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Bifurcation of positive solutions for nonlinear nonhomogeneous Robin and Neumann problems with competing nonlinearities
Partially hyperbolic diffeomorphisms with a trapping property
1. | CMAT, Facultad de Ciencias, Universidad de la República, Igua 4225, Montevideo 11400, Uruguay |
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. |
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