Tori with hyperbolic dynamics in 3-manifolds

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

Tori with hyperbolic dynamics in 3-manifolds

Federico Rodriguez Hertz ^1, , María Alejandra Rodriguez Hertz ^2, and Raúl Ures ^2,

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

Abstract
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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].

Keywords: irreducible manifold, mapping torus, JSJ decomposition, Anosov tori, atoroidal., Seifert fibration.

Mathematics Subject Classification: Primary: 37D20, 57M99; Secondary: 37D30, 57M5.

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. Google Scholar
[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. Google Scholar
[3]	M. Brin and Ya Pesin, Partially hyperbolic dynamical systems,, Math. USSR Izv., 8 (1974), 177. Google Scholar
[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. Google Scholar
[5]	D. Calegary, "Foliations and the Geometry of 3-Manifolds,'', Oxford Mathematical Monographs, (2007). Google Scholar
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[7]	D. Epstein, Periodic flows on 3-manifolds,, Ann. Math., 95 (1972), 66. Google Scholar
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[10]	M. Hirsch, C. Pugh and M. Shub, "Invariant Manifolds,'', Lecture Notes in Math. {\bf 583}, 583 (1977). Google Scholar
[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}, (). Google Scholar
[12]	W. Jaco and P. Shalen, "Seifert Fibered Spaces in 3-Manifolds,'', Memoirs AMS, 21 (1979). Google Scholar
[13]	K. Johannson, "Homotopy Equivalences of 3-Manifolds with Boundaries,'', Lecture Notes in Math. {\bf 761}, 761 (1977). Google Scholar
[14]	H. Kneser, Geschlossene Flächen in dreidimensionalen Mannigfaltigkeiten,, Jahresbericht der Deutschen Mathematiker-Vereinigung, 38 (1929), 248. Google Scholar
[15]	J. Milnor, A unique decomposition theorem for 3-manifolds,, Amer. J. of Math., 84 (1962), 1. doi: doi:10.2307/2372800. Google Scholar
[16]	H. Rosenberg, Foliations by planes,, Topology, 7 (1968), 131. doi: doi:10.1016/0040-9383(68)90021-9. Google Scholar
[17]	R. Roussarie, Sur les feuilletages des variétés de dimension trois,, (French), 21 (1971), 13. Google Scholar
[18]	F. Rodriguez Hertz, M. Rodriguez Hertz and R. Ures, "A Survey of Partially Hyperbolic Dynamics,'', Partially hyperbolic dynamics, (2007), 35. Google Scholar
[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. Google Scholar
[20]	F. Rodriguez Hertz, M. Rodriguez Hertz and R. Ures, Partial hyperbolicity and ergodicity in dimension three,, J. Mod. Dyn., 2 (2008), 187. Google Scholar
[21]	F. Rodriguez Hertz, M. Rodriguez Hertz and R. Ures, A non-dynamically coherent example in $\mathbbT^3$,, in preparation, (2009). Google Scholar
[22]	F. Waldhausen, Eine Klasse von $3$-dimensionalen Mannigfaltigkeiten. I, II,, Invent. Math., 3 (1967), 308. Google Scholar

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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. Google Scholar
[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. Google Scholar
[3]	M. Brin and Ya Pesin, Partially hyperbolic dynamical systems,, Math. USSR Izv., 8 (1974), 177. Google Scholar
[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. Google Scholar
[5]	D. Calegary, "Foliations and the Geometry of 3-Manifolds,'', Oxford Mathematical Monographs, (2007). Google Scholar
[6]	D. Calegary, M. Freedman and K. Walker, Positivity of the universal pairing in 3-dimensions,, Jounal of the AMS, 23 (2010), 107. Google Scholar
[7]	D. Epstein, Periodic flows on 3-manifolds,, Ann. Math., 95 (1972), 66. Google Scholar
[8]	J. Franks, Anosov diffeomorphisms,, 1970 Global Analysis (Proc. Sympos. Pure Math., (1968), 61. Google Scholar
[9]	A. Hammerlindl, "Leaf Conjugacies of the Torus,'', Phd Thesis, (2009). Google Scholar
[10]	M. Hirsch, C. Pugh and M. Shub, "Invariant Manifolds,'', Lecture Notes in Math. {\bf 583}, 583 (1977). Google Scholar
[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}, (). Google Scholar
[12]	W. Jaco and P. Shalen, "Seifert Fibered Spaces in 3-Manifolds,'', Memoirs AMS, 21 (1979). Google Scholar
[13]	K. Johannson, "Homotopy Equivalences of 3-Manifolds with Boundaries,'', Lecture Notes in Math. {\bf 761}, 761 (1977). Google Scholar
[14]	H. Kneser, Geschlossene Flächen in dreidimensionalen Mannigfaltigkeiten,, Jahresbericht der Deutschen Mathematiker-Vereinigung, 38 (1929), 248. Google Scholar
[15]	J. Milnor, A unique decomposition theorem for 3-manifolds,, Amer. J. of Math., 84 (1962), 1. doi: doi:10.2307/2372800. Google Scholar
[16]	H. Rosenberg, Foliations by planes,, Topology, 7 (1968), 131. doi: doi:10.1016/0040-9383(68)90021-9. Google Scholar
[17]	R. Roussarie, Sur les feuilletages des variétés de dimension trois,, (French), 21 (1971), 13. Google Scholar
[18]	F. Rodriguez Hertz, M. Rodriguez Hertz and R. Ures, "A Survey of Partially Hyperbolic Dynamics,'', Partially hyperbolic dynamics, (2007), 35. Google Scholar
[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. Google Scholar
[20]	F. Rodriguez Hertz, M. Rodriguez Hertz and R. Ures, Partial hyperbolicity and ergodicity in dimension three,, J. Mod. Dyn., 2 (2008), 187. Google Scholar
[21]	F. Rodriguez Hertz, M. Rodriguez Hertz and R. Ures, A non-dynamically coherent example in $\mathbbT^3$,, in preparation, (2009). Google Scholar
[22]	F. Waldhausen, Eine Klasse von $3$-dimensionalen Mannigfaltigkeiten. I, II,, Invent. Math., 3 (1967), 308. Google Scholar

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