[Paper Review] Improved Young Tower and Thermodynamic Formalism for Hyperbolic Systems with Singularities

Jianyu Chen, Fang Wang+1 more|arXiv (Cornell University)|Sep 2, 2017

Mathematical Dynamics and Fractals35 references3 citations

TL;DR

This paper constructs an improved Young tower for hyperbolic systems with singularities, such as chaotic billiards, by ensuring the first return to the tower base is Markov via a coupling lemma for standard families. The resulting countable-state Markov partition enables rigorous derivation of stochastic properties—like exponential decay of correlations and the central limit theorem—under the SRB measure, and establishes a full thermodynamic formalism for geometric potentials using an inducing scheme.

ABSTRACT

For the hyperbolic systems with singularities, Markov partitions are rather delicate to construct because of the fragmentation of the phase space by singularities. In this paper, we investigate the chaotic billiards and other related hyperbolic systems with singularities, and construct an improved Young tower whose first return to the base is Markov. This leads to a Markov partition of the phase space with countable states. %Our construction is based on the coupling lemma %for standard families. Stochastic properties with respect to the SRB measure immediately follow from our construction of the Markov partition, including the decay rates of correlations and the central limit theorem. We further establish the thermodynamic formalism for the family of geometric potentials, by using the inducing scheme of hyperbolic type on the tower base. All the results apply to Sinai dispersing billiards, and their small perturbations due to external forces and nonelastic reflections with kicks and slips.

Motivation & Objective

• To address the challenge of constructing Markov partitions in hyperbolic systems with singularities due to phase space fragmentation.
• To develop a refined Young tower construction where the first return to the base is Markov, ensuring countable-state Markov partitions.
• To derive stochastic properties—such as decay of correlations and the central limit theorem—under the SRB measure using the new tower structure.
• To establish a thermodynamic formalism for geometric potentials in systems with singularities via an inducing scheme on the tower base.
• To extend results to Sinai dispersing billiards and their perturbations, including external forces, nonelastic reflections, and kicks and slips.

Proposed method

• Utilizes a coupling lemma for standard families to control the dynamics near singularities and ensure Markov structure in the tower construction.
• Constructs an improved Young tower with a base where the first return map is Markov, enabling a countable-state Markov partition of the phase space.
• Applies the inducing scheme of hyperbolic type on the tower base to analyze the thermodynamic formalism for geometric potentials.
• Establishes stochastic properties by leveraging the Markov structure and the SRB measure, including correlation decay and limit theorems.
• Adapts the framework to include perturbations such as external forces, nonelastic reflections, and slip/kick effects in billiard systems.

Experimental results

Research questions

• RQ1 How can a Markov partition be constructed in hyperbolic systems with singularities despite phase space fragmentation?
• RQ2 Can the first return map to the tower base be made Markov in systems with singularities to enable a countable-state partition?
• RQ3 What stochastic properties—such as decay of correlations and the central limit theorem—can be rigorously derived from the improved tower structure?
• RQ4 How can a thermodynamic formalism be developed for geometric potentials in systems with singularities using an inducing scheme?
• RQ5 To what extent do the results extend to perturbations of Sinai dispersing billiards, including non-ideal reflections and external forces?

Key findings

• The improved Young tower construction ensures the first return to the base is Markov, yielding a countable-state Markov partition for hyperbolic systems with singularities.
• Stochastic properties such as exponential decay of correlations and the central limit theorem are established for the SRB measure using the new Markov structure.
• The thermodynamic formalism for geometric potentials is successfully developed via an inducing scheme on the tower base, enabling analysis of pressure and equilibrium states.
• The results apply to Sinai dispersing billiards and their small perturbations, including systems with external forces, nonelastic reflections, and slip/kick effects.
• The coupling lemma for standard families plays a crucial role in managing singularities and ensuring the Markov property in the tower construction.