2011, 29(3): 803-822. doi: 10.3934/dcds.2011.29.803

On integrable codimension one Anosov actions of $\RR^k$

1. 

Université d'Avignon et des pays de Vaucluse, LANLG, Faculté des Sciences, 33 rue Louis Pasteur, 84000 Avignon, France

2. 

Universidade de São Paulo - São Carlos, Instituto de Ciências Matemáticas e de Computação, Av. do Trabalhador São-Carlense 400, 13560-970 São Carlos, SP, Brazil

Received  January 2010 Revised  June 2010 Published  November 2010

In this paper, we consider codimension one Anosov actions of $\RR^k,\ k\geq 1,$ on closed connected orientable manifolds of dimension $n+k$ with $n\geq 3$. We show that the fundamental group of the ambient manifold is solvable if and only if the weak foliation of codimension one is transversely affine. We also study the situation where one $1$-parameter subgroup of $\RR^k$ admits a cross-section, and compare this to the case where the whole action is transverse to a fibration over a manifold of dimension $n$. As a byproduct, generalizing a Theorem by Ghys in the case $k=1$, we show that, under some assumptions about the smoothness of the sub-bundle $E^{ss}\oplus E$uu , and in the case where the action preserves the volume, it is topologically equivalent to a suspension of a linear Anosov action of $\mathbb{Z}^k$ on $\TT^{n}$.
Citation: Thierry Barbot, Carlos Maquera. On integrable codimension one Anosov actions of $\RR^k$. Discrete & Continuous Dynamical Systems - A, 2011, 29 (3) : 803-822. doi: 10.3934/dcds.2011.29.803
References:
[1]

T. Barbot, Actions de groupes sur les $1$-variétés non séparées et feuilletages de codimension un,, Ann. Fac. Sciences Toulouse, 7 (1998), 559.

[2]

T. Barbot and C. Maquera, Transitivity of codimension one Anosov actions of $\RR^k$ on closed manifolds,, to appear in Ergod. Th. & Dyn. Syst., (2010).

[3]

C. Bonatti and R. Langevin, Un exemple de flot d'Anosov transitif transverse à un tore et non conjugué à une suspension,, Ergod. Th. & Dyn. Sys., (1994), 633. doi: 10.1017/S0143385700008099.

[4]

M. I. Brin, Topological transitivity of one class of dynamical systems, and flows of frames on manifolds of negative curvature,, Funcional. Anal. i Prilozen, 9 (1975), 9.

[5]

M. I. Brin, The topology of group extensions of C-systems,, Math. Zametki, 3 (1975), 453.

[6]

J. Franks, Anosov diffeomorphisms,, in, 14 (1970), 61.

[7]

E. Ghys, Codimension one Anosov flows and suspensions,, Lecture Notes in Math. \textbf{1331}, 1331 (1988), 59.

[8]

E. Ghys, Groups acting on the circle,, Enseign. Math. (2), 47 (2001), 329.

[9]

A. Haefliger and G. Reeb, Variétés (non séparées) à une dimension et structures feuilletées du plan,, Enseign. Math., 3 (1957), 107.

[10]

G. Hector and U. Hirsch, "Introduction to the Geometry of the Foliations. Part B,", Aspects of Mathematics, (1981).

[11]

M. Hirsch, C. Pugh and M. Shub, Invariant manifolds,, Lecture Notes \textbf{583}, 583 (1977).

[12]

B. Kalinin and R. Spatzier, On the classification of Cartan actions,, G.A.F.A. Geom. Funct. Anal., 17 (2007), 468. doi: 10.1007/s00039-007-0602-2.

[13]

A. Katok and R.J. Spatzier, First cohomology of Anosov actions of higher rank abelian groups and applications to rigidity,, Inst. Hautes Études Sci. Publ. Math., 79 (1994), 131. doi: 10.1007/BF02698888.

[14]

A. Katok and J. Lewis, Global rigidity results for lattice actions on tori and new examples of volume-preserving actions,, Israel J. Math., 93 (1996), 253. doi: 10.1007/BF02761106.

[15]

S. Matsumoto, Codimension one foliations on solvable manifolds,, Comment. Math. Helv., 68 (1993), 633. doi: 10.1007/BF02565839.

[16]

S. E. Newhouse, On codimension one Anosov diffeomorphisms,, Amer. J. Math., 92 (1970), 761. doi: 10.2307/2373372.

[17]

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

[18]

J. Plante, Anosov flows, transversely affine foliations, and a conjecture of Verjovsky,, J. London Math. Soc. (2), 23 (1981), 359. doi: 10.1112/jlms/s2-23.2.359.

