The-Hausdorff-dimension-of-the-survivor-set

TL;DR

This paper determines the Hausdorff dimension of survivor sets in beta-transformations, linking it to the quasi-greedy beta-expansion and a specific equation involving the parameter t.

Contribution

It provides an explicit formula for the Hausdorff dimension of survivor sets in beta-transformations based on the quasi-greedy expansion and a unique solution to a power series equation.

Findings

Hausdorff dimension equals -ln(λ)/ln(β) under certain conditions

Dimension depends on the quasi-greedy beta-expansion of t

The local Hölder exponent exceeds the dimension value

Abstract

Let $ 1<\beta< 2 $, the sequence $\alpha(\beta)=\alpha(\beta)_1\alpha(\beta)_2\dotsb $ be the quasi-greedy $ \beta $-expansion of $ 1 $, and $ t\in [0,1) $ be a bifurcation parameter. The $\beta$-transformation is defined to be $T_{\beta}(x)=\beta x (mod 1) $ for $x\in [0,1)$. The Hausdorff dimension of the survivor set $K(t)=\{x\in [0,1)\colon T_{\beta}^k(x)\not\in (0,t), \forall k\geq0\} $ is equal to $ -\frac{\ln\lambda}{\ln\beta} $ under the condition that $ \sum_{i=k}^{\infty}\frac{\alpha(\beta)_i }{\beta^i}\geq t $ for any $ k\geq 1 $, where $ \lambda\in (0,1) $ is the smallest positive solution of the equation $\sum_{n=1}^{\infty}(\alpha(\beta)_n-t_n)x^n=1$ with $(t_n) $ being the quasi-greedy $\beta$-expansion of $t$. And the local H\"older exponent of the Hausdorff dimension function of $K(t) $ is larger than the value of the function itself.