Paper detail

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

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 $μ_a$. We establish sufficient conditions on such a one-parameter family such that a given point $x\in[0,1]$ is typical for $μ_a$ for a full Lebesgue measure set of parameters $a$, i.e. $$ \frac{1}{n}\sum_{i=0}^{n-1}δ_{T_a^i(x)} \overset{\text{weak-}*}{\longrightarrow}μ_a,\qquad\text{as} n\to\infty, $$ for Lebesgue almost every $a\in I$. In particular, we consider $C^{1,1}(L)$-versions of $β$-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.