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Rigorous computation of invariant measures and fractal dimension for maps with contracting fibers: 2D Lorenz-like maps

Published online by Cambridge University Press:  13 April 2015

STEFANO GALATOLO  and
ISAIA NISOLI
STEFANO GALATOLO
Affiliation:
Dipartimento di Matematica, Universita di Pisa, Via Buonarroti 1, Pisa, Italy email galatolo@dm.unipi.it
ISAIA NISOLI
Affiliation:
Instituto de Matemática, UFRJ Av. Athos da Silveira Ramos 149, Centro de Tecnologia, Bloco C Cidade Universitária, Ilha do Fundão, Caixa Postal 68530, 21941-909 Rio de Janeiro, RJ, Brasil email nisoli@im.ufrj.br
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Abstract

We consider a class of maps from the unit square to itself preserving a contracting foliation and inducing a one-dimensional map having an absolutely continuous invariant measure. We show how the physical measure of those systems can be rigorously approximated with an explicitly given bound on the error with respect to the Wasserstein distance. We present a rigorous implementation of our algorithm using interval arithmetics, and the result of the computation on a non-trivial example of a Lorenz-like two-dimensional map and its attractor, obtaining a statement on its local dimension.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 36 , Issue 6 , September 2016 , pp. 1865 - 1891
Copyright
© Cambridge University Press, 2015 

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