American Institute of Mathematical Sciences

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

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

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.
Citation: Kazumine Moriyasu, Kazuhiro Sakai, Kenichiro Yamamoto. Regular maps with the specification property. Discrete & Continuous Dynamical Systems - A, 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. 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. 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. Google Scholar [5] M. Denker, C. Grillenberger and K. Sigmund, "Ergodic Theory on Compact Spaces,", Lecture Notes in Math., 527 (1976). Google Scholar [6] L. J. Díaz, E. R. Pujals and R. Ures, Partial hyperbolicity and robust transitivity,, Acta Math., 183 (1999), 1. 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. Google Scholar [8] J. Franks, Necessary conditions for stability of diffeomorphisms,, Trans. Amer. Math. Soc., 158 (1971), 301. Google Scholar [9] M. Hirsch, C. Pugh and M. Shub, "Invariant Manifolds,", Lecture Notes in Math., 583 (1977). Google Scholar [10] K. Lee, K. Moriyasu and K. Sakai, $C^1$-stable shadowing diffeomorphisms,, Discrete and Continuous Dynam. Sys., 22 (2008), 683. 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. doi: 10.4134/BKMS.2012.49.2.263. Google Scholar [12] D. A. Lind, Ergodic group automorphisms and specification,, Ergodic Theory (Proc. Conf., 729 (1979), 93. Google Scholar [13] R. Mañé, An ergodic closing lemma,, Annals of Math., 116 (1982), 503. 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. Google Scholar [15] K. Moriyasu, The ergodic closing lemma for $C^1$ regular maps,, Tokyo J. Math., 15 (1992), 171. doi: 10.3836/tjm/1270130259. Google Scholar [16] F. Przytycki, Anosov endomorphisms,, Studia Math., 58 (1976), 249. Google Scholar [17] C. Robinson, "Dynamical Systems: Stability, Symbolic Dynamics, and Chaos (2-nd Ed.),", Studies in Advanced Mathematics, (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. 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. doi: 10.1090/S0002-9939-09-10085-0. Google Scholar [20] M. Shub, Endomorphisms of compact differentiable manifolds,, Amer. J. Math., 91 (1969), 175. Google Scholar [21] L. Wen, The $C^1$ closing lemma for non-singular endomorphisms,, Ergodic Theory Dynam. Systems, 11 (1991), 393. 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. doi: 10.1007/s00574-004-0023-x. Google Scholar

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References:
 [1] N. Aoki, M. Dateyama and M. Komuro, Solenoidal automorphisms with specification,, Monatsh. Math., 93 (1982), 79. 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. 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. Google Scholar [5] M. Denker, C. Grillenberger and K. Sigmund, "Ergodic Theory on Compact Spaces,", Lecture Notes in Math., 527 (1976). Google Scholar [6] L. J. Díaz, E. R. Pujals and R. Ures, Partial hyperbolicity and robust transitivity,, Acta Math., 183 (1999), 1. 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. Google Scholar [8] J. Franks, Necessary conditions for stability of diffeomorphisms,, Trans. Amer. Math. Soc., 158 (1971), 301. Google Scholar [9] M. Hirsch, C. Pugh and M. Shub, "Invariant Manifolds,", Lecture Notes in Math., 583 (1977). Google Scholar [10] K. Lee, K. Moriyasu and K. Sakai, $C^1$-stable shadowing diffeomorphisms,, Discrete and Continuous Dynam. Sys., 22 (2008), 683. 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. doi: 10.4134/BKMS.2012.49.2.263. Google Scholar [12] D. A. Lind, Ergodic group automorphisms and specification,, Ergodic Theory (Proc. Conf., 729 (1979), 93. Google Scholar [13] R. Mañé, An ergodic closing lemma,, Annals of Math., 116 (1982), 503. 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. Google Scholar [15] K. Moriyasu, The ergodic closing lemma for $C^1$ regular maps,, Tokyo J. Math., 15 (1992), 171. doi: 10.3836/tjm/1270130259. Google Scholar [16] F. Przytycki, Anosov endomorphisms,, Studia Math., 58 (1976), 249. Google Scholar [17] C. Robinson, "Dynamical Systems: Stability, Symbolic Dynamics, and Chaos (2-nd Ed.),", Studies in Advanced Mathematics, (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. 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. doi: 10.1090/S0002-9939-09-10085-0. Google Scholar [20] M. Shub, Endomorphisms of compact differentiable manifolds,, Amer. J. Math., 91 (1969), 175. Google Scholar [21] L. Wen, The $C^1$ closing lemma for non-singular endomorphisms,, Ergodic Theory Dynam. Systems, 11 (1991), 393. 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. doi: 10.1007/s00574-004-0023-x. Google Scholar
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