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October  2015, 35(10): 5037-5054. doi: 10.3934/dcds.2015.35.5037

Partially hyperbolic diffeomorphisms with a trapping property

Rafael Potrie 1,

1. 

CMAT, Facultad de Ciencias, Universidad de la República, Igua 4225, Montevideo 11400, Uruguay

Received  August 2014 Revised  January 2015 Published  April 2015

We study partially hyperbolic diffeomorphisms satisfying a trapping property which makes them look as if they were Anosov at large scale. We show that, as expected, they share several properties with Anosov diffeomorphisms. We construct an expansive quotient of the dynamics and study some dynamical consequences related to this quotient.
Keywords: quasi-attractors, classification, Partial hyperbolicity, dynamical coherence..
Mathematics Subject Classification: 37D30, 37C20, 37C7.
Citation: Rafael Potrie. Partially hyperbolic diffeomorphisms with a trapping property. Discrete & Continuous Dynamical Systems, 2015, 35 (10) : 5037-5054. doi: 10.3934/dcds.2015.35.5037
References:
[1]

A. Artigue, J. Brum and R. Potrie, Local product structure for expansive homeomorphisms, Topology and its Applications, 156 (2009), 674-685. doi: 10.1016/j.topol.2008.09.004.  Google Scholar

[2]

C. Bonatti, L. Diaz and M. Viana, Dynamics Beyond Uniform Hyperbolicity. A Global Geometric and Probabilistic Perspective, Encyclopaedia of Mathematical Sciences, {102}, Mathematical Physics III, Springer-Verlag, 2005.  Google Scholar

[3]

C. Bonatti and M. Viana, SRB measures for partially hyperbolic diffeomorphisms whose central direction is mostly contracting, Israel J. of Math., 115 (2000), 157-193. doi: 10.1007/BF02810585.  Google Scholar

[4]

C. Bonatti and A. Wilkinson, Transitive partially hyperbolic diffeomorphisms on 3-manifolds, Topology, 44 (2005), 475-508. doi: 10.1016/j.top.2004.10.009.  Google Scholar

[5]

D. Bonhet, Codimension-1 partially hyperbolic diffeomorphisms with a uniformly compact center foliation, Journal of Modern Dynamics, 7 (2013), 565-604. doi: 10.3934/jmd.2013.7.565.  Google Scholar

[6]

M. Brin, D. Burago and S. Ivanov, Dynamical coherence of partially hyperbolic diffeomorphisms of the 3-torus, Journal of Modern Dynamics, 3 (2009), 1-11. doi: 10.3934/jmd.2009.3.1.  Google Scholar

[7]

K. Burns and A. Wilkinson, Dynamical coherence and center bunching, Discrete and Continuous Dynamical Systems A (Pesin birthday issue), 22 (2008), 89-100. doi: 10.3934/dcds.2008.22.89.  Google Scholar

[8]

J. Buzzi and T. Fisher, Entropic stability beyond partial hyperbolicity, Journal of Modern Dynamics, 7 (2013), 527-552. doi: 10.3934/jmd.2013.7.527.  Google Scholar

[9]

A. Candel and L. Conlon, Foliations I and II, Graduate studies in Mathematics, 60, American Math. Society, Providence, RI, 2003.  Google Scholar

[10]

P. Carrasco, Compact Dynamical Foliations, Ph.D. Thesis, University of Toronto (Canada), 2011.  Google Scholar

[11]

M. Carvalho, Sinai-Ruelle-Bowen measures for N-dimensional derived from Anosov diffeomorphisms, Ergodic Theory and Dynamical Systems, 13 (1993), 21-44. doi: 10.1017/S0143385700007185.  Google Scholar

[12]

S. Crovisier and E. Pujals, Essential hyperbolicity and homoclinic bifurcations: A dichotomy phenomenon/mechanism for diffeomorphisms,, to appear in Inventiones Math., ().  doi: 10.1007/s00222-014-0553-9.  Google Scholar

[13]

R. Daverman, Decompositions of Manifolds, Pure and Applied Mathematics, 124, Academic Press, Inc., Orlando, FL, 1986.  Google Scholar

[14]

T. Fisher, R. Potrie and M. Sambarino, Dynamical coherence for partially hyperbolic diffeomorphisms isotopic to Anosov on tori, Mathematische Zeitchcrift, 278 (2014), 149-168. doi: 10.1007/s00209-014-1310-x.  Google Scholar

[15]

