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March  2020, 40(3): 1517-1554. doi: 10.3934/dcds.2020085

On the Patterson-Sullivan measure for geodesic flows on rank 1 manifolds without focal points

Fei Liu 1, , Fang Wang 2,, and  Weisheng Wu 3,

1. 

College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao 266590, China
2. 

School of Mathematical Sciences, Capital Normal University, Beijing 100048, China
3. 

Department of Applied Mathematics, College of Science, China Agricultural University, Beijing 100083, China

* Corresponding author: Fang Wang

Received  February 2019 Published  December 2019

Fund Project: The first author is partially supported by NSFC under Grant Nos.11301305 and 11571207. The second authoris partially supported by NSFC under Grant No.11571387 and by the State Scholarship Fund from China Scholarship Council (CSC). The third author is partially supported by NSFC under Grant Nos.11701559 and 11571387.

In this article, we consider the geodesic flow on a compact rank $ 1 $ Riemannian manifold $ M $ without focal points, whose universal cover is denoted by $ X $. On the ideal boundary $ X(\infty) $ of $ X $, we show the existence and uniqueness of the Busemann density, which is realized via the Patterson-Sullivan measure. Based on the the Patterson-Sullivan measure, we show that the geodesic flow on $ M $ has a unique invariant measure of maximal entropy. We also obtain the asymptotic growth rate of the volume of geodesic spheres in $ X $ and the growth rate of the number of closed geodesics on $ M $. These results generalize the work of Margulis and Knieper in the case of negative and nonpositive curvature respectively.

Keywords: Geodesic flows, no focal points, Patterson-Sullivan measure, measure of maximal entropy.
Mathematics Subject Classification: Primary: 37D40; Secondary: 37B40.
Citation: Fei Liu, Fang Wang, Weisheng Wu. On the Patterson-Sullivan measure for geodesic flows on rank 1 manifolds without focal points. Discrete and Continuous Dynamical Systems, 2020, 40 (3) : 1517-1554. doi: 10.3934/dcds.2020085
References:
[1]

W. Ballmann, Axial isometries of manifolds of nonpositive curvature, Math. Ann., 259 (1982), 131-144.  doi: 10.1007/BF01456836.

[2]

W. Ballmann, Nonpositively curved manifolds of higher rank, Ann. of Math. (2), 122 (1985), 597–609. doi: 10.2307/1971331.

[3]

W. Ballmann, Lectures on Spaces of Nonpositive Curvature, DMV Seminar, 25, Birkhäuser Verlag, Basel, 1995. doi: 10.1007/978-3-0348-9240-7.

[4]

W. Ballmann, M. Brin and P. Eberlein, Structure of manifolds of nonpositive curvature. I, Ann. of Math. (2), 122 (1985), 171–203. doi: 10.2307/1971373.

[5]

W. Ballmann, M. Brin and R. Spatzier, Structure of manifolds of nonpositive curvature. Ⅱ, Ann. of Math.(2), 122 (1985), 205–235. doi: 10.2307/1971303.

[6]

R. Bowen, Periodic orbits for hyperbolic flows, Amer. J. Math., 94 (1972), 1-30.  doi: 10.2307/2373590.

[7]

R. Bowen, Entropy-expansive maps, Trans. Amer. Math. Soc., 164 (1972), 323-331.  doi: 10.1090/S0002-9947-1972-0285689-X.

[8]

R. Bowen, Maximizing entropy for a hyperbolic flow, Math. Systems Theory, 7 (1974), 300-303.  doi: 10.1007/BF01795948.

[9]

K. Burns, V. Climenhaga, T. Fisher and D. J. Thompson, Unique equilibrium states for geodesic flows in nonpositive curvature, Geom. Funct. Anal., 28 (2018), 1209-1259.  doi: 10.1007/s00039-018-0465-8.

[10]

K. Burns and A. Katok, Manifolds with non-positive curvature, Ergodic Theory Dynam. Systems, 5 (1985), 307-317.  doi: 10.1017/S0143385700002935.

[11]

K. Burns and R. Spatzier, Manifolds of nonpositive curvature and their buildings, Inst. Hautes Études Sci. Publ. Math., 65 (1987), 35–59. doi: 10.1007/BF02698934.

