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The Khinchin Theorem for interval-exchange transformations
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 |
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].
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}, (). Google Scholar |
[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. Google Scholar |
[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). Google Scholar |
[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}, (). Google Scholar |
[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. Google Scholar |
[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). Google Scholar |
[22] |
F. Waldhausen, Eine Klasse von $3$-dimensionalen Mannigfaltigkeiten. I, II,, Invent. Math., 3 (1967), 308.
|
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