J. Franks, Anosov Diffeomorphisms, Proc. Sympos. Pure Math., 14 (1970), 61-93.  Google Scholar

[16]

J. Franks, Homology and Dynamical Systems, CBMS Regional Conference Series in Mathematics, 49, Published for the Conference Board of the Mathematical Sciences, Washington, D.C.; by the American Mathematical Society, Providence, R. I., 1982.  Google Scholar

[17]

A. Gogolev, Partially hyperbolic diffeomorphisms with compact center foliations, Journal of Modern Dynamics, 5 (2011), 747-769. doi: 10.3934/jmd.2011.5.747.  Google Scholar

[18]

A. Gogolev and F. Rodriguez Hertz, Manifolds with higher homotopy which do not support Anosov diffeomorphisms, Bulletin of the London Math Society, 46 (2014), 349-366. doi: 10.1112/blms/bdt100.  Google Scholar

[19]

A. Hammerlindl and R. Potrie, Pointwise partial hyperbolicity in three dimensional nilmanifolds, Journal of the London Math. Society, 89 (2014), 853-875. doi: 10.1112/jlms/jdu013.  Google Scholar

[20]

A. Hammerlindl and R. Potrie, Classification of partially hyperbolic diffeomorphisms in three dimensional manifolds with solvable fundamental group,, to appear in Journal of Topology, ().   Google Scholar

[21]

M. Hirsch, C. Pugh and M. Shub, Invariant Manifolds, Springer Lecture Notes in Math., Vol. 583, Springer-Verlag, Berlin-New York, 1977.  Google Scholar

[22]

W. Hsiang and C. T. C. Wall, On homotopy tori II, Bull. London Math. Soc., 1 (1969), 341-342. doi: 10.1112/blms/1.3.341.  Google Scholar

[23]

S. L. Jones, The impossibility of filling $E^n$ with arcs, Bull. Amer. Math. Soc., 74 (1968), 155-159. doi: 10.1090/S0002-9904-1968-11919-6.  Google Scholar

[24]

A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems, With a supplementary chapter by Katok and Leonardo Mendoza, Encyclopedia of Mathematics and its Applications, 54, Cambridge University Press, Cambridge, 1995. doi: 10.1017/CBO9780511809187.  Google Scholar

[25]

A. Manning, There are no new anosov on tori, Amer. Jour. of Math., 96 (1974), 422-429. doi: 10.2307/2373551.  Google Scholar

[26]

R. Mañe, Contributions to the stability conjecture, Topology, 17 (1978), 383-396. doi: 10.1016/0040-9383(78)90005-8.  Google Scholar

[27]

R. Mañe, Expansive homeomorphisms and topological dimension, Trans. Amer. Math. Soc., 252 (1979), 313-319. doi: 10.1090/S0002-9947-1979-0534124-9.  Google Scholar

[28]

S. Newhouse, On codimension one Anosov diffeomorphisms, Amer. Jour. of Math., 92 (1970), 761-770. doi: 10.2307/2373372.  Google Scholar

[29]

S. Newhouse, Hyperbolic limit sets, Transactions of the A.M.S, 167 (1972), 125-150. doi: 10.1090/S0002-9947-1972-0295388-6.  Google Scholar

[30]

J. Ombach, Equivalent conditions for hyperbolic coordinates, Topology and its Applications, 23 (1986), 87-90. doi: 10.1016/0166-8641(86)90019-2.  Google Scholar

[31]

R. Potrie, Wild Milnor attractors accumulated by lower dimensional dynamics, Ergodic Theory and Dynamical Systems, 34 (2014), 236-262. doi: 10.1017/etds.2012.124.  Google Scholar

[32]

R. Potrie, Partially Hyperbolicity and Attracting Regions in 3-Dimensional Manifolds, Ph.D Thesis, 2012, arXiv:1207.1822. Google Scholar

[33]

R. Potrie, Partial hyperbolicity and foliations in $\mathbbT^3$, Journal of Modern Dynamics, 2014. Google Scholar

[34]

R. Potrie, A few remarks on partially hyperbolic diffeomorphisms of $\mathbbT^3$ isotopic to Anosov, Journal of Dynamics and Differential Equations, 26 (2014), 805-815. doi: 10.1007/s10884-014-9362-5.  Google Scholar

[35]

J. H. Roberts, Collections filling the plane, Duke Math. J., 2 (1936), 10-19. doi: 10.1215/S0012-7094-36-00202-8.  Google Scholar

[36]

