Some Ergodic Theory

Let us show that the map $f = f_4$ has an absolutely continuous invariant probability measure $\mu$. (Remember that a measure is a map from a certain collection (an algebra) of subsets $A$ of $[0, 1]$ to $\mathbb{R}^+$. In this case the algebra is the Borel $\sigma$-algebra). This map $\mu$ is additive, maps the empty set to $0$ and the set $A$ to $1$.) For $\mu$ to be invariant means $\mu (f^{-1} (A) ) = \mu (A)$ for all measurable $A$, while absolute continuity means $\mu (A) > 0 \Rightarrow A$ has positive Lebesgue measure (i.e. it is not a null set). The measure $\mu$ is also ergodic, which means that if $f^{-1} (A) \triangle A$ (disjoint union of sets) has $\mu$ measure zero, then either $\mu (A) = 0$ or $\mu (A^C) = 0$.

In order to see that one has such a measure, first notice that Lebesgue measure is invariant for the map $g$. It is clearly absolutely continuous. Let us first show that the Lebesgue measure is also ergodic. For ergodicity we need to show that if $g^{-1} (A) = A$ then either $\lambda (A) = 0$ or $\lambda (A) = 1$. Let us assume $g^{-1} (A) = A$ and $\lambda (A) = 1$. To see this we use the following.

Theorem 2.3 (The Lebesgue Density Theorem)   Let $A \subset [0, 1]$ be Borel measurable and have $\lambda (A) > 0$. Then there exists $X \subset A$ such that $\lambda (A \backslash X) = 0$, and for all $x \in X$, $\lim_{\epsilon \to 0} \lambda (A \cap [x - \epsilon, x + \epsilon]) / 2 \epsilon = 1$.

So assume that $A \subset S^1$, that $g^{-1} (A) = A$ and that $A$ has positive Lebesgue measure. Let $\lambda$ be Lebesgue measure. Let $X$ be the subset of $A$ from the previous theorem. Next take $x \in X$ and $\epsilon = 2^{-(n + 1)}$. Then $\lambda (A \cap [x - \epsilon, x + \epsilon]) / 2 \epsilon = \lambda (g^n (A) \... ...n (x) + 1 / 2) / 1 \leq \lambda (A \cap [g^n (x) - 1 / 2, g^n (x) + 1 / 2]) / 1$, so $\lambda (A) / 1 = 1$. This proves $A$ has measure $1$ and so that Lebesgue measure is ergodic for $g$.

Using the semi-conjugacy to $f_4$ gives that the corresponding measure on $[0, 1]$ defined by $\mu (B) = \lambda (h^{-1} (B))$ for all Borel sets $B$, is also an absolutely continuous invariant probability measure which is ergodic.

The following theorem can be applied when one has an invariant measure.

Theorem 2.4 (The Birkhoff Ergodic Theorem)   Let $f: [0, 1] \to [0, 1]$ have an ergodic invariant Borel measure $\mu$. Then for each continuous function $\phi : [0, 1] \to \mathbb{R}$, $\sum_{i=0}^{n - 1} \phi (f^i (x)) / n \to \int \phi d \mu$ for $\mu$-almost all $x$.

Corollary 2.5   If $\mu$ is absolutely continuous then for Lebesgue-almost all $x$ (as $\mu$ is absolutely continuous we can write Lebesgue instead of $\mu$), $lim_{n \to \infty} \char93 \{0 \leq i \leq n \vert f^i (x) \in A \} / n = \mu (A)$.