[Paper Review] Mixing and decay of correlations in non-uniformly expanding maps: a survey of recent results

Stefano Luzzatto|ArXiv.org|Jan 27, 2003

Mathematical Dynamics and Fractals77 references3 citations

TL;DR

This survey investigates the rate of decay of correlations in non-uniformly expanding maps, linking it to geometric and dynamical properties such as expansion rates, critical points, and singularities. It synthesizes functional-analytic and probabilistic coupling methods to explain why different systems exhibit exponential, polynomial, or sub-exponential decay, with key results showing that geometric structures like the presence of indifferent fixed points or critical points fundamentally determine the decay rate.

ABSTRACT

We discuss recent results on decay of correlations for non-uniformly expanding maps. Throughout the discussion, we address the question of why different dynamical systems have different rates of decay of correlations and how this may reflect underlying geometrical characteristics of the system.

Motivation & Objective

• To understand why different non-uniformly expanding maps exhibit varying rates of decay of correlations.
• To identify the underlying geometric and dynamical characteristics that determine the rate of mixing in such systems.
• To survey recent advances in estimating decay rates using functional-analytic and probabilistic coupling techniques.
• To clarify the connection between spectral properties of the Perron-Frobenius operator and correlation decay.
• To examine the role of singularities, critical points, and indifferent fixed points in shaping correlation decay behavior.

Proposed method

• Uses the Perron-Frobenius operator on function spaces of densities to analyze invariant absolutely continuous measures and correlation decay.
• Applies spectral gap arguments in functional-analytic frameworks to deduce exponential decay of correlations.
• Employs the Birkhoff metric and invariant cones to handle discontinuities and improve estimates in non-uniformly expanding systems.
• Utilizes probabilistic coupling techniques to estimate convergence rates without relying on spectral analysis.
• Combines geometric estimates of expansion and smoothness with regularity conditions on observables (e.g., Hölder or weaker summability).
• Analyzes specific classes: uniformly expanding maps, maps with indifferent fixed points, one-dimensional maps with critical points, and two-dimensional Viana maps.

Experimental results

Research questions

• RQ1 What geometric features of a non-uniformly expanding map determine the rate of decay of correlations?
• RQ2 Why do some systems exhibit exponential, others polynomial or sub-exponential decay of correlations?
• RQ3 How do functional-analytic and probabilistic coupling methods compare in estimating decay rates?
• RQ4 To what extent can lower bounds for correlation decay be established, and what do they reveal about system structure?
• RQ5 How do critical points and indifferent fixed points affect the mixing behavior of dynamical systems?

Key findings

• Exponential decay of correlations is linked to the existence of a spectral gap in the Perron-Frobenius operator.
• The coupling method provides flexible, geometrically grounded estimates that can handle weaker regularity conditions on observables than Hölder continuity.
• Maps with indifferent fixed points or critical points exhibit slower, often polynomial or stretched exponential, decay of correlations.
• The Viana map class demonstrates that non-uniform expansion with non-uniformly bounded distortion leads to polynomial decay of correlations.
• The rate of decay is fundamentally tied to the geometry of the system, particularly the distribution and strength of expansion and singularities.
• Lower bounds for decay rates are rare but have been established in recent work, indicating deeper connections between geometry and statistical behavior.