Rigorous approximation of stationary measures and convergence to equilibrium for iterated function systems *All the authors were partially supported by ICTP and by EU Marie-Curie IRSES Brazilian–European partnership in Dynamical Systems (FP7-PEOPLE-2012-IRSES 318999 BREUDS), SG thanks The Leverhulme Trust for support through Network Grant IN-2014-021. (27th May 2016)

• Record Type: Journal Article
• Authors: Galatolo, Stefano; Monge, Maurizio; Nisoli, Isaia
• Is Part Of: Journal of physics. Volume 49:Number 27(2016)
• Publication Date: 2016-05-27
• Subjects: rigorous numerical methods -- stationary measure -- iterated function system -- convergence to equilibrium; 37M25 -- 37H; Mathematical physics -- Periodicals; Statistical physics -- Periodicals; Quantum theory -- Periodicals; Matter -- Properties -- Periodicals; 530.105
• DOI: 10.1088/1751-8113/49/27/274001
• Languages: English
• ISSNs: 1751-8113

Abstract

Abstract: We study the problem of the rigorous computation of the stationary measure and of the rate of convergence to equilibrium of an iterated function system described by a stochastic mixture of two or more dynamical systems that are either all uniformly expanding on the interval, either all contracting. In the expanding case, the associated transfer operators satisfy a Lasota–Yorke inequality, we show how to compute a rigorous approximations of the stationary measure in the L 1 norm and an estimate for the rate of convergence. The rigorous computation requires a computer-aided proof of the contraction of the transfer operators for the maps, and we show that this property propagates to the transfer operators of the IFS. In the contracting case we perform a rigorous approximation of the stationary measure in the Wasserstein–Kantorovich distance and rate of convergence, using the same functional analytic approach. We show that a finite computation can produce a realistic computation of all contraction rates for the whole parameter space. We conclude with a description of the implementation and numerical experiments.