcal B$$ . Since $$f^{-1}(B_1 \cup B_2 \cup \ldots B_n \cup \ldots ) = \cup _{i=1}^\infty f^{-1}(B_i)$$ , and the open sets generate the σ-algebra of measurable sets, the result follows. □
Example 2.5
One-dimensional Continuous and Nonsingular Dynamical Systems.
  1. 1.

    We consider $$(\mathbb R,\mathcal B,m)$$ with the usual Euclidean metric and Lebesgue measure. We fix an $$\alpha \in \mathbb R$$ and define f(x) = x + α. This gives a continuous map with predictable dynamical properties; that is, knowing the value of α, we can predict limn→∞ f n(x) for every x. (See Exercise 1 below).

     
  2. 2.

    We now fix α ∈ (0, 1) and define R α(x) = x + α ( mod 1) on $$\mathbb T^1= \mathbb R/\mathbb Z$$ . Again the dynamical properties are predictable, but they are more complicated in the sense that for an irrational α, given an open set $$U \subset \mathbb T^1$$ and an initial point x 0, we can find an integer N such that $$R_{\alpha }^N(x_0) \in U$$ . Moreover, $$\lim _{n \rightarrow \infty } R_\alpha ^n(x)$$ does not exist. (This example was studied in Chapter 1.)

     
  3. 3.
    Let $$\mathbb T^1=\mathbb R/\mathbb Z$$ denote the circle and $$\pi : \mathbb R \rightarrow \mathbb R/\mathbb Z$$ denote the standard quotient map. It is a classical result that if $$f: \mathbb T^1 \rightarrow \mathbb T^1$$ is continuous, then there exists a lift of f, a continuous map $$F:\mathbb R \rightarrow \mathbb R$$ such that
    $$\displaystyle \begin{aligned} f \circ \pi = \pi \circ F.\end{aligned}$$
    If F 1 and F 2 are lifts of f, then $$F_1(x) -F_2(x) = j \in \mathbb Z$$ and j does not depend on x (by continuity of f and F i). If f is a homeomorphism (orientation preserving), then its lift satisfies F(x + 1) = F(x) + 1 for all $$x \in \mathbb R$$ . More generally for each continuous f on $$\mathbb T^1$$ we define the degree of f to be the unique integer d such that
    $$\displaystyle \begin{aligned} F(x+1) = F(x) + d. \end{aligned} $$
    (2.5)

    The uniqueness of d and its independence from x and from the choice of lift F are standard results and can be found for example in [106]. Example 1.​2 in Chapter 1 gives a degree 2 measure-preserving map on $$\mathbb T^1$$ .

     
  4. 4.
    We consider the map on the real line given by
    $$\displaystyle \begin{aligned} f(x) = \frac 14 \left( x + \frac 1x+2 \right ),\end{aligned}$$

    which is defined on a set of full Lebesgue measure of $$\mathbb R$$ , but has a pole at 0. This is not a problem from the measurable point of view; if we add the point {∞} to $$\mathbb R$$ , then the map f is measurable and onto the space $$\mathbb R_{\infty }=\mathbb R \cup \{\infty \}$$ . Moreover we can extend f to be a continuous map on $$\mathbb R_{\infty }$$ by defining f(∞) = ∞; this can be seen since f(x) = f(1∕x) for all $$x \in \mathbb R$$ , and f(0) = ∞.

    For this map some points have predictable forward orbits. There are critical points at − 1 and 1; we have f(1) = 1 and $$\mathcal {O}^+(-1)=\{-1,0, \infty \}$$ . Therefore both forward critical orbits are finite. By considering the derivative at 0 of the map g(x) = 1∕f(1∕z) = 1∕f(z), we have reversed the roles of 0 and ∞ and we see that g′(0) = 4, and therefore ∞ is a repelling fixed point of f (see Definition 12.​7), with the orbit of − 1 terminating there. However a randomly chosen negative number will have a forward orbit that is dense in the interval (−∞, 0). The point x = 1 is an attractor (as defined in Chapter 3), and for every x > 0, limk→∞ f k(x) = 1 (see Exercise 5). This map extends to an analytic map of the Riemann sphere and can be studied using techniques from Chapter 12.

     

The next example gives a hands-on method for understanding attractors, which will be defined rigorously in Chapter 3.

Example 2.6

We consider the map $$g(x) = \pi \cos {}(x)$$ on $$\mathbb R$$ . Since we are interested in long-term behavior as we iterate g, our first observation is that the range of g is [−π, π], and the map is periodic with [−π, π] a fundamental period; therefore after one application of g we need only restrict our attention to the interval [−π, π].

Algorithm for Detecting the Attractor

  • Using a calculator or computer, enter a “random” number a by using an 8-digit decimal number with a decimal point somewhere in it. It does not matter where the decimal point is placed, since when we apply our map once, $$g(a)=\pi \cos {}(a) \in [-\pi ,\pi ]$$ .

