A Point is Normal for Almost All Maps $βx + α\mod 1$ or Generalized $β$-Maps

Details A Point is Normal for Almost All Maps $βx + α\mod 1$ or Generalized $β$-Maps A Point is Normal for Almost All Maps $βx + α\mod 1$ or Generalized $β$-Maps

2008-06-18

arxiv.org/abs/0806.2922v1

Ergod. Th & Dynam. Sys. 29 (2009) 1529-1547

Mathematics

Dynamical Systems

Latex, 16 pages

Scientific paper

We consider the map $T_{\alpha,\beta}(x):= \beta x + \alpha \mod 1$, which admits a unique probability measure of maximal entropy $\mu_{\alpha,\beta}$. For $x \in [0,1]$, we show that the orbit of $x$ is $\mu_{\alpha,\beta}$-normal for almost all $(\alpha,\beta)\in[0,1)\times(1,\infty)$ (Lebesgue measure). Nevertheless we construct analytic curves in $[0,1)\times(1,\infty)$ along them the orbit of $x=0$ is at most at one point $\mu_{\alpha,\beta}$-normal. These curves are disjoint and they fill the set $[0,1)\times(1,\infty)$. We also study the generalized $\beta$-maps (in particular the tent map). We show that the critical orbit $x=1$ is normal with respect to the measure of maximal entropy for almost all $\beta$.

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