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On the mixing and Bernoulli properties for geodesic flows on rank 1 manifolds without focal points
1. | College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao, 266590, China |
2. | Department of Mathematics, Southern University of Science and Technology, Shenzhen, 518055, China |
3. | School of Mathematical Sciences, Capital Normal University, Beijing, 100048, China |
If $ (M,g) $ is a smooth compact rank $ 1 $ Riemannian manifold without focal points, it is shown that the measure $ \mu_{\max} $ of maximal entropy for the geodesic flow is unique. In this article, we study the statistic properties and prove that this unique measure $ \mu_{\max} $ is mixing. Stronger conclusion that the geodesic flow on the unit tangent bundle $ SM $ with respect to $ \mu_{\max} $ is Bernoulli is acquired provided $ M $ is a compact surface with genus greater than one and no focal points.
References:
[1] |
M. Babillot,
On the mixing property for hyperbolic systems, Israel J. Math., 129 (2002), 61-76.
doi: 10.1007/BF02773153. |
[2] |
W. Ballmann,
Axial isometries of manifolds of nonpositive curvature, Math. Ann., 259 (1982), 131-144.
doi: 10.1007/BF01456836. |
[3] |
W. Ballmann, M. Brin and P. Eberlein,
Structure of manifolds of nonpositive curvature. I, Ann. of Math., 122 (1985), 171-203.
doi: 10.2307/1971373. |
[4] |
K. Burns and A. Katok,
Manifolds with non-positive curvature, Ergodic Theory Dynam. Systems, 5 (1985), 307-317.
doi: 10.1017/S0143385700002935. |
[5] |
E. I. Dinaburg,
On the relations among various entropy characteristics of dynamical systems, Math. USSR Izv., 5 (1971), 337-378.
doi: 10.1070/IM1971v005n02ABEH001050. |
[6] |
P. B. Eberlein, Geometry of Nonpositively Curved Manifolds, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 1996.
![]() |
[7] |
R. Gulliver,
On the variety of manifolds without conjugate points, Trans. Amer. Math. Soc., 210 (1975), 185-201.
doi: 10.1090/S0002-9947-1975-0383294-0. |
[8] |
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. |
[9] |
G. Knieper, Hyperbolic dynamics and Riemannian geometry, in Handbook of Dynamical Systems, 1A, North-Holland, Amsterdam, 2002,453–545.
doi: 10.1016/S1874-575X(02)80008-X. |
[10] |
F. Ledrappier, Y. Lima and O. Sarig,
Ergodic properties of equilibrium measures for smooth three dimensional flows, Comment. Math. Helvetici, 91 (2016), 65-106.
doi: 10.4171/CMH/378. |
[11] |
F. Liu and F. Wang,
Entropy-expansiveness of geodesic flows on closed manifolds without conjugate points, Acta Math. Sin. (Engl. Ser.), 32 (2016), 507-520.
doi: 10.1007/s10114-016-5200-5. |
[12] |
F. Liu, F. Wang and W. Wu,
On the Patterson-Sullivan measure for geodesic flows on rank $1$ manifolds without focal points, Discrete Contin. Dyn. Syst., 40 (2020), 1517-1554.
doi: 10.3934/dcds.2020085. |
[13] |
F. Liu, F. Wang and W. Wu,
The topological entropy for autonomous Lagrangian systems on compact manifolds whose fundamental groups have exponential growth, Sci. China Math., 63 (2020), 1323-1338.
doi: 10.1007/s11425-018-9408-8. |
[14] |
F. Liu and X. Zhu,
The transitivity of geodesic flows on rank $1$ manifolds without focal points, Differential Geom. Appl., 60 (2018), 49-53.
doi: 10.1016/j.difgeo.2018.05.007. |
[15] |
J. J. O'Sullivan,
Riemannian manifolds without focal points, J. Differential Geometry, 11 (1976), 321-333.
doi: 10.4310/jdg/1214433590. |
[16] |
G. P. Paternain, Geodesic Flows, Progress in Mathematics, 180, Birkhäuser Boston, Inc., Boston, MA, 1999.
doi: 10.1007/978-1-4612-1600-1. |
[17] |
R. O. Ruggiero,
Expansive geodesic flows in manifolds with no conjugate points, Ergodic Theory Dynam. Systems, 17 (1997), 211-225.
doi: 10.1017/S0143385797060963. |
[18] |
P. Walters, An Introduction to Ergodic Theory, Graduate Texts in Mathematics, 79, Springer-Verlag, New York-Berlin, 1982. |
[19] |
J. Watkins,
The higher rank rigidity theorem for manifolds with no focal points, Geom. Dedicata, 164 (2013), 319-349.