  • Continue applying g to your answer; in other words, construct a sequence a 0 = a, a 1 = g(a 0), a 2 = g(g(a)) = g(a 1), …, a n+1 = g(a n).

The result can be predicted; for almost every starting point, this process will quickly approach a number very close to − π ≈−3.1415926⋯. This follows from the fact that g(−π) = −π, and g′(−π) = 0, so − π attracts nearby points (see Exercise 6 in Chapter 1). Also since $$\cos {}(x)$$ is an even function, the other end of the interval of interest, (π − ε, π] gets mapped by g onto [−π, −π + ε 1), so points at the other end of the interval are quickly attracted to − π as well. Then we observe that g(0) = π, so a small interval around 0 gets mapped onto [−π, −π + ε 2) in two iterations. Meanwhile small intervals around ± π∕2 are mapped to an interval containing π in two iterations. This is illustrated in Figure 2.2 and gives a good example of a dissipative dynamical system with an attractor. We note that there is a fixed point at x 0 ≈−2.48, so not all points are attracted to 0, and the dynamics get quite complicated under iteration at certain points, as shown in Figure 2.2.
../images/491076_1_En_2_Chapter/491076_1_En_2_Fig2_HTML.png
Fig. 2.2

(a) The graph of $$\pi \cos (x)$$ . (b) The graph restricted to a fundamental domain $$\mathcal {J}$$ . (c) The graph on $$\mathcal {J}$$ after 3 iterations. (d) After 6 iterations many points are attracted to − π.

Example 2.7 (Cellular Automata)

In Section 1.​1 we discussed the dynamical system $$(\Sigma _n, \mathcal B, \rho ,\sigma )$$ with the left shift map $$\sigma (x)_i = x_{i+1}, \, i \in \mathbb Z$$ ; this is defined similarly on $$\Sigma _n^+$$ . Let X n denote either Σn or $$\Sigma _n^+$$ . Since the preimage of a cylinder set is the finite union of cylinder sets, the map σ is continuous with respect to the topology generated by the cylinder sets $$C_{k}^w$$ on X n, and therefore is measurable. There is a metric inducing the topology on X n, namely d X(x, v) = 1∕2m where $$m=\min \left \{ |i|\;:\;x_{i}\neq v_{i}\right \}$$ .

The simplest definition of a cellular automaton (CA) is that it is a continuous shift-commuting map on a shift space X n. We elaborate on this here and devote Chapter 14 to this topic.

A one-dimensional cellular automaton (CA) is a continuous map F on X n such that F ∘ σ = σ ∘ F. For each x ∈ X n and $$i \in \mathbb Z$$ , by x i or [x]i we denote the ith coordinate of x, and by x {k,l}, k < l we denote the block of coordinates from x k to x l; i.e., x {k,l}∈ A l−k+1.

By work of Curtis, Hedlund, and Lyndon in the late 1960s [93], the following result allows us to characterize CAs by a local rule, which was the motivating property of these systems [180, 181]. The proof of this result follows from the definition of continuity in the metric topology on Σn and is given in Chapter 14, Theorem 14.​1.

Theorem 2.8

The map F on X n is a CA if and only if there exists an integer r ≥ 0 and a local rule f : A 2r+1 → A such that for every $$x \in A^{\mathbb Z}$$ ,
$$\displaystyle \begin{aligned} F(x)_i = f(x_{\{i-r,i+r\}}).\end{aligned}$$

Many mathematical and physical consequences of Theorem 2.8 are given in Chapter 14.

We assume that every dynamical system $$(X,\mathcal B,\mu ,f)$$ satisfies several standing assumptions in addition to the measurability of the map $$f:(X, \mathcal B, \mu ) \rightarrow (X, \mathcal B, \mu ).$$ The push-forward measure of μ is defined by f ∗ μ(B) = μ(f −1 B); see Exercise 1 below to show that f ∗ μ is a measure on $$(X,\mathcal B)$$ .

Standing Assumptions
From now on, without explicit statements to that effect, we assume every dynamical system $$(X,\mathcal B,\mu ,f)$$ is measurable and satisfies the next two additional properties. Every dynamical system that we consider in this book satisfies these conditions.
  1. 1.

    Nonsingularity: f is nonsingular with respect to μ if for every set $$B \in \mathcal B$$ , μ(B) = 0 if and only if f ∗ μ(B) = 0.

     
  2. 2.

    Forward measurability and nonsingularity: For each set $$B \in \mathcal B$$ , we have $$f(B)\in \mathcal B$$ ; furthermore μ(B) = 0 if and only if μ(f(B)) = 0.

     
While f ∗ μ is always a measure, if f −1 does not exist (as a single-valued map μ-a.e.), then $$f^{-1}_*\mu (A) = \mu (f(A))$$ does not define a measure. Condition 1 implies that f is surjective up to a set of measure 0 (see Exercise 2). These standing assumptions are quite natural (see Appendix, Remark A.21).

