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

Daniel Schnellmann (1)

1 DMA - Département de Mathématiques et Applications - ENS Paris

Let $\I\subset\real$ 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 $\be$-transformations, piecewise expanding unimodal maps, and Markov structure preserving one-parameter families. For families of piecewise expanding unimodal maps we show that the turning point is almost surely typical whenever the family is transversal.

Mots clés: Typical points; piecewise expanding maps of the interval; absolutely continuous invariant measure

Domaines: Systèmes dynamiques [math.DS]

Daniel Schnellmann. Typical points for one-parameter families of piecewise expanding maps of the interval. 2011. ⟨hal-00609175⟩