Dynamical entropies are measures of the complexity of orbit structures. The topological entropy considers all the orbits, whereas the measure theoretic entropy focuses on those ``relevant" to a given invariant probability measure. The variational principle says that the topological entropy of a continuous self-map of a compact metrizable space is the supremum of the measure theoretic entropy over the set of invariant probability measures for the map.

A well known fact is that every transitive hyperbolic (Anosov) diffeomorphism has a unique invariant probability measure whose entropy equals the topological entropy. We analyze a class of deformations of Anosov diffeomorphisms containing many of the known nonhyperbolic robustly transitive diffeomorphisms. We show that these $C0$-small, but $C1$-macroscopic, deformations leave all the high entropy dynamics of the Anosov system unchanged, and that there is a partial conjugacy identifying all invariant probability measures with entropy close to the maximum for the deformation with those of the original Anosov system.

Additionally, we show that these results apply to a class of nonpartially hyperbolic, robustly transitive diffeomorphisms described by Bonatti and Viana and a class originally described by Mane. In fact these methods apply to several classes of systems which are similarly derived from Anosov, i.e., produced by an isotopy from an Anosov system.