The following condition, stronger than nonsingularity, sometimes holds but is not a standing assumption. Chapter 8 is devoted to this topic.

Definition 2.9

The dynamical system $$(X,\mathcal B,\mu ,f)$$ is measure preserving if μ = f ∗ μ. Equivalently, we say that f is measure preserving if for every $$A \in \mathcal B$$ , μ(A) = μ(f −1 A).

We now turn to some dynamical properties of interest. The simplest property perhaps is that of periodicity. If a dynamical system always returns to its starting state after a fixed time interval, then it is completely predictable. Alternatively, it could happen that there are some periodic points for the dynamical system, but the transformation is not periodic. The second phenomenon (Definition 2.10) is more interesting than the first. We say that a map $$f:(X,\mathcal B,\mu ) \rightarrow (X,\mathcal B,\mu )$$ is periodic if there exists a $$k\in \mathbb N$$ such that f k(x) = x for μ-a.e. x ∈ X. Otherwise f is aperiodic. Usually if a map is periodic, it is periodic for every x ∈ X, but our definition allows us to ignore a set of measure 0. Earlier, Example 1.​1 of a rational rotation on the circle, R p∕q(x) = x + p∕q ( mod 1) on $$\mathbb R/\mathbb Z$$ , showed a simple example of a periodic map.

Definition 2.10

Let $$(X,\mathcal B,\mu ,f)$$ be a nonsingular dynamical system. A point x 0 ∈ X is a periodic point of f of period k if f k(x 0) = x 0 and $$k \in \mathbb N$$ is minimal. A fixed point x 0 is a periodic point of period 1. The cycle or periodic orbit containing a periodic point x 0 of period k is the set of k points {x 0, f(x 0), …, f k−1(x 0)}.

Measure Theoretic Conventions
We state some mathematical conventions used throughout this book. We assume that $$(X,\mathcal B,\mu ,f)$$ is a nonsingular dynamical system, and is not necessarily invertible or measure preserving.
  1. 1.
    For sets $$A,B \in \mathcal B$$ , A△B = (A ∖ B) ∪ (B ∖ A). We say
    ../images/491076_1_En_2_Chapter/491076_1_En_2_Equf_HTML.png
     
  2. 2.

    For x ∈ X, $$A \in \mathcal B$$ , f 0(x) = x for all x, so A = f 0 A.

     
  3. 3.
    If P and Q are two statements about points in X, then these statements are equivalent, and written interchangeably as needed:
    1. (a)

      P = Q a.e., or, P = Q μ-a.e.

       
    2. (b)

      N = {x ∈ X : P ≠ Q} satisfies μ(N) = 0.

       
    3. (c)

      for μ-a.e. x ∈ X, P(x) = Q(x).

       
     
  4. 4.
    We note that the standing assumptions on a dynamical system $$(X,\mathcal B,\mu ,f)$$ imply that for every set $$A \in \mathcal B$$ ,
    $$\displaystyle \begin{aligned} f(f^{-1}(A))=A \ (\mu \bmod 0). \end{aligned} $$
    (2.6)

    This statement about sets holds even when f is not invertible.

     

2.2 Recurrent, Conservative, and Dissipative Systems

We turn to some basic recurrence and conservation laws in this section to examine properties that hold for nonsingular and noninvertible maps. The next property is one of the earliest studied in ergodic theory, by Poincaré in 1899.

Definition 2.11

A nonsingular dynamical system $$(X,\mathcal B,\mu ,f)$$ is recurrent if for every set $$A \subset \mathcal B$$ , for μ-a.e. x ∈ A there exists $$n \in \mathbb N$$ , dependent on x, for which f n x ∈ A. The transformation f is infinitely recurrent if there exist infinitely many $$n \in \mathbb N$$ for which f n x ∈ A.

First we show the two types of recurrence are the same when f preserves a probability measure.

Theorem 2.12 (Poincaré Recurrence Theorem)
If f is measure preserving and μ(X) < ∞, then for every $$A \in \mathcal B$$ , f is infinitely recurrent; i.e.,
$$\displaystyle \begin{aligned} A \cap \left( \bigcap_{n=0}^{\infty}\bigcup_{i=n}^{\infty}f^{-i} A \right ) = A \ (\mu \bmod 0). \end{aligned}$$
Proof
Define, for each nonnegative integer n, $$A_n=\bigcup _{i=n}^{\infty }f^{-i} A.$$ An induction argument on n shows that for each n, A n+1 = A n ∖ f −n A, and therefore
$$\displaystyle \begin{aligned} A_0 \supset A_1 \supset \cdots \end{aligned} $$
(2.7)
and f −n A 0 = A n. Therefore since f is measure preserving, μ(A n) = μ(A 0) for each n; (2.7) and the finiteness of μ imply that
../images/491076_1_En_2_Chapter/491076_1_En_2_Equ8_HTML.png
(2.8)
Moreover A ⊂ A 0, so μ(A ∩ A 0) = μ(A). Finally it follows from (2.8) that
$$\displaystyle \begin{aligned} \bigcap_{n=0}^{\infty} A_n = A_0 \ (\mu \bmod 0),\end{aligned}$$
so
$$\displaystyle \begin{aligned} A \cap \bigcap_{n=0}^{\infty} A_n = A \cap A_0 = A \ (\mu \bmod 0), \end{aligned}$$
and this proves the result. □

