A Point is Normal for Almost All Maps $\beta x+\alpha \mathrm{mod}1$ or Generalized $\beta$-Maps

Abstract

We consider the map $T_{\alpha ,\beta }(x):=\beta x+\alpha \mathrm{mod}1$, 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 $\beta$.

Latex, 16 pages