2011, 5(1): 185-202. doi: 10.3934/jmd.2011.5.185

Tori with hyperbolic dynamics in 3-manifolds

1. 

IMERL-Facultad de Ingeniería, Universidad de la República, ulio Herrera y Reissig 565, CC 30, 11300 Montevideo, Uruguay

2. 

IMERL-Facultad de Ingeniería, Universidad de la República, CC 30 Montevideo, Uruguay

Received  August 2010 Revised  February 2011 Published  April 2011

Let $M$ be a closed orientable irreducible $3$-dimensional manifold. An embedded $2$-torus $\mathbb{T}$ is an Anosov torus if there exists a diffeomorphism $f$ over $M$ for which $\T$ is $f$-invariant and $f_\#|_\mathbb{T}:\pi_1(\mathbb{T})\to \pi_1(\mathbb{T})$ is hyperbolic. We prove that only few irreducible $3$-manifolds admit Anosov tori: (1) the $3$-torus $\mathbb{T}^3$; (2) the mapping torus of $-\Id$; and (3) the mapping tori of hyperbolic automorphisms of $\mathbb{T}^2$.
   This has consequences for instance in the context of partially hyperbolic dynamics of $3$-manifolds: if there is an invariant foliation $\mathcal{F}^{cu}$ tangent to the center-unstable bundle $E^c\oplus E^u$, then $\mathcal{F}^{cu}$ has no compact leaves [21]. This has led to the first example of a non-dynamically coherent partially hyperbolic diffeomorphism with one-dimensional center bundle [21].
Citation: Federico Rodriguez Hertz, María Alejandra Rodriguez Hertz, Raúl Ures. Tori with hyperbolic dynamics in 3-manifolds. Journal of Modern Dynamics, 2011, 5 (1) : 185-202. doi: 10.3934/jmd.2011.5.185
References:
[1]

M. Brin, D. Burago and S. Ivanov, On partially hyperbolic diffeomorphisms of 3-manifolds with commutative fundamental group,, Modern Dynamical Systems and Applications, (2004), 307.

[2]

M. Brin, D. Burago and S. Ivanov, Dynamical coherence of partially hyperbolic diffeomorphisms of the $3$-torus,, J. Mod. Dyn., 3 (2009), 1. doi: doi:10.3934/jmd.2009.3.1.

[3]

M. Brin and Ya Pesin, Partially hyperbolic dynamical systems,, Math. USSR Izv., 8 (1974), 177.

[4]

D. Burago and S. Ivanov, Partially hyperbolic diffeomorphisms of 3-manifolds with abelian fundamental groups,, J. Mod. Dyn., 2 (2008), 541. doi: doi:10.3934/jmd.2008.2.541.

[5]

D. Calegary, "Foliations and the Geometry of 3-Manifolds,'', Oxford Mathematical Monographs, (2007).

[6]

D. Calegary, M. Freedman and K. Walker, Positivity of the universal pairing in 3-dimensions,, Jounal of the AMS, 23 (2010), 107.

[7]

D. Epstein, Periodic flows on 3-manifolds,, Ann. Math., 95 (1972), 66.

[8]

J. Franks, Anosov diffeomorphisms,, 1970 Global Analysis (Proc. Sympos. Pure Math., (1968), 61.

[9]

A. Hammerlindl, "Leaf Conjugacies of the Torus,'', Phd Thesis, (2009).

[10]

M. Hirsch, C. Pugh and M. Shub, "Invariant Manifolds,'', Lecture Notes in Math. {\bf 583}, 583 (1977).

[11]

A. Hatcher, "Notes on Basic 3-Manifold Topology,'', author's webpage at Cornell Math. Dep. \url{http://www.math.cornell.edu/ hatcher/3M/3M.pdf}, ().

[12]

W. Jaco and P. Shalen, "Seifert Fibered Spaces in 3-Manifolds,'', Memoirs AMS, 21 (1979).

[13]

K. Johannson, "Homotopy Equivalences of 3-Manifolds with Boundaries,'', Lecture Notes in Math. {\bf 761}, 761 (1977).

[14]

H. Kneser, Geschlossene Flächen in dreidimensionalen Mannigfaltigkeiten,, Jahresbericht der Deutschen Mathematiker-Vereinigung, 38 (1929), 248.

[15]

J. Milnor, A unique decomposition theorem for 3-manifolds,, Amer. J. of Math., 84 (1962), 1. doi: doi:10.2307/2372800.

[16]

H. Rosenberg, Foliations by planes,, Topology, 7 (1968), 131. doi: doi:10.1016/0040-9383(68)90021-9.

[17]

R. Roussarie, Sur les feuilletages des variétés de dimension trois,, (French), 21 (1971), 13.

[18]

F. Rodriguez Hertz, M. Rodriguez Hertz and R. Ures, "A Survey of Partially Hyperbolic Dynamics,'', Partially hyperbolic dynamics, (2007), 35.

