Hostname: page-component-8448b6f56d-tj2md Total loading time: 0 Render date: 2024-04-16T09:38:30.968Z Has data issue: false hasContentIssue false
  • English
  • Français

Geodesic Flows Modelled by Expansive Flows

Part of: Global differential geometry Dynamical systems with hyperbolic behavior Measure-theoretic ergodic theory Symplectic geometry, contact geometry

Published online by Cambridge University Press:  28 August 2018

Katrin Gelfert  and
Rafael O. Ruggiero
Katrin Gelfert*
Affiliation:
Instituto de Matemática, Universidade Federal do Rio de Janeiro, Cidade Universitária – Ilha do Fundão, Rio de Janeiro 21945-909, Brazil (gelfert@im.ufrj.br)
Rafael O. Ruggiero
Affiliation:
Departamento de Matemática PUC-Rio, Rua Marqués de São Vicente 225, Rio de Janeiro 22543-900, Brazil (rorr@mat.puc-rio.br)
*
*Corresponding author.
Get access Rights & Permissions [Opens in a new window]

Abstract

Given a smooth compact surface without focal points and of higher genus, it is shown that its geodesic flow is semi-conjugate to a continuous expansive flow with a local product structure such that the semi-conjugation preserves time parametrization. It is concluded that the geodesic flow has a unique measure of maximal entropy.

Keywords

expansive flowgeodesic flowconjugate pointsfocal pointstime-preserving semi-conjugacymeasure of maximal entropy

