Dynamics of Hyperbolic Meromorphic Functions

TL;DR

This paper defines hyperbolic meromorphic functions and explores their dynamical behavior, thermodynamic formalism, and geometric properties on Julia sets, establishing key formulas and expanding properties.

Contribution

It introduces a formal definition of hyperbolic meromorphic functions and proves fundamental properties like the Bowen formula and expansion on Julia sets.

Findings

Proves the Bowen formula for hyperbolic functions on the complex plane.

Establishes expanding properties of hyperbolic functions with respect to the Euclidean metric.

Shows that Walters' theory applies to hyperbolic functions on the Riemann sphere.

Abstract

A definition of hyperbolic meromorphic functions is given and then we discuss the dynamical behavior and the thermodynamic formalism of hyperbolic functions on the Julia set. We prove the important expanding properties for hyperbolic functions on the complex plane and with respect to the euclidean metric. We establish the Bowen formula for hyperbolic functions on the complex plane, that is, the Poincare exponent equals to the Hausdorff dimension of the radial Julia set and furthermore, we prove that all the results in the Walters' theory hold for hyperbolic functions on the Riemann sphere.