Introduction to dynamics

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Dynamics is the study of iterations. Given a map $T:X\to X$ one seeks in dynamics to understand for points $x$ in $X$ the sequence $$x,T(x),T(T(x)),T(T(T(x))),T(T(T(T(x)))),\dots$$ obtained by repeated application of $T$. The above sequence is known as the orbit of $x$. For brevity we write $T^n(x)$ for the point obtained from $x$ by $n$ application of $T$.

Exactly what it means to "understand" the sequence depends on the context. For example, if $X$ is a topological space then one might want to know how the closure $$\overline{\{T^n(x):n\in \mathbb{N}\}}$$ of the orbit of $x$ fluctuates as $x$ varies, or how it it influenced by the properties of the map $T$.

Ergodic theory is about the probabilistic and statistical properties of dynamical systems. The prototypical quantity in the statistical analysis of dynamical systems is the frequency $${\displaystyle \frac{|\{1\le n\le N:T^n(x)\in E\}|}{N}}$$ with which the orbit belongs to a given set $E\subset X$. The beginnings of ergodic theory were in the analysis of such frequencies in situations where the map $T$ was related to problems in statistical physics.

The standard context in ergodic theory is that $X$ is a space equipped with a probability measure. Thus we have a σ-algebra $\mathcal{B}$ on $X$ and a measure $\mu$ on $\mathcal{B}$ with the property that $\mu (X)=1$. One assumes that the map $T:X\to X$ is $(\mathcal{B},\mathcal{B})$ measurable and, crucially, that the measure $\mu$ is $T$-invariant. By this is meant that $T$ does not change the sizes of subsets of $X$ in the sense that $$\mu (T^{-1}(B))=\mu (B)$$ for all $B\in \mathcal{B}$. If one imagines the application of $T$ as advancing the status of all points in $X$ by a single unit of time and attributes a probabilistic meaning to the output of $\mu$ then we can understand the above invariance in the following terms: an event is no more or less likely today than it was yesterday.

Over the past sixty years there has been a great deal of progress in abstract ergodic theory and ergodic theory has made tremendous contributions to other areas of mathematics. Here are some highlights.

• Furstenberg's proof of Szemerédi's theorem and the subsequent progress made on problems in density combinatorics by dynamical methods.
• Margulis's resolution of the Oppenheim conjecture, and Ratner's work on the classification of invariant measures for unipotent flows on homogeneous spaces.
• Progress on the Hardy-Littlewood conjecture by Einsiedler, Katok and Lindenstrauss.
• Eskin and Mirzakhani's work on $\mathsf{SL}(2,\mathbb{R})$ invariant measures on the moduli spaces of Abelian differentials on flat surfaces.

Our first dynamical system

Let us begin to study dynamical systems with a specific example. We will take $X=[0,1)$ and define $$T(x)=x+\alpha mod1$$ where we fix the real paramater $\alpha$ in advance. We can immediately calculate that $$T^n(x)=x+n\alpha mod1$$ for all $n\in \mathbb{N}$. We see the orbits of $T$ are closely related to the number-theoretic properties of the sequence $n\alpha mod1$.

Before we delve into ergodic theory, let us understand these orbits topologically. We will do this by equipping $X$ with the metric $$\mathsf{d}(x,y)=min\{|x-y|,1-|x-y|\}$$ in which the sequence $n\mapsto 1-\frac{1}{n}$ converges to zero.

First let us ask whether the orbit $$\mathsf{orb}(x,T)=\{T^n(x):n\in \mathbb{N}\}$$ can ever contain the point $x$ itself. Suppose therefore that $x=T^n(x)$ for some $n\in \mathbb{N}$. This means $$x=x+n\alpha mod1$$ and there must then be some $k\in \mathbb{Z}$ with $n\alpha =k$. It must then be the case that $\alpha$ is rational and that $\mathsf{orb}(x,T)$ is finite.

From the above, if $\alpha$ is irrational then the orbit $\mathsf{orb}(x,T)$ does not contain $x$. What can we say about it? In fact it is always dense.

Theorem

Fix $\alpha$ irrational and set $T(x)=x+\alpha mod1$ on $[0,1)$. For every $x$ in $[0,1)$ the orbit $\mathsf{orb}(x,T)$ is dense in $[0,1)$.

Proof:

Fix $x\in [0,1)$. First notice that $T^n(x)$ and $T^m(x)$ are distinct whenever $n\ne m$. Fixing $k\in \mathbb{N}$ there must therefore be $m<n$ with $$\mathsf{d}(T^n(x),T^m(x))<\frac{1}{k}$$ which is to say that $(n-m)\alpha mod1$ is quite small. If we look then at the sequence $$T^{n-m}(x),T^{2(n-m)}(x),T^{3(n-m)}(x),\dots$$ we see that every point in $[0,1)$ is within $\frac{1}{k}$ of a point in this sequence. Since $k\in \mathbb{N}$ was arbitrary the orbit $\mathsf{orb}(x,T)$ is dense in $[0,1)$.▮

An example of invariance

Is there any connection between the Lebesgue measure $\mu$ on $[0,1)$ and the dynamics of $T(x)=x+\alpha mod1$? Specifically, is $\mu$ invariant for $T$? By this we mean that $\mu (B)$ and $\mu (T^{-1}B)$ are the same for every Borel set $B\subset [0,1)$ where $$T^{-1}(B)=\{x\in X:T(x)\in B\}$$ as before.

How can we verify an equality of measures? Fortunately, it suffices to verify equality on the intervals.

Theorem

Let $\mu$ and $\nu$ be Borel measures on $\mathbb{R}$. If $\mu (I)=\nu (I)$ for every open interval $I\subset \mathbb{R}$ then $\mu =\nu$.

Proof:

This is a consequence of the π-𝜆 theorem discussed in the third worksheet.▮

Let us use this to verify $\mu$ is $T$-invariant. Define $\nu (B)=\mu (T^{-1}(B))$. It is straightforward to check that $\nu$ is also a Borel measure on $X$. Fix an open interval $(a,b)\subset [0,1)$. We know that $\mu ((a,b))=b-a$. To finish we must determine $T^{-1}((a,b))$ and calculate is size with respect to $\mu$. There are two possibilities: either $T^{-1}((a,b))$ is an interval of length $b-a$ or it is a disjoint union of two intervals whose lengths sum to $b-a$. Which possibility arises depends on whether the interval $(a,b)-\alpha$ contains an integer or not. Either way, the set $(a,b)$ has measure $b-a$ with respect to $\nu$ and we are done.