[19]

F. Rodriguez Hertz, M. Rodriguez Hertz and R. Ures, Accessibility and stable ergodicity for partially hyperbolic diffeomorphisms with 1D-center bundle,, Invent. Math., 172 (2008), 353. doi: doi:10.1007/s00222-007-0100-z.

[20]

F. Rodriguez Hertz, M. Rodriguez Hertz and R. Ures, Partial hyperbolicity and ergodicity in dimension three,, J. Mod. Dyn., 2 (2008), 187.

[21]

F. Rodriguez Hertz, M. Rodriguez Hertz and R. Ures, A non-dynamically coherent example in $\mathbbT^3$,, in preparation, (2009).

[22]

F. Waldhausen, Eine Klasse von $3$-dimensionalen Mannigfaltigkeiten. I, II,, Invent. Math., 3 (1967), 308.

show all references

References:
[1]

M. Brin, D. Burago and S. Ivanov, On partially hyperbolic diffeomorphisms of 3-manifolds with commutative fundamental group,, Modern Dynamical Systems and Applications, (2004), 307.

[2]

M. Brin, D. Burago and S. Ivanov, Dynamical coherence of partially hyperbolic diffeomorphisms of the $3$-torus,, J. Mod. Dyn., 3 (2009), 1. doi: doi:10.3934/jmd.2009.3.1.

[3]

M. Brin and Ya Pesin, Partially hyperbolic dynamical systems,, Math. USSR Izv., 8 (1974), 177.

[4]

D. Burago and S. Ivanov, Partially hyperbolic diffeomorphisms of 3-manifolds with abelian fundamental groups,, J. Mod. Dyn., 2 (2008), 541. doi: doi:10.3934/jmd.2008.2.541.

[5]

D. Calegary, "Foliations and the Geometry of 3-Manifolds,'', Oxford Mathematical Monographs, (2007).

[6]

D. Calegary, M. Freedman and K. Walker, Positivity of the universal pairing in 3-dimensions,, Jounal of the AMS, 23 (2010), 107.

[7]

D. Epstein, Periodic flows on 3-manifolds,, Ann. Math., 95 (1972), 66.

[8]

J. Franks, Anosov diffeomorphisms,, 1970 Global Analysis (Proc. Sympos. Pure Math., (1968), 61.

[9]

A. Hammerlindl, "Leaf Conjugacies of the Torus,'', Phd Thesis, (2009).

[10]

M. Hirsch, C. Pugh and M. Shub, "Invariant Manifolds,'', Lecture Notes in Math. {\bf 583}, 583 (1977).

[11]

A. Hatcher, "Notes on Basic 3-Manifold Topology,'', author's webpage at Cornell Math. Dep. \url{http://www.math.cornell.edu/ hatcher/3M/3M.pdf}, ().

[12]

W. Jaco and P. Shalen, "Seifert Fibered Spaces in 3-Manifolds,'', Memoirs AMS, 21 (1979).

[13]

K. Johannson, "Homotopy Equivalences of 3-Manifolds with Boundaries,'', Lecture Notes in Math. {\bf 761}, 761 (1977).

[14]

H. Kneser, Geschlossene Flächen in dreidimensionalen Mannigfaltigkeiten,, Jahresbericht der Deutschen Mathematiker-Vereinigung, 38 (1929), 248.

[15]

J. Milnor, A unique decomposition theorem for 3-manifolds,, Amer. J. of Math., 84 (1962), 1. doi: doi:10.2307/2372800.

[16]

H. Rosenberg, Foliations by planes,, Topology, 7 (1968), 131. doi: doi:10.1016/0040-9383(68)90021-9.

[17]

R. Roussarie, Sur les feuilletages des variétés de dimension trois,, (French), 21 (1971), 13.

[18]

F. Rodriguez Hertz, M. Rodriguez Hertz and R. Ures, "A Survey of Partially Hyperbolic Dynamics,'', Partially hyperbolic dynamics, (2007), 35.

[19]

F. Rodriguez Hertz, M. Rodriguez Hertz and R. Ures, Accessibility and stable ergodicity for partially hyperbolic diffeomorphisms with 1D-center bundle,, Invent. Math., 172 (2008), 353. doi: doi:10.1007/s00222-007-0100-z.

[20]

F. Rodriguez Hertz, M. Rodriguez Hertz and R. Ures, Partial hyperbolicity and ergodicity in dimension three,, J. Mod. Dyn., 2 (2008), 187.

[21]

F. Rodriguez Hertz, M. Rodriguez Hertz and R. Ures, A non-dynamically coherent example in $\mathbbT^3$,, in preparation, (2009).

[22]

F. Waldhausen, Eine Klasse von $3$-dimensionalen Mannigfaltigkeiten. I, II,, Invent. Math., 3 (1967), 308.

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