We point out that μ(X) < ∞ was used in the proof, to establish (2.8). We give a different proof here strengthening the result by giving an upper bound on the first return time for a set of positive measure. This proof uses the pigeonhole principle that says that j sets the same size as A must overlap if j > 1∕μ(A).

Proposition 2.13
If $$(X,\mathcal B,\mu ,f)$$ is measure preserving and μ(X) = 1, then for every measurable set A with μ(A) > 0, there exists some 0 < i ≤ 1∕μ(A) such that
$$\displaystyle \begin{aligned} \mu( A \cap f^{-i} A) &gt;0, \end{aligned} $$
(2.9)

and f is recurrent.

Proof
Assume μ(A ∩ f −i A) = 0 for all i ≤ 1∕μ(A), and set j = ⌊1∕μ(A)⌋, the integer part of 1∕μ(A). It follows that for all 0 ≤ i < k ≤ j,
$$\displaystyle \begin{aligned}\mu(f^{-i}A \cap f^{-k} A) =0 = \mu(A \cap f^{(-k+i)}A),\end{aligned}$$
using the forward nonsingularity of f. Then
$$\displaystyle \begin{aligned} 1= \mu(X) \geq \sum_{i=0}^{j}\mu(f^{-i}A)= \mu(A) \left( 1+j \right) &gt; 1.\end{aligned}$$
This contradiction implies that (2.9) holds and f is recurrent. □

In Exercise 6 and Proposition 2.20 below, we extend recurrence results to nonsingular maps on infinite spaces, assuming some additional property replaces preservation of a probability measure. On the other hand, an easy example to show that not all nonsingular transformations on finite measure spaces are recurrent is the map: f(x) = x 2 on [0, 1]. No points in the interval A = (1∕2, 7∕10) ever get mapped back into A under f.

2.2.1 Ergodicity

Definition 2.14
Given a nonsingular dynamical system $$(X, \mathcal B, \mu ,f)$$ , with μ a σ-finite measure, the next statements depend on μ.
  1. 1.
    A set $$A \in \mathcal B$$ is invariant under f or f-invariant if
    $$\displaystyle \begin{aligned} f^{-1}A= A\ (\mu \bmod 0). \end{aligned}$$
     
  2. 2.

    Using (2.6), f −1 A = A (μ mod  0) ⇒ f(A) = A (μ mod  0) as well, and sometimes it is useful to say that A is completely invariant to stress this point, which uses the nonsingularity of f.

     
  3. 3.

    A transformation f is (measurably) decomposable if X = A ∪ B, μ(A) > 0, μ(B) > 0, and both A and B are f-invariant. Otherwise f is indecomposable.

     
  4. 4.

    A nonsingular transformation f is ergodic if it is indecomposable. Equivalently, f is ergodic if f −1 A = A (μ mod  0) implies either μ(A) = 0 or μ(X ∖ A) = 0.

     
  5. 5.

    If μ(X) = 1, and if $$A \in \mathcal B$$ , μ(A) > 0, is invariant under an ergodic transformation f, then μ(A) = 1.

     
  6. 6.

    Let $$\mathcal B_+ = \{B \in \mathcal B : \, \mu (B)&gt;0\} \subset \mathcal B$$ . Then f is ergodic if and only if every f-invariant set $$A \in \mathcal B_+$$ satisfies A = X (μ mod  0).

     

The last statement holds since if $$A \in \mathcal B$$ is completely invariant, the same holds for X ∖ A. Therefore every invariant set either has full measure or measure 0. If $$A \in \mathcal B_+$$ , then the smallest set containing A, call it A ∗, such that f −1(A ∗) ⊂ A ∗ is $$\displaystyle A^*= \cup _{j=0}^{\infty } f^{-j}A$$ . The indecomposability of an ergodic dynamical system is further reflected in the next result.

