Invariant measure for double base expansions

TL;DR

This paper investigates the dynamical properties of greedy and lazy maps associated with double base expansions, establishing the existence of unique invariant measures and their implications for the measure of points with unique expansions.

Contribution

It proves the existence and uniqueness of absolutely continuous invariant measures for the greedy and lazy maps in double base expansions and analyzes their dynamical properties.

Findings

Both maps have unique absolutely continuous invariant measures.

The invariant measures are equivalent to Lebesgue measure on specific intervals.

Under certain conditions, the set of points with unique expansions has Lebesgue measure zero.

Abstract

Given a pair $Q=(q_0,q_1)\in(1,\infty)^2$ with $q_0+q_1\ge q_0q_1$, a sequence $(c_i)\in\set{0,1}^\infty$ is called a $Q$-expansion of $x$ if<br/>\begin{equation*}<br/>x=\sum_{i=1}^{\infty}\frac{c_i}{q_{c_1}\cdots q_{c_i}}.<br/>\end{equation*}<br/>We primarily study the dynamical properties of the greedy and lazy maps, which are the piecewise-linear maps on the interval $I_Q=[0,\,1/(q_1-1)]$ defined by the corresponding algorithms for $Q$-expansions. <br/>We show that the greedy and lazy maps each of which has a unique absolutely continuous invariant probability measure, equivalent to the Lebesgue measure on the intervals<br/>\begin{equation*}<br/>\left[0,\frac{q_0}{q_1}\right)\qtq{and}\left(\frac{q_1}{q_0(q_1-1)}-1,\frac{1}{q_1-1}\right],<br/>\end{equation*}<br/>respectively. <br/>Furthermore, the corresponding dynamical systems are exact on $I_Q$. <br/>As a dynamical consequence, under the stronger condition $q_0+q_1>q_0q_1$ the set of points having unique $Q$-expansions has Lebesgue measure zero, and almost every $x\in I_{Q}$ admits a continuum of $Q$-expansions.