MSC classification

Primary: 53D25: Geodesic flows
Secondary: 53C22: Geodesics 37D40: Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.) 37D25: Nonuniformly hyperbolic systems (Lyapunov exponents, Pesin theory, etc.) 37D35: Thermodynamic formalism, variational principles, equilibrium states 28D20: Entropy and other invariants 28D99: None of the above, but in this section
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 62 , Issue 1 , February 2019 , pp. 61 - 95
Copyright
Copyright © Edinburgh Mathematical Society 2018 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1.Abramov, L. M., On the entropy of a flow, Doklad. Acad. Nauk. 128 (1959), 873–875.Google Scholar
2.Barbosa Gomes, J. and Ruggiero, R. O., Uniqueness of central foliations of geodesic flows for compact surfaces without conjugate points, Nonlinearity 20 (2007), 497–515.Google Scholar
3.Bing, R. H., An alternative proof that 3-manifolds can be triangulated, Ann. Math. (2) 59 (1959), 37–65.Google Scholar
4.Bowen, R., Entropy for group endomorphisms and homogeneous spaces, Trans. Amer. Math. Soc. 153 (1971), 401–414.Google Scholar
5.Bowen, R., Periodic orbits for hyperbolic flows, Amer. J. Math. 94 (1972), 1–30.Google Scholar
6.Bowen, R., Maximizing entropy for hyperbolic flows, Math. Syst. Theory 7 (1973), 300–303.Google Scholar
7.Bowen, R., Some systems with unique equilibrium states, Math. Syst. Theory 8 (1974/75), 193–202.Google Scholar
8.Bowen, R. and Walters, P., Expansive one-parameter flows, J. Diff. Equ. 12 (1972), 180–193.Google Scholar
9.Burns, K., Hyperbolic behavior of geodesic flows on manifolds with no focal points, Ergodic Theory Dynam. Systems 3 (1983), 1–12.Google Scholar
10.Burns, K., The flat strip theorem fails for surfaces with no conjugate points, Proc. Amer. Math. Soc. 115 (1992), 199–206.Google Scholar
11.Buzzi, J., Fisher, T., Sambarino, M. and Vásquez, C., Maximal entropy measures for certain partially hyperbolic, derived from Anosov systems, Ergodic Theory Dynam. Systems 32 (2012), 63–79.Google Scholar
12.Coudene, Y. and Schapira, B., Generic measures for geodesic flows on nonpositively curved manifolds, J. Éc. Polytech. Math. 1 (2014), 387–408.Google Scholar
13.Croke, C., Rigidity for surfaces of nonpositive curvature, Comment. Math. Helv. 65 (1990), 150–169.Google Scholar
14.Croke, C. and Fathi, A., An inequality between energy and intersection, Bull. Lond. Math. Soc. 22 (1990), 489–494.Google Scholar
15.Eberlein, P., Geodesic flows in certain manifolds without conjugate points, Trans. Amer. Math. Soc. 167 (1972), 151–170.Google Scholar
16.Eberlein, P., When is a geodesic flow of Anosov type: I, J. Diff. Geom. 8 (1973), 437–463.Google Scholar
17.Eberlein, P., Geodesic flows on negatively curved manifolds II, Trans. Amer. Math. Soc. 178 (1973), 57–82.Google Scholar
18.Eberlein, P. and O'Neill, B., Visibility manifolds, Pacific J. Math. 46 (1973), 45–109.Google Scholar
19.Eschenburg, J., Horospheres and the stable part of the geodesic flow, Math. Z. 153 (1977), 237–251.Google Scholar
20.Franco, E., Flows with unique equilibrium states, Amer. J. Math. 99 (1977), 486–514.Google Scholar
21.Gallot, S., Hulin, D. and Lafontaine, J., Riemannian geometry, Universitext (Springer, Berlin, 2004).Google Scholar
22.Ghys, E., Flots d'Anosov sur les 3-variétés fibrées en cercles, Ergodic Theory Dynam. Systems 4 (1984), 67–80.Google Scholar
23.Green, L., Surfaces without conjugate points, Trans. Amer. Math. Soc. 76 (1954), 529–546.Google Scholar
24.Gromov, M., Hyperbolic groups, In Essays in group theory (ed. S. M. Gersten), pp. 75–263, Mathematical Sciences Research Institute Publications, Volume 8 (Springer, New York, 1987).Google Scholar
25.Hiraide, K., Expansive homeomorphisms of compact surfaces are pseudo-Anosov, Osaka J. Math. 27 (1990), 117–162.Google Scholar
26.Inaba, T. and Matsumoto, S., Nonsingular expansive flows on 3-manifolds and foliations with circle prone singularities, Japan. J. Math. (N.S.) 16 (1990), 329–340.Google Scholar
27.Ito, S., On the topological entropy of a dynamical system, Proc. Japan Acad. 45 (1969), 383–840.Google Scholar
28.Katok, A. and Hasselblatt, B., Introduction to the modern theory of dynamical systems, Encyclopedia of Mathematics and its Applications, Volume 54 (Cambridge University Press, Cambridge, 1995).Google Scholar
29.Knieper, G., The uniqueness of the measure of maximal entropy for geodesic flows on rank 1 manifolds, Ann. Math. (2) 148 (1998), 291–314.Google Scholar
30.Kwietniak, D. and Oprocha, P., A note on the average shadowing property for expansive maps, Topology Appl. 159 (2012), 19–27.Google Scholar
31.Lewowicz, J., Expansive homeomorphisms of surfaces, Bull. Braz. Math. Soc. (N.S.) 20 (1989), 113–133.Google Scholar
32.Moise, E., Geometric topology in dimensions 2 and 3, Graduate Texts in Mathematics, Volume 47 ( Springer, New York–Heidelberg, 1977).Google Scholar
33.Morse, M., A fundamental class of geodesics on any closed surface of genus greater than one, Trans. Amer. Math. Soc. 26 (1924), 25–60.Google Scholar
34.Newhouse, S., Continuity of the entropy, Ann. Math. (2) 129 (1989), 215–235.Google Scholar
35.Oka, M., Hyperbolic expansivity and equilibrium states, Southeast Asian Bull. Math. 1 (1997), 159–165.Google Scholar
36.Otal, J.-P., Le spectre marqué des longueurs des surfaces a courbure négative, Ann. Math. (2) 131 (1990), 151–162.Google Scholar
37.Paternain, M., Expansive geodesic flows on surfaces, Ergodic Theory Dynam. Systems 13 (1993), 153–165.Google Scholar
38.Pesin, Ya., Geodesic flows in closed Riemannian manifolds without focal points, Izv. Acad. Nauk SSSR Ser. Mat. 41 (1977), 1252–1288.Google Scholar
39.Pesin, Ya., Characteristic Lyapunov exponents, and smooth ergodic theory, Russ. Math. Surveys 32 (1977), 55–114.Google Scholar
40.Ruelle, D., An inequality for the entropy of differentiable maps, Bol. Soc. Brasil. Mat. 9 (1978), 83–87.Google Scholar
41.Ruggiero, R., Expansive dynamics and hyperbolic geometry, Bull. Braz. Math. Soc. (N.S.) 25 (1994), 139–172.Google Scholar
42.Ruggiero, R., Expansive geodesic flows in manifolds with no conjugate points, Ergodic Theory Dynam. Systems 17 (1997), 211–225.Google Scholar
43.Schröder, J. Ph., Minimal rays on closed surfaces, Israel J. Math. 217 (2017), 197–229.Google Scholar
44.Thomas, R. F., Canonical coordinates and the pseudo orbit tracing property, J. Diff. Equ. 90 (1991), 316–343.Google Scholar
45.Walters, P., An introduction to ergodic theory, Graduate Texts in Mathematics, Volume 79 ( Springer, New York–Berlin, 1982).Google Scholar