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The multifractal spectra of certain planar recurrence sets in the continued fraction dynamical system. (English) Zbl 1336.37023

The aim of the present paper is to study some particular planar recurrence sets described by the lower and upper quantitative hitting time indicators in the continued fraction dynamical system from the point of view of multifractal analysis. By considering the relations between the quantitative hitting time indicators and the quantitative recurrence time indicators the authors managed to generalize the result given by L. Peng et al. [Sci. China, Math. 55, No. 1, 131–140 (2012; Zbl 1241.28007)], into a higher-dimensional case. To this direction, the authors by introducing the planar recurrence level sets \[ S(\alpha,\beta):= \{(x,y)\in[0,1)\times[0, 1)/\underline R(x,y)= \alpha,\,\overline R(x,y)= \beta\},\quad 0\leq\alpha\leq\beta\leq+\propto, \] managed to find out that the multifractal spectrum is always degenerate in a sense that it is constant independent of the choice of the level set. Here, \(\underline R(x,y)\), \(\overline R(x,y)\) are the lower and upper quantitative hitting time indicators of \(y\) and \(x\), respectively in the continued fraction dynamical system.

MSC:

37C45 Dimension theory of smooth dynamical systems
37A45 Relations of ergodic theory with number theory and harmonic analysis (MSC2010)

Citations:

Zbl 1241.28007
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References:

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