Lemma 2.15
If f : X → X is an ergodic measure-preserving transformation of a probability space $$(X, \mathcal B, \mu )$$ , then for every set $$A \in \mathcal B_+$$ , we have
$$\displaystyle \begin{aligned} X= \bigcup_{j=0}^{\infty} f^{-j}A \ (\mu \bmod 0).\end{aligned} $$
(2.10)
Proof

Given a set $$A \in \mathcal B$$ , the set $$A^*= \bigcup _{j=0}^{\infty } f^{-j}A$$ satisfies f −1 A ∗⊆ A ∗ and by hypothesis μ(A ∗) = μ(f −1 A ∗). Therefore f −1 A ∗ = A ∗ (μ mod  0), and by ergodicity, μ(A ∗) = 0 or 1. If μ(A) > 0, then μ(A ∗) = μ(X) = 1, and therefore μ(A ∗△X) = 0, which is equivalent to (2.10). □

2.2.2 Kac’s Lemma

One can obtain an estimate on how long it might take for a point in a set A to return to A under an ergodic measure-preserving transformation f. The next result is an extension of Proposition 2.13; we do not need to assume that f is invertible, but we assume invertibility in the proof we give below.

Lemma 2.16 (Kac’s Lemma)
Let $$(X, \mathcal B, \mu ,f)$$ be an ergodic probability measure-preserving invertible dynamical system and let $$A \in \mathcal B_+$$ . Define a function $$n_A: X \rightarrow \mathbb N$$ by
$$\displaystyle \begin{aligned}n_A(x) = \inf \{ n \geq 1\ : \ f^n(x) \in A\}.\end{aligned}$$

Then n A is defined for μ-a.e. x ∈ X and ∫A n A(x)dμ(x) = 1.

Proof

Since f is a finite measure-preserving ergodic transformation, Lemma 2.15 and Theorem 2.12 imply that n A is defined for μ-a.e. x ∈ X. Define A n = {x ∈ A  :  n A(x) = n}; then A n is the set of points in A whose first return time to A under iterations of f is n.

Then X ∖ A can be written as the disjoint union of countably many sets that are the images under f of the A n sets, before their first return into the set A. Specifically, writing for each k = 0, …, n − 1, f −k A n,k = A n, this defines the sets A n,k, and we note that A n = A n,0. Since μ is preserved, for each $$n \in \mathbb N$$ and k = 0, 1, …, n − 1, μ(A n,k) = μ(A n). Moreover by the ergodicity of f, Lemma 2.15 implies that μ-a.e. x ∈ X lies in some A n,k; this set structure is illustrated in Figure 2.3. Then
$$\displaystyle \begin{aligned} \int_A n_A d\mu &amp;= \sum_{n=1}^\infty \int_{A_n} n_A d\mu \\ &amp;= \sum_{n=1}^ \infty n \mu(A_n) \\ &amp;= \sum_{n=1}^\infty \sum_{k=0}^{n-1}\mu(A_{n,k}) = \mu(X)=1, \end{aligned} $$
(2.11)
which proves the result. □
../images/491076_1_En_2_Chapter/491076_1_En_2_Fig3_HTML.png
Fig. 2.3

Illustration of the proof of Kac’s Lemma

Kac’s Lemma holds if $$(X,\mathcal B,\mu ,f)$$ is as above but noninvertible; a similar proof in this case appears in ([116], Chapter 1, Theorem 3.6). Therefore if a set has small measure, one should expect it to take longer to return to that set under f on the average. We also note that even if μ is preserved and f is invertible, f can fail to be recurrent if μ is infinite, as the next example shows.

Example 2.17
We give an example of an invertible ergodic transformation which is nonrecurrent. We let ν denote counting measure on $$\mathbb N$$ , so $$\nu (\mathbb N)= \infty $$ . On $$(\mathbb N, \mathcal B, \nu )$$ we consider the transformation:
$$\displaystyle \begin{aligned} f(k) = \left\{ \begin{array}{ll} k+2 &amp; \text{ if } k \text{ is even,} \\ k-2 &amp; \text{ if } k \text{ is odd, } k \neq 1, \\ 2 &amp; \text{ if } k=1. \end{array} \right. \end{aligned}$$
Figure 2.4 shows a diagram of the dynamics of f. It is straightforward to show that f is invertible, measure preserving, and ergodic. The one point set {1} is a set of positive measure that does not return to itself. This shows that the hypothesis of finite measure cannot be relaxed in Proposition 2.13 to a σ-finite infinite measure.
../images/491076_1_En_2_Chapter/491076_1_En_2_Fig4_HTML.png
Fig. 2.4

A nonrecurrent ergodic transformation f on $$ \mathbb N$$

2.2.3 Conservativity and Hopf Decomposition

A measurable set W ⊂ X is wandering or backward wandering for a nonsingular dynamical system $$(X,\mathcal B,\mu ,f)$$ if the sets $$\{f^{-n}W\}_{n=0}^\infty $$ are all disjoint. Equivalently, no point in a wandering set W ever returns to W under f. We use the convention that wandering set always refers to a backward wandering set. A measurable set V  is forward wandering if the sets $$\{f^{n}V\}_{n=0}^\infty $$ are all disjoint. Every forward wandering set is also wandering, but the converse does not always hold. When f is invertible the concepts are identical since f −n(f n x) = x μ -a.e.

