Inverse SRB measures for endomorphisms on surfaces

Abstract

We extend D. Burguet's construction of SRB measures for the non-invertible scenario obtaining hyperbolic invariant measures with absolutely continuous disintegrations on stable manifolds for a certain class of endomorphisms on the two-torus. The constructed measures also maximize the folding entropy, and we call them equidistributed inverse SRB measures. For dissipative perturbations of the examples given by M. Andersson, P. Carrasco and R. Saghin we obtain existence, uniqueness and continuity with respect to the map of the equidistributed inverse SRB measures as well as the SPR property of the stable geometric potential for such maps.