For conservative diffeomorphims on closed surfaces one has the following dichotomy: either they are approximated in the C¹ topology by diffeomorphisms with zero exponents, or they are uniformly hyperbolic; in the later case the surface is the two-torus. This is consequence of the by now classical theorem of Bochi-Mañé, and reveals some rigidity phenomena for these systems.
But what about endomorphisms? One may think that for local diffeomorphisms the situation should be somewhat similar, but until recently no proof of this was found. The reason is that in the endomorphism case the above dichotomy is actually false.
Together with Martin Andersson (UFF) and Radu Saghin (PUCV) we started researching this interesting topic and we found, for example, that in essentially all isotopy classes in the two-torus one can find examples of C¹ robust conservative endomorphisms that have one positive and one negative Lyapunov exponents.