Problem 36

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If a geodesic flow is expansive, is it an Anosov flow?



Not exactly the answer, but almost; [1]


  1. [bolton1979conditions] Bolton J.  1979.  Conditions under which a geodesic flow is Anosov. Mathematische Annalen. 240:103–113.

A counterexample is given by a nonpositively curved connected compact surface, such that the set of points with zero curvature is exactly a closed geodesic. By results of Eberlein, or just by looking at the Poincare return map along the geodesic, the geodesic flow on the unitary bundle of the surface is not Anosov. On the other hand, we still have expansiveness because there is no flat strip. In fact, there is still a product structure so the usual topological properties of the flow hold.

Topologically, in surfaces, it is. Paternain, Miguel. Expansive geodesic flows on surfaces. Ergodic Theory Dynam. Systems 13 (1993), no. 1, 153--165.

The answer is no, there are many examples of nonpositively curved compact surfaces where the curvature vanishes along a simple closed geodesic. Such examples can be found in Contreras-Ruggiero, "Non hyperbolic surfaces having all ideal triangles with finite area", Bol. Soc. Mat. Bras. vol. 28, 1 (1997) 43-71. One of the goals of this article was to explore the asymptotics of the orbits in the stable manifold of a flat geodesic without strip. It is shown that for every a>0 there exists a compact surface as described above such that the distance from the closed geodesic with vanishing curvature to an asymptotic one is of the order of T-a where T is the arc length parameter of the closed geodesic. An interesting feature came out from this analysis, if the function T-a is integrable in the set T>1 then the best regularity for the Riemannian metric in the surface is C3. This regularity constraint that seemed to arise from the construction of the surfaces was examined in the article "Flatness of Gaussian curvature and area of ideal triangles", by R. Ruggiero, Bol. Soc. Bras. Mat. vol. 28, 1 (1997) 73-87, where it is proved that if a compact surface with nonpositive curvature is C4 and the area of ideal triangles in the universal covering is bounded above by some constant then the geodesic flow is actually Anosov. This statement should extend to compact surfaces without conjugate points and higher genus.

So Bowen's question involves subtle issues about the regularity of the dynamics and the feasible asymptotic behaviors of non-hyperbolic dynamics. In the case of geodesic flows of compact nonpositively curved surfaces the above results imply that when the surface is of class C4 then the area of ideal triangles is finite if and only if the geodesic flow is Anosov, so an expansive non Anosov geodesic flow in this family of surfaces must have ideal triangles with arbitrarily large area.  If the regularity of the surface is less than C4 there are examples of expansive non Anosov geodesic flows all of whose ideal triangles have finite area in the first mentioned reference. Such examples share many special  properties with Anosov flows, like the existence of solutions of cohomological and subcohomological equations (see "Cohomology and subcohomology problems for expansive, non Anosov geodesic flows", by A. Lopes, W. Rosas and R. Ruggiero, Disc. Cont. Dyn. Sys. vol. 17, 2 (2007) 403-422). with applications to the thermodynamics formalism of the dynamics. 

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