Invariant densities and escape rates: Rigorous and computable approximations in the L^∞-norm

2010/08/03 by Wael Bahsoun, Bahsoun, Wael, Christopher Bose +1 · 2 citations
Mathematics · Physics and Astronomy · #37A05 #37E05 #Dynamical Systems (math.DS) #FOS: Mathematics #FOS: Physical sciences #Markov Chains and Monte Carlo Methods #Mathematical Dynamics and Fractals #Mathematical Physics (math-ph) #Model Reduction and Neural Networks

paper · pdf · doi:10.48550/arxiv.1008.0556

openalex publication_date 2010/08/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this article we study a piecewise linear discretization schemes for transfer operators (Perron-Frobenius operators) associated with interval maps. We show how these can be used to provide rigorous \bf pointwise approximations for invariant densities of Markov interval maps. We also derive the order of convergence of the approximate invariant density to the real one in the L^∞-norm. The outcome of this paper complements rigorous results on L¹ approximations of invariant densities \citeKMY and recent results on the formulae of escape rates of open dynamical systems \citeKL2. We implement our computations on two examples (one rigorous and one non-rigorous) to illustrate the feasibility and efficiency of our schemes.

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• Efficient computation of stationary measures and the Lyapunov Landscape for families random dynamical systems with smooth additive noise
• A pseudospectral approach to rigorous numerical estimation of Ruelle resonances of transfer operators

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