# American Institute of Mathematical Sciences

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

## Partially hyperbolic diffeomorphisms with a trapping property

 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.
Citation: Rafael Potrie. Partially hyperbolic diffeomorphisms with a trapping property. Discrete and 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. [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. [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. [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. [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. [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. [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. [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. [9] A. Candel and L. Conlon, Foliations I and II, Graduate studies in Mathematics, 60, American Math. Society, Providence, RI, 2003. [10] P. Carrasco, Compact Dynamical Foliations, Ph.D. Thesis, University of Toronto (Canada), 2011. [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. [12] S. Crovisier and E. Pujals, Essential hyperbolicity and homoclinic bifurcations: A dichotomy phenomenon/mechanism for diffeomorphisms, to appear in Inventiones Math., arXiv:1011.3836. doi: 10.1007/s00222-014-0553-9. [13] R. Daverman, Decompositions of Manifolds, Pure and Applied Mathematics, 124, Academic Press, Inc., Orlando, FL, 1986. [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. [15] J. Franks, Anosov Diffeomorphisms, Proc. Sympos. Pure Math., 14 (1970), 61-93. [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. [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. [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. [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. [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, arXiv:1307.4631. [21] M. Hirsch, C. Pugh and M. Shub, Invariant Manifolds, Springer Lecture Notes in Math., Vol. 583, Springer-Verlag, Berlin-New York, 1977. [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. [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. [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. [25] A. Manning, There are no new anosov on tori, Amer. Jour. of Math., 96 (1974), 422-429. doi: 10.2307/2373551. [26] R. Mañe, Contributions to the stability conjecture, Topology, 17 (1978), 383-396. doi: 10.1016/0040-9383(78)90005-8. [27] R. Mañe, Expansive homeomorphisms and topological dimension, Trans. Amer. Math. Soc., 252 (1979), 313-319. doi: 10.1090/S0002-9947-1979-0534124-9. [28] S. Newhouse, On codimension one Anosov diffeomorphisms, Amer. Jour. of Math., 92 (1970), 761-770. doi: 10.2307/2373372. [29] S. Newhouse, Hyperbolic limit sets, Transactions of the A.M.S, 167 (1972), 125-150. doi: 10.1090/S0002-9947-1972-0295388-6. [30] J. Ombach, Equivalent conditions for hyperbolic coordinates, Topology and its Applications, 23 (1986), 87-90. doi: 10.1016/0166-8641(86)90019-2. [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. [32] R. Potrie, Partially Hyperbolicity and Attracting Regions in 3-Dimensional Manifolds, Ph.D Thesis, 2012, arXiv:1207.1822. [33] R. Potrie, Partial hyperbolicity and foliations in $\mathbbT^3$, Journal of Modern Dynamics, 2014. [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. [35] J. H. Roberts, Collections filling the plane, Duke Math. J., 2 (1936), 10-19. doi: 10.1215/S0012-7094-36-00202-8. [36] M. Roldan, Hyperbolic sets and entropy at the homological level, preprint, 2014, arXiv:1407.5101. [37] K. Shiraiwa, Manifolds which do not admit Anosov diffeomorphisms, Nagoya Math J., 49 (1973), 111-115. [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.

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##### 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. [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. [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. [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. [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. [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. [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. [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. [9] A. Candel and L. Conlon, Foliations I and II, Graduate studies in Mathematics, 60, American Math. Society, Providence, RI, 2003. [10] P. Carrasco, Compact Dynamical Foliations, Ph.D. Thesis, University of Toronto (Canada), 2011. [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. [12] S. Crovisier and E. Pujals, Essential hyperbolicity and homoclinic bifurcations: A dichotomy phenomenon/mechanism for diffeomorphisms, to appear in Inventiones Math., arXiv:1011.3836. doi: 10.1007/s00222-014-0553-9. [13] R. Daverman, Decompositions of Manifolds, Pure and Applied Mathematics, 124, Academic Press, Inc., Orlando, FL, 1986. [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. [15] J. Franks, Anosov Diffeomorphisms, Proc. Sympos. Pure Math., 14 (1970), 61-93. [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. [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. [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. [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. [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, arXiv:1307.4631. [21] M. Hirsch, C. Pugh and M. Shub, Invariant Manifolds, Springer Lecture Notes in Math., Vol. 583, Springer-Verlag, Berlin-New York, 1977. [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. [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. [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. [25] A. Manning, There are no new anosov on tori, Amer. Jour. of Math., 96 (1974), 422-429. doi: 10.2307/2373551. [26] R. Mañe, Contributions to the stability conjecture, Topology, 17 (1978), 383-396. doi: 10.1016/0040-9383(78)90005-8. [27] R. Mañe, Expansive homeomorphisms and topological dimension, Trans. Amer. Math. Soc., 252 (1979), 313-319. doi: 10.1090/S0002-9947-1979-0534124-9. [28] S. Newhouse, On codimension one Anosov diffeomorphisms, Amer. Jour. of Math., 92 (1970), 761-770. doi: 10.2307/2373372. [29] S. Newhouse, Hyperbolic limit sets, Transactions of the A.M.S, 167 (1972), 125-150. doi: 10.1090/S0002-9947-1972-0295388-6. [30] J. Ombach, Equivalent conditions for hyperbolic coordinates, Topology and its Applications, 23 (1986), 87-90. doi: 10.1016/0166-8641(86)90019-2. [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. [32] R. Potrie, Partially Hyperbolicity and Attracting Regions in 3-Dimensional Manifolds, Ph.D Thesis, 2012, arXiv:1207.1822. [33] R. Potrie, Partial hyperbolicity and foliations in $\mathbbT^3$, Journal of Modern Dynamics, 2014. [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. [35] J. H. Roberts, Collections filling the plane, Duke Math. J., 2 (1936), 10-19. doi: 10.1215/S0012-7094-36-00202-8. [36] M. Roldan, Hyperbolic sets and entropy at the homological level, preprint, 2014, arXiv:1407.5101. [37] K. Shiraiwa, Manifolds which do not admit Anosov diffeomorphisms, Nagoya Math J., 49 (1973), 111-115. [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.
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