[19]

C. Pugh and M. Shub, Ergodicity of Anosov actions,, Invent. Math., 15 (1972), 1. doi: 10.1007/BF01418639.

[20]

S. Schwartzman, Asymptotic cycles,, Ann. of Math., 66 (1957), 270. doi: 10.2307/1969999.

[21]

S. Simic, Codimension one Anosov flows and a conjecture of Verjovsky,, Ergodic Thery Dynam. Systems, 17 (1997), 1211.

[22]

A. Verjovsky, Codimension one Anosov flows,, Bol. Soc. Mat. Mexicana, 19 (1974), 49.

[23]

C. T. C. Wall, Surgery on compact manifolds,, in, (1970).

show all references

References:
[1]

T. Barbot, Actions de groupes sur les $1$-variétés non séparées et feuilletages de codimension un,, Ann. Fac. Sciences Toulouse, 7 (1998), 559.

[2]

T. Barbot and C. Maquera, Transitivity of codimension one Anosov actions of $\RR^k$ on closed manifolds,, to appear in Ergod. Th. & Dyn. Syst., (2010).

[3]

C. Bonatti and R. Langevin, Un exemple de flot d'Anosov transitif transverse à un tore et non conjugué à une suspension,, Ergod. Th. & Dyn. Sys., (1994), 633. doi: 10.1017/S0143385700008099.

[4]

M. I. Brin, Topological transitivity of one class of dynamical systems, and flows of frames on manifolds of negative curvature,, Funcional. Anal. i Prilozen, 9 (1975), 9.

[5]

M. I. Brin, The topology of group extensions of C-systems,, Math. Zametki, 3 (1975), 453.

[6]

J. Franks, Anosov diffeomorphisms,, in, 14 (1970), 61.

[7]

E. Ghys, Codimension one Anosov flows and suspensions,, Lecture Notes in Math. \textbf{1331}, 1331 (1988), 59.

[8]

E. Ghys, Groups acting on the circle,, Enseign. Math. (2), 47 (2001), 329.

[9]

A. Haefliger and G. Reeb, Variétés (non séparées) à une dimension et structures feuilletées du plan,, Enseign. Math., 3 (1957), 107.

[10]

G. Hector and U. Hirsch, "Introduction to the Geometry of the Foliations. Part B,", Aspects of Mathematics, (1981).

[11]

M. Hirsch, C. Pugh and M. Shub, Invariant manifolds,, Lecture Notes \textbf{583}, 583 (1977).

[12]

B. Kalinin and R. Spatzier, On the classification of Cartan actions,, G.A.F.A. Geom. Funct. Anal., 17 (2007), 468. doi: 10.1007/s00039-007-0602-2.

[13]

A. Katok and R.J. Spatzier, First cohomology of Anosov actions of higher rank abelian groups and applications to rigidity,, Inst. Hautes Études Sci. Publ. Math., 79 (1994), 131. doi: 10.1007/BF02698888.

[14]

A. Katok and J. Lewis, Global rigidity results for lattice actions on tori and new examples of volume-preserving actions,, Israel J. Math., 93 (1996), 253. doi: 10.1007/BF02761106.

[15]

S. Matsumoto, Codimension one foliations on solvable manifolds,, Comment. Math. Helv., 68 (1993), 633. doi: 10.1007/BF02565839.

[16]

S. E. Newhouse, On codimension one Anosov diffeomorphisms,, Amer. J. Math., 92 (1970), 761. doi: 10.2307/2373372.

[17]

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

[18]

J. Plante, Anosov flows, transversely affine foliations, and a conjecture of Verjovsky,, J. London Math. Soc. (2), 23 (1981), 359. doi: 10.1112/jlms/s2-23.2.359.

[19]

C. Pugh and M. Shub, Ergodicity of Anosov actions,, Invent. Math., 15 (1972), 1. doi: 10.1007/BF01418639.

[20]

S. Schwartzman, Asymptotic cycles,, Ann. of Math., 66 (1957), 270. doi: 10.2307/1969999.

[21]

S. Simic, Codimension one Anosov flows and a conjecture of Verjovsky,, Ergodic Thery Dynam. Systems, 17 (1997), 1211.

[22]

A. Verjovsky, Codimension one Anosov flows,, Bol. Soc. Mat. Mexicana, 19 (1974), 49.

[23]

C. T. C. Wall, Surgery on compact manifolds,, in, (1970).

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