A nonsingular map f is conservative on $$C \in \mathcal B_+$$ if C contains no wandering set of positive measure; f is conservative if it is conservative on X. A nonconservative map is called dissipative; if f is not conservative on a set of positive measure, then f is completely dissipative. It is well-known that for a nonsingular dynamical system $$(X,\mathcal B,\mu , f)$$ with μ finite or infinite, f is conservative if and only if f is recurrent (see Proposition 2.20, Exercise 6, and Exercise 7 below).

There exists a maximal set C on which f is conservative; this result is called the Hopf Decomposition Theorem and is a classical result that dates back to the beginning of ergodic theory. The theorem says that a dynamical system on a σ-finite measure space admits a maximal subset on which the dynamics are conservative and recurrent.

Theorem 2.18 (Hopf Decomposition)
If $$f:(X, \mathcal B, \mu ) \rightarrow (X, \mathcal B, \mu )$$ is nonsingular and μ is σ-finite, there exists a decomposition of X into two disjoint measurable sets C and D such that:
  1. 1.

    C ⊂ f −1 C;

     
  2. 2.

    f|C (the restriction of f to the set C) is conservative;

     
  3. 3.

    D = X ∖ C is a union of at most countably many wandering sets.

     
Proof

If W is a wandering set, then every subset of W is wandering as well, so the property of being wandering is hereditary. Therefore let $$\mathcal {F}$$ denote the hereditary collection of measurable wandering sets. Now define $$D = U(\mathcal {F})$$ , its measurable union (see Appendix A.37). Using an exhaustion argument, D can be written as a countable union of wandering sets by Lemma A.39. Define C = X ∖ D; if W is wandering, then both W and f −1(W) are in D, so C ⊂ f −1 C (μ mod  0) and then since it contains no wandering sets, f|C : C → C is well-defined and conservative. □

Theorem 2.18 holds whether or not f is invertible, but the result is stronger if it is.

Corollary 2.19

Assume $$f:(X, \mathcal B, \mu ) \rightarrow (X, \mathcal B, \mu )$$ is nonsingular, μ is σ-finite, and f is invertible. Then C and D, the sets from Theorem 2.18 , are invariant. Moreover, there exists a wandering set $$\widetilde {W}$$ such that $$\displaystyle D = \cup _{j=- \infty }^{\infty } f^j(\widetilde {W})$$ .

Proof

By Theorem 2.18, using the notation from its proof, write the collection of wandering sets D by: $$D = \cup _{i=1}^{\infty } W_i,$$ where each W i is wandering. If W is a wandering set for f, then so is f(W) since f −1(f(W)) = W by the invertibility of f. Therefore the sets C and D are invariant for f.

Define the set $$M_1 = \sup \{ \mu (W)\,:\, W \in \mathcal {F} \},$$ and find $$V_1 \in \mathcal {F}$$ such that μ(V 1) ≥ M 1∕2. Now proceed inductively: letting
$$\displaystyle \begin{aligned}M_n = \sup\{ \mu(W)\,:\, W \in \mathcal{F}, W \cap V_j = \emptyset \, \mbox{for every} \, j&lt; n\},\end{aligned}$$
choose V n such that μ(V n) ≥ M n∕2 with V n ∩ V j = ∅ for all j < n.
For each $$j \in \mathbb N$$ , define $$\widetilde {V}_j = \cup _{k= - \infty }^{\infty } f^kV_j$$ . Finally, define
$$\displaystyle \begin{aligned}\widetilde{W} = \bigcup_{n=1}^{\infty} \left( V_n \setminus \cup_{j=1}^{n-1}\widetilde{V_j} \right ),\end{aligned}$$
which has the property claimed. □

We prove that every conservative map is recurrent; the next result holds for noninvertible maps and for infinite measures.

Proposition 2.20

If $$(X,\mathcal B,\mu ,f)$$ is nonsingular and conservative, then f is infinitely recurrent.

Proof
Let $$A \in \mathcal B_+$$ . Define the points in A that never return by
$$\displaystyle \begin{aligned} W = \{ x \, : f^nx \notin A \, \mbox{for every} \, n \in \mathbb N\} \cap A.\end{aligned}$$
Since W ⊂ A, by construction,
$$\displaystyle \begin{aligned} f^nW \cap W = \emptyset \, \mbox{for every} \, n \in \mathbb N. \end{aligned} $$
(2.12)
Therefore for all positive integers i, j, with i > j,
$$\displaystyle \begin{aligned} f^{-i}W \cap f^{-j}W = f^{-i}(W \cap f^{-j+i}W) = \emptyset \end{aligned}$$
by (2.12), so W is a wandering set. By the conservativity of f, μ(W) = 0, and therefore it follows that
$$\displaystyle \begin{aligned} \mu \left(\bigcup_{i=0}^{\infty}f^{-k}W \right )\leq \sum_{k=0}^{\infty}\mu(f^{-k}W)=0.\end{aligned}$$

