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The Relative Variational Principle by Ledrappier and Walters: A survey

Anthony Quas

TL;DR

This survey contextualizes Ledrappier and Walters' relative variational principle within Thermodynamic Formalism, tracing foundational entropy and pressure concepts and then detailing the relative framework for factor maps. It highlights the core results: relative topological entropy and relative pressure formulas that maximize over lift measures with respect to a base system, and the corresponding relative equilibrium states. The discussion extends to subsequent developments in extension and fiber entropies, partially hyperbolic dynamics, and connections to fractal geometry via compensation functions and weighted pressures. Overall, the article demonstrates how relative entropy/pressure unifies dynamical, geometric, and information-theoretic aspects across diverse settings.

Abstract

Ledrappier and Walters's article "A Relativised Variational Principle for Continuous Transformations", J. Lond. Math. Soc. (2) 16 (1977), no.3, 568-576) is a landmark in the development of Thermodynamic Formalism. This survey, aimed at newcomers to the field and experts in adjacent fields discusses the background, the Ledrappier-Walters article and some subsequent developments in the field.

The Relative Variational Principle by Ledrappier and Walters: A survey

TL;DR

This survey contextualizes Ledrappier and Walters' relative variational principle within Thermodynamic Formalism, tracing foundational entropy and pressure concepts and then detailing the relative framework for factor maps. It highlights the core results: relative topological entropy and relative pressure formulas that maximize over lift measures with respect to a base system, and the corresponding relative equilibrium states. The discussion extends to subsequent developments in extension and fiber entropies, partially hyperbolic dynamics, and connections to fractal geometry via compensation functions and weighted pressures. Overall, the article demonstrates how relative entropy/pressure unifies dynamical, geometric, and information-theoretic aspects across diverse settings.

Abstract

Ledrappier and Walters's article "A Relativised Variational Principle for Continuous Transformations", J. Lond. Math. Soc. (2) 16 (1977), no.3, 568-576) is a landmark in the development of Thermodynamic Formalism. This survey, aimed at newcomers to the field and experts in adjacent fields discusses the background, the Ledrappier-Walters article and some subsequent developments in the field.

Paper Structure

This paper contains 15 sections, 4 theorems, 23 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Measure-theoretic entropy
  4. Shannon-MacMillan-Breiman
  5. Topological entropy
  6. The Variational Principle for topological entropy
  7. Pressure
  8. The Variational Principle for pressure
  9. A simple heuristic motivating the Variational Principle
  10. The article of Ledrappier and Walters
  11. Subsequent developments
  12. Extension entropies, fiber entropies and entropy structures
  13. Partially hyperbolic dynamics
  14. Geometric measure theory and fractal dimensions
  15. Compensation functions and relative equilibrium states

Key Result

Theorem 1

Let $X$ be a compact metric space and let $T\colon X\to X$ be a continuous dynamical system. Let $\phi\colon X\to\mathbb R$ be a continuous function (often called a potential). Then where $M_T(X)$ denotes the collection of $T$-invariant Borel probability measures.

Theorems & Definitions (4)

  • Theorem 1: Variational Principle
  • Theorem 2: Ledrappier and Walters LedrappierWalters, relative topological entropy version
  • Theorem 3: LedrappierWalters, relative pressure version
  • Theorem 4: Bowen Bowen:EntropyforGroupEndomorphisms