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Zero entropy versus infinite entropy
A criterion for topological entropy to decrease under normalised Ricci flow
1. | Department of Mathematics, Pennsylvania State University, University Park, State College, PA 16802, United States |
References:
[1] |
G. Besson, G. Courtois and S. Gallot, Minimal entropy and Mostow's rigidity theorems, Ergodic Theory Dynam. Systems, 16 (1996), 623-649. |
[2] |
R. Bowen, Periodic orbits for hyperbolic flows, American J. Math., 94 (1972), 1-30.
doi: 10.2307/2373590. |
[3] |
K. Burns and G. Paternain, Anosov magnetic flows, critical values and topological entropy, Nonlinearity, 15 (2002), 281-314.
doi: 10.1088/0951-7715/15/2/305. |
[4] |
B. Chow and D. Knopf, "The Ricci Flow: An introduction," Mathematical Surveys and Monographs, 110, AMS, 2004. |
[5] |
G. Contreras, Regularity of topological entropy of hyperbolic flows, Math. Z., 210 (1992), 97-111.
doi: 10.1007/BF02571785. |
[6] |
F. T. Farrell and P. Ontaneda, A caveat on the convergence of the Ricci flow for pinched negatively curved manifolds, Asian J. Math., 9 (2005), 401-406. |
[7] |
L. Flaminio, Local entropy rigidity for hyperbolic manifolds, Comm. Anal. Geom., 3 (1995), 555-596. |
[8] |
A. Freire and R. Mané, On the entropy of geodesic flow in manifolds without conjugate points, Invent. Math., 69 (1982), 375-392.
doi: 10.1007/BF01389360. |
[9] |
R. Hamilton, The formation of singularities in the Ricci flow, Surveys in Differential Geometry, 2 (1995), 7-136. |
[10] |
D. Jane, An example of how the Ricci flow can increase topological entropy, Ergodic Theory Dynam. Systems, 27 (2007), 1919-1932.
doi: 10.1017/S0143385707000211. |
[11] |
A. Katok, Entropy and closed geodesics, Ergodic Theory Dynam. Systems, 2 (1982), 339-365.
doi: 10.1017/S0143385700001656. |
[12] |
A. Katok and B. Hasselblatt, "Introduction to the Modern Theory of Dynamical Systems," "Encyclopedia of Mathematics and its Applications," 54, Cambridge University Press, 1995. |
[13] |
A. Katok, G. Knieper, M. Pollicott and H. Weiss, Differentiability and analyticity of topological entropy for Anosov and geodesic flows, Invent. Math., 98 (1989), 581-597.
doi: 10.1007/BF01393838. |
[14] |
A. Katok, G. Knieper and H. Weiss, Formulas for the derivative and critical points of topological entropy for Anosov and geodesic flows, Comm. Math. Phys., 138 (1991), 19-31.
doi: 10.1007/BF02099667. |
[15] |
G. Knieper, A second derivative formula of the Liouville entropy at spaces of constant negative curvature, Ergodic Theory Dynam. Systems, 17 (1997), 1131-1135.
doi: 10.1017/S0143385797086446. |
[16] |
A. Manning, Topological entropy for geodesic flows, Ann. Math., 110 (1979), 567-573.
doi: 10.2307/1971239. |
[17] |
A. Manning, The volume entropy of a surface decreases along the Ricci flow, Ergodic Theory Dynam. Systems, 24 (2004), 171-176.
doi: 10.1017/S0143385703000415. |
[18] |
J. Morgan and G. Tian, "Ricci Flow and the Poincaré Conjecture," Clay Mathematics Monographs, 3, AMS, 2007. |
[19] |
R. Osserman and P. Sarnak, A new curvature invariant and entropy of geodesic flows, Invent. Math., 77 (1984), 455-462.
doi: 10.1007/BF01388833. |
[20] |
G. Paternain and J. Petean, The pressure of Ricci curvature, Geometriae Dedicata, 100 (2003), 93-102.
doi: 10.1023/A:1025842932050. |
[21] |
R. M. Schoen, Variational theory for the total scalar curvature functional for Riemannian metrics and related topics, in "Topics in the Calculus of Variations" (ed. M. Giaquinta), Lecture Notes in Math, 1365, Springer-Verlag, (1987), 120-154. |
[22] |
P. Topping, "Lectures on the Ricci Flow," LMS Lecture Note Series, 325, LMS, 2006. |
[23] |
P. Walters, "An Introduction to Ergodic Theory," Graduate Texts in Mathematics, 79, Springer, New York, 1982. |
[24] |
R. Ye, Ricci flow, Einstein metrics and space forms, Trans. Amer. Math. Soc., 338 (1993), 871-896.
doi: 10.2307/2154433. |
show all references
References:
[1] |
G. Besson, G. Courtois and S. Gallot, Minimal entropy and Mostow's rigidity theorems, Ergodic Theory Dynam. Systems, 16 (1996), 623-649. |
[2] |
R. Bowen, Periodic orbits for hyperbolic flows, American J. Math., 94 (1972), 1-30.
doi: 10.2307/2373590. |
[3] |
K. Burns and G. Paternain, Anosov magnetic flows, critical values and topological entropy, Nonlinearity, 15 (2002), 281-314.