Now define $$B = A \setminus \cup _{k=0}^{\infty } f^{-k}W$$ , so μ(B) = μ(A). If x ∈ B, then $$f^{n_1}x \in B$$ for some $$n_1 \in \mathbb N$$ . Then $$f^{n_1}x \in B$$ , so $$f^{n_2}(f^{n_1})x = f^{n_1+n_2}x \in B$$ for some $$n_2 \in \mathbb N$$ . Proceeding inductively, the result follows. □

It follows from the definitions that a nonsingular map f is conservative and ergodic if and only if for all sets $$A,B \in \mathcal B_+$$ , there is a positive integer n such that μ(B ∩ f −n A) > 0. If μ(X) = 1, f is conservative and ergodic if and only if given $$A\in \mathcal B_+$$ , μ(A) < 1, there is a positive integer n such that μ(A c ∩ f −n A) > 0 (see Exercise 9 below).

We next consider topological analogs of the sets C and D that are constructed in Theorem 2.18.

Definition 2.21
If f : X → X is a continuous transformation of a compact space, then x ∈ X is a wandering point if there is an open neighborhood V  of x that is a wandering set. The non-wandering set for f, denoted Ω(f), is the set of points that are not wandering; equivalently,
$$\displaystyle \begin{aligned} \Omega(f) {=} \{ x \in X : \, \mbox{for every neighborhood} \, V \, {\mbox{of}} \, x \, \mbox{there exists} \, n \in \mathbb N \, \mbox{s.t.} \, f^{-n}V \cap V {\neq} \emptyset \}. \end{aligned}$$

There are many connections among the wandering and recurrence sets defined so far and we mention one typical result. We note that since X ∖ Ω(f) is open, the non-wandering set is closed.

Theorem 2.22

If f : X → X is a continuous transformation of a locally compact space, and if μ is a Borel probability measure on X such that f ∗ μ = μ, then μ( Ω(f)) = 1. If μ(U) > 0 for every nonempty open set U, then letting C denote the set of conservativity in the Hopf Decomposition Theorem, C =  Ω(f) = X.

Proof

By Theorem 2.13, we have that μ( Ω(f)) = 1. In particular, if there is an open wandering set W, since μ(f −i W) = μ(W) for all i, μ(W) = 0. Since Ω is closed and μ( Ω) = 1, we must have Ω = X. □

2.3 Noninvertible Maps and Exactness

We introduce an important property of noninvertible maps which is stronger than ergodicity. Many details on exact maps can be found in [26] and [48], and we revisit it in later chapters. This section can be skipped at a first reading.

For a standard measure space, if $$\mathcal B$$ denotes the σ-algebra of Borel sets, then for a nonsingular dynamical system $$(X,\mathcal B,\mu ,f),$$ for each $$k \in \mathbb N$$ ,
$$\displaystyle \begin{aligned}f^{-k}\mathcal B = \{ f^{-k} A\, : \, A \in \mathcal B \}.\end{aligned}$$
We say f is exact if it has a trivial tail field, a σ-algebra of tail sets denoted by $${\mathcal Z} \subset \mathcal B$$ and defined by $$\displaystyle \mathcal {Z}= \displaystyle \cap _{n \geq 0}f^{-n}\mathcal B$$ . We note that $$f^{-1}\mathcal {Z} \subseteq \mathcal {Z} \ (\mu \bmod 0)$$ . Saying $$\mathcal {Z}$$ is trivial means that
$$\displaystyle \begin{aligned}\mathcal{Z} = \{X, \emptyset\} \ (\mu \bmod 0).\end{aligned}$$
Equivalently, f is exact if every set B with the property that
$$\displaystyle \begin{aligned}\mu (f^{-n} (f^n(B)) \bigtriangleup B)=0 \quad \mbox{for all} \ n \in \mathbb N,\end{aligned}$$
has either zero or full measure. Since for each $$A \in \mathcal B$$ ,
$$\displaystyle \begin{aligned}f^{-n}(f^n(A)) =\{x \in X \, : \, f^n(x)=f^n(y) \,\mathrm{for} \, \mathrm{some}\, y \in A\},\end{aligned}$$
the sets $$\{f^{-n}(f^n(A))\}_{n \in \mathbb N}$$ form an increasing sequence in $$\mathcal B$$ . Each set $$A \in \mathcal B_+$$ determines a tail set in $$\mathcal B_+$$ by
$$\displaystyle \begin{aligned}\mathit{Tail}(A) = \bigcup_{n\in \mathbb N}f^{-n}(f^n(A)). \end{aligned}$$

We note that exactness is only a property of noninvertible maps because if f is invertible, f −1(f(x)) = x for μ-a.e. x ∈ X so for every set $$A \in \mathcal B$$ , Tail(A) = A (μ mod  0). If f is exact, then for every set $$A \in \mathcal B_+$$ , Tail(A) = X (μ mod  0).

