If $f$ is Axiom A, is there an Axiom A $g$, $C^0$ near $f$ with dim$\Omega (g) = d$ and $h(f) = h(g) ?$
What is $d$? Is it $0$? In that case, see Shub, M.; Sullivan, D.
Homology theory and dynamical systems. Topology 14 (1975), 109–132.
d is 0 indeed.