doi: 10.1088/0951-7715/15/2/305. |
[4] |
B. Chow and D. Knopf, "The Ricci Flow: An introduction," Mathematical Surveys and Monographs, 110, AMS, 2004. |
[5] |
G. Contreras, Regularity of topological entropy of hyperbolic flows, Math. Z., 210 (1992), 97-111.
doi: 10.1007/BF02571785. |
[6] |
F. T. Farrell and P. Ontaneda, A caveat on the convergence of the Ricci flow for pinched negatively curved manifolds, Asian J. Math., 9 (2005), 401-406. |
[7] |
L. Flaminio, Local entropy rigidity for hyperbolic manifolds, Comm. Anal. Geom., 3 (1995), 555-596. |
[8] |
A. Freire and R. Mané, On the entropy of geodesic flow in manifolds without conjugate points, Invent. Math., 69 (1982), 375-392.
doi: 10.1007/BF01389360. |
[9] |
R. Hamilton, The formation of singularities in the Ricci flow, Surveys in Differential Geometry, 2 (1995), 7-136. |
[10] |
D. Jane, An example of how the Ricci flow can increase topological entropy, Ergodic Theory Dynam. Systems, 27 (2007), 1919-1932.
doi: 10.1017/S0143385707000211. |
[11] |
A. Katok, Entropy and closed geodesics, Ergodic Theory Dynam. Systems, 2 (1982), 339-365.
doi: 10.1017/S0143385700001656. |
[12] |
A. Katok and B. Hasselblatt, "Introduction to the Modern Theory of Dynamical Systems," "Encyclopedia of Mathematics and its Applications," 54, Cambridge University Press, 1995. |
[13] |
A. Katok, G. Knieper, M. Pollicott and H. Weiss, Differentiability and analyticity of topological entropy for Anosov and geodesic flows, Invent. Math., 98 (1989), 581-597.
doi: 10.1007/BF01393838. |
[14] |
A. Katok, G. Knieper and H. Weiss, Formulas for the derivative and critical points of topological entropy for Anosov and geodesic flows, Comm. Math. Phys., 138 (1991), 19-31.
doi: 10.1007/BF02099667. |
[15] |
G. Knieper, A second derivative formula of the Liouville entropy at spaces of constant negative curvature, Ergodic Theory Dynam. Systems, 17 (1997), 1131-1135.
doi: 10.1017/S0143385797086446. |
[16] |
A. Manning, Topological entropy for geodesic flows, Ann. Math., 110 (1979), 567-573.
doi: 10.2307/1971239. |
[17] |
A. Manning, The volume entropy of a surface decreases along the Ricci flow, Ergodic Theory Dynam. Systems, 24 (2004), 171-176.
doi: 10.1017/S0143385703000415. |
[18] |
J. Morgan and G. Tian, "Ricci Flow and the Poincaré Conjecture," Clay Mathematics Monographs, 3, AMS, 2007. |
[19] |
R. Osserman and P. Sarnak, A new curvature invariant and entropy of geodesic flows, Invent. Math., 77 (1984), 455-462.
doi: 10.1007/BF01388833. |
[20] |
G. Paternain and J. Petean, The pressure of Ricci curvature, Geometriae Dedicata, 100 (2003), 93-102.
doi: 10.1023/A:1025842932050. |
[21] |
R. M. Schoen, Variational theory for the total scalar curvature functional for Riemannian metrics and related topics, in "Topics in the Calculus of Variations" (ed. M. Giaquinta), Lecture Notes in Math, 1365, Springer-Verlag, (1987), 120-154. |
[22] |
P. Topping, "Lectures on the Ricci Flow," LMS Lecture Note Series, 325, LMS, 2006. |
[23] |
P. Walters, "An Introduction to Ergodic Theory," Graduate Texts in Mathematics, 79, Springer, New York, 1982. |
[24] |
R. Ye, Ricci flow, Einstein metrics and space forms, Trans. Amer. Math. Soc., 338 (1993), 871-896.
doi: 10.2307/2154433. |
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