A criterion for topological entropy to decrease under normalised Ricci flow

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November 2011, 30(4): 1243-1248. doi: 10.3934/dcds.2011.30.1243

A criterion for topological entropy to decrease under normalised Ricci flow

Daniel J. Thompson ^1,

1.	Department of Mathematics, Pennsylvania State University, University Park, State College, PA 16802, United States

Received July 2010 Revised December 2010 Published May 2011

Abstract
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In 2004, Manning showed that the topological entropy of the geodesic flow for a surface of negative curvature decreases as the metric evolves under the normalised Ricci flow. It is an interesting open problem, also due to Manning, to determine to what extent such behaviour persists for higher dimensional manifolds. In this short note, we describe the problem and give a curvature criterion under which monotonicity of the topological entropy can be established for a short time. In particular, the criterion applies to metrics of negative sectional curvature which are in the same conformal class as a metric of constant negative sectional curvature.

Keywords: Ricci flow, Topological entropy, geodesic flow.

Mathematics Subject Classification: Primary: 37D40, 53C44; Secondary: 37A3.

Citation: Daniel J. Thompson. A criterion for topological entropy to decrease under normalised Ricci flow. Discrete and Continuous Dynamical Systems, 2011, 30 (4) : 1243-1248. doi: 10.3934/dcds.2011.30.1243

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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