Ergodic Theory and Dynamical Systems

Research Article

On Hausdorff dimension of invariant sets for expanding maps of a circle

Mariusz Urbańskia1

a1 Institute of Mathematics, N. Copernicus University, Schopina 12/18, 87-100 Toruń, Poland

Abstract

Given an orientation preserving C2 expanding mapping g: S1Sl of a circle we consider the family of closed invariant sets Kg(ε) defined as those points whose forward trajectory avoids the interval (0, ε). We prove that topological entropy of g|Kg(ε) is a Cantor function of ε. If we consider the map g(z) = zq then the Hausdorff dimension of the corresponding Cantor set around a parameter ε in the space of parameters is equal to the Hausdorff dimension of Kg(ε). In § 3 we establish some relationships between the mappings g|Kg(ε) and the theory of β-transformations, and in the last section we consider DE-bifurcations related to the sets Kg(ε).

(Received November 05 1984)

(Revised July 01 1985)

(Revised October 10 1985)