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Ruelle's zeta function for non-Archimedean rational maps

Yunping Jiang, Chenxi Wu

Abstract

We studied the transfer operators defined over $\mathbb{C}_p$-valued analytic functions for subhyperbolic rational maps on $\mathbb{Q}_p$, and showed that the corresponding Ruelle's zeta functions are meromorphic on $\mathbb{C}_p$. We also used $\mathbb{R}$-valued transfer operators to study the shape of the corresponding Julia sets, and proved a Levin-Sodin-Yuditski type identity for general rational maps on $\mathbb{C}_p$. In all the results above, $\mathbb{Q}_p$ can be replaced with any non-Archimedean local field with characteristic $0$, and $\mathbb{C}_p$ the metric completion of its algebraic closure.

Ruelle's zeta function for non-Archimedean rational maps

Abstract

We studied the transfer operators defined over -valued analytic functions for subhyperbolic rational maps on , and showed that the corresponding Ruelle's zeta functions are meromorphic on . We also used -valued transfer operators to study the shape of the corresponding Julia sets, and proved a Levin-Sodin-Yuditski type identity for general rational maps on . In all the results above, can be replaced with any non-Archimedean local field with characteristic , and the metric completion of its algebraic closure.

Paper Structure

This paper contains 8 sections, 8 theorems, 24 equations.

Table of Contents

  1. Introduction
  2. Notification and Background
  3. Levin-Sodin-Yuditski Type Identity
  4. Cp-valued zeta functions for rational maps on Qp
  5. Nuclear Operators, Trace and Determinant
  6. Transfer Operator and Zeta Function, Hyperbolic Case
  7. Subhyperbolic maps
  8. Transfer operator for real valued weights, Hausdorff dimension

Key Result

Theorem 1.1

When $f$ is a polynomial map on $\mathbb{C}_p$ whose all critical points are non-degenerate and whose critical orbits on $\mathbb{P}^1(\mathbb{C}_p)$ are all disjoint. Then the following two formal power series are identical here $c_i$ are the critical points of $f$, and $\hbox{Fix}(f^{n})$ is the set of fixed points of $f^{n}$. In particular, when both the left-hand side and the right-hand side

Theorems & Definitions (23)

  • Theorem 1.1: Analogy of baladi2002dynamical
  • Theorem 1.2: Analogy of baladi2002dynamical
  • Definition 2.1
  • Remark 2.2
  • Example 2.3
  • Proposition 2.4: hsia2025zeta
  • Example 2.5
  • proof : Proof of Theorem \ref{['lsy']}
  • Definition 4.1
  • Remark 4.2
  • ...and 13 more