A Thermodynamic Definition of Topological Pressure for Non-Compact Sets

TL;DR

This paper introduces a new thermodynamic definition of topological pressure applicable to non-compact, non-invariant sets in metric spaces, providing a variational principle and exploring its properties through examples like Lyapunov level sets.

Contribution

It proposes a novel thermodynamic approach to define topological pressure for arbitrary Borel sets, extending classical notions and enabling analysis of non-compact, non-invariant sets.

Findings

New definition aligns with classical pressure in compact cases

Application to Lyapunov exponent level sets demonstrates usefulness

Comparison with existing notions shows advantages in non-compact settings

Abstract

We give a new definition of topological pressure for arbitrary (non-compact, non-invariant) Borel subsets of metric spaces. This new quantity is defined via a suitable variational principle, leading to an alternative definition of an equilibrium state. We study the properties of this new quantity and compare it with existing notions of topological pressure. We are particularly interested in the situation when the ambient metric space is assumed to be compact. We motivate the naturality of our definition by applying it to some interesting examples, including the level sets of the pointwise Lyapunov exponent for the Manneville-Pomeau family of maps.