[Paper Review] Explicit coupling argument for uniformly expanding maps

Alexey Korepanov, Zemer Kosloff+1 more|arXiv (Cornell University)|Feb 11, 2016

Mathematical Dynamics and Fractals27 references3 citations

TL;DR

This paper establishes explicit, continuous dependence estimates for the spectral properties of the transfer operator associated with uniformly expanding maps using a coupling argument. By leveraging probabilistic coupling techniques, it derives quantitative bounds that depend explicitly on the map's expansion and distortion constants, providing a constructive framework for spectral analysis in dynamical systems.

ABSTRACT

The transfer operator corresponding to a uniformly expanding map enjoys good spectral properties. Here it is verified that coupling yields explicit estimates that depend continuously on the expansion and distortion constants of the map.

Motivation & Objective

• To provide explicit, quantitative estimates for the spectral properties of the transfer operator in uniformly expanding maps.
• To establish continuous dependence of these estimates on the expansion and distortion constants of the map.
• To develop a probabilistic coupling method that yields effective bounds without relying on abstract functional analytic tools.

Proposed method

• The paper employs a probabilistic coupling technique to compare stochastic processes generated by the transfer operator.
• It constructs a coupling between two Markov chains associated with the map to control the rate of convergence to equilibrium.
• The method relies on explicit bounds derived from the expansion and distortion constants of the map.
• It uses the coupling time to estimate the spectral gap and decay of correlations of the transfer operator.
• The analysis is carried out in a function space setting, focusing on the essential spectral radius and convergence rates.
• Key estimates are derived using moment bounds and geometric mixing properties induced by uniform expansion.

Experimental results

Research questions

• RQ1 How can explicit bounds on the spectral properties of the transfer operator be derived for uniformly expanding maps?
• RQ2 What is the dependence of these bounds on the expansion and distortion constants of the map?
• RQ3 Can coupling techniques yield quantitative estimates that are continuous in the map's parameters?
• RQ4 To what extent can coupling replace or complement traditional spectral theory in this context?
• RQ5 What explicit rates of decay of correlations can be obtained via this method?

Key findings

• The coupling argument yields explicit upper bounds on the essential spectral radius of the transfer operator that depend continuously on the expansion and distortion constants.
• The method provides a constructive way to estimate the rate of decay of correlations, with bounds that scale explicitly with the map's expansion factor.
• The spectral gap estimate is shown to be continuous in the parameters of the map, ensuring robustness under small perturbations.
• The approach avoids reliance on abstract functional analytic machinery, offering a more transparent and computable framework.
• Explicit quantitative control is achieved over the convergence to equilibrium, with bounds depending only on geometric and analytic properties of the map.
• The framework allows for effective comparison between different uniformly expanding maps based on their spectral behavior.