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. The dimension of the following set
E$(\{\ell_n\}_{n\geq 1},x_0)$ = $\{\beta>1: T_\beta^n1x_0<\beta^{\ell_n}$, for infinitely many n$\in\mathbb{N}}$
is determined, where $x_0$ is a given point in $[0,1]$ and $\{\ell_n\}_{n\geq 1}$ 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 $\{\beta\in\mathbb{R}: \beta>1\}$.
