Generalized physical and SRB measures for hyperbolic diffeomorphisms

Abstract. In this paper we introduce the notion of generalized phys-ical and SRB measures. These measures naturally generalize classical physical and SRB measures to measures which are supported on invari-ant sets that are not necessarily attractors. We then perform a detailed case study of these measures for hyperbolic Hénon maps. For this class of systems we are able to develop a complete theory about the exis-tence, uniqueness, finiteness, and properties of these natural measures. Moreover, we derive a classification for the existence of a measure of full dimension. We also consider general hyperbolic surface diffeomorphisms and discuss possible extensions of, as well as the differences to, the re-sults for Hénon maps. Finally, we study the regular dependence of the dimension of the generalized physical/SRB measure on the diffeomor-phism. For the proofs we apply various techniques from smooth ergodic theory including the thermodynamic formalism. 1