Existence and Uniqueness of SRB Measure on C 1 Generic Hyperbolic Attractors

Abstract

Let M be a smooth Riemannian manifold. We show that for C1 generic \({f\in {\rm Diff}^1(M)}\), if f has a hyperbolic attractor Λf, then there exists a unique SRB measure supported on Λf. Moreover, the SRB measure happens to be the unique equilibrium state of potential function \({\psi_f\in C^0(\Lambda_f)}\) defined by \({\psi_f(x)=-\log|\det(Df|E^u_x)|, x\in \Lambda_f}\), where \({E^u_x}\) is the unstable space of TxM.