Quasiconformal Anosov flows and quasisymmetric rigidity of Hamenst\ddot{a}dt distances

Previous Article
Dynamics of Klein-Gordon on a compact surface near a homoclinic orbit
DCDS Home
This Issue
Next Article
Periodic solutions of El Niño model through the Vallis differential system

September 2014, 34(9): 3471-3483. doi: 10.3934/dcds.2014.34.3471

Quasiconformal Anosov flows and quasisymmetric rigidity of Hamenst$\ddot{a}$dt distances

1.	Département de Mathématiques, Université de Cergy-Pontoise, avenue Adolphe Chauvin, 95302, Cergy-Pontoise Cedex

Received May 2013 Revised November 2013 Published March 2014

Abstract
Related Papers

This paper deals with interactions between metric quasiconformal geometry and the rigidity of Anosov flows. In the first part of this article, we study a canonical time change of Anosov flows. Then we use it to obtain the thorough classification of volume-preserving quasiconformal Anosov flows.
Given a $C^\infty$ Anosov flow $\varphi$, it is well-known that along the leaves of its strong unstable foliation there are two natural distances: the Hamenst$\ddot {\rm a}$dt distance and the induced Riemannian distance. In general these two distances are H$\ddot{\rm o}$lder equivalent. We prove the following rigidity result about quasisymmetric equivalence:
Let $\varphi: M\to M$ be a $C^\infty$ transversely symplectic Anosov flow with $C^1$ weak distributions. We suppose that ${\rm dim}M\geq 5$.
If over a leaf of the strong unstable foliation, the Hamenst$\ddot{\rm a}$dt distance is quasisymmetric equivalent to the Riemannian distance, then up to finite covers $\varphi$ is $C^\infty$ orbit equivalent either to the geodesic flow of a closed hyperbolic manifold, or to the suspension of an Anosov automorphism.

Keywords: entropy, Anosov flow, quasisymmetry..

Mathematics Subject Classification: Primary: 37A35, 34D20; Secondary: 37D35, 37D4.

Citation: Yong Fang. Quasiconformal Anosov flows and quasisymmetric rigidity of Hamenst$\ddot{a}$dt distances. Discrete and Continuous Dynamical Systems, 2014, 34 (9) : 3471-3483. doi: 10.3934/dcds.2014.34.3471

References:

[1]	V. D. Anosov, Geodesic flows on closed Riemannian manifolds with negative curvature, Pros. Inst. Steklov, 90 (1967), 1-235.
[2]	Y. Benoist, P. Foulon and F. Labourie, Flots d'Anosov à distributions stable et instable différentiables, J. Amer. Math. Soc., 5 (1992), 33-74. doi: 10.2307/2152750.
[3]	Y. Fang, Smoth rigidity of uniformly quasiconformal Anosov flows, Ergodic theory and Dynam. Systems, 24 (2004), 1937-1959. doi: 10.1017/S0143385704000264.
[4]	Y. Fang, On the rigidity of quasiconformal Anosov flows, Ergodic Theory Dynam. Systems, 27 (2007), 1773-1802. doi: 10.1017/S0143385707000326.
[5]	Y. Fang, Thermodynamic invariants of Anosov flows and rigidity, Continuous and Discrete Dynamical Systems, 24 (2009), 1185-1204. doi: 10.3934/dcds.2009.24.1185.
[6]	Y. Fang, Geometric Anosov flows of dimension five with smooth distributions, Journal of the Inst. of Math. Jussieu, 4 (2005), 1-30. doi: 10.1017/S1474748005000083.
[7]	P. Foulon, Entropy rigidity of Anosov flows in dimension three, Ergodic Theory and Dynam. Systems, 21 (2001), 1101-1112. doi: 10.1017/S0143385701001523.
[8]	E. Ghys, Déformation des flots d'Anosov et e groupes fuchsiens, Ann. Inst. Fourier, 42 (1992), 209-247. doi: 10.5802/aif.1290.
[9]	E. Ghys, Flots d'Anosov dont les feuilletages stable et instable sont différentiables, Ann. Scient. Ec. Norm. Sup., 20 (1987), 251-270.
[10]	U. Hamenstädt, A new description of the Bowen-Margulis measure, Ergodic Theory and Dynam. Systems, 9 (1989), 455-464. doi: 10.1017/S0143385700005095.
[11]	B. Hasselblatt, A new construction of the Margulis measure for Anosov flows, Ergodic Theory and Dynam. Systems, 9 (1989), 465-468. doi: 10.1017/S0143385700005101.
[12]	J. Heinonen, What is a quasiconformal mapping?, Notices of the AMS, 53 (2006), 1334-1335.
[13]	J. Heinonen, Lectures on Analysis on metric spaces, Universitext, Springer, 2001. doi: 10.1007/978-1-4613-0131-8.
[14]	B. Hasselblatt and A. Katok, Introduction to the Modern Theory of Dynamical Systems, Encyclopedia of Mathematics and its Applications, 54, Cambridge University Press, Cambridge, 1995.
[15]	M. W. Hirsch and C. C. Pugh, Smoothness of horocycle foliations, J. Differential Geometry, 10 (1975), 225-238.
[16]	M. Kanai, Differential-geometric studies on dynamics of geodesic and frame flows, Japan. J. Math., 19 (1993), 1-30.
[17]	W. Parry, Synchronization of canonical measures for hyperbolic attractors, Commun.Math. Phys., 106 (1986), 267-275. doi: 10.1007/BF01454975.
[18]	V. Sadovskaya, On uniformly quasiconformal Anosov flows, Math. Res. Lett., 12 (2005), 425-441. doi: 10.4310/MRL.2005.v12.n3.a12.
[19]	V. Sadovskaya and B. Kalinin, On local and global rigidity of quasiconformal Anosov diffeomorphisms, J. Inst. Math. Jussieu, 2 (2003), 567-582. doi: 10.1017/S1474748003000161.
[20]	C. Yue, Smooth rigidity of rank-1 lattice actions on the sphere at infinity, Math. Res. Lett., 2 (1995), 327-338. doi: 10.4310/MRL.1995.v2.n3.a10.

