INVARIANT-MEASURES OF FULL DIMENSION FOR SOME EXPANDING MAPS

It is an open problem to determine for which maps f, any compact invar iant 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 co nsequence 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 \Df(1)\ greater than or equal to sup \Df(2)\, then for a large class of invariant sets K, w e 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.