Surface diffeomorphisms hold a particularly important position in the study of differentiable dynamical systems, as surfaces are the lowest-dimensional setting in which sufficient complexity emerges to illustrate the intricacies of diffeomorphisms.

Recently there are many progress in the ergodic theory of surface diffeomorphisms, such as Buzzi-Crovisier-Sarig’s results on Newhouse conjecture and Burguet’s results on Viana conjecture.

In this talk, we will give an overview of this theory and report our recent results in this direction.