[12]

C. B. Croke and V. Schroeder, The fundamental group of compact manifolds without conjugate points, Comment. Math. Helv., 61 (1986), 161-175. 

[13]

F. Dal'bo, M. Peigné and A. Sambusetti, On the horoboundary and the geometry of rays of negatively curved manifolds, Pacific J. Math., 259 (2012), 55-100.  doi: 10.2140/pjm.2012.259.55.

[14] P. Eberlein, Geometry of Nonpositively Curved Manifolds, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 1996. 
[15]

P. Eberlein and B. O'Neill, Visibility manifolds, Pacific J. Math., 46 (1973), 45-109.  doi: 10.2140/pjm.1973.46.45.

[16]

A. Freire and R. Mañé, On the entropy of the geodesic flow in manifolds without conjugate points, Invent. Math., 69 (1982), 375-392.  doi: 10.1007/BF01389360.

[17]

K. Gelfert and R. Riggiero, Geodesic flows modelled by expansive flows, Proc. Edinb. Math. Soc. (2), 62 (2019), 61–95. doi: 10.1017/S0013091518000160.

[18]

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.

[19]

A. Katok, Entropy and closed geodesics, Ergodic Theory Dynam. Systems, 2 (1982), 339-365.  doi: 10.1017/S0143385700001656.

[20] 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.  doi: 10.1017/CBO9780511809187.
[21]

G. Knieper, Das Wachstum der Äquivalenzklassen geschlossener Geodätischer in kompakten Mannigfaltigkeiten, Arch. Math. (Basel), 40 (1983), 559-568.  doi: 10.1007/BF01192824.

[22]

G. Knieper, On the asymptotic geometry of nonpositively curved manifolds, Geom. Funct. Anal., 7 (1997), 755-782.  doi: 10.1007/s000390050025.

[23]

G. Knieper, The uniqueness of the measure of maximal entropy for geodesic flows on rank 1 manifolds, Ann. of Math. (2), 148 (1998), 291–314. doi: 10.2307/120995.

[24]

G. Knieper, Closed geodesics and the uniqueness of the maximal measure for rank 1 geodesic flows, in Smooth Ergodic Theory and Its Applications, Proc. Sympos. Pure Math., 69, Amer. Math. Soc., Providence, RI, 2001, 573–590.

[25]

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.

[26]

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.

[27]

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.

[28]

A. Manning, Topological entropy for geodesic flows, Ann. of Math. (2), 110 (1979), 567–573. doi: 10.2307/1971239.

[29]

G. A. Margulis, Certain applications of ergodic theory to the investigation of manifolds of negative curvature, Funkcional. Anal. i Priložen, 3 (1969), 89–90.

[30]

G. A. Margulis, Certain measures that are associated with $\gamma$-flows on compact manifolds, Funkcional. Anal. i Priložen, 4 (1970), 62–76.

[31]

G. A. Margulis, On Some Aspects of the Theory of Anosov Systems, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2004. doi: 10.1007/978-3-662-09070-1.

[32]

J. O'Sullivan, Riemannian manifolds without focal points, J. Differential Geometry, 11 (1976), 321-333.  doi: 10.4310/jdg/1214433590.

[33]

G. P. Paternain, Geodesic Flows, Progress in Mathematics, 180, Birkhäuser Boston, Inc., Boston, MA, 1999. doi: 10.1007/978-1-4612-1600-1.

[34]

S. J. Patterson, The limit set of a Fuchsian group, Acta Math., 136 (1976), 241-273.  doi: 10.1007/BF02392046.

[35]

R. Ruggiero, Expansive geodesic flows in manifolds with no conjugate points, Ergodic Theory Dynam. Systems, 17 (1997), 211-225.  doi: 10.1017/S0143385797060963.

[36]

R. Ruggiero, Dynamics and Global Geometry of Manifolds Without Conjugate Points, Ensaios Matemáticos, 12, Sociedade Brasileira de Matemática, Rio de Janeiro, 2007.

