Quantitative recurrence properties for beta-dynamical system. (English) Zbl 1284.11113
Summary: This note is concerned with the quantitative recurrence properties of beta dynamical system \(([0,1],T_\beta)\) for general \(\beta >1\); the size of points with the prescribed recurrence rate is determined. More precisely, Hausdorff dimensions of the sets
\[ \{x\in [0,1]: | T_\beta^n x-x|<\psi(n)\;\text{for infinitely many}\;n\in\mathbb N\} \] and \[ \{x\in [0,1]: | T_\beta^n x-x|<e^{-\sum_{j=0}^{n-1} f(T_\beta^j x)}\;\text{for infinitely many}\;n\in\mathbb N\} \] are obtained completely, where \(\psi \) is a positive function defined on \(\mathbb N\) and \(f\) is a positive continuous function on \([0,1]\). Besides, the pressure function \(P(f,T_\beta )\) with a continuous potential \(f\) is proven to be continuous with respect to \(\beta \).
\[ \{x\in [0,1]: | T_\beta^n x-x|<\psi(n)\;\text{for infinitely many}\;n\in\mathbb N\} \] and \[ \{x\in [0,1]: | T_\beta^n x-x|<e^{-\sum_{j=0}^{n-1} f(T_\beta^j x)}\;\text{for infinitely many}\;n\in\mathbb N\} \] are obtained completely, where \(\psi \) is a positive function defined on \(\mathbb N\) and \(f\) is a positive continuous function on \([0,1]\). Besides, the pressure function \(P(f,T_\beta )\) with a continuous potential \(f\) is proven to be continuous with respect to \(\beta \).
MSC:
| 11K55 | Metric theory of other algorithms and expansions; measure and Hausdorff dimension |
| 28A80 | Fractals |
| 37A45 | Relations of ergodic theory with number theory and harmonic analysis (MSC2010) |
Keywords:
\(\beta \)-expansion; quantitative recurrence properties; pressure function; Hausdorff dimensionReferences:
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