SRB Measures Of Parthally Hyperholic Flows

SRB measures are one of the most important classes of natural invariant measures with chaotic behavior in dynamical systems.We are interested in the existence and finiteness of SRB measures on partially hyperbolic attractors of flows.Suppose that ?is a flow generated by a C1 vector field X on a compact smooth Riemannian manifold M.If an attractor A of ? has a partially hyperbolic splitting then there is an SRB measure supported on A.In this thesis,we actually prove the following stronger result:for Lebesgue almost every point x in a neighborhood of ?,? is a limit of(?),one of whose ergodic components is an SRB measure.Applying the above results,we prove t,hat if ? is a C2 flow and Ec is Gibb-s non-uniformly expanding,then there are finitely many SRB(physical)measures?1,?2,…,?k on ? such that#12The proof of this result has to deal with the difficulties that do not occur in the case of diffeomorphisms.