In recent years, many authors have sought to establish in the hyperbolic setting the functional analytic approach developed for one-dimensional piecewise expanding maps. The approach is, very roughly speaking, to directly study the action of the transfer operator on appropriate spaces of distributions.
This approach was successfully extended to multi-dimensional expanding maps, but its application to hyperbolic systems with discontinuities has been lacking until recently, see the references of C. Liverani [Dynamical systems. Part II. Topological, geometrical and ergodic properties of dynamics, Selected papers from the Research Trimester held in Pisa, Italy, February 4-April 26, 2002, Pisa: Scuola Normale Superiore, Pubblicazioni del Centro di Ricerca Matematica Ennio de Giorgi, Proceedings, 185–237 (2003; Zbl 1066.37013)]. As far as hyperbolic systems with discontinuities are concerned, the only available approaches are Ya. B. Pesin [Ergodic Theory Dyn. Syst. 12, 123–151 (1992; Zbl 0774.58029)] and L.-S. Young [Ann. Math. (2) 147, No. 3, 585–650 (1998; Zbl 0945.37009)]. Such approaches require a very deep preliminary understanding of the regularity properties of the invariant foliations and are not well-suited to the study of perturbations of the systems under considerations.
The present paper under review makes a first step in overcoming the difficulty of discontinuities by showing that in the two-dimensional case the functional analytic approach can be carried out successfully. In this paper, for two-dimensional piecewise hyperbolic maps with uniformly bounded second derivatives, the authors obtain a complete description of the SRB measures, their statistical properties and their stability with respect to many types of perturbations including deterministic and random perturbations and holes.