To see that exactness is strictly stronger than ergodicity, we assume f is exact and suppose that f −1 A = A for some $$A \in \mathcal B_+$$ . Then A = f(f −1 A) = f(A), and for all $$n \in \mathbb N$$ , f −n(f n(A)) = A (μ mod  0). Therefore $$A \in \mathit {Tail}(A) \cap \mathcal B_+$$ , so μ(X ∖ A) = 0 and f is ergodic. Since many invertible maps are ergodic, the notions are not equivalent.

Remark 2.23
  1. 1.

    Every one-sided Bernoulli shift is exact, but the converse is false; however every exact map exhibits highly mixing behavior as we will see in later chapters.

     
  2. 2.

    If $$(X,\mathcal B,\mu ,f)$$ is exact, then for each $$n\in \mathbb N$$ , f n is ergodic.

    To prove this, if f n is not ergodic, then there exists a set A, with μ(A) > 0 and μ(X ∖ A) > 0 such that A = f −n(f n(A)) (μ mod  0). Then A ⊂Tail(A), so f cannot be exact.

     
  3. 3.

    For an example of an ergodic nonexact dynamical system, consider f(x) = 2x ( mod 1) with respect to Lebesgue measure m on [0, 1), and the invertible Bernoulli shift σ on Σn with respect to μ p determined by the probability vector p. The map we want is f × σ on the product space [0, 1) × Σn with the product measure m × μ p. Let $$\mathcal B_{I \times \Sigma _n}$$ denote the collection of Borel sets on the product space. Every set of the form [0, 1) × C, C ⊂ Σn Borel, is in $$\displaystyle \cap _{n \geq 0}(f\times \sigma )^{-n}\mathcal B_{I \times \Sigma _n}$$ . Therefore there are many tail sets in $$\mathcal B_{I \times \Sigma _n}$$ for f × σ, so the map is not exact. It is an exercise to show that the map is ergodic.

     

We give another characterization of exactness, useful in the setting of complex dynamics, in Chapter 5, Proposition 5.​28.

Exercises
  1. 1.

    Show that for a Borel space $$(X,\mathcal B,\mu )$$ , if f is measurable, then f ∗ μ is a measure.

     
  2. 2.

    Show that the standing assumption (1) of nonsingularity, preceding Definition 2.9, implies the following: μ(X ∖ f(X)) = 0, and therefore f is surjective in the sense that f(X) = X (μ mod  0).

     
  3. 3.

    Show that if A is f-invariant, then X ∖ A is f-invariant as well.

     
  4. 4.

    Show that Example 2.5 (1) is a completely dissipative dynamical system if α ≠ 0, and f is not ergodic.

     
  5. 5.

    Show that for x > 0, limk→∞ f k(x) = 1 in Example 2.5 (4), and that this is false for all other $$x \in \mathbb R$$ .

     
  6. 6.

    Show that if $$f:(X,\mathcal B,\mu ) \rightarrow (X,\mathcal B,\mu )$$ is nonsingular, and for every $$A \in \mathcal B$$ , such that μ(A) ∈ (0, ∞), there exist some integers 0 < i < j ≤ 1∕μ(A) such that μ(f −j A ∩ f −i A) > 0, then f is recurrent. (Do not assume f preserves μ or that μ(X) = 1.)

     
  7. 7.

    Show that if a nonsingular dynamical system $$(X,\mathcal B,\mu , f)$$ (μ finite or infinite) is recurrent, then f is conservative.

     
  8. 8.

    Assume $$f:(X, \mathcal B, \mu ) \rightarrow (X, \mathcal B, \mu )$$ is nonsingular, μ is σ-finite, and f is invertible. Show that there exists a wandering set W such that $$D = \cup _{j=- \infty }^{\infty } f^j(W)$$ , where D is the dissipative set from Theorem 2.18.

     
  9. 9.
    Assume $$(X,\mathcal B,\mu ,f) $$ is a nonsingular dynamical system.
    1. (a)

      Prove that f is conservative and ergodic if and only if for all sets $$A,B \in \mathcal B_+$$ , there is a positive integer n such that μ(B ∩ f −n A) > 0.

       
    2. (b)

      If μ(X) = 1, show f is conservative and ergodic if and only if for every measurable A, with μ(A) ∈ (0, 1), there is a positive integer n such that μ(A c ∩ f −n A) > 0

       
     
  10. 10.

    Show that if $$f:(X,\mathcal B,\mu ) \rightarrow (X,\mathcal B,\mu )$$ is an invertible, dissipative, ergodic, nonsingular transformation of a σ-finite space, then f is isomorphic to the map x↦x + 1 on $$\mathbb Z$$ with an appropriately weighted counting measure on $$\mathbb Z$$ .