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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

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 & 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.  Google Scholar

[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.  Google Scholar

[3]

P. Berger and A. Rovella, On the inverse limit stability of endomorphisms,, preprint, ().   Google Scholar

[4]

R. Bowen, Periodic points and measures for Axiom A diffeomorphisms, Trans. Amer. Math. Soc., 154 (1971), 377-397.  Google Scholar

[5]

M. Denker, C. Grillenberger and K. Sigmund, "Ergodic Theory on Compact Spaces," Lecture Notes in Math., 527, Springer-Verlag, Berlin, 1976.  Google Scholar

[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.  Google Scholar

[7]

A. Eizenberg, Y. Kifer and B. Weiss, Large deviations for $\mathbbZ^d$-actions, Comm. Math. Phys., 164 (1994), 433-454.  Google Scholar

[8]

J. Franks, Necessary conditions for stability of diffeomorphisms, Trans. Amer. Math. Soc., 158 (1971), 301-308.  Google Scholar

[9]

M. Hirsch, C. Pugh and M. Shub, "Invariant Manifolds," Lecture Notes in Math., 583, Springer-Verlag, Berlin, 1977.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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).  Google Scholar

[13]

R. Mañé, An ergodic closing lemma, Annals of Math., 116 (1982), 503-540. doi: 10.2307/2007021.  Google Scholar

[14]

R. Mañé, A proof of the $C^1$ stability conjecture, Inst. Hautes Etudes Sci. Publ. Math., 66 (1988), 161-210.  Google Scholar

[15]

K. Moriyasu, The ergodic closing lemma for $C^1$ regular maps, Tokyo J. Math., 15 (1992), 171-183. doi: 10.3836/tjm/1270130259.  Google Scholar

[16]

F. Przytycki, Anosov endomorphisms, Studia Math., 58 (1976), 249-285.  Google Scholar

[17]

C. Robinson, "Dynamical Systems: Stability, Symbolic Dynamics, and Chaos (2-nd Ed.)," Studies in Advanced Mathematics, CRC Press, 1999.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[20]

M. Shub, Endomorphisms of compact differentiable manifolds, Amer. J. Math., 91 (1969), 175-199.  Google Scholar

[21]

L. Wen, The $C^1$ closing lemma for non-singular endomorphisms, Ergodic Theory Dynam. Systems, 11 (1991), 393-412. doi: 10.1017/S0143385700006210.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[3]

P. Berger and A. Rovella, On the inverse limit stability of endomorphisms,, preprint, ().   Google Scholar

[4]

R. Bowen, Periodic points and measures for Axiom A diffeomorphisms, Trans. Amer. Math. Soc., 154 (1971), 377-397.  Google Scholar

[5]

M. Denker, C. Grillenberger and K. Sigmund, "Ergodic Theory on Compact Spaces," Lecture Notes in Math., 527, Springer-Verlag, Berlin, 1976.  Google Scholar

[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.  Google Scholar

[7]

A. Eizenberg, Y. Kifer and B. Weiss, Large deviations for $\mathbbZ^d$-actions, Comm. Math. Phys., 164 (1994), 433-454.  Google Scholar

[8]

J. Franks, Necessary conditions for stability of diffeomorphisms, Trans. Amer. Math. Soc., 158 (1971), 301-308.  Google Scholar

[9]

M. Hirsch, C. Pugh and M. Shub, "Invariant Manifolds," Lecture Notes in Math., 583, Springer-Verlag, Berlin, 1977.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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).  Google Scholar

[13]

R. Mañé, An ergodic closing lemma, Annals of Math., 116 (1982), 503-540. doi: 10.2307/2007021.  Google Scholar

[14]

R. Mañé, A proof of the $C^1$ stability conjecture, Inst. Hautes Etudes Sci. Publ. Math., 66 (1988), 161-210.  Google Scholar

[15]

K. Moriyasu, The ergodic closing lemma for $C^1$ regular maps, Tokyo J. Math., 15 (1992), 171-183. doi: 10.3836/tjm/1270130259.  Google Scholar

[16]

F. Przytycki, Anosov endomorphisms, Studia Math., 58 (1976), 249-285.  Google Scholar

[17]

C. Robinson, "Dynamical Systems: Stability, Symbolic Dynamics, and Chaos (2-nd Ed.)," Studies in Advanced Mathematics, CRC Press, 1999.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[20]

M. Shub, Endomorphisms of compact differentiable manifolds, Amer. J. Math., 91 (1969), 175-199.  Google Scholar

[21]

L. Wen, The $C^1$ closing lemma for non-singular endomorphisms, Ergodic Theory Dynam. Systems, 11 (1991), 393-412. doi: 10.1017/S0143385700006210.  Google Scholar

[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.  Google Scholar

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