October  2021, 41(10): 4791-4804. doi: 10.3934/dcds.2021057

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

* Corresponding author: Fang Wang

Received  September 2020 Published  October 2021 Early access  March 2021

Fund Project: F. Liu is partially supported by Natural Science Foundation of Shandong Province under Grant No. ZR2020MA017, and NSFC under Grant Nos. 11301305, 11571207. F. Wang is partially supported by NSFC under Grant No. 11871045 and the State Scholarship Fund from China Scholarship Council (CSC). The research is also partially supported by key research project of the Academy for Multidisciplinary Studies, Capital Normal University

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.

Citation: Fei Liu, Xiaokai Liu, Fang Wang. On the mixing and Bernoulli properties for geodesic flows on rank 1 manifolds without focal points. Discrete and Continuous Dynamical Systems, 2021, 41 (10) : 4791-4804. doi: 10.3934/dcds.2021057
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. BallmannM. 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. LedrappierY. 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. LiuF. 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. LiuF. 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.

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. BallmannM. 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. LedrappierY. 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. LiuF. 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. LiuF. 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.

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