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

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. $$\frac{1}{n}\sum _{i=0}^{n-1}{\delta }_{T_a^i(x)}\stackrel{\text{weak-}\ast }{⟶}{\mu }_a,\text{as}n\to \infty ,$$ 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.

33 pages, 3 figures; inclusion of a new section about almost sure typicality in transversal families of piecewise expanding unimodal maps; in the first part of the paper the conditions in order to obtain almost sure typicality are weakened; several other (small) improvements