Multifractal analysis of the growth rate of digits in Schneider's $p$-adic continued fraction dynamical system

Abstract

Let ${\mathbb{Z}}_p$ be the ring of $p$-adic integers and $a_n(x)$ be the $n$-th digit of Schneider's $p$-adic continued fraction of $x\in p{\mathbb{Z}}_p$. We study the growth rate of the digits $\{a_n(x){\}}_{n\ge 1}$ from the viewpoint of multifractal analysis. The Hausdorff dimension of the set \[E_{\sup}(ψ)=\Big\{x\in p\mathbb{Z}_p:\ \limsup\limits_{n\to\infty}\frac{a_n(x)}{ψ(n)}=1\Big\}\] 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 \[E^{\sup}_{\inf}(ψ,α_1,α_2)=\left\{x\in p\mathbb{Z}_p:\liminf_{n\rightarrow\infty}\dfrac{a_n(x)}{ψ(n)}=α_1,~\limsup_{n\rightarrow\infty}\dfrac{a_n(x)}{ψ(n)}=α_2\right\}\] for the above function $\psi$ and $0\le {\alpha }_1<{\alpha }_2\le \infty$.