- Previous Article
- JMD Home
- This Issue
-
Next Article
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, 307-312, Cambridge Univ. Press, Cambridge, (2004). |
[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. |
[3] |
M. Brin and Ya Pesin, Partially hyperbolic dynamical systems, Math. USSR Izv., 8 (1974), 177-218. |
[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. |
[5] |
D. Calegary, "Foliations and the Geometry of 3-Manifolds,'' Oxford Mathematical Monographs, Oxford University Press, Oxford, 2007. |
[6] |
D. Calegary, M. Freedman and K. Walker, Positivity of the universal pairing in 3-dimensions, Jounal of the AMS, 23 (2010), 107-188. |
[7] |
D. Epstein, Periodic flows on 3-manifolds, Ann. Math., 2nd Series, 95 (1972), 66-82. |
[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. |
[9] |
A. Hammerlindl, "Leaf Conjugacies of the Torus,'' Phd Thesis, University of Toronto, 2009. |
[10] |
M. Hirsch, C. Pugh and M. Shub, "Invariant Manifolds,'' Lecture Notes in Math. 583, Springer Verlag, 1977. |
[11] |
A. Hatcher, "Notes on Basic 3-Manifold Topology,'' author's webpage at Cornell Math. Dep. 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), viii+192 pp. |
[13] |
K. Johannson, "Homotopy Equivalences of 3-Manifolds with Boundaries,'' Lecture Notes in Math. 761, Spriinger Verlag, Berlin-New York, 1977. |
[14] |
H. Kneser, Geschlossene Flächen in dreidimensionalen Mannigfaltigkeiten, Jahresbericht der Deutschen Mathematiker-Vereinigung, 38 (1929), 248-260. |
[15] |
J. Milnor, A unique decomposition theorem for 3-manifolds, Amer. J. of Math., 84 (1962), 1-7.
doi: doi:10.2307/2372800. |
[16] |
H. Rosenberg, Foliations by planes, Topology, 7 (1968), 131-138.
doi: doi:10.1016/0040-9383(68)90021-9. |
[17] |
R. Roussarie, Sur les feuilletages des variétés de dimension trois, (French), Ann. Inst. Fourier (Grenoble), 21 (1971), 13-82. |
[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. |
[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. |
[20] |
F. Rodriguez Hertz, M. Rodriguez Hertz and R. Ures, Partial hyperbolicity and ergodicity in dimension three, J. Mod. Dyn., 2 (2008), 187-208. |
[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) 308-333; ibid., 4 (1967), 87-117. |
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). |
[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. |
[3] |
M. Brin and Ya Pesin, Partially hyperbolic dynamical systems, Math. USSR Izv., 8 (1974), 177-218. |
[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. |
[5] |
D. Calegary, "Foliations and the Geometry of 3-Manifolds,'' Oxford Mathematical Monographs, Oxford University Press, Oxford, 2007. |
[6] |
D. Calegary, M. Freedman and K. Walker, Positivity of the universal pairing in 3-dimensions, Jounal of the AMS, 23 (2010), 107-188. |
[7] |
D. Epstein, Periodic flows on 3-manifolds, Ann. Math., 2nd Series, 95 (1972), 66-82. |
[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. |
[9] |
A. Hammerlindl, "Leaf Conjugacies of the Torus,'' Phd Thesis, University of Toronto, 2009. |
[10] |
M. Hirsch, C. Pugh and M. Shub, "Invariant Manifolds,'' Lecture Notes in Math. 583, Springer Verlag, 1977. |
[11] |
A. Hatcher, "Notes on Basic 3-Manifold Topology,'' author's webpage at Cornell Math. Dep. 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), viii+192 pp. |
[13] |
K. Johannson, "Homotopy Equivalences of 3-Manifolds with Boundaries,'' Lecture Notes in Math. 761, Spriinger Verlag, Berlin-New York, 1977. |
[14] |
H. Kneser, Geschlossene Flächen in dreidimensionalen Mannigfaltigkeiten, Jahresbericht der Deutschen Mathematiker-Vereinigung, 38 (1929), 248-260. |