[37]

R. Ruggiero and V. Rosas Meneses, On the Pesin set of expansive geodesic flows in manifolds with no conjugate points, Bull. Braz. Math. Soc. (N. S.), 34 (2003), 263-274.  doi: 10.1007/s00574-003-0012-5.

[38]

D. Sullivan, The density at infinity of a discrete group of hyperbolic motions, Inst. Hautes Études Sci. Publ. Math., 50 (1979), 171–202. doi: 10.1007/BF02684773.

[39]

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.

[40]

W. Wu, On the ergodicity of geodesic flows on surfaces of nonpositive curvature, Ann. Fac. Sci. Toulouse Math. (6), 24 (2015), 625–639. doi: 10.5802/afst.1457.

[41]

W. Wu, Higher rank rigidity for Berwald spaces, preprint, Ergodic Theory Dynam. Systems. doi: 10.1017/etds.2018.130.

[42]

W. Wu, F. Liu and F. Wang, On the ergodicity of geodesic flows on surfaces without focal points, preprint, arXiv: math/1812.04409. doi: 10.5802/afst.1457.

show all references

References:
[1]

W. Ballmann, Axial isometries of manifolds of nonpositive curvature, Math. Ann., 259 (1982), 131-144.  doi: 10.1007/BF01456836.

[2]

W. Ballmann, Nonpositively curved manifolds of higher rank, Ann. of Math. (2), 122 (1985), 597–609. doi: 10.2307/1971331.

[3]

W. Ballmann, Lectures on Spaces of Nonpositive Curvature, DMV Seminar, 25, Birkhäuser Verlag, Basel, 1995. doi: 10.1007/978-3-0348-9240-7.

[4]

W. Ballmann, M. Brin and P. Eberlein, Structure of manifolds of nonpositive curvature. I, Ann. of Math. (2), 122 (1985), 171–203. doi: 10.2307/1971373.

[5]

W. Ballmann, M. Brin and R. Spatzier, Structure of manifolds of nonpositive curvature. Ⅱ, Ann. of Math.(2), 122 (1985), 205–235. doi: 10.2307/1971303.

[6]

R. Bowen, Periodic orbits for hyperbolic flows, Amer. J. Math., 94 (1972), 1-30.  doi: 10.2307/2373590.

[7]

R. Bowen, Entropy-expansive maps, Trans. Amer. Math. Soc., 164 (1972), 323-331.  doi: 10.1090/S0002-9947-1972-0285689-X.

[8]

R. Bowen, Maximizing entropy for a hyperbolic flow, Math. Systems Theory, 7 (1974), 300-303.  doi: 10.1007/BF01795948.

[9]

K. Burns, V. Climenhaga, T. Fisher and D. J. Thompson, Unique equilibrium states for geodesic flows in nonpositive curvature, Geom. Funct. Anal., 28 (2018), 1209-1259.  doi: 10.1007/s00039-018-0465-8.

[10]

K. Burns and A. Katok, Manifolds with non-positive curvature, Ergodic Theory Dynam. Systems, 5 (1985), 307-317.  doi: 10.1017/S0143385700002935.

[11]

K. Burns and R. Spatzier, Manifolds of nonpositive curvature and their buildings, Inst. Hautes Études Sci. Publ. Math., 65 (1987), 35–59. doi: 10.1007/BF02698934.

[12]

C. B. Croke and V. Schroeder, The fundamental group of compact manifolds without conjugate points, Comment. Math. Helv., 61 (1986), 161-175. 

[13]

F. Dal'bo, M. Peigné and A. Sambusetti, On the horoboundary and the geometry of rays of negatively curved manifolds, Pacific J. Math., 259 (2012), 55-100.  doi: 10.2140/pjm.2012.259.55.

[14] P. Eberlein, Geometry of Nonpositively Curved Manifolds, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 1996. 
[15]

P. Eberlein and B. O'Neill, Visibility manifolds, Pacific J. Math., 46 (1973), 45-109.  doi: 10.2140/pjm.1973.46.45.

[16]

A. Freire and R. Mañé, On the entropy of the geodesic flow in manifolds without conjugate points, Invent. Math., 69 (1982), 375-392.  doi: 10.1007/BF01389360.

