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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 |
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.
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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