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Constrained Ergodic optimization for generic continuous functions

Shoya Motonaga, Mao Shinoda

TL;DR

The analogical result that for any dynamical system on a compact metric space with the specification property and for a generic continuous function f every invariant probability measure that maximizes the space average of f must have zero entropy is established.

Abstract

One of the fundamental results of ergodic optimization asserts that for any dynamical system on a compact metric space with the specification property and for a generic continuous function $f$ every invariant probability measure that maximizes the space average of $f$ must have zero entropy. We establish the analogical result in the context of constraint ergodic optimization, which is introduced by Garibaldi and Lopes (2007).

Constrained Ergodic optimization for generic continuous functions

TL;DR

The analogical result that for any dynamical system on a compact metric space with the specification property and for a generic continuous function f every invariant probability measure that maximizes the space average of f must have zero entropy is established.

Abstract

One of the fundamental results of ergodic optimization asserts that for any dynamical system on a compact metric space with the specification property and for a generic continuous function every invariant probability measure that maximizes the space average of must have zero entropy. We establish the analogical result in the context of constraint ergodic optimization, which is introduced by Garibaldi and Lopes (2007).
Paper Structure (9 sections, 15 theorems, 55 equations)

This paper contains 9 sections, 15 theorems, 55 equations.

Table of Contents

  1. Introduction
  2. Structure of rotation classes
  3. Density of convex combinations of periodic measures
  4. Symbolic dynamics and density of periodic measures
  5. Generic property for constraint ergodic optimization
  6. Characterization by tangency
  7. Generic zero entropy for a symbolic dynamics
  8. On positive entropy
  9. Prevalent uniqueness

Key Result

Theorem 1

Let $(\Omega, \sigma)$ be an irreducible subshift of finite type. Let $\varphi=(\varphi_1, \ldots, \varphi_d)\in C(\Omega, \mathbb{Q}^d)$ be a locally constant function and $h\in {\rm int}({\rm Rot}(\varphi))\cap \mathbb{Q}^d$. Then the set ${\rm rv}_{\varphi}^{-1}(h)\cap \mathcal{M}_\sigma^p(\Omega

Theorems & Definitions (34)

  • Theorem 1
  • Remark 1.1
  • Remark 1.2
  • Remark 1.3
  • Theorem 2
  • Proposition 2.1
  • proof
  • Remark 2.2
  • Corollary 2.3
  • proof
  • ...and 24 more