Rotational beta expansion: Ergodicity and Soficness

TL;DR

This paper investigates the ergodic and sofic properties of a family of piecewise expanding maps on the plane generated by rotations and similitudes, establishing conditions for unique invariant measures and soficness related to algebraic properties of parameters.

Contribution

It introduces new constants determining ergodic properties and characterizes when the system is sofic based on algebraic conditions involving Pisot numbers and rationality.

Findings

Unique absolutely continuous invariant measure for large 2

System is equivalent to Lebesgue measure for 2 above a threshold

Soficness characterized by algebraic conditions, with counterexamples for certain parameters

Abstract

We study a family of piecewise expanding maps on the plane, generated by composition of a rotation and an expansive similitude of expansion constant $\beta$. We give two constants $B_1$ and $B_2$ depending only on the fundamental domain that if $\beta>B_1$ then the expanding map has a unique absolutely continuous invariant probability measure, and if $\beta>B_2$ then it is equivalent to $2$-dimensional Lebesgue measure. Restricting to a rotation generated by $q$-th root of unity $\zeta$ with all parameters in $\mathbb{Q}(\zeta,\beta)$, it gives a sofic system when $\cos(2\pi/q) \in \mathbb{Q}(\beta)$ and $\beta$ is a Pisot number. It is also shown that the condition $\cos(2\pi/q) \in \mathbb{Q}(\beta)$ is necessary by giving a family of non-sofic systems for $q=5$.