The multifractal spectra for the recurrence rates of beta-transformations

In this paper, we show a handy approximate approach to provide a lower bound of the Hausdorff dimension of a given subset in [0,1) related to β-transformation dynamical system. Here approximation means from special class with β-shift satisfying the specification property or being subshift of finite type to general β>1. As an application, we obtain the multifractal spectra for the recurrence rate of the first return time of β-transformation, including the cases returning to the ball and cylinder.