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

TL;DR

This paper proves that for most parameters, the orbit of any point under certain linear mod 1 maps is normal with respect to the measure of maximal entropy, and explores the structure of exceptional parameter sets.

Contribution

It establishes that almost all points have normal orbits for a broad class of maps and constructs analytic curves where normality is rare, revealing intricate parameter-dependent behavior.

Findings

Almost all points are normal for the map $T_{eta,eta}$ with respect to the measure of maximal entropy.

Constructs disjoint analytic curves where the orbit of zero is at most one point normal.

Shows the critical orbit $x=1$ is normal for almost all $eta$ with respect to the measure of maximal entropy.

Abstract

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 $\beta$.