Newhouse stated that unfoldings of homoclinic tangencies of a surface $C^2$-diffeomorphisms yield open sets where the diffeomorphisms with infinitely many sinks/sources are locally generic. There is a version of this result for parametrized families of diffeomorphisms. Palis conjectured that the set of parameters corresponding to diffeomorphisms with infinitely many sinks has measure zero. Gorodetski-Kaloshin gave a partial answer to this conjecture.

Motivated by these results, we study a formulation of this result in the partially hyperbolic setting, where sinks/sources are replaced by homoclinic classes and homoclinic tangencies by heterodimensional cycles. Our result is that it is not possible to generate infinitely many different homoclinic classes using a renormalization-like construction.

This is a join work with J. Rocha (Porto, Portugal).