Typical points for one-parameter families of piecewise expanding maps of the interval

TL;DR

This paper provides conditions under which a point is typical for the invariant measure in one-parameter families of piecewise expanding maps, including specific cases like skew tent maps, ensuring statistical properties hold for almost all parameters.

Contribution

It establishes sufficient conditions for typicality of points in parameterized families of piecewise expanding maps, covering various classes such as $eta$-transformations and skew tent maps.

Findings

Turning point is almost surely typical for skew tent maps.

Conditions ensure pointwise convergence of empirical measures for almost all parameters.

Applicable to $C^{1,1}(L)$-regular families of maps.

Abstract

Let $I\subset\mathbb{R}$ be an interval and $T_a:[0,1]\to[0,1]$, $a\in I$, a one-parameter family of piecewise expanding maps such that for each $a\in I$ the map $T_a$ admits a unique absolutely continuous invariant probability measure $\mu_a$. We establish sufficient conditions on such a one-parameter family such that a given point $x\in[0,1]$ is typical for $\mu_a$ for a full Lebesgue measure set of parameters $a$, i.e. $$ \frac{1}{n}\sum_{i=0}^{n-1}\delta_{T_a^i(x)} \overset{\text{weak-}*}{\longrightarrow}\mu_a,\qquad\text{as} n\to\infty, $$ for Lebesgue almost every $a\in I$. In particular, we consider $C^{1,1}(L)$-versions of $\beta$-transformations, skew tent maps, and Markov structure preserving one-parameter families. For the skew tent maps we show that the turning point is almost surely typical.