Dissipative surface dynamics: critical points vs. induced maps.
Abstract. We compare two fundamentally different approaches to prove the existence of SRB measures in parametrized area decreasing surface diffeomorphisms. The first, pioneered by Benedicks-Carleson and Benedicks-Young for the Henon family, and subsequently extended by Mora-Viana, Luzatto-Viana, and Wang-Young, seeks to find "good" parameters for which one can define a "critical set" whose forward orbits determine the dynamics. The second, which is part of on-going work of Michael Jakobson and myself, seeks to find "good" parameters where there is a certain power or induced map which is uniformly hyperbolic. Thus, in a certain sense, the critical points have been eliminated from consideration. This latter approach is inspired by Jakobson's original proof of the abundance of absolutely continuous invariant measures for quadratic interval maps.