doi: 10.1007/s10711-012-9776-3. |
[20] |
W. Wu, F. Liu and F. Wang, On the ergodicity of geodesic flows on surfaces without focal points, preprint, arXiv:1812.04409. Google Scholar |
show all references
References:
[1] |
M. Babillot,
On the mixing property for hyperbolic systems, Israel J. Math., 129 (2002), 61-76.
doi: 10.1007/BF02773153. |
[2] |
W. Ballmann,
Axial isometries of manifolds of nonpositive curvature, Math. Ann., 259 (1982), 131-144.
doi: 10.1007/BF01456836. |
[3] |
W. Ballmann, M. Brin and P. Eberlein,
Structure of manifolds of nonpositive curvature. I, Ann. of Math., 122 (1985), 171-203.
doi: 10.2307/1971373. |
[4] |
K. Burns and A. Katok,
Manifolds with non-positive curvature, Ergodic Theory Dynam. Systems, 5 (1985), 307-317.
doi: 10.1017/S0143385700002935. |
[5] |
E. I. Dinaburg,
On the relations among various entropy characteristics of dynamical systems, Math. USSR Izv., 5 (1971), 337-378.
doi: 10.1070/IM1971v005n02ABEH001050. |
[6] |
P. B. Eberlein, Geometry of Nonpositively Curved Manifolds, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 1996.
![]() |
[7] |
R. Gulliver,
On the variety of manifolds without conjugate points, Trans. Amer. Math. Soc., 210 (1975), 185-201.
doi: 10.1090/S0002-9947-1975-0383294-0. |
[8] |
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. |
[9] |
G. Knieper, Hyperbolic dynamics and Riemannian geometry, in Handbook of Dynamical Systems, 1A, North-Holland, Amsterdam, 2002,453–545.
doi: 10.1016/S1874-575X(02)80008-X. |
[10] |
F. Ledrappier, Y. Lima and O. Sarig,
Ergodic properties of equilibrium measures for smooth three dimensional flows, Comment. Math. Helvetici, 91 (2016), 65-106.
doi: 10.4171/CMH/378. |
[11] |
F. Liu and F. Wang,
Entropy-expansiveness of geodesic flows on closed manifolds without conjugate points, Acta Math. Sin. (Engl. Ser.), 32 (2016), 507-520.
doi: 10.1007/s10114-016-5200-5. |
[12] |
F. Liu, F. Wang and W. Wu,
On the Patterson-Sullivan measure for geodesic flows on rank $1$ manifolds without focal points, Discrete Contin. Dyn. Syst., 40 (2020), 1517-1554.
doi: 10.3934/dcds.2020085. |
[13] |
F. Liu, F. Wang and W. Wu,
The topological entropy for autonomous Lagrangian systems on compact manifolds whose fundamental groups have exponential growth, Sci. China Math., 63 (2020), 1323-1338.
doi: 10.1007/s11425-018-9408-8. |
[14] |
F. Liu and X. Zhu,
The transitivity of geodesic flows on rank $1$ manifolds without focal points, Differential Geom. Appl., 60 (2018), 49-53.
doi: 10.1016/j.difgeo.2018.05.007. |
[15] |
J. J. O'Sullivan,
Riemannian manifolds without focal points, J. Differential Geometry, 11 (1976), 321-333.
doi: 10.4310/jdg/1214433590. |
[16] |
G. P. Paternain, Geodesic Flows, Progress in Mathematics, 180, Birkhäuser Boston, Inc., Boston, MA, 1999.
doi: 10.1007/978-1-4612-1600-1. |
[17] |
R. O. Ruggiero,
Expansive geodesic flows in manifolds with no conjugate points, Ergodic Theory Dynam. Systems, 17 (1997), 211-225.
doi: 10.1017/S0143385797060963. |
[18] |
P. Walters, An Introduction to Ergodic Theory, Graduate Texts in Mathematics, 79, Springer-Verlag, New York-Berlin, 1982. |
[19] |
J. Watkins,
The higher rank rigidity theorem for manifolds with no focal points, Geom. Dedicata, 164 (2013), 319-349.
doi: 10.1007/s10711-012-9776-3. |
[20] |
W. Wu, F. Liu and F. Wang, On the ergodicity of geodesic flows on surfaces without focal points, preprint, arXiv:1812.04409. Google Scholar |
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