Discretization of circle maps

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 C ²-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.