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A variational principle for holomorphic correspondences

Subith Gopinathan, Shrihari Sridharan

Abstract

In this paper, we consider a dynamical system on the Riemann sphere that evolves through a set-valued map, namely a holomorphic correspondence. Analogous to the investigation of the dynamics effected by a continuous map defined on a compact metric space, wherein the concept of measure-theoretic entropy of the map and its utility in defining the pressure of a function are well-studied, we define the measure-theoretic entropy of a holomorphic correspondence and use the same to define the pressure of continuous functions. These ideas naturally lead to the formulation of a variational principle in the context of the dynamics of a holomorphic correspondence.

A variational principle for holomorphic correspondences

Abstract

In this paper, we consider a dynamical system on the Riemann sphere that evolves through a set-valued map, namely a holomorphic correspondence. Analogous to the investigation of the dynamics effected by a continuous map defined on a compact metric space, wherein the concept of measure-theoretic entropy of the map and its utility in defining the pressure of a function are well-studied, we define the measure-theoretic entropy of a holomorphic correspondence and use the same to define the pressure of continuous functions. These ideas naturally lead to the formulation of a variational principle in the context of the dynamics of a holomorphic correspondence.
Paper Structure (11 sections, 10 theorems, 51 equations)

This paper contains 11 sections, 10 theorems, 51 equations.

Table of Contents

  1. Introduction
  2. Some notations
  3. The push-forward of a measure
  4. Proof:
  5. An intermediate entropy
  6. Proof:
  7. Measure-theoretic entropy
  8. Proof:
  9. Proof:
  10. The Ruelle operator
  11. Proof:

Key Result

Lemma 3.2

$\mathscr{S}^{\Gamma}$ is a weak*-compact, convex subset of $\mathscr{M} \left( \widehat{\mathbb{C}} \right)$.

Theorems & Definitions (16)

  • Definition 1.1
  • Definition 2.1
  • Definition 2.2
  • Definition 3.1
  • Lemma 3.2
  • Theorem 4.1
  • Lemma 4.2
  • Definition 5.1
  • Theorem 5.2: Variational principle for a holomorphic correspondence
  • Corollary 5.3
  • ...and 6 more