Quantitative recurrence properties for beta-dynamical system

This note is concerned with the quantitative recurrence properties of beta dynamical system ([0,1],Tβ)([0,1],Tβ) for general β>1β>1; the size of points with the prescribed recurrence rate is determined. More precisely, Hausdorff dimensions of the sets{x∈[0,1]:|Tβnx−x|<ψ(n) for infinitely many n∈N} and{x∈[0,1]:|Tβnx−x|