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Dynamical properties of spatial discretizations of a generic homeomorphism

Published online by Cambridge University Press:  23 April 2014

PIERRE-ANTOINE GUIHÉNEUF
PIERRE-ANTOINE GUIHÉNEUF*
Affiliation:
Laboratoire de mathématiques CNRS UMR 8628, Université Paris-Sud 11, Bât. 425, 91405 Orsay Cedex, France email pierre-antoine.guiheneuf@math.u-psud.fr
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Abstract

This paper concerns the link between the dynamical behaviour of a dynamical system and the dynamical behaviour of its numerical simulations. Here, we model numerical truncation as a spatial discretization of the system. Some previous works on well-chosen examples (such as Gambaudo and Tresser [Some difficulties generated by small sinks in the numerical study of dynamical systems: two examples. Phys. Lett. A 94(9) (1983), 412–414]) show that the dynamical behaviours of dynamical systems and of their discretizations can be quite different. We are interested in generic homeomorphisms of compact manifolds. So our aim is to tackle the following question: can the dynamical properties of a generic homeomorphism be detected on the spatial discretizations of this homeomorphism? We will prove that the dynamics of a single discretization of a generic conservative homeomorphism does not depend on the homeomorphism itself, but rather on the grid used for the discretization. Therefore, dynamical properties of a given generic conservative homeomorphism cannot be detected using a single discretization. Nevertheless, we will also prove that some dynamical features of a generic conservative homeomorphism (such as the set of the periods of all periodic points) can be read on a sequence of finer and finer discretizations.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 35 , Issue 5 , August 2015 , pp. 1474 - 1523
Copyright
© Cambridge University Press, 2014 

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