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The splitting lemmas for nonsmooth functionals on Hilbert spaces I
Regular maps with the specification property
1. | Department of Mathematics, Tokushima University, Tokushima 770-8502 |
2. | Department of Mathematics, Utsunomiya University, Utsunomiya 321-8505 |
3. | School of Information Environment, Tokyo Denki University, 2-1200 Buseigakuendai, Inzai-shi, Chiba 270-1382, Japan |
References:
[1] |
N. Aoki, M. Dateyama and M. Komuro, Solenoidal automorphisms with specification, Monatsh. Math., 93 (1982), 79-110.
doi: 10.1007/BF01301397. |
[2] |
N. Aoki, K. Moriyasu and N. Sumi, $C^1$-maps having hyperbolic periodic points, Fund. Math., 169 (2001), 1-49.
doi: 10.4064/fm169-1-1. |
[3] |
P. Berger and A. Rovella, On the inverse limit stability of endomorphisms, preprint, arXiv:1006.4302. |
[4] |
R. Bowen, Periodic points and measures for Axiom A diffeomorphisms, Trans. Amer. Math. Soc., 154 (1971), 377-397. |
[5] |
M. Denker, C. Grillenberger and K. Sigmund, "Ergodic Theory on Compact Spaces," Lecture Notes in Math., 527, Springer-Verlag, Berlin, 1976. |
[6] |
L. J. Díaz, E. R. Pujals and R. Ures, Partial hyperbolicity and robust transitivity, Acta Math., 183 (1999), 1-43.
doi: 10.1007/BF02392945. |
[7] |
A. Eizenberg, Y. Kifer and B. Weiss, Large deviations for $\mathbbZ^d$-actions, Comm. Math. Phys., 164 (1994), 433-454. |
[8] |
J. Franks, Necessary conditions for stability of diffeomorphisms, Trans. Amer. Math. Soc., 158 (1971), 301-308. |
[9] |
M. Hirsch, C. Pugh and M. Shub, "Invariant Manifolds," Lecture Notes in Math., 583, Springer-Verlag, Berlin, 1977. |
[10] |
K. Lee, K. Moriyasu and K. Sakai, $C^1$-stable shadowing diffeomorphisms, Discrete and Continuous Dynam. Sys., 22 (2008), 683-697.
doi: 10.3934/dcds.2008.22.683. |
[11] |
K. Lee and X. Wen, Shadowable chain transitive sets of $C^1$-generic diffeomorphisms, Bull. Korean Math. Soc., 49 (2012), 263-270.
doi: 10.4134/BKMS.2012.49.2.263. |
[12] |
D. A. Lind, Ergodic group automorphisms and specification, Ergodic Theory (Proc. Conf., Math. Forschungsinst., Oberwolfach, 1978), 93-104, Lecture Notes in Math., 729, Springer-Verlag, Berlin, (1979). |
[13] |
R. Mañé, An ergodic closing lemma, Annals of Math., 116 (1982), 503-540.
doi: 10.2307/2007021. |
[14] |
R. Mañé, A proof of the $C^1$ stability conjecture, Inst. Hautes Etudes Sci. Publ. Math., 66 (1988), 161-210. |
[15] |
K. Moriyasu, The ergodic closing lemma for $C^1$ regular maps, Tokyo J. Math., 15 (1992), 171-183.
doi: 10.3836/tjm/1270130259. |
[16] |
F. Przytycki, Anosov endomorphisms, Studia Math., 58 (1976), 249-285. |
[17] |
C. Robinson, "Dynamical Systems: Stability, Symbolic Dynamics, and Chaos (2-nd Ed.)," Studies in Advanced Mathematics, CRC Press, 1999. |
[18] |
A. Rovella and M. Sambarino, The $C^1$ closing lemma for generic $C^1$ endomorphisms, Ann. I. H. Poincaré AN, 27 (2010), 1461-1469.
doi: 10.1016/j.anihpc.2010.09.003. |
[19] |
K. Sakai, N. Sumi and K. Yamamoto, Diffeomorphisms satisfying the specification property, Proc. Amer. Math. Soc., 138 (2010), 315-321.
doi: 10.1090/S0002-9939-09-10085-0. |
[20] |
M. Shub, Endomorphisms of compact differentiable manifolds, Amer. J. Math., 91 (1969), 175-199. |
[21] |
L. Wen, The $C^1$ closing lemma for non-singular endomorphisms, Ergodic Theory Dynam. Systems, 11 (1991), 393-412.
doi: 10.1017/S0143385700006210. |
[22] |
L. Wen, Generic diffeomorphisms away from homoclinic tangencies and heterodimensional cycles, Bull. Braz. Math. Soc. (N.S.), 35 (2004), 419-452.
