[Paper Review] Multifractal analysis of irregular sets for weak Gibbs measures

Thiago Bomfim, Paulo Varandas|arXiv (Cornell University)|May 11, 2014

Mathematical Dynamics and Fractals44 references3 citations

TL;DR

This paper establishes precise estimates for the topological pressure of irregular sets—points whose Birkhoff averages deviate from the equilibrium state's space average—under weak Gibbs measures in expanding repeller systems. For H"older continuous potentials, it shows the pressure of such sets over an interval I is determined by the full system's pressure and a large deviations rate function, proving most irregular sets have strictly lower pressure than the full system when the specification property holds.

ABSTRACT

In this article we prove estimates for the topological pressure of the set of points whose Birkhoff time averages are far from the space averages corresponding to the unique equilibrium state that has a weak Gibbs property. In particular, if $f$ has an expanding repeller and $\phi$ is an Holder continuous potential we prove that the topological pressure of the set of points whose accumulation values of Birkhoff averages belong to some interval $I\subset \mathbb R$ can be expressed in terms of the topological pressure of the whole system and the large deviations rate function. As a byproduct we deduce that most irregular sets for maps with the specification property have topological pressure strictly smaller than the whole system. Some extensions to a non-uniformly hyperbolic setting, level-2 irregular sets and hyperbolic flows are also given.

Motivation & Objective

• To analyze the topological pressure of irregular sets where Birkhoff time averages deviate from space averages under weak Gibbs measures.
• To establish a quantitative relationship between the pressure of such irregular sets and the large deviations rate function.
• To extend results to non-uniformly hyperbolic settings and hyperbolic flows.
• To demonstrate that irregular sets under the specification property have strictly smaller topological pressure than the full system.

Proposed method

• Utilizes multifractal analysis techniques to study the level sets of Birkhoff averages.
• Applies the weak Gibbs property to control the measure-theoretic behavior of typical orbits.
• Employs large deviations principles to express the pressure of irregular sets in terms of the full system's pressure and a rate function.
• Uses the specification property to construct orbits with desired ergodic averages and estimate pressure.
• Relies on H"older continuity of the potential to ensure regularity in pressure estimates.
• Extends results from uniformly hyperbolic systems to non-uniformly hyperbolic and hyperbolic flow settings.

Experimental results

Research questions

• RQ1 How does the topological pressure of the set of points with Birkhoff averages in a given interval I compare to the pressure of the full system under weak Gibbs measures?
• RQ2 What is the precise functional relationship between the pressure of irregular sets and the large deviations rate function in expanding repeller systems?
• RQ3 To what extent do irregular sets under the specification property have strictly smaller pressure than the full system?
• RQ4 Can the results on multifractal pressure be extended to non-uniformly hyperbolic systems and hyperbolic flows?
• RQ5 How does the weak Gibbs property influence the pressure estimates of irregular sets?

Key findings

• The topological pressure of the set of points with Birkhoff averages in an interval I is expressed as the sum of the full system's pressure and a term involving the large deviations rate function.
• For maps with the specification property, most irregular sets have topological pressure strictly less than that of the full system.
• The results hold for H"older continuous potentials on expanding repellers with weak Gibbs equilibrium states.
• Extensions to non-uniformly hyperbolic systems and hyperbolic flows are established, showing robustness of the pressure estimates.
• The pressure of level-2 irregular sets is characterized via the same large deviations framework.
• The weak Gibbs property enables precise control over measure-theoretic and topological pressure estimates for irregular sets.