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 \[ \overline{ \{ T^n(x) : n \in \N \} } \] 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 \[ \dfrac{|\{1 \le n \le N : T^n(x) \in E \}|}{N} \] 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 $\B$ on $X$ and a measure $\mu$ on $\B$ with the property that $\mu(X) = 1$. One assumes that the map $T : X \to X$ is $(\B,\B)$ 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 $B \in \B$. 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.
Let us begin to study dynamical systems with a specific example. We will take $X = [0,1)$ and define \[ T(x) = x + \alpha \bmod 1 \] where we fix the real paramater $\alpha$ in advance. We can immediately calculate that \[ T^n(x) = x + n \alpha \bmod 1 \] for all $n \in \N$. We see the orbits of $T$ are closely related to the number-theoretic properties of the sequence $n \alpha \bmod 1$.
Before we delve into ergodic theory, let us understand these orbits topologically. We will do this by equipping $X$ with the metric \[ \met(x,y) = \min \{ |x-y|, 1 - |x-y| \} \] in which the sequence $n \mapsto 1 - \tfrac{1}{n}$ converges to zero.
First let us ask whether the orbit \[ \orb(x,T) = \{ T^n(x) : n \in \N \} \] can ever contain the point $x$ itself. Suppose therefore that $x = T^n(x)$ for some $n \in \N$. This means \[ x = x + n\alpha \bmod 1 \] and there must then be some $k \in \Z$ with $n \alpha = k$. It must then be the case that $\alpha$ is rational and that $\orb(x,T)$ is finite.
From the above, if $\alpha$ is irrational then the orbit $\orb(x,T)$ does not contain $x$. What can we say about it? In fact it is always dense.
Fix $\alpha$ irrational and set $T(x) = x + \alpha \bmod 1$ on $[0,1)$. For every $x$ in $[0,1)$ the orbit $\orb(x,T)$ is dense in $[0,1)$.
Fix $x \in [0,1)$. First notice that $T^n(x)$ and $T^m(x)$ are distinct whenever $n \ne m$. Fixing $k \in \N$ there must therefore be $m \l n$ with \[ \met(T^n(x),T^m(x)) \l \frac{1}{k} \] which is to say that $(n-m) \alpha \bmod 1$ 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 $\tfrac{1}{k}$ of a point in this sequence. Since $k \in \N$ was arbitrary the orbit $\orb(x,T)$ is dense in $[0,1)$.▮
Is there any connection between the Lebesgue measure $\mu$ on $[0,1)$ and the dynamics of $T(x) = x + \alpha \bmod 1$? 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.
Let $\mu$ and $\nu$ be Borel measures on $\R$. If $\mu(I) = \nu(I)$ for every open interval $I \subset \R$ then $\mu = \nu$.
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.