show all references

References:

[1]	V. D. Anosov, Geodesic flows on closed Riemannian manifolds with negative curvature, Pros. Inst. Steklov, 90 (1967), 1-235.
[2]	Y. Benoist, P. Foulon and F. Labourie, Flots d'Anosov à distributions stable et instable différentiables, J. Amer. Math. Soc., 5 (1992), 33-74. doi: 10.2307/2152750.
[3]	Y. Fang, Smoth rigidity of uniformly quasiconformal Anosov flows, Ergodic theory and Dynam. Systems, 24 (2004), 1937-1959. doi: 10.1017/S0143385704000264.
[4]	Y. Fang, On the rigidity of quasiconformal Anosov flows, Ergodic Theory Dynam. Systems, 27 (2007), 1773-1802. doi: 10.1017/S0143385707000326.
[5]	Y. Fang, Thermodynamic invariants of Anosov flows and rigidity, Continuous and Discrete Dynamical Systems, 24 (2009), 1185-1204. doi: 10.3934/dcds.2009.24.1185.
[6]	Y. Fang, Geometric Anosov flows of dimension five with smooth distributions, Journal of the Inst. of Math. Jussieu, 4 (2005), 1-30. doi: 10.1017/S1474748005000083.
[7]	P. Foulon, Entropy rigidity of Anosov flows in dimension three, Ergodic Theory and Dynam. Systems, 21 (2001), 1101-1112. doi: 10.1017/S0143385701001523.
[8]	E. Ghys, Déformation des flots d'Anosov et e groupes fuchsiens, Ann. Inst. Fourier, 42 (1992), 209-247. doi: 10.5802/aif.1290.
[9]	E. Ghys, Flots d'Anosov dont les feuilletages stable et instable sont différentiables, Ann. Scient. Ec. Norm. Sup., 20 (1987), 251-270.
[10]	U. Hamenstädt, A new description of the Bowen-Margulis measure, Ergodic Theory and Dynam. Systems, 9 (1989), 455-464. doi: 10.1017/S0143385700005095.
[11]	B. Hasselblatt, A new construction of the Margulis measure for Anosov flows, Ergodic Theory and Dynam. Systems, 9 (1989), 465-468. doi: 10.1017/S0143385700005101.
[12]	J. Heinonen, What is a quasiconformal mapping?, Notices of the AMS, 53 (2006), 1334-1335.
[13]	J. Heinonen, Lectures on Analysis on metric spaces, Universitext, Springer, 2001. doi: 10.1007/978-1-4613-0131-8.
[14]	B. Hasselblatt and A. Katok, Introduction to the Modern Theory of Dynamical Systems, Encyclopedia of Mathematics and its Applications, 54, Cambridge University Press, Cambridge, 1995.
[15]	M. W. Hirsch and C. C. Pugh, Smoothness of horocycle foliations, J. Differential Geometry, 10 (1975), 225-238.
[16]	M. Kanai, Differential-geometric studies on dynamics of geodesic and frame flows, Japan. J. Math., 19 (1993), 1-30.
[17]	W. Parry, Synchronization of canonical measures for hyperbolic attractors, Commun.Math. Phys., 106 (1986), 267-275. doi: 10.1007/BF01454975.
[18]	V. Sadovskaya, On uniformly quasiconformal Anosov flows, Math. Res. Lett., 12 (2005), 425-441. doi: 10.4310/MRL.2005.v12.n3.a12.
[19]	V. Sadovskaya and B. Kalinin, On local and global rigidity of quasiconformal Anosov diffeomorphisms, J. Inst. Math. Jussieu, 2 (2003), 567-582. doi: 10.1017/S1474748003000161.
[20]	C. Yue, Smooth rigidity of rank-1 lattice actions on the sphere at infinity, Math. Res. Lett., 2 (1995), 327-338. doi: 10.4310/MRL.1995.v2.n3.a10.

