Geodesic flows on negatively curved manifolds. II

Abstract

Let M be a complete Riemannian manifold with sectional curvature K ≤ 0 K \leq 0 , SM the unit tangent bundle of M, T t {T_t} the geodesic flow on SM and Ω ⊆ S M \Omega \subseteq SM the set of nonwandering points relative to T t {T_t} . T t {T_t} is topologically mixing (respectively topologically transitive) on SM if for any open sets 0, U of SM there exists A > 0 A > 0 such that | t | ≥ A \left | t \right | \geq A implies T t ( O ) ∩ U ≠ ∅ {T_t}(O) \cap U \ne \emptyset (respectively there exists t ε R t\;\varepsilon \;R such that T t ( O ) ∩ U ≠ ∅ {T_t}(O) \cap U \ne \emptyset ). For each vector v ε S M v\;\varepsilon \;SM we define stable and unstable sets W s ( v ) , W s s ( v ) , W u ( v ) {W^s}(v),{W^{ss}}(v),{W^u}(v) and W u u ( v ) {W^{uu}}(v) , and we relate topological mixing (respectively topological transitivity) of T t {T_t} to the existence of a vector v ∈ S M v\; \in \;SM such that W s s ( v ) {W^{ss}}(v) (respectively W s ( v ) {W^s}(v) ) is dense in SM. If M is a Visibility manifold (implied by K ≤ c > 0 K \leq c > 0 ) and if Ω = S M \Omega = SM then T t {T_t} is topologically mixing on SM. Let S n = {S_n} = {Visibility manifolds M of dimension n such that T t {T_t} is topologically mixing on SM}. For each n ≥ 2 n \geq 2 , S n {S_n} is closed under normal (Galois) Riemannian coverings. If M ∈ S n M\; \in \;{S_n} we classify { v ∈ S M : W s s ( v ) v\; \in \;SM:\;{W^{ss}}(v) is dense in SM}, and M is compact if and only if this set = SM. We also consider the case where Ω \Omega is a proper subset of SM.