# American Institute of Mathematical Sciences

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  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 & Continuous Dynamical Systems, 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.  Google Scholar [2] W. Ballmann, Axial isometries of manifolds of nonpositive curvature, Math. Ann., 259 (1982), 131-144.  doi: 10.1007/BF01456836.  Google Scholar [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.  Google Scholar [4] K. Burns and A. Katok, Manifolds with non-positive curvature, Ergodic Theory Dynam. Systems, 5 (1985), 307-317.  doi: 10.1017/S0143385700002935.  Google Scholar [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.  Google Scholar [6] P. B. Eberlein, Geometry of Nonpositively Curved Manifolds, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 1996.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [15] J. J. O'Sullivan, Riemannian manifolds without focal points, J. Differential Geometry, 11 (1976), 321-333.  doi: 10.4310/jdg/1214433590.  Google Scholar [16] G. P. Paternain, Geodesic Flows, Progress in Mathematics, 180, Birkhäuser Boston, Inc., Boston, MA, 1999. doi: 10.1007/978-1-4612-1600-1.  Google Scholar [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.  Google Scholar [18] P. Walters, An Introduction to Ergodic Theory, Graduate Texts in Mathematics, 79, Springer-Verlag, New York-Berlin, 1982.  Google Scholar [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.  Google Scholar [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.  Google Scholar [2] W. Ballmann, Axial isometries of manifolds of nonpositive curvature, Math. Ann., 259 (1982), 131-144.  doi: 10.1007/BF01456836.  Google Scholar [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.  Google Scholar [4] K. Burns and A. Katok, Manifolds with non-positive curvature, Ergodic Theory Dynam. Systems, 5 (1985), 307-317.  doi: 10.1017/S0143385700002935.  Google Scholar [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.  Google Scholar [6] P. B. Eberlein, Geometry of Nonpositively Curved Manifolds, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 1996.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [15] J. J. O'Sullivan, Riemannian manifolds without focal points, J. Differential Geometry, 11 (1976), 321-333.  doi: 10.4310/jdg/1214433590.  Google Scholar [16] G. P. Paternain, Geodesic Flows, Progress in Mathematics, 180, Birkhäuser Boston, Inc., Boston, MA, 1999. doi: 10.1007/978-1-4612-1600-1.  Google Scholar [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.  Google Scholar [18] P. Walters, An Introduction to Ergodic Theory, Graduate Texts in Mathematics, 79, Springer-Verlag, New York-Berlin, 1982.  Google Scholar [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.  Google Scholar [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
 [1] A. Kochergin. Well-approximable angles and mixing for flows on T^2 with nonsingular fixed points. Electronic Research Announcements, 2004, 10: 113-121. [2] Jihoon Lee, Ngocthach Nguyen. Flows with the weak two-sided limit shadowing property. Discrete & Continuous Dynamical Systems, 2021  doi: 10.3934/dcds.2021040 [3] Misha Bialy, Andrey E. Mironov. Rich quasi-linear system for integrable geodesic flows on 2-torus. Discrete & Continuous Dynamical Systems, 2011, 29 (1) : 81-90. doi: 10.3934/dcds.2011.29.81 [4] Yunjuan Jin, Aifang Qu, Hairong Yuan. Radon measure solutions for steady compressible hypersonic-limit Euler flows passing cylindrically symmetric conical bodies. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021048 [5] Ondrej Budáč, Michael Herrmann, Barbara Niethammer, Andrej Spielmann. On a model for mass aggregation with maximal size. Kinetic & Related Models, 2011, 4 (2) : 427-439. doi: 10.3934/krm.2011.4.427 [6] Kiyoshi Igusa, Gordana Todorov. Picture groups and maximal green sequences. Electronic Research Archive, , () : -. doi: 10.3934/era.2021025 [7] Bing Gao, Rui Gao. On fair entropy of the tent family. Discrete & Continuous Dynamical Systems, 2021, 41 (8) : 3797-3816. doi: 10.3934/dcds.2021017 [8] Takao Komatsu, Bijan Kumar Patel, Claudio Pita-Ruiz. Several formulas for Bernoulli numbers and polynomials. Advances in Mathematics of Communications, 2021  doi: 10.3934/amc.2021006 [9] Elena Bonetti, Pierluigi Colli, Gianni Gilardi. Singular limit of an integrodifferential system related to the entropy balance. Discrete & Continuous Dynamical Systems - B, 2014, 19 (7) : 1935-1953. doi: 10.3934/dcdsb.2014.19.1935 [10] Zemer Kosloff, Terry Soo. The orbital equivalence of Bernoulli actions and their Sinai factors. Journal of Modern Dynamics, 2021, 17: 145-182. doi: 10.3934/jmd.2021005 [11] Michael Schmidt, Emmanuel Trélat. Controllability of couette flows. Communications on Pure & Applied Analysis, 2006, 5 (1) : 201-211. doi: 10.3934/cpaa.2006.5.201 [12] Charlene Kalle, Niels Langeveld, Marta Maggioni, Sara Munday. Matching for a family of infinite measure continued fraction transformations. Discrete & Continuous Dynamical Systems, 2020, 40 (11) : 6309-6330. doi: 10.3934/dcds.2020281 [13] Ugo Bessi. Another point of view on Kusuoka's measure. Discrete & Continuous Dynamical Systems, 2021, 41 (7) : 3241-3271. doi: 10.3934/dcds.2020404 [14] Pablo D. Carrasco, Túlio Vales. A symmetric Random Walk defined by the time-one map of a geodesic flow. Discrete & Continuous Dynamical Systems, 2021, 41 (6) : 2891-2905. doi: 10.3934/dcds.2020390 [15] Gioconda Moscariello, Antonia Passarelli di Napoli, Carlo Sbordone. Planar ACL-homeomorphisms : Critical points of their components. Communications on Pure & Applied Analysis, 2010, 9 (5) : 1391-1397. doi: 10.3934/cpaa.2010.9.1391 [16] Clara Cufí-Cabré, Ernest Fontich. Differentiable invariant manifolds of nilpotent parabolic points. Discrete & Continuous Dynamical Systems, 2021  doi: 10.3934/dcds.2021053 [17] Ruchika Sehgal, Aparna Mehra. Worst-case analysis of Gini mean difference safety measure. Journal of Industrial & Management Optimization, 2021, 17 (4) : 1613-1637. doi: 10.3934/jimo.2020037 [18] Daniele Cassani, Luca Vilasi, Jianjun Zhang. Concentration phenomena at saddle points of potential for Schrödinger-Poisson systems. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021039 [19] Manfred Einsiedler, Elon Lindenstrauss. On measures invariant under diagonalizable actions: the Rank-One case and the general Low-Entropy method. Journal of Modern Dynamics, 2008, 2 (1) : 83-128. doi: 10.3934/jmd.2008.2.83 [20] Marcel Braukhoff, Ansgar Jüngel. Entropy-dissipating finite-difference schemes for nonlinear fourth-order parabolic equations. Discrete & Continuous Dynamical Systems - B, 2021, 26 (6) : 3335-3355. doi: 10.3934/dcdsb.2020234

2019 Impact Factor: 1.338

Article outline