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Limits of equilibrium states for coupled weakly interacting systems. Application to the measure of maximal entropy

Renaud Leplaideur

TL;DR

The paper analyzes metastability in coupled weakly interacting systems via symbolic dynamics and thermodynamic formalism. By representing two independent subshifts of finite type connected through a shrinking coupling hole, it derives how the global equilibrium state converges to a convex combination of the independent equilibria, with weights determined by dwell-time parameters encoded in θ. The authors develop inducing schemes, transfer-operator methods, and extensions of eigenfunctions to compare local and global equilibria, obtaining precise limiting distributions for both the conformal measure and the eigenfunction, as well as the measure of maximal entropy in a geometric setting. A key contribution is showing that, under two different hole-reduction schemes, the limit is either (1/2)(μ_A+μ_D) or (1/(1+θ))(θ μ_A + μ_D), highlighting robust metastable behavior and enabling a bridge to geometrical systems via a conjugacy that yields convergence of the measure of maximal entropy to the average of the two component measures. These results deepen the understanding of how small couplings shape long-term statistical behavior in dynamical systems and provide tools for extending the analysis to more complex, multi-component arrangements.

Abstract

We study metastability for symbolic dynamic. We prove that for a global system given by two independent sub-systems linked by a hole, and for a Lipschitz continuous potential, the global equilibrium state converges, as the hole shrinks, to a convex combination of the two independent equilibria in each component. Two kinds of convergence occur, depending on the assumptions on how long an orbit has to stay in each well. As a by-product, we show that this can be applied to a geometrical system inspired from [11] and for the measure of maximal entropy.

Limits of equilibrium states for coupled weakly interacting systems. Application to the measure of maximal entropy

TL;DR

The paper analyzes metastability in coupled weakly interacting systems via symbolic dynamics and thermodynamic formalism. By representing two independent subshifts of finite type connected through a shrinking coupling hole, it derives how the global equilibrium state converges to a convex combination of the independent equilibria, with weights determined by dwell-time parameters encoded in θ. The authors develop inducing schemes, transfer-operator methods, and extensions of eigenfunctions to compare local and global equilibria, obtaining precise limiting distributions for both the conformal measure and the eigenfunction, as well as the measure of maximal entropy in a geometric setting. A key contribution is showing that, under two different hole-reduction schemes, the limit is either (1/2)(μ_A+μ_D) or (1/(1+θ))(θ μ_A + μ_D), highlighting robust metastable behavior and enabling a bridge to geometrical systems via a conjugacy that yields convergence of the measure of maximal entropy to the average of the two component measures. These results deepen the understanding of how small couplings shape long-term statistical behavior in dynamical systems and provide tools for extending the analysis to more complex, multi-component arrangements.

Abstract

We study metastability for symbolic dynamic. We prove that for a global system given by two independent sub-systems linked by a hole, and for a Lipschitz continuous potential, the global equilibrium state converges, as the hole shrinks, to a convex combination of the two independent equilibria in each component. Two kinds of convergence occur, depending on the assumptions on how long an orbit has to stay in each well. As a by-product, we show that this can be applied to a geometrical system inspired from [11] and for the measure of maximal entropy.

Paper Structure

This paper contains 47 sections, 8 theorems, 86 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Background
  3. Settings for symbolic dynamics
  4. Settings at initial stage
  5. The coupling hole.
  6. First reduction of the coupling hole and Main Theorem \ref{['th-main1-2']}
  7. Second reduction of the coupling hole and Main Theorem \ref{['th-main2-1']}
  8. Settings for geometrical dynamical systems
  9. Open questions, Plan of the paper and Aknowledgment
  10. Open questions
  11. Plan of the paper
  12. Acknowledgment
  13. Recalls and technical results on thermodynamical formalism
  14. Equilibrium states and transfer operator
  15. Extension of the eigenfunctions for transfer operators
  16. ...and 32 more sections

Key Result

Proposition 2.1

The transfer operator ${\mathcal{L}}_{E'}$ associated to the potential $\varphi$ can be extended to an operator acting on continuous and on Lipschitz functions defined on $\Sigma_{E}$. The spectral properties, including decompositions still hold for that extended operator. In particular, the eigenfu

Figures (8)

  • Figure 1: The family of maps $T_{\varepsilon}$
  • Figure 2: Partition of $\Omega_{A}$
  • Figure 3: Orbits leaving the hole in $\Omega_{A}$ to return in the hole in $\Omega_{D}$
  • Figure 4: Map $T_{\varepsilon_{0}}$ and Markov partition
  • Figure 5: construction of the right-hand side of the map $T_{\varepsilon}$ at step $m$
  • ...and 3 more figures

Theorems & Definitions (18)

  • Remark 1
  • Proposition 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • Lemma 2.4
  • proof
  • Proposition 2.5
  • proof
  • ...and 8 more