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The specification approach to equilibrium states for parabolic rational maps

Katelynn Huneycutt, Daniel J. Thompson

Abstract

We develop the specification and orbit-decomposition approach to equilibrium states for parabolic rational maps of the Riemann Sphere. Our result extends the well-known results on uniqueness of equilibrium states in this setting, notably the results of Denker, Przytycki and Urbański. We extend the class of potentials from Hölder to those with the Bowen property on 'good orbits' . We obtain uniqueness of the equilibrium state for potentials satisfying a pressure gap condition which is sharp in the class of potentials we consider. We show that our equilibrium state has the $K$-property, and in particular it has positive entropy. When the potential is Hölder, the theory of equilibrium states is already highly developed. Nevertheless, several interesting results on equilibrium states for Hölder potentials follow readily from our approach. In the family of geometric potentials, we obtain a simple proof of uniqueness of equilibrium states up to the phase transition that occurs at the Hausdorff dimension of the Julia set. For Hölder potentials on parabolic rational maps, we show that hyperbolicity of the potential is equivalent to having a unique equilibrium state which is fully supported. This does not appear to have been stated in the literature before, although it may be considered folklore.

The specification approach to equilibrium states for parabolic rational maps

Abstract

We develop the specification and orbit-decomposition approach to equilibrium states for parabolic rational maps of the Riemann Sphere. Our result extends the well-known results on uniqueness of equilibrium states in this setting, notably the results of Denker, Przytycki and Urbański. We extend the class of potentials from Hölder to those with the Bowen property on 'good orbits' . We obtain uniqueness of the equilibrium state for potentials satisfying a pressure gap condition which is sharp in the class of potentials we consider. We show that our equilibrium state has the -property, and in particular it has positive entropy. When the potential is Hölder, the theory of equilibrium states is already highly developed. Nevertheless, several interesting results on equilibrium states for Hölder potentials follow readily from our approach. In the family of geometric potentials, we obtain a simple proof of uniqueness of equilibrium states up to the phase transition that occurs at the Hausdorff dimension of the Julia set. For Hölder potentials on parabolic rational maps, we show that hyperbolicity of the potential is equivalent to having a unique equilibrium state which is fully supported. This does not appear to have been stated in the literature before, although it may be considered folklore.
Paper Structure (13 sections, 28 theorems, 57 equations, 1 figure)

This paper contains 13 sections, 28 theorems, 57 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Background
  3. Thermodynamic Formalism
  4. Dynamics of rational maps
  5. Hyperbolicity and the hyperbolic metric
  6. Metric expansion properties and the Milnor metric
  7. Proof of main result
  8. Specification
  9. Choice of $\lambda$-decomposition and the Bowen Property
  10. Pressure Estimate
  11. Completing the proof of Theorem \ref{['thm:mainA']}
  12. Family of geometric potentials
  13. Positive entropy and the hyperbolicity criterion

Key Result

Theorem 1.1

Let $f:\widehat{\mathbb{C}}\rightarrow \widehat{\mathbb{C}}$ be a parabolic rational map of degree $d\geq 2$ and $\Omega = \{w_{1},\dots,w_{k}\}$ be the set of its rationally indifferent periodic points. If $\varphi: J(f) \to {\mathbb R}$ is Hölder and $A(\Omega, \varphi)< P(J(f),\varphi)$, then $\v

Figures (1)

  • Figure 1: Pressure of the potential $\phi_{t}(x)=-t\log|f'(x)|$ for a parabolic rational map, $f$. The Hausdorff dimension of $J(f)$ is $h$.

Theorems & Definitions (47)

  • Theorem 1.1
  • Corollary 1.2
  • Theorem 1.3
  • Corollary 1.4
  • Corollary 1.5
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • ...and 37 more