[1]	Huyi Hu, Miaohua Jiang, Yunping Jiang. Infimum of the metric entropy of volume preserving Anosov systems. Discrete and Continuous Dynamical Systems, 2017, 37 (9) : 4767-4783. doi: 10.3934/dcds.2017205
[2]	Marcelo R. R. Alves. Positive topological entropy for Reeb flows on 3-dimensional Anosov contact manifolds. Journal of Modern Dynamics, 2016, 10: 497-509. doi: 10.3934/jmd.2016.10.497
[3]	César J. Niche. Topological entropy of a magnetic flow and the growth of the number of trajectories. Discrete and Continuous Dynamical Systems, 2004, 11 (2&3) : 577-580. doi: 10.3934/dcds.2004.11.577
[4]	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
[5]	Gunhild A. Reigstad. Numerical network models and entropy principles for isothermal junction flow. Networks and Heterogeneous Media, 2014, 9 (1) : 65-95. doi: 10.3934/nhm.2014.9.65
[6]	João P. Almeida, Albert M. Fisher, Alberto Adrego Pinto, David A. Rand. Anosov diffeomorphisms. Conference Publications, 2013, 2013 (special) : 837-845. doi: 10.3934/proc.2013.2013.837
[7]	Zheng Sun, José A. Carrillo, Chi-Wang Shu. An entropy stable high-order discontinuous Galerkin method for cross-diffusion gradient flow systems. Kinetic and Related Models, 2019, 12 (4) : 885-908. doi: 10.3934/krm.2019033
[8]	Tong Li, Kun Zhao. Global existence and long-time behavior of entropy weak solutions to a quasilinear hyperbolic blood flow model. Networks and Heterogeneous Media, 2011, 6 (4) : 625-646. doi: 10.3934/nhm.2011.6.625
[9]	Raimund Bürger, Kenneth H. Karlsen, John D. Towers. On some difference schemes and entropy conditions for a class of multi-species kinematic flow models with discontinuous flux. Networks and Heterogeneous Media, 2010, 5 (3) : 461-485. doi: 10.3934/nhm.2010.5.461
[10]	Raimund Bürger, Antonio García, Kenneth H. Karlsen, John D. Towers. Difference schemes, entropy solutions, and speedup impulse for an inhomogeneous kinematic traffic flow model. Networks and Heterogeneous Media, 2008, 3 (1) : 1-41. doi: 10.3934/nhm.2008.3.1
[11]	Meera G. Mainkar, Cynthia E. Will. Examples of Anosov Lie algebras. Discrete and Continuous Dynamical Systems, 2007, 18 (1) : 39-52. doi: 10.3934/dcds.2007.18.39
[12]	Dominic Veconi. Equilibrium states of almost Anosov diffeomorphisms. Discrete and Continuous Dynamical Systems, 2020, 40 (2) : 767-780. doi: 10.3934/dcds.2020061
[13]	Tracy L. Payne. Anosov automorphisms of nilpotent Lie algebras. Journal of Modern Dynamics, 2009, 3 (1) : 121-158. doi: 10.3934/jmd.2009.3.121
[14]	Gareth Ainsworth. The magnetic ray transform on Anosov surfaces. Discrete and Continuous Dynamical Systems, 2015, 35 (5) : 1801-1816. doi: 10.3934/dcds.2015.35.1801
[15]	Yong Fang. Thermodynamic invariants of Anosov flows and rigidity. Discrete and Continuous Dynamical Systems, 2009, 24 (4) : 1185-1204. doi: 10.3934/dcds.2009.24.1185
[16]	Dan Jane, Gabriel P. Paternain. On the injectivity of the X-ray transform for Anosov thermostats. Discrete and Continuous Dynamical Systems, 2009, 24 (2) : 471-487. doi: 10.3934/dcds.2009.24.471
[17]	Maria Carvalho. First homoclinic tangencies in the boundary of Anosov diffeomorphisms. Discrete and Continuous Dynamical Systems, 1998, 4 (4) : 765-782. doi: 10.3934/dcds.1998.4.765
[18]	Chris Johnson, Martin Schmoll. Pseudo-Anosov eigenfoliations on Panov planes. Electronic Research Announcements, 2014, 21: 89-108. doi: 10.3934/era.2014.21.89
[19]	Carlangelo Liverani. Fredholm determinants, Anosov maps and Ruelle resonances. Discrete and Continuous Dynamical Systems, 2005, 13 (5) : 1203-1215. doi: 10.3934/dcds.2005.13.1203
[20]	Christian Bonatti, Stanislav Minkov, Alexey Okunev, Ivan Shilin. Anosov diffeomorphism with a horseshoe that attracts almost any point. Discrete and Continuous Dynamical Systems, 2020, 40 (1) : 441-465. doi: 10.3934/dcds.2020017

2020 Impact Factor: 1.392

Tools

Metrics

Other articles
by authors

on AIMS
Yong Fang
on Google Scholar
Yong Fang

[Back to Top]