Multifractal analysis of the growth rate of digits in Schneider's $ p $ -adic continued fraction dynamical system
Year: 2024
Type: article
Abstract: Let $ \mathbb{Z}_p $ be the ring of $ p $-adic integers, and $ a_n(x) $ be the $ n $-th digit of the Schneider's $ p $-adic continued fraction of $ x\in p\mathbb{Z}_p $. We study the growth rate of the digits $ \{a_n(x)\}_{n\geq1} $ from the viewpoint of multifractal analysis. The Hausdorff dimension of the set$ E_{\sup}(\psi) = \Big\{x\in p\mathbb{Z}_p:\ \limsup\limits_{n\to\infty}\frac{a_n(x)}{\psi(n)} = 1\Big\} $is completely determined for any $ \psi:\mathbb{N}\to\mathbb{R}^{+} $ satisfying $ \psi(n)\to \infty $ as $ n\to\infty $. As an application, we also calculate the Hausdorff dimension of the intersection sets$ E^{\sup}_{\inf}(\psi,\alpha_1,\alpha_2) = \left\{x\in p\mathbb{Z}_p:\liminf\limits_{n\rightarrow \infty}\dfrac{a_n(x)}{\psi(n)} = \alpha_1,\; \limsup\limits_{n\rightarrow \infty}\dfrac{a_n(x)}{\psi(n)} = \alpha_2\right\} $for the above function $ \psi $ and $ 0\leq\alpha_1<\alpha_2\leq\infty $.
DOI: 10.3934/dcds.2024170
Authors: Kunkun Song, Wanlou Wu, Yueli Yu, Sainan Zeng
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: diamond
OpenAlex ID: W4405219509