Ergodic Theory and Dynamical Systems



Exactness and maximal automorphic factors of unimodal interval maps


HENK BRUIN a1 and JANE HAWKINS a2
a1 Mathematics Department 253-37, California Institute of Technology, Pasadena, CA 91125, USA
a2 Mathematics Department CB #3250, University of North Carolina at Chapel Hill, Chapel Hill, NC 27599, USA (e-mail: jmh@math.unc.edu)

Abstract

We study exactness and maximal automorphic factors of C^3 unimodal maps of the interval. We show that for a large class of infinitely renormalizable maps, the maximal automorphic factor is an odometer with an ergodic non-singular measure. We give conditions under which maps with absorbing Cantor sets have an irrational rotation on a circle as a maximal automorphic factor, as well as giving exact examples of this type. We also prove that every C^3 S-unimodal map with no attractor is exact with respect to Lebesgue measure. Additional results about measurable attractors in locally compact metric spaces are given.

(Received June 18 1999)
(Revised March 14 2000)