One Day Ergodic Theory Meeting
October 2020
Fix $(X,\mathscr{B},\mu,T)$ an ergodic system. Given $f : X \to \R$ measurable, the skew-product $T_f$ over $X$ is defined by \[ T_f(x,t) = (Tx, t + f(x)) \] on $X \times \R$.
Write $\haar$ for Lebesgue measure on $\R$. The product measure $\mu \otimes \haar$ is an infinite $T_f$ invariant measure.
Main Question: Is $\mu \otimes \haar$ ergodic for $T_f$?
First we should ask if $T_f$ is recurrent.
Definition: The skew-product $T_f$ is recurrent if, for every $B \in \mathscr{B}$ and every $\epsilon > 0$, one can find $n \in \N$ with \[ \mu(B \cap (T^n)^{-1} B \cap \{ x \in X : \left| \sum_{j=0}^{n-1} f(T^n x) \right| < \epsilon \} ) > 0 \]
Theorem (Atkinson): If $\displaystyle\int\!\! f \intd \mu = 0$ then $T_f$ is recurrent.
The converse is true. The theorem is not true for skew products using other fiber groups.
Example: $X = [0,1)$ and $T(x) = x + \alpha \bmod 1$ ($\alpha \notin \mathbb{Q}$)
Today $T$ will be an interval exchange transformation.
Interval exchange transformations are defined by a partition $I_1 \sqcup \cdots \sqcup I_d$ of $[0,1)$ into half-open intervals and translators $\omega_1,\dots,\omega_d$. The associated interval map sends $x$ to $x + \omega_i$ for all $x \in I_i$. The images of the intervals must not overlap.
Example ($d = 2$): $I_1 = [0,1 - \alpha)$ and $I_2 = [1 - \alpha,1)$. We must have $\omega_1 = \alpha$ and $\omega_2 = 1 - \alpha$. Thus exchanges of two intervals are nothing but rotations.
Example ($d = 3$): There are several ways to rearrange the intervals $I_1,I_2,I_3$. All but \[ \omega_1 = |I_2| + |I_3| \qquad \omega_2 = - |I_1| + |I_3| \qquad \omega_3 = - |I_1| - |I_2| \] give degenerate maps.
We will be interested in IET that are linearly recurrent.
Definition: An IET is linearly recurrent if there is a constant $c > 0$ such that every orbit segment $x,\dots,T^n x$ is $c/n$ dense.
For $d = 2$ intervals, linear recurrence is characterized by $\alpha$ having bounded partial quotients.
For the purposes of this talk, linear recurrence gives us the following picture of an IET: every interval $J$ of length at most $c/n$ participates in a tower of at least $n$ intervals, none of which contain a discontinuity of $T$.
A step function is any function of the form \[ f = y_1 1_{[0,x_1)} + y_2 1_{[x_1,x_1 + x_2)} + \cdots + y_{d+1} 1_{[x_1 + \cdots + x_d,1)} \] for $y_i \ne y_{i+1}$ in $\R$ and $x_1 + \cdots + x_{d+1} = 1$ in $(0,1)$.
For zero-mean step functions almost every refers to sets of full Lebesgue measure on the surface \[ x_1 y_1 + \cdots + x_{d+1} y_{d+1} = 0 \] of mean-zero step functions.
Theorem (with J. Chaika): Let $T$ be a linearly recurrent IET. For almost every mean-zero step function $f$ the skew-product $T_f$ is ergodic with respect to $\mu \otimes \haar$.
If the jumps of our step function are essential values (Schmidt) and generate a dense subgroup of $\R$ then $T_f$ is ergodic.
Since $T_f$ is recurrent (Atkinson) we can look at the induced transformation $S_{f,Q}$ on $[0,1) \times [-Q,Q]$.
If we are in a very nice situation then we can get invariance under jump discontinuities of our step function.
The very nice situation can be obtained almost surely by perturbation.