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

Abstract

We consider the map $T_{α,β}(x):= βx + α\mod 1$, which admits a unique probability measure of maximal entropy $μ_{α,β}$. For $x \in [0,1]$, we show that the orbit of $x$ is $μ_{α,β}$-normal for almost all $(α,β)\in[0,1)\times(1,\infty)$ (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 $μ_{α,β}$-normal. These curves are disjoint and they fill the set $[0,1)\times(1,\infty)$. We also study the generalized $β$-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 $β$.