Hostname: page-component-7f64f4797f-fz5kh Total loading time: 0 Render date: 2025-11-04T00:16:56.165Z Has data issue: false hasContentIssue false
  • English
  • Français

Equilibrium measures for certain isometric extensions of Anosov systems

Published online by Cambridge University Press:  22 September 2016

RALF SPATZIER  and
DANIEL VISSCHER
RALF SPATZIER
Affiliation:
University of Michigan, Department of Mathematics, 530 Church Street, Ann Arbor, Michigan 48109, USA email spatzier@umich.edu, davissch@umich.edu
DANIEL VISSCHER
Affiliation:
University of Michigan, Department of Mathematics, 530 Church Street, Ann Arbor, Michigan 48109, USA email spatzier@umich.edu, davissch@umich.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

We prove that for the frame flow on a negatively curved, closed manifold of odd dimension other than 7, and a Hölder continuous potential that is constant on fibers, there is a unique equilibrium measure. Brin and Gromov’s theorem on the ergodicity of frame flows follows as a corollary. Our methods also give a corresponding result for automorphisms of the Heisenberg manifold fibering over the torus.

Information

Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 38 , Issue 3 , May 2018 , pp. 1154 - 1167
Copyright
© Cambridge University Press, 2016 

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.)

Article purchase

Temporarily unavailable

References

Avila, A., Viana, M. and Wilkinson, A.. Absolute continuity, Lyapunov exponents and rigidity I: geodesic flows. J. Eur. Math. Soc. (JEMS) 17(6) (2015), 1435–1462.Google Scholar
Berg, K. R.. Convolution of invariant measures, maximal entropy. Math. Syst. Theory 3 (1969), 146–150.Google Scholar
Bonatti, C., Díaz, L. J. and Viana, M.. Dynamics Beyond Uniform Hyperbolicity: A Global Geometric and Probabilistic Perspective (Encyclopaedia of Mathematical Sciences, 102, Mathematical Physics, III) . Springer, Berlin, 2005.Google Scholar
Bowen, R.. Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms (Lecture Notes in Mathematics, 470) . Springer, New York, 1975.Google Scholar
Bowen, R. and Ruelle, D.. The ergodic theory of Axiom A flows. Invent. Math. 29(3) (1975), 181–202.Google Scholar
Brin, M. and Gromov, M.. On the ergodicity of frame flows. Invent. Math. 60(1) (1980), 1–7.Google Scholar
Brin, M. and Karcher, H.. Frame flows on manifolds with pinched negative curvature. Compos. Math. 52(3) (1984), 275–297.Google Scholar
Burns, K., Climenhaga, V., Fisher, T. and Thompson, D.. Unique equilibrium states for geodesic flows in nonpositive curvature. Personal communication, 2015.Google Scholar
Burns, K. and Pollicott, M.. Stable ergodicity and frame flows. Geom. Dedicata 98 (2003), 189–210.Google Scholar
Climenhaga, V., Fisher, T. and Thompson, D.. Unique equilibrium states for some robustly transitive systems. Preprint, 2015, arXiv:1505.06371.Google Scholar
Climenhaga, V. and Thompson, D.. Unique equilibrium states for flows and homeomorphisms with non-uniform structure. Preprint, 2015, arXiv:1505.03803.Google Scholar
Einsiedler, M., Katok, A. and Lindenstrauss, E.. Invariant measures and the set of exceptions to Littlewood’s conjecture. Ann. of Math. (2) 164(2) (2006), 513–560.CrossRefGoogle Scholar
Einsiedler, M. and Lindenstrauss, E.. On measures invariant under tori on quotients of semisimple groups. Ann. of Math. (2) 181(3) (2015), 993–1031.Google Scholar
Einsiedler, M., Lindenstrauss, E. and Ward, T.. Entropy in ergodic theory and homogeneous dynamics.http://maths.dur.ac.uk/∼tpcc68/entropy/welcome.html.Google Scholar
Fisher, T.. Recent developments in entropy and equilibrium states for diffeomorphisms. https://math.byu.edu/∼tfisher/documents/papers/notes.pdf.Google Scholar
Goetze, E. R. and Spatzier, R. J.. On Livšic’s theorem, superrigidity, and Anosov actions of semisimple Lie groups. Duke Math. J. 88(1) (1997), 1–27.Google Scholar
Katok, A. and Hasselblatt, B.. Introduction to the Modern Theory of Dynamical Systems (Encyclopedia of Mathematics and its Applications, 54) . Cambridge University Press, Cambridge, 1995.Google Scholar
Katok, A. and Spatzier, R. J.. Invariant measures for higher-rank hyperbolic abelian actions. Ergod. Th. & Dynam. Sys. 16(4) (1996), 751–778.Google Scholar
Knieper, G.. The uniqueness of the measure of maximal entropy for geodesic flows on rank 1 manifolds. Ann. of Math. (2) 148(1) (1998), 291–314.Google Scholar
Knieper, G.. The uniqueness of the maximal measure for geodesic flows on symmetric spaces of higher rank. Israel J. Math. 149 (2005), 171–183.CrossRefGoogle Scholar
Ledrappier, F. and Walters, P.. A relativised variational principle for continuous transformations. J. Lond. Math. Soc. (2) 16(3) (1977), 568–576.Google Scholar
Leplaideur, R.. Local product structure for equilibrium states. Trans. Amer. Math. Soc. 352(4) (2000), 1889–1912.Google Scholar
Lindenstrauss, E. and Schmidt, K.. Invariant sets and measures of nonexpansive group automorphisms. Israel J. Math. 144 (2004), 29–60.Google Scholar
Livšic, A. N.. Cohomology of dynamical systems. Izv. Akad. Nauk SSSR Ser. Mat. 36 (1972), 1296–1320.Google Scholar
Newhouse, S. E.. Continuity properties of entropy. Ann. of Math. (2) 129(2) (1989), 215–235.Google Scholar
Quas, A. and Soo, T.. Ergodic universality of some topological dynamical systems. Trans. Amer. Math. Soc. 368(6) (2016), 4137–4170.Google Scholar
Rodriguez Hertz, F., Rodriguez Hertz, M. A., Tahzibi, A. and Ures, R.. Maximizing measures for partially hyperbolic systems with compact center leaves. Ergod. Th. & Dynam. Sys. 32(2) (2012), 825–839.Google Scholar
Ures, R.. Intrinsic ergodicity of partially hyperbolic diffeomorphisms with a hyperbolic linear part. Proc. Amer. Math. Soc. 140(6) (2012), 1973–1985.CrossRefGoogle Scholar