Diophantine approximation of the orbit of 1 in the dynamical system of bete expansions

Abstract

We consider the distribution of the orbits of the number 1 under the $\beta$-transformations $T_{\beta }$ as $\beta$ varies. Mainly, the size of the set of $\beta >1$ for which a given point can be well approximated by the orbit of 1 is measured by its Hausdorff dimension. That is, the dimension of the following set $$ E\big({\ell_n}_{n\ge 1}, x_0\big)=\Big{β>1: |T^n_β1-x_0|<β^{-\ell_n}, {for infinitely many} n\in \N\Big} $$ is determined, where $x_0$ is a given point in $[0,1]$ and ${{\ell }_n}_{n\ge 1}$ is a sequence of integers tending to infinity as $n\to \infty$. For the proof of this result, the notion of the recurrence time of a word in symbolic space is introduced to characterize the lengths and the distribution of cylinders (the set of $\beta$ with a common prefix in the expansion of 1) in the parameter space $\beta \in \mathbb{R}:\beta >1$.