Canonical $C^0$ perturbation of Anosov diffeo to 0-dimensional $\Omega _i$'s with the same entropy
Is the following an answer to this?
M.Shub and D.Sulivan, Homology and dynamical systems, Topology 14 (1975), 109-132.
(see also Chapter 4 in J. Franks, Homology and dynamical systems, CBMS Regional Conference Series in Mathematics
Volume: 49; 1982; 120 pp.)
Maller,M and M.Shub, The Integral Homology of Smale Diffeomorphisms Topology, Vol. 24 (1985) pp. 153-164 may be relevant. The paper considers when Axiom A no cycles diffeorphisms with zero dimensional omega may be isotoped to omega conjugate fitted examples.