Ergodic Theory and Dynamical Systems



Comparing measures and invariant line fields


VOLKER MAYER a1
a1 UFR de Mathématiques, Université de Lille I, UMR 8524 du CNRS, 59655 Villeneuve d'Ascq Cedex, France (e-mail: volker.mayer@univ-lille1.fr)

Abstract

We give elementary proofs of two rigidity results. The first one asserts that the maximal entropy measure \mu_f of a rational map f is singular with respect to any given conformal measure excepted if f is a power, Tchebychev or Lattès map. This is a variation of a result of Zdunik. Our second result is an improvement of a theorem of Fisher and Urbanski. It gives a sharp description of the exceptional functions that admit invariant line fields which are defined with respect to certain invariant measures

(Received February 15 2000)
(Revised February 28 2001)