M. Roldan, Hyperbolic sets and entropy at the homological level, preprint, 2014, arXiv:1407.5101. Google Scholar

[37]

K. Shiraiwa, Manifolds which do not admit Anosov diffeomorphisms, Nagoya Math J., 49 (1973), 111-115.  Google Scholar

[38]

J. L. Vieitez, Expansive homeomorphisms and hyperbolic diffeomorphisms on three manifolds, Ergodic Theory and Dynamical Systems, 16 (1996), 591-622. doi: 10.1017/S0143385700008981.  Google Scholar

show all references

References:
[1]

A. Artigue, J. Brum and R. Potrie, Local product structure for expansive homeomorphisms, Topology and its Applications, 156 (2009), 674-685. doi: 10.1016/j.topol.2008.09.004.  Google Scholar

[2]

C. Bonatti, L. Diaz and M. Viana, Dynamics Beyond Uniform Hyperbolicity. A Global Geometric and Probabilistic Perspective, Encyclopaedia of Mathematical Sciences, {102}, Mathematical Physics III, Springer-Verlag, 2005.  Google Scholar

[3]

C. Bonatti and M. Viana, SRB measures for partially hyperbolic diffeomorphisms whose central direction is mostly contracting, Israel J. of Math., 115 (2000), 157-193. doi: 10.1007/BF02810585.  Google Scholar

[4]

C. Bonatti and A. Wilkinson, Transitive partially hyperbolic diffeomorphisms on 3-manifolds, Topology, 44 (2005), 475-508. doi: 10.1016/j.top.2004.10.009.  Google Scholar

[5]

D. Bonhet, Codimension-1 partially hyperbolic diffeomorphisms with a uniformly compact center foliation, Journal of Modern Dynamics, 7 (2013), 565-604. doi: 10.3934/jmd.2013.7.565.  Google Scholar

[6]

M. Brin, D. Burago and S. Ivanov, Dynamical coherence of partially hyperbolic diffeomorphisms of the 3-torus, Journal of Modern Dynamics, 3 (2009), 1-11. doi: 10.3934/jmd.2009.3.1.  Google Scholar

[7]

K. Burns and A. Wilkinson, Dynamical coherence and center bunching, Discrete and Continuous Dynamical Systems A (Pesin birthday issue), 22 (2008), 89-100. doi: 10.3934/dcds.2008.22.89.  Google Scholar

[8]

J. Buzzi and T. Fisher, Entropic stability beyond partial hyperbolicity, Journal of Modern Dynamics, 7 (2013), 527-552. doi: 10.3934/jmd.2013.7.527.  Google Scholar

[9]

A. Candel and L. Conlon, Foliations I and II, Graduate studies in Mathematics, 60, American Math. Society, Providence, RI, 2003.  Google Scholar

[10]

P. Carrasco, Compact Dynamical Foliations, Ph.D. Thesis, University of Toronto (Canada), 2011.  Google Scholar

[11]

M. Carvalho, Sinai-Ruelle-Bowen measures for N-dimensional derived from Anosov diffeomorphisms, Ergodic Theory and Dynamical Systems, 13 (1993), 21-44. doi: 10.1017/S0143385700007185.  Google Scholar

[12]

S. Crovisier and E. Pujals, Essential hyperbolicity and homoclinic bifurcations: A dichotomy phenomenon/mechanism for diffeomorphisms,, to appear in Inventiones Math., ().  doi: 10.1007/s00222-014-0553-9.  Google Scholar

[13]

R. Daverman, Decompositions of Manifolds, Pure and Applied Mathematics, 124, Academic Press, Inc., Orlando, FL, 1986.  Google Scholar

[14]

T. Fisher, R. Potrie and M. Sambarino, Dynamical coherence for partially hyperbolic diffeomorphisms isotopic to Anosov on tori, Mathematische Zeitchcrift, 278 (2014), 149-168. doi: 10.1007/s00209-014-1310-x.  Google Scholar

[15]

J. Franks, Anosov Diffeomorphisms, Proc. Sympos. Pure Math., 14 (1970), 61-93.  Google Scholar

[16]

J. Franks, Homology and Dynamical Systems, CBMS Regional Conference Series in Mathematics, 49, Published for the Conference Board of the Mathematical Sciences, Washington, D.C.; by the American Mathematical Society, Providence, R. I., 1982.  Google Scholar

[17]

A. Gogolev, Partially hyperbolic diffeomorphisms with compact center foliations, Journal of Modern Dynamics, 5 (2011), 747-769. doi: 10.3934/jmd.2011.5.747.  Google Scholar

