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Zeitschrift für angewandte Mathematik und Physik

, Volume 49, Issue 6, pp 869–895 | Cite as

Discretization of circle maps

  • N. Nicolaisen
  • B. Werner
Article

Abstract.

To investigate the dynamical behavior of a discrete dynamical system given by a map f, it is nowadays a standard method to look at the discretization of the Frobenius-Perron operator w.r.t. to a box-partition of the state space resulting in a transition matrix M(f). The aim is to obtain characteristics of f by that of M(f) - e.g. invariant measures and their densities.¶In this paper we will treat the special case of circle maps \(f\colon S^1\rightarrow S^1\) which we assume to be orientation preserving C2-diffeomorphisms. They appear in several applications as for instance in investigations of the dynamic on invariant curves or tori.¶We will find a number of relations between f and M(f) concerning the rotation number and the ergodic measure of f presented by the Denjoy conjugacy map. Approximations for the rotation number \(\rho(f)\) based on M(f) and an a posteriori error estimation for the invariant measure of ergodic circle maps given by the Frobenius eigenvector of M(f) will be constructed.

Key words. Circle map, rotation number, ergodic measure, Denjoy conjugacy, density. 

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Copyright information

© Birkhäuser Verlag, Basel, 1998

Authors and Affiliations

  • N. Nicolaisen
    • 1
  • B. Werner
    • 1
  1. 1.Institut für Angewandte Mathematik, Universität Hamburg, D-20146 Hamburg, e-mail: http://www.math.uni-hamburg.de/angmath/Germany

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