[17]

K. Gelfert and R. Riggiero, Geodesic flows modelled by expansive flows, Proc. Edinb. Math. Soc. (2), 62 (2019), 61–95. doi: 10.1017/S0013091518000160.

[18]

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.

[19]

A. Katok, Entropy and closed geodesics, Ergodic Theory Dynam. Systems, 2 (1982), 339-365.  doi: 10.1017/S0143385700001656.

[20] 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.  doi: 10.1017/CBO9780511809187.
[21]

G. Knieper, Das Wachstum der Äquivalenzklassen geschlossener Geodätischer in kompakten Mannigfaltigkeiten, Arch. Math. (Basel), 40 (1983), 559-568.  doi: 10.1007/BF01192824.

[22]

G. Knieper, On the asymptotic geometry of nonpositively curved manifolds, Geom. Funct. Anal., 7 (1997), 755-782.  doi: 10.1007/s000390050025.

[23]

G. Knieper, The uniqueness of the measure of maximal entropy for geodesic flows on rank 1 manifolds, Ann. of Math. (2), 148 (1998), 291–314. doi: 10.2307/120995.

[24]

G. Knieper, Closed geodesics and the uniqueness of the maximal measure for rank 1 geodesic flows, in Smooth Ergodic Theory and Its Applications, Proc. Sympos. Pure Math., 69, Amer. Math. Soc., Providence, RI, 2001, 573–590.

[25]

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.

[26]

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.

[27]

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.

[28]

A. Manning, Topological entropy for geodesic flows, Ann. of Math. (2), 110 (1979), 567–573. doi: 10.2307/1971239.

[29]

G. A. Margulis, Certain applications of ergodic theory to the investigation of manifolds of negative curvature, Funkcional. Anal. i Priložen, 3 (1969), 89–90.

[30]

G. A. Margulis, Certain measures that are associated with $\gamma$-flows on compact manifolds, Funkcional. Anal. i Priložen, 4 (1970), 62–76.

[31]

G. A. Margulis, On Some Aspects of the Theory of Anosov Systems, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2004. doi: 10.1007/978-3-662-09070-1.

[32]

J. O'Sullivan, Riemannian manifolds without focal points, J. Differential Geometry, 11 (1976), 321-333.  doi: 10.4310/jdg/1214433590.

[33]

G. P. Paternain, Geodesic Flows, Progress in Mathematics, 180, Birkhäuser Boston, Inc., Boston, MA, 1999. doi: 10.1007/978-1-4612-1600-1.

[34]

S. J. Patterson, The limit set of a Fuchsian group, Acta Math., 136 (1976), 241-273.  doi: 10.1007/BF02392046.

[35]

R. Ruggiero, Expansive geodesic flows in manifolds with no conjugate points, Ergodic Theory Dynam. Systems, 17 (1997), 211-225.  doi: 10.1017/S0143385797060963.

[36]

R. Ruggiero, Dynamics and Global Geometry of Manifolds Without Conjugate Points, Ensaios Matemáticos, 12, Sociedade Brasileira de Matemática, Rio de Janeiro, 2007.

[37]

R. Ruggiero and V. Rosas Meneses, On the Pesin set of expansive geodesic flows in manifolds with no conjugate points, Bull. Braz. Math. Soc. (N. S.), 34 (2003), 263-274.  doi: 10.1007/s00574-003-0012-5.

[38]

D. Sullivan, The density at infinity of a discrete group of hyperbolic motions, Inst. Hautes Études Sci. Publ. Math., 50 (1979), 171–202. doi: 10.1007/BF02684773.

[39]

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.

[40]

W. Wu, On the ergodicity of geodesic flows on surfaces of nonpositive curvature, Ann. Fac. Sci. Toulouse Math. (6), 24 (2015), 625–639. doi: 10.5802/afst.1457.

[41]

W. Wu, Higher rank rigidity for Berwald spaces, preprint, Ergodic Theory Dynam. Systems. doi: 10.1017/etds.2018.130.

[42]

W. Wu, F. Liu and F. Wang, On the ergodicity of geodesic flows on surfaces without focal points, preprint, arXiv: math/1812.04409. doi: 10.5802/afst.1457.

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