Non-trivial wandering domains and homoclinic bifurcations

We prove that on any surface there is a C^∞ diffeomorphism exhibiting a wandering domain D with the following ergodic property: for any orbit starting in D the corresponding Birkhoff mean of Dirac measures converges to the invariant measure supported on a hyperbolic horseshoe Λ which is equivalent to the unique non-trivial Hausdorff measure in Λ. The construction is obtained by perturbation of a diffeomorphism such that the unstable and stable foliations of this horseshoe Λ are relatively thick and in tangential position. We describe, in addition, the set of accumulation points of orbits starting in D.