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(x₁,x₂)=(f₁(x₁),f₂(x₂)), where f₁ and f₂ are conformal and expanding maps satisfying inf Df₁≥ Df₂, 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.
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Gatzouras et al. (1997) studied this question.