Ergodic Theory and Dynamical Systems



Invariant measures of full dimension for some expanding maps


DIMITRIOS GATZOURAS a1c1 and YUVAL PERES a2a3
a1 Department of Mathematics, University of Crete, 714 09 Iraklion, Crete, Greece (e-mail: gatzoura@edu.uch.gr)
a2 Institute of Mathematics, The Hebrew University, Israel
a3 Department of Statistics, University of California, Berkeley, CA 94720, U.S.A (e-mail: peres@stat.berkeley.edu)

Abstract

It is an open problem to determine for which maps $f$, any compact invariant set $K$ carries an ergodic invariant measure of the same Hausdorff dimension as $K$. If $f$ is conformal and expanding, then it is a known consequence of the thermodynamic formalism that such measures do exist. (We give a proof here under minimal smoothness assumptions.) If $f$ has the form $f(x_1,x_2)=(f_1(x_1),f_2(x_2))$, where $f_1$ and $f_2$ are conformal and expanding maps satisfying $\inf \vert Df_1\vert\geq\sup\vert Df_2\vert$, then for a large class of invariant sets $K$, we show that ergodic invariant measures of dimension arbitrarily close to the dimension of $K$ do exist. The proof is based on approximating $K$ by self-affine sets.

(Received July 15 1994)
(Revised October 1 1995)


Correspondence:
c1 Current address: Statistical Laboratory, University of Cambridge, 16 Mill Lane, Cambridge CB2 1SB, UK (e-mail: d.gatzouras@statslab.cam.ac.uk)