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 ${a_n(x)}{n\geq1}$ 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{n\rightarrow\infty}\dfrac{a_n(x)}{\psi(n)}=\alpha_1,~\limsup_{n\rightarrow\infty}\dfrac{a_n(x)}{\psi(n)}=\alpha_2\right}] for the above function $\psi$ and $0\leq\alpha_1<\alpha_2\leq\infty$.