Ergodic Theory and Dynamical Systems

Invariant measures of full dimension for some expanding maps

a1 Department of Mathematics, University of Crete, 714 09 Iraklion, Crete, Greece (e-mail:
a2 Institute of Mathematics, The Hebrew University, Israel
a3 Department of Statistics, University of California, Berkeley, CA 94720, U.S.A (e-mail:


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)

c1 Current address: Statistical Laboratory, University of Cambridge, 16 Mill Lane, Cambridge CB2 1SB, UK (e-mail: