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

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

Received  July 2010 Revised  December 2010 Published  May 2011

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.
Citation: Daniel J. Thompson. A criterion for topological entropy to decrease under normalised Ricci flow. Discrete & Continuous Dynamical Systems - A, 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.   Google Scholar

[2]

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

[3]

K. Burns and G. Paternain, Anosov magnetic flows, critical values and topological entropy,, Nonlinearity, 15 (2002), 281.  doi: 10.1088/0951-7715/15/2/305.  Google Scholar

[4]

B. Chow and D. Knopf, "The Ricci Flow: An introduction,", Mathematical Surveys and Monographs, 110 (2004).   Google Scholar

[5]

G. Contreras, Regularity of topological entropy of hyperbolic flows,, Math. Z., 210 (1992), 97.  doi: 10.1007/BF02571785.  Google Scholar

[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.   Google Scholar

[7]

L. Flaminio, Local entropy rigidity for hyperbolic manifolds,, Comm. Anal. Geom., 3 (1995), 555.   Google Scholar

[8]

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

[9]

R. Hamilton, The formation of singularities in the Ricci flow,, Surveys in Differential Geometry, 2 (1995), 7.   Google Scholar

[10]

D. Jane, An example of how the Ricci flow can increase topological entropy,, Ergodic Theory Dynam. Systems, 27 (2007), 1919.  doi: 10.1017/S0143385707000211.  Google Scholar

[11]

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

[12]

A. Katok and B. Hasselblatt, "Introduction to the Modern Theory of Dynamical Systems,", "Encyclopedia of Mathematics and its Applications,", 54 (1995).   Google Scholar

[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.  doi: 10.1007/BF01393838.  Google Scholar

[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.  doi: 10.1007/BF02099667.  Google Scholar

[15]

G. Knieper, A second derivative formula of the Liouville entropy at spaces of constant negative curvature,, Ergodic Theory Dynam. Systems, 17 (1997), 1131.  doi: 10.1017/S0143385797086446.  Google Scholar

[16]

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

[17]

A. Manning, The volume entropy of a surface decreases along the Ricci flow,, Ergodic Theory Dynam. Systems, 24 (2004), 171.  doi: 10.1017/S0143385703000415.  Google Scholar

[18]

J. Morgan and G. Tian, "Ricci Flow and the Poincaré Conjecture,", Clay Mathematics Monographs, 3 (2007).   Google Scholar

[19]

R. Osserman and P. Sarnak, A new curvature invariant and entropy of geodesic flows,, Invent. Math., 77 (1984), 455.  doi: 10.1007/BF01388833.  Google Scholar

[20]

G. Paternain and J. Petean, The pressure of Ricci curvature,, Geometriae Dedicata, 100 (2003), 93.  doi: 10.1023/A:1025842932050.  Google Scholar

[21]

R. M. Schoen, Variational theory for the total scalar curvature functional for Riemannian metrics and related topics,, in, 1365 (1987), 120.   Google Scholar

[22]

P. Topping, "Lectures on the Ricci Flow,", LMS Lecture Note Series, 325 (2006).   Google Scholar

[23]

P. Walters, "An Introduction to Ergodic Theory,", Graduate Texts in Mathematics, 79 (1982).   Google Scholar

[24]

R. Ye, Ricci flow, Einstein metrics and space forms,, Trans. Amer. Math. Soc., 338 (1993), 871.  doi: 10.2307/2154433.  Google Scholar

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.   Google Scholar

[2]

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

[3]

K. Burns and G. Paternain, Anosov magnetic flows, critical values and topological entropy,, Nonlinearity, 15 (2002), 281.  doi: 10.1088/0951-7715/15/2/305.  Google Scholar

[4]

B. Chow and D. Knopf, "The Ricci Flow: An introduction,", Mathematical Surveys and Monographs, 110 (2004).   Google Scholar

[5]

G. Contreras, Regularity of topological entropy of hyperbolic flows,, Math. Z., 210 (1992), 97.  doi: 10.1007/BF02571785.  Google Scholar

[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.   Google Scholar

[7]

L. Flaminio, Local entropy rigidity for hyperbolic manifolds,, Comm. Anal. Geom., 3 (1995), 555.   Google Scholar

[8]

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

[9]

R. Hamilton, The formation of singularities in the Ricci flow,, Surveys in Differential Geometry, 2 (1995), 7.   Google Scholar

[10]

D. Jane, An example of how the Ricci flow can increase topological entropy,, Ergodic Theory Dynam. Systems, 27 (2007), 1919.  doi: 10.1017/S0143385707000211.  Google Scholar

[11]

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

[12]

A. Katok and B. Hasselblatt, "Introduction to the Modern Theory of Dynamical Systems,", "Encyclopedia of Mathematics and its Applications,", 54 (1995).   Google Scholar

[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.  doi: 10.1007/BF01393838.  Google Scholar

[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.  doi: 10.1007/BF02099667.  Google Scholar

[15]

G. Knieper, A second derivative formula of the Liouville entropy at spaces of constant negative curvature,, Ergodic Theory Dynam. Systems, 17 (1997), 1131.  doi: 10.1017/S0143385797086446.  Google Scholar

[16]

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

[17]

A. Manning, The volume entropy of a surface decreases along the Ricci flow,, Ergodic Theory Dynam. Systems, 24 (2004), 171.  doi: 10.1017/S0143385703000415.  Google Scholar

[18]

J. Morgan and G. Tian, "Ricci Flow and the Poincaré Conjecture,", Clay Mathematics Monographs, 3 (2007).   Google Scholar

[19]

R. Osserman and P. Sarnak, A new curvature invariant and entropy of geodesic flows,, Invent. Math., 77 (1984), 455.  doi: 10.1007/BF01388833.  Google Scholar

[20]

G. Paternain and J. Petean, The pressure of Ricci curvature,, Geometriae Dedicata, 100 (2003), 93.  doi: 10.1023/A:1025842932050.  Google Scholar

[21]

R. M. Schoen, Variational theory for the total scalar curvature functional for Riemannian metrics and related topics,, in, 1365 (1987), 120.   Google Scholar

[22]

P. Topping, "Lectures on the Ricci Flow,", LMS Lecture Note Series, 325 (2006).   Google Scholar

[23]

P. Walters, "An Introduction to Ergodic Theory,", Graduate Texts in Mathematics, 79 (1982).   Google Scholar

[24]

R. Ye, Ricci flow, Einstein metrics and space forms,, Trans. Amer. Math. Soc., 338 (1993), 871.  doi: 10.2307/2154433.  Google Scholar

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