Partially hyperbolic diffeomorphisms with a trapping property

Previous Article
Computing Mather's $\beta$-function for Birkhoff billiards
DCDS Home
This Issue
Next Article
Bifurcation of positive solutions for nonlinear nonhomogeneous Robin and Neumann problems with competing nonlinearities

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

Abstract
Related Papers

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 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.

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.
[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.

[1]	Keith Burns, Amie Wilkinson. Dynamical coherence and center bunching. Discrete and Continuous Dynamical Systems, 2008, 22 (1&2) : 89-100. doi: 10.3934/dcds.2008.22.89
[2]	Sergey Kryzhevich, Sergey Tikhomirov. Partial hyperbolicity and central shadowing. Discrete and Continuous Dynamical Systems, 2013, 33 (7) : 2901-2909. doi: 10.3934/dcds.2013.33.2901
[3]	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
[4]	Federico Rodriguez Hertz, María Alejandra Rodriguez Hertz, Raúl Ures. Partial hyperbolicity and ergodicity in dimension three. Journal of Modern Dynamics, 2008, 2 (2) : 187-208. doi: 10.3934/jmd.2008.2.187
[5]	Jérôme Buzzi, Todd Fisher. Entropic stability beyond partial hyperbolicity. Journal of Modern Dynamics, 2013, 7 (4) : 527-552. doi: 10.3934/jmd.2013.7.527
[6]	Yakov Pesin. On the work of Dolgopyat on partial and nonuniform hyperbolicity. Journal of Modern Dynamics, 2010, 4 (2) : 227-241. doi: 10.3934/jmd.2010.4.227
[7]	Michael Brin, Dmitri Burago, Sergey Ivanov. Dynamical coherence of partially hyperbolic diffeomorphisms of the 3-torus. Journal of Modern Dynamics, 2009, 3 (1) : 1-11. doi: 10.3934/jmd.2009.3.1
[8]	Andy Hammerlindl. Partial hyperbolicity on 3-dimensional nilmanifolds. Discrete and Continuous Dynamical Systems, 2013, 33 (8) : 3641-3669. doi: 10.3934/dcds.2013.33.3641
[9]	Eleonora Catsigeras, Xueting Tian. Dominated splitting, partial hyperbolicity and positive entropy. Discrete and Continuous Dynamical Systems, 2016, 36 (9) : 4739-4759. doi: 10.3934/dcds.2016006
[10]	Rafael Potrie. Partial hyperbolicity and foliations in $\mathbb{T}^3$. Journal of Modern Dynamics, 2015, 9: 81-121. doi: 10.3934/jmd.2015.9.81
[11]	Lu Yang, Meihua Yang, Peter E. Kloeden. Pullback attractors for non-autonomous quasi-linear parabolic equations with dynamical boundary conditions. Discrete and Continuous Dynamical Systems - B, 2012, 17 (7) : 2635-2651. doi: 10.3934/dcdsb.2012.17.2635
[12]	Radosław Czaja. Pullback attractors via quasi-stability for non-autonomous lattice dynamical systems. Discrete and Continuous Dynamical Systems - B, 2021 doi: 10.3934/dcdsb.2021276
[13]	Yejuan Wang, Chengkui Zhong, Shengfan Zhou. Pullback attractors of nonautonomous dynamical systems. Discrete and Continuous Dynamical Systems, 2006, 16 (3) : 587-614. doi: 10.3934/dcds.2006.16.587
[14]	Bernd Aulbach, Martin Rasmussen, Stefan Siegmund. Approximation of attractors of nonautonomous dynamical systems. Discrete and Continuous Dynamical Systems - B, 2005, 5 (2) : 215-238. doi: 10.3934/dcdsb.2005.5.215
[15]	Alfredo Marzocchi, Sara Zandonella Necca. Attractors for dynamical systems in topological spaces. Discrete and Continuous Dynamical Systems, 2002, 8 (3) : 585-597. doi: 10.3934/dcds.2002.8.585
[16]	Anna-Lena Horlemann-Trautmann, Alessandro Neri. A complete classification of partial MDS (maximally recoverable) codes with one global parity. Advances in Mathematics of Communications, 2020, 14 (1) : 69-88. doi: 10.3934/amc.2020006
[17]	Miguel Ângelo De Sousa Mendes. Quasi-invariant attractors of piecewise isometric systems. Discrete and Continuous Dynamical Systems, 2003, 9 (2) : 323-338. doi: 10.3934/dcds.2003.9.323
[18]	Corrado Falcolini, Laura Tedeschini-Lalli. A numerical renormalization method for quasi–conservative periodic attractors. Journal of Computational Dynamics, 2020, 7 (2) : 461-468. doi: 10.3934/jcd.2020018
[19]	Antonio Algaba, Estanislao Gamero, Cristóbal García. The reversibility problem for quasi-homogeneous dynamical systems. Discrete and Continuous Dynamical Systems, 2013, 33 (8) : 3225-3236. doi: 10.3934/dcds.2013.33.3225
[20]	Aaron W. Brown. Nonexpanding attractors: Conjugacy to algebraic models and classification in 3-manifolds. Journal of Modern Dynamics, 2010, 4 (3) : 517-548. doi: 10.3934/jmd.2010.4.517

2021 Impact Factor: 1.588

Tools

Metrics

Other articles
by authors

on AIMS
Rafael Potrie
on Google Scholar
Rafael Potrie

[Back to Top]