[18]

A. Gogolev and F. Rodriguez Hertz, Manifolds with higher homotopy which do not support Anosov diffeomorphisms, Bulletin of the London Math Society, 46 (2014), 349-366. doi: 10.1112/blms/bdt100.  Google Scholar

[19]

A. Hammerlindl and R. Potrie, Pointwise partial hyperbolicity in three dimensional nilmanifolds, Journal of the London Math. Society, 89 (2014), 853-875. doi: 10.1112/jlms/jdu013.  Google Scholar

[20]

A. Hammerlindl and R. Potrie, Classification of partially hyperbolic diffeomorphisms in three dimensional manifolds with solvable fundamental group,, to appear in Journal of Topology, ().   Google Scholar

[21]

M. Hirsch, C. Pugh and M. Shub, Invariant Manifolds, Springer Lecture Notes in Math., Vol. 583, Springer-Verlag, Berlin-New York, 1977.  Google Scholar

[22]

W. Hsiang and C. T. C. Wall, On homotopy tori II, Bull. London Math. Soc., 1 (1969), 341-342. doi: 10.1112/blms/1.3.341.  Google Scholar

[23]

S. L. Jones, The impossibility of filling $E^n$ with arcs, Bull. Amer. Math. Soc., 74 (1968), 155-159. doi: 10.1090/S0002-9904-1968-11919-6.  Google Scholar

[24]

A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems, With a supplementary chapter by Katok and Leonardo Mendoza, Encyclopedia of Mathematics and its Applications, 54, Cambridge University Press, Cambridge, 1995. doi: 10.1017/CBO9780511809187.  Google Scholar

[25]

A. Manning, There are no new anosov on tori, Amer. Jour. of Math., 96 (1974), 422-429. doi: 10.2307/2373551.  Google Scholar

[26]

R. Mañe, Contributions to the stability conjecture, Topology, 17 (1978), 383-396. doi: 10.1016/0040-9383(78)90005-8.  Google Scholar

[27]

R. Mañe, Expansive homeomorphisms and topological dimension, Trans. Amer. Math. Soc., 252 (1979), 313-319. doi: 10.1090/S0002-9947-1979-0534124-9.  Google Scholar

[28]

S. Newhouse, On codimension one Anosov diffeomorphisms, Amer. Jour. of Math., 92 (1970), 761-770. doi: 10.2307/2373372.  Google Scholar

[29]

S. Newhouse, Hyperbolic limit sets, Transactions of the A.M.S, 167 (1972), 125-150. doi: 10.1090/S0002-9947-1972-0295388-6.  Google Scholar

[30]

J. Ombach, Equivalent conditions for hyperbolic coordinates, Topology and its Applications, 23 (1986), 87-90. doi: 10.1016/0166-8641(86)90019-2.  Google Scholar

[31]

R. Potrie, Wild Milnor attractors accumulated by lower dimensional dynamics, Ergodic Theory and Dynamical Systems, 34 (2014), 236-262. doi: 10.1017/etds.2012.124.  Google Scholar

[32]

R. Potrie, Partially Hyperbolicity and Attracting Regions in 3-Dimensional Manifolds, Ph.D Thesis, 2012, arXiv:1207.1822. Google Scholar

[33]

R. Potrie, Partial hyperbolicity and foliations in $\mathbbT^3$, Journal of Modern Dynamics, 2014. Google Scholar

[34]

R. Potrie, A few remarks on partially hyperbolic diffeomorphisms of $\mathbbT^3$ isotopic to Anosov, Journal of Dynamics and Differential Equations, 26 (2014), 805-815. doi: 10.1007/s10884-014-9362-5.  Google Scholar

[35]

J. H. Roberts, Collections filling the plane, Duke Math. J., 2 (1936), 10-19. doi: 10.1215/S0012-7094-36-00202-8.  Google Scholar

[36]

M. Roldan, Hyperbolic sets and entropy at the homological level, preprint, 2014, arXiv:1407.5101. Google Scholar

[37]

K. Shiraiwa, Manifolds which do not admit Anosov diffeomorphisms, Nagoya Math J., 49 (1973), 111-115.  Google Scholar

[38]

J. L. Vieitez, Expansive homeomorphisms and hyperbolic diffeomorphisms on three manifolds, Ergodic Theory and Dynamical Systems, 16 (1996), 591-622. doi: 10.1017/S0143385700008981.  Google Scholar

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