2016, 36(3): 1271-1278. doi: 10.3934/dcds.2016.36.1271

Rigidity of Hamenstädt metrics of Anosov flows

1. 

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

Received  January 2015 Revised  May 2015 Published  August 2015

Let $\varphi$ be a $C^\infty$ transversely symplectic topologically mixing Anosov flow such that $dim E^{su}\geq 2$. We suppose that the weak distributions of $\varphi$ are $C^1$. If the length Hamenstädt metrics of $\varphi$ are sub-Riemannian then we prove that the weak distributions of $\varphi$ are necessarily $C^\infty$. Combined with our previous rigidity result in [5] we deduce the classification of such Anosov flows with $C^1$ weak distributions provided that the length Hamenstädt metrics are sub-Riemannian.
Citation: Yong Fang. Rigidity of Hamenstädt metrics of Anosov flows. Discrete & Continuous Dynamical Systems - A, 2016, 36 (3) : 1271-1278. doi: 10.3934/dcds.2016.36.1271
References:
[1]

D. V. Anosov, Geodesic flows on closed Riemannian manifolds with negative curvature,, Proc. Inst. Steklov, 90 (1967).

[2]

G. Besson, G. Courtois and S. Gallot, Entropies et rigidités des espaces localement symétriques de courbure strictement négative,, GAFA, 5 (1995), 731. doi: 10.1007/BF01897050.

[3]

M. G. Cowling and A. Ottazzi, Conformal maps of Carnot groups,, to appear in Ann. Acad. Sci. Fenn. Math., (). doi: 10.5186/aasfm.2015.4008.

[4]

P. Eberlein, U. Hamenstädt and V. Schroeder, Manifolds of nonpositive curvature,, in Differential Geometry: Riemannian Geometry (Los Angeles CA, (1990), 179.

[5]

Y. Fang, A dynamical-geometric characterization of the geodesic flows of negatively curved locally symmetric spaces,, to appear in Ergodic Theory and Dynamical Systems., (). doi: 10.1017/S0143385711000010.

[6]

Y. Fang, Smooth rigidity of uniformly quasiconformal Anosov flows,, Ergod. Th. and Dynam. Sys., 24 (2004), 1937. doi: 10.1017/S0143385704000264.

[7]

Y. Fang, Quasiconformal Anosov flows and quasisymmetric rigidity of Hamenstädt distances,, Discrete and Continuous Dynamical Systems, 34 (2014), 3471. doi: 10.3934/dcds.2014.34.3471.

[8]

M. Gromov, J. Lafontaine and P. Pansu, Structures Métriques Pour Les Variétés Riemanniennes,, Cedic-Fernand Nathan, (1981).

[9]

M. Guysinsky, The theory of non-stationary normal forms,, Ergod. Theory and Dyn. Syst., 22 (2002), 845. doi: 10.1017/S0143385702000421.

[10]

M. Guysinsky and A. Katok, Normal forms and invariant geometric structures for dynamical systems with invariant contracting foliations,, Math. Research Letters, 5 (1998), 149. doi: 10.4310/MRL.1998.v5.n2.a2.

[11]

U. Hamenstädt, A new description of the Bowen-Margulis measure,, Ergod. Th. and Dynam. Sys., 9 (1989), 455. doi: 10.1017/S0143385700005095.

[12]

U. Hamenstädt, Some regularity theorems for Carnot-Carathéodory metrics,, J. Differential Geometry, 32 (1990), 819.

[13]

U. Hamenstädt, Entropy-rigidity of locally symmetric spaces of negative curvature,, Annals of Mathematics, 131 (1990), 35. doi: 10.2307/1971507.

[14]

B. Hasselblatt, A new construction of the Margulis measure for Anosov flows,, Ergod. Th. and Dynam. Sys., 9 (1989), 465. doi: 10.1017/S0143385700005101.

[15]

B. Hasselblatt and A. Katok, Introduction to the Modern Theory of Dynamical Systems,, Encyclopedia of Mathematics and its Applications, 54 (1995). doi: 10.1017/CBO9780511809187.

[16]

J. L. Journée, A regularity lemma for functions of several variables,, Revista Mate-mática Iberoamericana, 19 (1988), 187. doi: 10.4171/RMI/69.

[17]

A. Katok, Entropy and closed geodesics,, Ergod. Th. and Dynam. Sys., 2 (1982), 339. doi: 10.1017/S0143385700001656.

[18]

J. Mitchell, On Carnot-Carathéodory metrics,, J. Differential Geometry, 21 (1985), 35.

[19]

G. A. Margulis and G. D. Mostow, The differential of a quasi-conformal mapping of a Carnot-Caratheodory space,, GAFA, 5 (1995), 402. doi: 10.1007/BF01895673.

