Entropic stability

Andronov and Pontryagin suggested that the study of dynamical systems should focus on stable systems. It turns out that topologically C1-stable dynamics (also called structurally stable systems) can be analyzed; indeed, they are exactly the uniformly hyperbolic systems that satisfy an additional assumption. However, the structurally stable diffeomorphisms are not dense and therefore the study of such systems is insufficient. A natural approach is to consider weaker forms of stability. In this talk we will introduce an entropy-based notion of stability. Furthermore, we analyze a class of deformations of Anosov diffeomorphisms: these deformations break the topological conjugacy class but leave the high entropy dynamics unchanged. More precisely, there is a partial conjugacy between the deformation and the original Anosov system that identifies all invariant probability measures with entropy close to the maximum.