doi: 10.1007/s00574-004-0023-x. |
show all references
References:
[1] |
N. Aoki, M. Dateyama and M. Komuro, Solenoidal automorphisms with specification, Monatsh. Math., 93 (1982), 79-110.
doi: 10.1007/BF01301397. |
[2] |
N. Aoki, K. Moriyasu and N. Sumi, $C^1$-maps having hyperbolic periodic points, Fund. Math., 169 (2001), 1-49.
doi: 10.4064/fm169-1-1. |
[3] |
P. Berger and A. Rovella, On the inverse limit stability of endomorphisms, preprint, arXiv:1006.4302. |
[4] |
R. Bowen, Periodic points and measures for Axiom A diffeomorphisms, Trans. Amer. Math. Soc., 154 (1971), 377-397. |
[5] |
M. Denker, C. Grillenberger and K. Sigmund, "Ergodic Theory on Compact Spaces," Lecture Notes in Math., 527, Springer-Verlag, Berlin, 1976. |
[6] |
L. J. Díaz, E. R. Pujals and R. Ures, Partial hyperbolicity and robust transitivity, Acta Math., 183 (1999), 1-43.
doi: 10.1007/BF02392945. |
[7] |
A. Eizenberg, Y. Kifer and B. Weiss, Large deviations for $\mathbbZ^d$-actions, Comm. Math. Phys., 164 (1994), 433-454. |
[8] |
J. Franks, Necessary conditions for stability of diffeomorphisms, Trans. Amer. Math. Soc., 158 (1971), 301-308. |
[9] |
M. Hirsch, C. Pugh and M. Shub, "Invariant Manifolds," Lecture Notes in Math., 583, Springer-Verlag, Berlin, 1977. |
[10] |
K. Lee, K. Moriyasu and K. Sakai, $C^1$-stable shadowing diffeomorphisms, Discrete and Continuous Dynam. Sys., 22 (2008), 683-697.
doi: 10.3934/dcds.2008.22.683. |
[11] |
K. Lee and X. Wen, Shadowable chain transitive sets of $C^1$-generic diffeomorphisms, Bull. Korean Math. Soc., 49 (2012), 263-270.
doi: 10.4134/BKMS.2012.49.2.263. |
[12] |
D. A. Lind, Ergodic group automorphisms and specification, Ergodic Theory (Proc. Conf., Math. Forschungsinst., Oberwolfach, 1978), 93-104, Lecture Notes in Math., 729, Springer-Verlag, Berlin, (1979). |
[13] |
R. Mañé, An ergodic closing lemma, Annals of Math., 116 (1982), 503-540.
doi: 10.2307/2007021. |
[14] |
R. Mañé, A proof of the $C^1$ stability conjecture, Inst. Hautes Etudes Sci. Publ. Math., 66 (1988), 161-210. |
[15] |
K. Moriyasu, The ergodic closing lemma for $C^1$ regular maps, Tokyo J. Math., 15 (1992), 171-183.
doi: 10.3836/tjm/1270130259. |
[16] |
F. Przytycki, Anosov endomorphisms, Studia Math., 58 (1976), 249-285. |
[17] |
C. Robinson, "Dynamical Systems: Stability, Symbolic Dynamics, and Chaos (2-nd Ed.)," Studies in Advanced Mathematics, CRC Press, 1999. |
[18] |
A. Rovella and M. Sambarino, The $C^1$ closing lemma for generic $C^1$ endomorphisms, Ann. I. H. Poincaré AN, 27 (2010), 1461-1469.
doi: 10.1016/j.anihpc.2010.09.003. |
[19] |
K. Sakai, N. Sumi and K. Yamamoto, Diffeomorphisms satisfying the specification property, Proc. Amer. Math. Soc., 138 (2010), 315-321.
doi: 10.1090/S0002-9939-09-10085-0. |
[20] |
M. Shub, Endomorphisms of compact differentiable manifolds, Amer. J. Math., 91 (1969), 175-199. |
[21] |
L. Wen, The $C^1$ closing lemma for non-singular endomorphisms, Ergodic Theory Dynam. Systems, 11 (1991), 393-412.
doi: 10.1017/S0143385700006210. |
[22] |
L. Wen, Generic diffeomorphisms away from homoclinic tangencies and heterodimensional cycles, Bull. Braz. Math. Soc. (N.S.), 35 (2004), 419-452.
doi: 10.1007/s00574-004-0023-x. |
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