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Bowen's formula for a rational graph-directed Markov system

Tadashi Arimitsu, Johannes Jaerisch, Hiroki Sumi, Takayuki Watanabe

Abstract

We establish Bowen's formula for the Julia set of a non-elementary, expanding, irreducible and aperiodic rational graph-directed Markov system satisfying the backward separating condition. Towards this end, we shall prove that the associated skew product map is topologically exact on the skew product Julia set, and satisfies the density of repelling periodic points. Moreover, we give a criterion for expandingness in terms of hyperbolicity.

Bowen's formula for a rational graph-directed Markov system

Abstract

We establish Bowen's formula for the Julia set of a non-elementary, expanding, irreducible and aperiodic rational graph-directed Markov system satisfying the backward separating condition. Towards this end, we shall prove that the associated skew product map is topologically exact on the skew product Julia set, and satisfies the density of repelling periodic points. Moreover, we give a criterion for expandingness in terms of hyperbolicity.
Paper Structure (8 sections, 18 theorems, 46 equations)

This paper contains 8 sections, 18 theorems, 46 equations.

Table of Contents

  1. Introduction and statement of results
  2. Rational graph directed Markov systems
  3. Main results
  4. Proof of main results
  5. The density of repelling periodic points and the topological exactness of a skew product map
  6. Hyperbolicity implies expandingness
  7. Bowen's formula
  8. Example

Key Result

Proposition 2.1

Let $S$ be a rational GDMS and $\tilde{f}$ be the skew product map associated with $S$. Then, we have the following.

Theorems & Definitions (47)

  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Definition 1.4
  • Proposition 2.1
  • proof
  • Proposition 2.2
  • proof
  • Claim
  • Definition 2.3
  • ...and 37 more