Dynamical Resistance Day

Title: A model factor map for surface diffeomorphisms (video)

We show that if a C1+α diffeomorphism f of T2 homotopic to the identity has a rotation set with nonempty interior, then it is monotonically semiconjugate to a homeomorphism F of T2 which is area-preserving, topologically mixing, has dense periodic points and every point has a nontrivial stable and unstable set. Moreover, it has a strong form of continuum-wise expansiveness. Further, F has the same rotation set as $f$, so one consequence is that every rotation set realizable by a C1+α diffeomorphism is also realizable by an area-preserving homeomorphism with all these properties.
We also obtain a similar result on surfaces of higher genus, with the condition on the rotation set replaced by the existence of certain types of periodic orbits.
Joint work with A. de Carvalho and F. A. Tal.