With Sylvain Crovisier and Todd Fisher, we have just finished the paper Local perturbations of conservative C1-diffeomorphisms. It is now available on arxiv.
While studying the entropy of diffeomorphisms in this paper, we needed a number of lemmas for perturbing the dynamics of periodic orbits with support contained in a small neighborhood and preserving given homoclinic relations. Such results were known in the dissipative setting (though some are quite recent, see Nicolas Gourmelon‘s work). However, the conservative (i.e., volume-preserving or symplectic) versions were often weaker or even lacking. The new preprint fills this gap.