Special Session 73: Entropy and statistical properties for smooth dynamics
Contents
We prove that there is a $C^1$-open and dense subset of the $C^1$-open set of diffeomorphisms with
robust cycles consisting of diffeomorpphisms with non-hyperbolic ergodic measures with positive entropy.
Examples of diffeomorphisms with $C^1$-robust cycles are those having a chain recurrence class or a
homoclinic class containing hyperbolic saddles of different index (dimension of the stable bundle).
As a consequence we get that there is a $C^1$-open and dense subset of
the set of non-Anosov robustly transitive diffeomorphisms of systems with
non-hyperbolic ergodic measures with positive entropy.
The main technical ingredient of our results is a semi-local mechanism called
flip-flop families that allows us to construct compact sets with
controlled Birkhoff averages. We use this mechanism to control the average of the central
Jacobian of partially hyperbolic sets with one-dimensional center direction.