Regular maps with the specification property

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July 2013, 33(7): 2991-3009. doi: 10.3934/dcds.2013.33.2991

Regular maps with the specification property

Kazumine Moriyasu ^1, , Kazuhiro Sakai ^2, and Kenichiro Yamamoto ^3,

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

Received March 2012 Revised August 2012 Published January 2013

Abstract
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Let $f$ be a $C^1$-regular map of a closed $C^{\infty}$ manifold $M$ and $\Lambda$ be a locally maximal closed invariant set of $f$. We show that $f|_{\Lambda}$ satisfies the $C^1$-stable specification property if and only if $\Lambda$ is a hyperbolic elementary set. We also prove that there exists a residual subset $\mathcal{R}$ in the space of $C^1$-regular maps endowed with the $C^1$-topology such that for $f \in \mathcal{R}$, $f|_{\Lambda}$ satisfies the specification property if and only if $\Lambda$ is a hyperbolic elementary set.

Keywords: $C^1$-topology., Specification property, regular maps, Axiom A, elementary sets, $C^1$-stable specification property.

Mathematics Subject Classification: Primary: 37A25, 37B35, 37C20, 37D20, 37D3.

Citation: Kazumine Moriyasu, Kazuhiro Sakai, Kenichiro Yamamoto. Regular maps with the specification property. Discrete and Continuous Dynamical Systems, 2013, 33 (7) : 2991-3009. doi: 10.3934/dcds.2013.33.2991

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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