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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

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.

Citation: Fei Liu, Fang Wang, Weisheng Wu. On the Patterson-Sullivan measure for geodesic flows on rank 1 manifolds without focal points. Discrete & Continuous Dynamical Systems - A, 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.  Google Scholar

[2]

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

[3]

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

[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.  Google Scholar

[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.  Google Scholar

[6]

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

[7]

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

[8]

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

[9]

K. BurnsV. ClimenhagaT. 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.  Google Scholar

[10]

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

[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.  Google Scholar

[12]

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

[13]

F. Dal'boM. 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.  Google Scholar

[14] P. Eberlein, Geometry of Nonpositively Curved Manifolds, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 1996.   Google Scholar
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P. Eberlein and B. O'Neill, Visibility manifolds, Pacific J. Math., 46 (1973), 45-109.  doi: 10.2140/pjm.1973.46.45.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

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

[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.  Google Scholar
[21]

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

[22]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[28]

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

[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.  Google Scholar

[30]

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

[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.  Google Scholar

[32]

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

[33]

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

[34]

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

[35]

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

[36]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[41]

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

[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.  Google Scholar

show all references

References:
[1]

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

[2]

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

[3]

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

[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.  Google Scholar

[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.  Google Scholar

[6]

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

[7]

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

[8]

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

[9]

K. BurnsV. ClimenhagaT. 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.  Google Scholar

[10]

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

[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.  Google Scholar

[12]

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

[13]

F. Dal'boM. 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.  Google Scholar

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

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

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

[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.  Google Scholar
[21]

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

[22]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[28]

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

[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.  Google Scholar

[30]

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

[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.  Google Scholar

[32]

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

[33]

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

[34]

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

[35]

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

[36]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[41]

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

[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.  Google Scholar

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