Expanding property and statistical laws for $p$-adic subhyperbolic rational maps

TL;DR

The paper develops a non-archimedean (p-adic) theory of expanding dynamics for subhyperbolic rational maps. It constructs an admissible, singular metric capturing expansion near the Julia postcritical set and proves the equivalence between subhyperbolicity and expansion in this metric. Building on this, it proves the existence of exponentially contracting metrics, a Hölder semiconjugacy to a symbolic shift, and a thermodynamic formalism that yields a Central Limit Theorem, Law of the Iterated Logarithm, exponential decay of correlations, and a Large Deviation Principle for invariant measures on the p-adic Julia set under mild compactness and density assumptions. The approach combines non-archimedean analysis with symbolic coding to transfer dynamical properties to statistical laws, providing a p-adic analogue of complex-analytic results for subhyperbolic maps. These results deepen the understanding of ergodic properties in $p$-adic dynamics and open avenues for further investigations into spectral data and invariant measures in Berkovich settings.

Abstract

Let $K$ be a finite extension of the field $\mathbb{Q}_p$ of $p$-adic numbers. A rational map $φ\in K(z)$ of degree at least $2$ is subhyperbolic if each critical point in the $\mathbb{C}_p$-Julia set of $φ$ is eventually periodic. We show that subhyperbolic maps in $K(z)$ exhibit expanding property with respect to some (singular) metric. As an application, under a mild assumption, we establish several statistical laws for such maps in $K(z)$ with compact $\mathbb{C}_p$-Julia sets.

Theorems & Definitions (40)

• Definition 1.1
• Definition 1.2
• Theorem 1.1
• Theorem 1.2
• Lemma 2.1: Fan21
• Lemma 2.2
• Lemma 2.3
• Lemma 2.4: Benedetto19
• Lemma 2.5: Benedetto19
• Proposition 2.6