Lyapunov spectrum for geodesic flows of rank 1 surfaces

Keith Burns; Katrin Gelfert

doi:10.3934/dcds.2014.34.1841

Article Contents

2014, Volume 34, Issue 5: 1841-1872. Doi: 10.3934/dcds.2014.34.1841

This issue Previous Article On a functional satisfying a weak Palais-Smale condition Next Article A note on integrable mechanical systems on surfaces

Lyapunov spectrum for geodesic flows of rank 1 surfaces

Keith Burns^1, and
Katrin Gelfert^2,

1.
Department of Mathematics, Northwestern University, Evanston, IL 60208-2730
2.
Instituto de Matemática, Universidade Federal do Rio de Janeiro, Cidade Universitária - Ilha do Fundão, Rio de Janeiro 21945-909

Received: May 2012

Revised: August 2013

Published: May 2014

Abstract / Introduction Related Papers Cited by

Abstract

We give estimates on the Hausdorff dimension of the levels sets of the Lyapunov exponent for the geodesic flow of a compact rank 1 surface. We show that the level sets of points with small (but non-zero) exponents has full Hausdorff dimension, but carries small topological entropy.

Keywords:

Mathematics Subject Classification: Primary: 37D25, 37D35, 28D20, 37C45.

Citation:

Related Papers

Cited by

References

[1]	L. M. Abramov, On the entropy of a flow, Doklad. Acad. Nauk., 128 (1959), 873-875.
[2]	D. Anosov, private communications.
[3]	W. Ballmann, Axial isometries of manifolds of non-positive curvature, Math. Annal., 259 (1982), 131-144.doi: 10.1007/BF01456836.
[4]	L. Barreira and P. Doutor, Birkhoff averages for hyperbolic flows: Variational principles and applications, J. Statist. Phys., 115 (2004), 1567-1603.doi: 10.1023/B:JOSS.0000028069.64945.65.
[5]	R. Bowen, Topological entropy for noncompact sets, Trans. Amer. Math. Soc., 184 (1973), 125-136.doi: 10.1090/S0002-9947-1973-0338317-X.
[6]	R. Bowen and D. Ruelle, The ergodic theory of Axiom A flows, Invent. Math., 29 (1975), 181-202.doi: 10.1007/BF01389848.
[7]	Y. Coudene and B. Schapira, Generic measures for hyperbolic flows on no-compact spaces, Israel. J. Math., 179 (2010), 157-172.doi: 10.1007/s11856-010-0076-z.
[8]	S. Crovisier, Une remarque sur les ensembles hyperboliques localement maximaux. C. R. Math. Acad. Sci. Paris, 334 (2001), 401-404.doi: 10.1016/S1631-073X(02)02274-4.
[9]	P. Eberlein, Geodesic flow in certain manifolds without conjugate points, Trans. Amer. Math. Soc., 167 (1972), 151-170.doi: 10.1090/S0002-9947-1972-0295387-4.
[10]	P. Eberlein, When is a geodesic flow of Anosov-type? I,II, J. Diff. Geom., 8 (1973), 437-463.
[11]	P. Eberlein, Geodesic flows on negatively curved manifolds, Trans. Amer. Math. Soc., 178 (1973), 57-82.doi: 10.1090/S0002-9947-1973-0314084-0.
[12]	P. Eberlein, Geometry of Nonpositively Curved Manifolds, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 1996.
[13]	P. Eberlein and B. O'Neill, Visibility manifolds, Pacific J. Math., 46 (1973), 45-109.doi: 10.2140/pjm.1973.46.45.
[14]	T. Fisher, Hyperbolic sets that are not locally maximal, Ergodic Theory Dynam. Systems, 26 (2006), 1491-1509.doi: 10.1017/S0143385706000411.
[15]	K. Gelfert and M. Rams, The Lyapunov spectrum of some parabolic systems, Ergodic Theory Dynam. Systems, 29 (2009), 919-940.doi: 10.1017/S0143385708080462.
[16]	M. Gromov, Manifolds of negative curvature, J. Diff. Geom., 13 (1978), 223-230.
[17]	S. Ito, On the topological entropy of a dynamical system, Proc. Japan Acad., 45 (1969), 383-840.doi: 10.3792/pja/1195520543.
[18]	A. Katok, Lyapunov exponents, entropy and periodic orbits for diffeomorphisms, Publ. Math. Inst. Hautes Études Sci., 51 (1980), 137-173.
[19]	A. Katok, Entropy and closed geodesics, Ergodic Theory Dynam. Systems, 2 (1982), 339-365.doi: 10.1017/S0143385700001656.
[20]	A. Katok, Nonuniform Hyperbolicity and Structure of Smooth Dynamical Systems, (Warszawa, 1983) (Proceedings of the International Congress of Mathematicians). Eds. Z. Ciesielski and C. Olech, North-Holland, 1984, pp. 1245-1253.
[21]	A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems, Encyclopedia of Mathematics and Its Applications 54, Cambridge University Press, Cambridge, 1995.
[22]	G. Knieper, The uniqueness of the measure of maximal entropy for geodesic flows on rank $1$ manifolds, Ann. of Math., 148 (1998), 291-314.doi: 10.2307/120995.
[23]	A. Manning, A relation between Lyapunov exponents, Hausdorff dimension and entropy, Ergodic Theory Dynam. Systems, 1 (1981), 451-459.
[24]	S. Newhouse, Entropy and volume, Ergodic Theory Dynam. Systems, 8 (1988), 283-299.doi: 10.1017/S0143385700009469.
[25]	S. Newhouse, Continuity of the entropy, Ann. of Math., 129 (1989), 215-235.doi: 10.2307/1971492.
[26]	T. Ohno, A weak equivalence and topological entropy, Publ. Res. Inst. Math. Sci., 16 (1980), 289-298.doi: 10.2977/prims/1195187508.
[27]	Ya. Pesin, Geodesic flows in closed Riemannian manifolds without focal points, Izv. Acad. Nauk SSSR Ser. Mat., 41 (1977), 1252-1288.
[28]	Ya. Pesin, Dimension Theory in Dynamical Systems: Contemporary Views and Applications, Chicago Lectures in Math., University of Chicago Press, Chicago, IL, 1997.
[29]	Ya. Pesin and B. Pitskel, Topological pressure and the variational principle for noncompact sets, Funct. Anal. Appl., 18 (1984), 50-63.
[30]	Ya. Pesin and V. Sadovskaya, Multifractal analysis of conformal axiom A flows, Commun. Math. Phys., 216 (2001), 277-312.doi: 10.1007/s002200000329.
[31]	D. Ruelle, Thermodynamic Formalism. The Mathematical Structures of Equilibrium Statistical Mechanics, Sec. ed., Cambridge University Press, Cambridge, 2004.doi: 10.1017/CBO9780511617546.
[32]	P. Walters, An Introduction to Ergodic Theory, Graduate Texts in Mathematics 79, Springer, New York, 1981.