[20]

J. F. Plante, Anosov flows,, Amer. J. Math., 94 (1972), 729. doi: 10.2307/2373755.

[21]

V. Sadovskaya, On uniformly quasiconformal Anosov flows,, Math. Res. Lett., 12 (2005), 425. doi: 10.4310/MRL.2005.v12.n3.a12.

show all references

References:
[1]

D. V. Anosov, Geodesic flows on closed Riemannian manifolds with negative curvature,, Proc. Inst. Steklov, 90 (1967).

[2]

G. Besson, G. Courtois and S. Gallot, Entropies et rigidités des espaces localement symétriques de courbure strictement négative,, GAFA, 5 (1995), 731. doi: 10.1007/BF01897050.

[3]

M. G. Cowling and A. Ottazzi, Conformal maps of Carnot groups,, to appear in Ann. Acad. Sci. Fenn. Math., (). doi: 10.5186/aasfm.2015.4008.

[4]

P. Eberlein, U. Hamenstädt and V. Schroeder, Manifolds of nonpositive curvature,, in Differential Geometry: Riemannian Geometry (Los Angeles CA, (1990), 179.

[5]

Y. Fang, A dynamical-geometric characterization of the geodesic flows of negatively curved locally symmetric spaces,, to appear in Ergodic Theory and Dynamical Systems., (). doi: 10.1017/S0143385711000010.

[6]

Y. Fang, Smooth rigidity of uniformly quasiconformal Anosov flows,, Ergod. Th. and Dynam. Sys., 24 (2004), 1937. doi: 10.1017/S0143385704000264.

[7]

Y. Fang, Quasiconformal Anosov flows and quasisymmetric rigidity of Hamenstädt distances,, Discrete and Continuous Dynamical Systems, 34 (2014), 3471. doi: 10.3934/dcds.2014.34.3471.

[8]

M. Gromov, J. Lafontaine and P. Pansu, Structures Métriques Pour Les Variétés Riemanniennes,, Cedic-Fernand Nathan, (1981).

[9]

M. Guysinsky, The theory of non-stationary normal forms,, Ergod. Theory and Dyn. Syst., 22 (2002), 845. doi: 10.1017/S0143385702000421.

[10]

M. Guysinsky and A. Katok, Normal forms and invariant geometric structures for dynamical systems with invariant contracting foliations,, Math. Research Letters, 5 (1998), 149. doi: 10.4310/MRL.1998.v5.n2.a2.

[11]

U. Hamenstädt, A new description of the Bowen-Margulis measure,, Ergod. Th. and Dynam. Sys., 9 (1989), 455. doi: 10.1017/S0143385700005095.

[12]

U. Hamenstädt, Some regularity theorems for Carnot-Carathéodory metrics,, J. Differential Geometry, 32 (1990), 819.

[13]

U. Hamenstädt, Entropy-rigidity of locally symmetric spaces of negative curvature,, Annals of Mathematics, 131 (1990), 35. doi: 10.2307/1971507.

[14]

B. Hasselblatt, A new construction of the Margulis measure for Anosov flows,, Ergod. Th. and Dynam. Sys., 9 (1989), 465. doi: 10.1017/S0143385700005101.

[15]

B. Hasselblatt and A. Katok, Introduction to the Modern Theory of Dynamical Systems,, Encyclopedia of Mathematics and its Applications, 54 (1995). doi: 10.1017/CBO9780511809187.

[16]

J. L. Journée, A regularity lemma for functions of several variables,, Revista Mate-mática Iberoamericana, 19 (1988), 187. doi: 10.4171/RMI/69.

[17]

A. Katok, Entropy and closed geodesics,, Ergod. Th. and Dynam. Sys., 2 (1982), 339. doi: 10.1017/S0143385700001656.

[18]

J. Mitchell, On Carnot-Carathéodory metrics,, J. Differential Geometry, 21 (1985), 35.

[19]

G. A. Margulis and G. D. Mostow, The differential of a quasi-conformal mapping of a Carnot-Caratheodory space,, GAFA, 5 (1995), 402. doi: 10.1007/BF01895673.

[20]

J. F. Plante, Anosov flows,, Amer. J. Math., 94 (1972), 729. doi: 10.2307/2373755.

[21]

V. Sadovskaya, On uniformly quasiconformal Anosov flows,, Math. Res. Lett., 12 (2005), 425. doi: 10.4310/MRL.2005.v12.n3.a12.

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