On the escape rate of unique beta-expansions

TL;DR

This paper investigates the escape rate of points with unique beta-expansions, establishing a formula linking Hausdorff dimension and escape rate, and shows that the escape rate varies as a devil's staircase with respect to beta.

Contribution

It introduces a formula connecting Hausdorff dimension and escape rate for unique beta-expansions and proves the escape rate's devil's staircase behavior as a function of beta.

Findings

Derived a formula linking Hausdorff dimension and escape rate.

Proved the escape rate forms a devil's staircase function.

Established the zero Lebesgue measure of the univoque set.

Abstract

Let $1<\beta \leq 2$. It is well-known that the set of points in $% [0,1/(\beta -1)]$ having unique $\beta $-expansion, in other words, those points whose orbits under greedy $\beta $-transformation escape a hole depending on $\beta $, is of zero Lebesgue measure. The corresponding escape rate is investigated in this paper. A formula which links the Hausdorff dimension of univoque set and escape rate is established in this study. Then we also proved that such rate forms a devil's staircase function with respect to $\beta $.