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A Caltech Library Repository Feedhttp://www.rssboard.org/rss-specificationpython-feedgenenMon, 15 Apr 2024 15:29:27 +0000Anosov flows with smooth foliations and rigidity of geodesic flows on three-dimensional manifolds of negative curvature
https://resolver.caltech.edu/CaltechAUTHORS:20170408-163655442
Authors: {'items': [{'id': 'Feres-R', 'name': {'family': 'Feres', 'given': 'Renato'}}, {'id': 'Katok-A', 'name': {'family': 'Katok', 'given': 'Anatole'}}]}
Year: 1990
DOI: 10.1017/S0143385700005836
We consider Anosov flows on a 5-dimensional smooth manifold V that possesses an invariant symplectic form (transverse to the flow) and a smooth invariant probability measure λ. Our main technical result is the following: If the Anosov foliations are C∞, then either (1) the manifold is a transversely locally symmetric space, i.e. there is a flow-invariant C∞ affine connection ∇ on V such that ∇R ≡ 0, where R is the curvature tensor of ∇, and the torsion tensor T only has nonzero component along the flow direction, or (2) its Oseledec decomposition extends to a C∞ splitting of TV (defined everywhere on V) and for any invariant ergodic measure μ, there exists χ_μ > 0 such that the Lyapunov exponents are −2χ_μ, −χ_μ, 0, χ_μ, and 2χ_μ, μ-almost everywhere.
As an application, we prove: Given a closed three-dimensional manifold of negative curvature, assume the horospheric foliations of its geodesic flow are C∞. Then, this flow is C∞ conjugate to the geodesic flow on a manifold of constant negative curvature.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/r4egt-44r08Formulas for the derivative and critical points of topological entropy for Anosov and geodesic flows
https://resolver.caltech.edu/CaltechAUTHORS:20170408-143320653
Authors: {'items': [{'id': 'Katok-A', 'name': {'family': 'Katok', 'given': 'Anatole'}}, {'id': 'Knieper-G', 'name': {'family': 'Knieper', 'given': 'Gerhard'}}, {'id': 'Weiss-H', 'name': {'family': 'Weiss', 'given': 'Howard'}}]}
Year: 1991
DOI: 10.1007/BF02099667
This paper represents part of a program to understand the behavior of topological entropy for Anosov and geodesic flows. In this paper, we have two goals. First we obtain some regularity results for C^1 perturbations. Second, and more importantly, we obtain explicit formulas for the derivative of topological entropy. These formulas allow us to characterize the critical points of topological entropy on the space of negatively curved metrics.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/8vaks-8e191