A pseudospectral approach to rigorous numerical estimation of Ruelle resonances of transfer operators

2025/07/11 by Blumenthal, Alex, Nisoli, Isaia, Taylor-Crush, Toby · 1 citation
#65G99 #Dynamical Systems (math.DS) #FOS: Mathematics #Functional Analysis (math.FA) #Primary: 37C30 #Secondary: 47A10

paper · doi:10.48550/arxiv.2507.09021

Abstract

Resonances, isolated eigenvalues of a transfer operator acting on suitably chosen Banach spaces, play a fundamental role in understanding the statistical properties of chaotic dynamical systems. In this paper, we introduce a pseudospectral approach, inspired by Householder's theorem, for the rigorous, computer-assisted estimation of resonances, providing regions where resonances must exist and precluding the presence of resonances elsewhere. The approach is general, and applies to the transfer operators of a wide variety of chaotic systems, including Anosov/ Axiom A diffeomorphisms and piecewise expanding maps. We implement this approach computationally for a class of analytic uniformly expanding maps of the circle. We anticipate that the pseudospectral framework developed here will be broadly applicable to other spectral problems in dynamical systems and beyond.

Citations

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Cited by

• Efficient computation of stationary measures and the Lyapunov Landscape for families random dynamical systems with smooth additive noise

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• Emergence of quantum dynamics from chaos: The case of prequantum cat maps