January  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, 307-312, Cambridge Univ. Press, Cambridge, (2004).  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), 1-11. 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), 177-218.  Google Scholar

[4]

D. Burago and S. Ivanov, Partially hyperbolic diffeomorphisms of 3-manifolds with abelian fundamental groups, J. Mod. Dyn., 2 (2008), 541-580. doi: doi:10.3934/jmd.2008.2.541.  Google Scholar

[5]

D. Calegary, "Foliations and the Geometry of 3-Manifolds,'' Oxford Mathematical Monographs, Oxford University Press, Oxford, 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), 107-188.  Google Scholar

[7]

D. Epstein, Periodic flows on 3-manifolds, Ann. Math., 2nd Series, 95 (1972), 66-82.  Google Scholar

[8]

J. Franks, Anosov diffeomorphisms, 1970 Global Analysis (Proc. Sympos. Pure Math., Vol. XIV, Berkeley, Calif., 1968) pp. 61-93, Amer. Math. Soc., Providence, R.I.  Google Scholar

[9]

A. Hammerlindl, "Leaf Conjugacies of the Torus,'' Phd Thesis, University of Toronto, 2009.  Google Scholar

[10]

M. Hirsch, C. Pugh and M. Shub, "Invariant Manifolds,'' Lecture Notes in Math. 583, Springer Verlag, 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), viii+192 pp.  Google Scholar

[13]

K. Johannson, "Homotopy Equivalences of 3-Manifolds with Boundaries,'' Lecture Notes in Math. 761, Spriinger Verlag, Berlin-New York, 1977.  Google Scholar

[14]

H. Kneser, Geschlossene Flächen in dreidimensionalen Mannigfaltigkeiten, Jahresbericht der Deutschen Mathematiker-Vereinigung, 38 (1929), 248-260. Google Scholar

[15]

J. Milnor, A unique decomposition theorem for 3-manifolds, Amer. J. of Math., 84 (1962), 1-7. doi: doi:10.2307/2372800.  Google Scholar

[16]

H. Rosenberg, Foliations by planes, Topology, 7 (1968), 131-138. doi: doi:10.1016/0040-9383(68)90021-9.  Google Scholar

[17]

R. Roussarie, Sur les feuilletages des variétés de dimension trois, (French), Ann. Inst. Fourier (Grenoble), 21 (1971), 13-82.  Google Scholar

[18]

F. Rodriguez Hertz, M. Rodriguez Hertz and R. Ures, "A Survey of Partially Hyperbolic Dynamics,'' Partially hyperbolic dynamics, laminations, and Teichmüller flow, 35-87, Fields Inst. Commun., 51, Amer. Math. Soc., Providence, RI, 2007.  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), 353-381. 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), 187-208.  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) 308-333; ibid., 4 (1967), 87-117.  Google Scholar

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, 307-312, Cambridge Univ. Press, Cambridge, (2004).  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), 1-11. 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), 177-218.  Google Scholar

[4]

D. Burago and S. Ivanov, Partially hyperbolic diffeomorphisms of 3-manifolds with abelian fundamental groups, J. Mod. Dyn., 2 (2008), 541-580. doi: doi:10.3934/jmd.2008.2.541.  Google Scholar

[5]

D. Calegary, "Foliations and the Geometry of 3-Manifolds,'' Oxford Mathematical Monographs, Oxford University Press, Oxford, 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), 107-188.  Google Scholar

[7]

D. Epstein, Periodic flows on 3-manifolds, Ann. Math., 2nd Series, 95 (1972), 66-82.  Google Scholar

[8]

J. Franks, Anosov diffeomorphisms, 1970 Global Analysis (Proc. Sympos. Pure Math., Vol. XIV, Berkeley, Calif., 1968) pp. 61-93, Amer. Math. Soc., Providence, R.I.  Google Scholar

[9]

A. Hammerlindl, "Leaf Conjugacies of the Torus,'' Phd Thesis, University of Toronto, 2009.  Google Scholar

[10]

M. Hirsch, C. Pugh and M. Shub, "Invariant Manifolds,'' Lecture Notes in Math. 583, Springer Verlag, 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), viii+192 pp.  Google Scholar

[13]

K. Johannson, "Homotopy Equivalences of 3-Manifolds with Boundaries,'' Lecture Notes in Math. 761, Spriinger Verlag, Berlin-New York, 1977.  Google Scholar

[14]

H. Kneser, Geschlossene Flächen in dreidimensionalen Mannigfaltigkeiten, Jahresbericht der Deutschen Mathematiker-Vereinigung, 38 (1929), 248-260. Google Scholar

[15]

J. Milnor, A unique decomposition theorem for 3-manifolds, Amer. J. of Math., 84 (1962), 1-7. doi: doi:10.2307/2372800.  Google Scholar

[16]

H. Rosenberg, Foliations by planes, Topology, 7 (1968), 131-138. doi: doi:10.1016/0040-9383(68)90021-9.  Google Scholar

[17]

R. Roussarie, Sur les feuilletages des variétés de dimension trois, (French), Ann. Inst. Fourier (Grenoble), 21 (1971), 13-82.  Google Scholar

[18]

F. Rodriguez Hertz, M. Rodriguez Hertz and R. Ures, "A Survey of Partially Hyperbolic Dynamics,'' Partially hyperbolic dynamics, laminations, and Teichmüller flow, 35-87, Fields Inst. Commun., 51, Amer. Math. Soc., Providence, RI, 2007.  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), 353-381. 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), 187-208.  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) 308-333; ibid., 4 (1967), 87-117.  Google Scholar

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