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.