Invariant measure of rotational beta expansion and a problem of Tarski

TL;DR

This paper investigates the existence of absolutely continuous invariant measures for a class of piecewise expanding maps in Euclidean space, providing new proofs and extending previous results using Tarski's problem.

Contribution

It offers a new proof and improvements for the existence of absolutely continuous invariant measures in rotational beta expansions, connecting Tarski's problem to dynamical systems.

Findings

When the similarity ratio is at least m+1, an absolutely continuous invariant measure exists.

The method applies to 2D cases, improving previous results by Akiyama-Caalim.

The approach links invariant measure theory with Tarski's problem in geometry.

Abstract

We study invariant measures of a piecewise expanding map in $\mathbb{R}^m$ defined by an expanding similitude modulo lattice. Using the result of Bang on a problem of Tarski, we show that when the similarity ratio is not less than $m+1$, it has an absolutely continuous invariant measure equivalent to the $m$-dimensional Lebesgue measure, under some mild assumption on the fundamental domain. Applying the method to the case $m=2$, we obtain an alternative proof of the result in Akiyama-Caalim:2015 together with some improvement.