Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Entropy formula of folding type for $C^{1+α}$ maps

Gang Liao, Shirou Wang

TL;DR

The paper addresses characterizing inverse SRB measures for non-invertible $C^{1+\alpha}$ maps admitting degeneracy, by establishing the entropy formula of folding type $h_(f)= F_(f)-\int\sum_{\lambda_i(x)<0}\lambda_i(x)\,d\mu(x)$ under a Jacobian-integrability condition. It develops Pesin theory for general $C^{1+\alpha}$ maps with degeneracy, introducing Lyapunov neighborhoods and Pesin blocks $\widetilde{\Lambda}_{a,b,k}$ and constructing a measurable partition subordinate to stable manifolds to relate entropy to stable dynamics. The necessity and sufficiency parts of the main theorem are proved via inverse limit techniques, disintegration along stable leaves, and analysis of the Jacobian-series along $\lambda^s$ directions, yielding a full criterion for inverse SRB measures. Applications include conservative systems where the entropy production vanishes and dissipative degenerate systems where inverse SRB measures exist and satisfy the folding formula, thus broadening the scope of entropy-production analysis in non-invertible dynamics.

Abstract

In the study of non-equilibrium statistical mechanics, Ruelle derived explicit formulae for entropy production of smooth dynamical systems. The vanishing or strict positivity of entropy production is determined by the {\it entropy formula of folding type} \[h_μ(f)= F_μ(f)-\displaystyle\int\sum\nolimits_{λ_i(x)<0} λ_i(x)dμ(x), \] which relates the metric entropy, folding entropy and negative Lyapunov exponents. This paper establishes the formula for all inverse SRB measures of $C^{1+α}$ maps, including those with degeneracy (i.e., zero Jacobian). More specifically, we establish the equivalence that $μ$ is an inverse SRB measure if and only if the folding-type entropy formula holds and the Jacobian series is integrable. To overcome the degeneracy, we develop Pesin theory for general $C^{1+α}$ maps.

Entropy formula of folding type for $C^{1+α}$ maps

TL;DR

The paper addresses characterizing inverse SRB measures for non-invertible maps admitting degeneracy, by establishing the entropy formula of folding type under a Jacobian-integrability condition. It develops Pesin theory for general maps with degeneracy, introducing Lyapunov neighborhoods and Pesin blocks and constructing a measurable partition subordinate to stable manifolds to relate entropy to stable dynamics. The necessity and sufficiency parts of the main theorem are proved via inverse limit techniques, disintegration along stable leaves, and analysis of the Jacobian-series along directions, yielding a full criterion for inverse SRB measures. Applications include conservative systems where the entropy production vanishes and dissipative degenerate systems where inverse SRB measures exist and satisfy the folding formula, thus broadening the scope of entropy-production analysis in non-invertible dynamics.

Abstract

In the study of non-equilibrium statistical mechanics, Ruelle derived explicit formulae for entropy production of smooth dynamical systems. The vanishing or strict positivity of entropy production is determined by the {\it entropy formula of folding type} which relates the metric entropy, folding entropy and negative Lyapunov exponents. This paper establishes the formula for all inverse SRB measures of maps, including those with degeneracy (i.e., zero Jacobian). More specifically, we establish the equivalence that is an inverse SRB measure if and only if the folding-type entropy formula holds and the Jacobian series is integrable. To overcome the degeneracy, we develop Pesin theory for general maps.

Paper Structure

This paper contains 9 sections, 18 theorems, 209 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Pesin theory for general $C^{1+\alpha}$ maps
  3. Lyapunov neighborhoods for $C^{1+\alpha}$ maps with degeneracy
  4. Measurable partitions subordinate to stable manifolds
  5. Proof of Theorem \ref{['thm1']} -- necessity
  6. Proof of Theorem \ref{['thm1']}-- sufficiency
  7. Applications
  8. Conservative systems using the "only if " part
  9. Dissipative systems using the "if" part

Key Result

Theorem 1.1

Let $M$ be a compact Riemannian manifold without boundary, $f:M\to M$ be a $C^{1+\alpha}$ map, and $\mu$ be an $f$-invariant Borel probability measure satisfying the integrability condition integrable for the Jacobian. Then $\mu$ is an inverse SRB measure if and only if the entropy formula of foldin and condition (H)" hold.

Figures (1)

  • Figure 1: Geometric decomposition of unit vector $v \in E_m(x)$ into $v_i \in Df^m F_i(x)$ and $v_i^c \in Df^m F_i^c(x)$.

Theorems & Definitions (38)

  • Definition 1.1
  • Remark 1.1
  • Theorem 1.1
  • Remark 1.2
  • Theorem 2.1
  • proof
  • Lemma 2.1: Pesin sets in degenerate setting
  • proof : Proof of Lemma \ref{['Pesin-set']}
  • Proposition 2.1
  • proof
  • ...and 28 more