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

Abstract

Let $ 1<β< 2 $, the sequence $α(β)=α(β)_1α(β)_2\dotsb $ be the quasi-greedy $ β$-expansion of $ 1 $, and $ t\in [0,1) $ be a bifurcation parameter. The $β$-transformation is defined to be $T_β(x)=βx (mod 1) $ for $x\in [0,1)$. The Hausdorff dimension of the survivor set $K(t)=\{x\in [0,1)\colon T_β^k(x)\not\in (0,t), \forall k\geq0\} $ is equal to $ -\frac{\lnλ}{\lnβ} $ under the condition that $ \sum_{i=k}^{\infty}\frac{α(β)_i }{β^i}\geq t $ for any $ k\geq 1 $, where $ λ\in (0,1) $ is the smallest positive solution of the equation $\sum_{n=1}^{\infty}(α(β)_n-t_n)x^n=1$ with $(t_n) $ being the quasi-greedy $β$-expansion of $t$. And the local Hölder exponent of the Hausdorff dimension function of $K(t) $ is larger than the value of the function itself.