[15] |
J. Milnor, A unique decomposition theorem for 3-manifolds, Amer. J. of Math., 84 (1962), 1-7.
doi: doi:10.2307/2372800. |
[16] |
H. Rosenberg, Foliations by planes, Topology, 7 (1968), 131-138.
doi: doi:10.1016/0040-9383(68)90021-9. |
[17] |
R. Roussarie, Sur les feuilletages des variétés de dimension trois, (French), Ann. Inst. Fourier (Grenoble), 21 (1971), 13-82. |
[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. |
[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. |
[20] |
F. Rodriguez Hertz, M. Rodriguez Hertz and R. Ures, Partial hyperbolicity and ergodicity in dimension three, J. Mod. Dyn., 2 (2008), 187-208. |
[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) 308-333; ibid., 4 (1967), 87-117. |
[1] |
Andrey Gogolev, Misha Guysinsky. $C^1$-differentiable conjugacy of Anosov diffeomorphisms on three dimensional torus. Discrete and Continuous Dynamical Systems, 2008, 22 (1&2) : 183-200. doi: 10.3934/dcds.2008.22.183 |
[2] |
Andrey Gogolev. Smooth conjugacy of Anosov diffeomorphisms on higher-dimensional tori. Journal of Modern Dynamics, 2008, 2 (4) : 645-700. doi: 10.3934/jmd.2008.2.645 |
[3] |
Saikat Mazumdar. Struwe's decomposition for a polyharmonic operator on a compact Riemannian manifold with or without boundary. Communications on Pure and Applied Analysis, 2017, 16 (1) : 311-330. doi: 10.3934/cpaa.2017015 |
[4] |
Andy Hammerlindl, Jana Rodriguez Hertz, Raúl Ures. Ergodicity and partial hyperbolicity on Seifert manifolds. Journal of Modern Dynamics, 2020, 0: 331-348. doi: 10.3934/jmd.2020012 |
[5] |
Bernhard Ruf, P. N. Srikanth. Hopf fibration and singularly perturbed elliptic equations. Discrete and Continuous Dynamical Systems - S, 2014, 7 (4) : 823-838. doi: 10.3934/dcdss.2014.7.823 |
[6] |
Andy Hammerlindl, Rafael Potrie, Mario Shannon. Seifert manifolds admitting partially hyperbolic diffeomorphisms. Journal of Modern Dynamics, 2018, 12: 193-222. doi: 10.3934/jmd.2018008 |
[7] |
João P. Almeida, Albert M. Fisher, Alberto Adrego Pinto, David A. Rand. Anosov diffeomorphisms. Conference Publications, 2013, 2013 (special) : 837-845. doi: 10.3934/proc.2013.2013.837 |
[8] |
Fabrizio Colombo, Irene Sabadini, Frank Sommen. The inverse Fueter mapping theorem. Communications on Pure and Applied Analysis, 2011, 10 (4) : 1165-1181. doi: 10.3934/cpaa.2011.10.1165 |
[9] |
K. H. Kim and F. W. Roush. The Williams conjecture is false for irreducible subshifts. Electronic Research Announcements, 1997, 3: 105-109. |
[10] |
John Banks. Topological mapping properties defined by digraphs. Discrete and Continuous Dynamical Systems, 1999, 5 (1) : 83-92. doi: 10.3934/dcds.1999.5.83 |
[11] |
Mads Kyed. On a mapping property of the Oseen operator with rotation. Discrete and Continuous Dynamical Systems - S, 2013, 6 (5) : 1315-1322. doi: 10.3934/dcdss.2013.6.1315 |
[12] |
Viktor L. Ginzburg and Basak Z. Gurel. On the construction of a $C^2$-counterexample to the Hamiltonian Seifert Conjecture in $\mathbb{R}^4$. Electronic Research Announcements, 2002, 8: 11-19. |
[13] |
Rémi Carles, Erwan Faou. Energy cascades for NLS on the torus. Discrete and Continuous Dynamical Systems, 2012, 32 (6) : 2063-2077. doi: 10.3934/dcds.2012.32.2063 |
[14] |
Simon Lloyd. On the Closing Lemma problem for the torus. Discrete and Continuous Dynamical Systems, 2009, 25 (3) : 951-962. doi: 10.3934/dcds.2009.25.951 |
[15] |
Peter Seibt. A period formula for torus automorphisms. Discrete and Continuous Dynamical Systems, 2003, 9 (4) : 1029-1048. doi: 10.3934/dcds.2003.9.1029 |
[16] |
Aaron W. Brown. Smooth stabilizers for measures on the torus. Discrete and Continuous Dynamical Systems, 2015, 35 (1) : 43-58. doi: 10.3934/dcds.2015.35.43 |
[17] |
Meera G. Mainkar, Cynthia E. Will. Examples of Anosov Lie algebras. Discrete and Continuous Dynamical Systems, 2007, 18 (1) : 39-52. doi: 10.3934/dcds.2007.18.39 |
[18] |
Anuradha Sharma, Saroj Rani. Trace description and Hamming weights of irreducible constacyclic codes. Advances in Mathematics of Communications, 2018, 12 (1) : 123-141. doi: 10.3934/amc.2018008 |
[19] |
Thierry Coulbois. Fractal trees for irreducible automorphisms of free groups. Journal of Modern Dynamics, 2010, 4 (2) : 359-391. doi: 10.3934/jmd.2010.4.359 |
[20] |
Mike Boyle, Sompong Chuysurichay. The mapping class group of a shift of finite type. Journal of Modern Dynamics, 2018, 13: 115-145. doi: 10.3934/jmd.2018014 |
2020 Impact Factor: 0.848
Tools
Metrics
Other articles
by authors
[Back to Top]