Ergodic Properties of Geodesic Flows on Closed Riemannian Manifolds of Negative Curvature

By D. Anosov. 2020

520 citations recorded by the source corpus.

Citation in format AMSBIB \Bibitem{Ano67} \by D.~V.~Anosov \paper Geodesic flows on closed Riemannian manifolds of negative curvature \serial Trudy Mat. Inst. Steklov. \yr 1967 \vol 90 \pages 3--210 \mathnet{http://mi.mathnet.ru/tm2795} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=0224110} \zmath{https://zbmath.org/?q=an:0163.43604|0176.19101} \transl \jour Proc. Steklov Inst. Math. \yr 1967 \vol 90 \pages 1--235

Subfields: Applied Mathematics, Astronomy and Astrophysics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/04d8388c6fbeec75f84cc9f2bd0c71a45f67679d

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Citations in the corpus

On structurally stable diffeomorphisms with codimension one expanding attractors (2004, background): “In [1], it was proved that Anosov diffeomorphisms on a compact manifold are structurally stable.”
Absolute continuity, Lyapunov exponents and rigidity I : geodesic flows (2011, background): “The classical definition of absolute continuity [1, 2] goes as follows.”
Physical measures and absolute continuity for one-dimensional center direction (2013, background): “By Anosov’s closing lemma [11], periodic points are dense in the non-wandering set of fc.”
Chapter 11 Dynamics of subgroup actions on homogeneous spaces of lie groups and applications to number theory (2002, background): “On the other hand, geodesic flows on homeomorphic compact surfaces of constant negative curvature are always topologically equivalent, though the surfaces may not be isometric [4].”
Cross ratios, surface groups, PSL(n,ℝ) and diffeomorphisms of the circle (2007, background): “More generally, Otal’s construction can be extended to Anosov flows on unit tangent bundle of surfaces of whom geodesic flows of a negatively curved metrics are special cases [1].”
Frontiers of chaotic advection (2014, background): “Then the flow preserves some modified mass and is mixing by Anosov’s alternative (Anosov, 1967), i.e., C(τ) = ∫ ψ(φτx)χ(x) → 0 as τ →∞ for all suitable pairs of functions ψ, χ.”
On Deformation Boundary Rigidity and Spectral Rigidity of Riemannian Surfaces with No Focal Points (2000, background): “On the other hand, it is well known [1] that the stable distribution of an Anosov flow is continuous.”
Stable ergodicity of certain linear automorphisms of the torus (2002, background): “As any ergodic linear automorphism of T4 is either Anosov or pseudo-Anosov (see Corollary A.7 of Appendix A) and any ergodic linear automorphism of T5 is Anosov (see Corollary A.5 of Appendix A), we get Corollary 1.3.”
Entropy along expanding foliations (2016, methodology): “Introduction Partially hyperbolic diffeomorphisms were proposed by Brin, Pesin [11] and Pugh, Shub [28] independently at the early 1970’s, as an extension of the class of Anosov diffeomorphisms [3, 4].”
Spectral rigidity and invariant distributions on Anosov surfaces (2012, background): “Note that there are no L(2) solutions to Xw = 0 (not even L(1)) due to the ergodicity of the geodesic flow [2, 22], so H−1 is the optimal regularity in the H Sobolev scale.”
Banach spaces adapted to Anosov systems (2004, background): “Introduction The study of the statistical properties of Anosov systems dates back almost half a century [1] and many approaches have been developed to investigate various aspects of the field (the most historically relevant one being based on the introduction of Markov partitions [2, 8, 24, 31]).”
Expanding maps on Cantor sets and analytic continuation of zeta functions (2005, background): “In the case of Anosov flows, it is enough to divide U0 into closed segments of equal lengths, but in our case we need in addition a partition “well ordered” with respect to the Cantor set.”