Abstract
We give a new proof of the existence of compact surfaces embedded in ℝ3 with Anosov geodesic flows. This proof starts with a noncompact model surface whose geodesic flow is shown to be Anosov using a uniformly strictly invariant cone condition. Using a sequence of explicit maps based on the standard torus embedding, we produce compact embedded surfaces that can be seen as small perturbations of the Anosov model system and hence are themselves Anosov.
Similar content being viewed by others
References
Anosov, D.V., Geodesic Flows on Closed Riemannian Manifolds of Negative Curvature, Proc. Steklov Inst. Math., 1967, vol. 90, pp. 1–235; see also: Tr. Mat. Inst. Steklova, 1967, vol. 90, pp. 3–210.
Dolgopyat, D., On Decay of Correlations in Anosov Flows, Ann. of Math. (2), 1998, vol. 147, no. 2, pp. 357–390.
Donnay, V. J. and Pugh, Ch., Finite Horizon Riemann Structures and Ergodicity, Ergodic Theory Dynam. Systems, 2004, vol. 24, no. 1, pp. 89–106.
Donnay, V. J. and Pugh, Ch., Anosov Geodesic Flows for Embedded Surfaces, in Geometric Methods in Dynamics: 2, W. de Melo, M.Viana, J.-Ch.Yoccoz (Eds.), Astérisque, vol. 287, Paris: Soc. Math. France, 2003, pp. 61–69.
Katok, A. and Hasselblatt, B., Introduction to the Modern Theory of Dynamical Systems, Encyclopedia Math. Appl., vol. 54, Cambridge: Cambridge Univ. Press, 1995.
Klingenberg, W., Riemannian Manifolds with Geodesic Flow of Anosov Type, Ann. of Math. (2), 1974, vol. 99, pp. 1–13.
Kourganoff, M., Embedded Surfaces with Anosov Geodesic Flows, Approximating Spherical Billiards, arXiv: 1612.05430 (2016).
Kourganoff, M., Uniform hyperbolicity in nonflat billiards, Discrete Cont. Dyn.-A, 2018, vol. 38, no. 3, pp. 1145–1160.
Liverani, C., On Contact Anosov Flows, Ann. of Math. (2), 2004, vol. 159, no. 3, pp. 1275–1312.
Mañé, R., On a Theorem of Klingenberg, in Dynamical Systems and Bifurcation Theory (Rio de Janeiro, 1985), Pitman Res. Notes Math. Ser., vol. 160, Harlow: Longman Sci. Tech., 1987, pp. 319–345.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Donnay, V., Visscher, D. A New Proof of the Existence of Embedded Surfaces with Anosov Geodesic Flow. Regul. Chaot. Dyn. 23, 685–694 (2018). https://doi.org/10.1134/S1560354718060047
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S1560354718060047