The SRB Measures For Non-uniformly Hyperbolic Systems

The research of SRB measures was motivated by the work of Sinai,Ruelle and Bowen in 1970s,where the SRB measure was established in uniformly hyperbolic systems.Since then,it becomes a central topic in smooth dynamical systems.In this paper,we study the existence and finiteness of SRB measures or physical measures for some non-uniformly hyperbolic systems.The results of this paper are divided by two chapters.In Chapter 2,we consider the existence of SRB measures and physical measures for C2 dynamical systems with continuous invariant splittings,one of the subbundles has the non-uniformly expanding property,the other direction has the weak contracting property.More precisely,let f be a C2 diffeomorphism on compact Riemannian manifold M,K is an.attractor of f and U is the open neighborhood of A.Assume that there exists an invariant Holder continuous splitting E(?)F over U,We prove that:If there is a set with positive Lebesgue measure such that every x of it is non-uniformly expanding,i.e.,(?)and there exists a full measure set(see Definition 1)such that for every x in it,(?)Then there exists some SRB measure supported on the attractor K.If we assume "<0"instead of the "<0" in the second inequality,then we obtain the existence of physical measures.Non-uniformly expanding was introduced by Alves,Bonatti and Viana in[2],which as a tool,they proved the existence of SRB measures and physical measures under the dominated splitting assumption.Due to the lack of domination in our case,we need to construct some technical results to obtain the weak domination property.With this weak domination,one can use the method from the papers[44,2]to get the SRB measures and physical measures.We also obtain a local disk version of the above result.In Chapter 3,we consider the existence and finiteness of SRB measures and physical measures for some kinds of partially hyperbolic systems.Let f be a C2 partially hyperbolic diffeomorphism on compact Riemannian manifold M,with splitting TM = Eu(?)Ecu(?)Ecs,where Eu is the unstable direction,Ecu is the strong central direction and Ecs is the weak central direction.We prove that:If we assume that the Lyapunov exponents along the strong central direction Ecu are positive and along weak central direction Ecs are negative for every Gibbs u-measure,then there are finitely many SRB measures,they are physical measures whose basins can cover a full Lebesgue measure subset.We obtain the same results under the assumptions similar to the above case,with the modification on strong central direction Ecu,where we assume that the Lyapunov exponents along Eau are positive for every Gibbs u-measure in the integrable sense.We mainly use the properties and assumptions of Gibbs iu-measures to get the non-uniformly expanding condition along Ecu on full Lebesgue measure subset which enable us to construct Gibbs cu-measures.This implies the existence of SRB measures.Then with some uniform properties obtained by non-uniformly expanding and using absolute continuity of Pesin stable foliation,we can show the finiteness and basin covering property of these SRB measures.