Hyperbolicity, transitivity and the two-sided limit shadowing property

Carvalho, Bernardo

doi:10.1090/S0002-9939-2014-12250-7

Hyperbolicity, transitivity and the two-sided limit shadowing property
HTML articles powered by AMS MathViewer

by Bernardo Carvalho PDF

Proc. Amer. Math. Soc. 143 (2015), 657-666 Request permission

Abstract:

We explore the notion of the two-sided limit shadowing property introduced by Pilyugin in 2007. Indeed, we characterize the $C^1$-interior of the set of diffeomorphisms with such a property on closed manifolds as the set of transitive Anosov diffeomorphisms. As a consequence we obtain that all codimension-one Anosov diffeomorphisms have the two-sided limit shadowing property. We also prove that every diffeomorphism $f$ with such a property has neither sinks nor sources and is transitive Anosov (in the Axiom A case). In particular, no Morse-Smale diffeomorphism has the two-sided limit shadowing property. Finally, we prove that $C^1$-generic diffeomorphisms with the two-sided limit shadowing property are transitive Anosov. All these results allow us to reduce the well-known conjecture about the transitivity of Anosov diffeomorphisms to prove that the set of diffeomorphisms with the two-sided limit shadowing property coincides with the set of Anosov diffeomorphisms.

References

Flavio Abdenur and Lorenzo J. Díaz, Pseudo-orbit shadowing in the $C^1$ topology, Discrete Contin. Dyn. Syst. 17 (2007), no. 2, 223–245. MR 2257429, DOI 10.3934/dcds.2007.17.223
Manfred Denker, Christian Grillenberger, and Karl Sigmund, Ergodic theory on compact spaces, Lecture Notes in Mathematics, Vol. 527, Springer-Verlag, Berlin-New York, 1976. MR 0457675
Timo Eirola, Olavi Nevanlinna, and Sergei Yu. Pilyugin, Limit shadowing property, Numer. Funct. Anal. Optim. 18 (1997), no. 1-2, 75–92. MR 1442020, DOI 10.1080/01630569708816748
John Franks, Anosov diffeomorphisms, Global Analysis (Proc. Sympos. Pure Math., Vol. XIV, Berkeley, Calif., 1968) Amer. Math. Soc., Providence, R.I., 1970, pp. 61–93. MR 0271990
John Franks, Necessary conditions for stability of diffeomorphisms, Trans. Amer. Math. Soc. 158 (1971), 301–308. MR 283812, DOI 10.1090/S0002-9947-1971-0283812-3
Shuhei Hayashi, Diffeomorphisms in $\scr F^1(M)$ satisfy Axiom A, Ergodic Theory Dynam. Systems 12 (1992), no. 2, 233–253. MR 1176621, DOI 10.1017/S0143385700006726
Anatole Katok and Boris Hasselblatt, Introduction to the modern theory of dynamical systems, Encyclopedia of Mathematics and its Applications, vol. 54, Cambridge University Press, Cambridge, 1995. With a supplementary chapter by Katok and Leonardo Mendoza. MR 1326374, DOI 10.1017/CBO9780511809187
Keonhee Lee, Kazumine Moriyasu, and Kazuhiro Sakai, $C^1$-stable shadowing diffeomorphisms, Discrete Contin. Dyn. Syst. 22 (2008), no. 3, 683–697. MR 2429859, DOI 10.3934/dcds.2008.22.683
Jorge Lewowicz, Lyapunov functions and topological stability, J. Differential Equations 38 (1980), no. 2, 192–209. MR 597800, DOI 10.1016/0022-0396(80)90004-2
Ricardo Mañé, Expansive homeomorphisms and topological dimension, Trans. Amer. Math. Soc. 252 (1979), 313–319. MR 534124, DOI 10.1090/S0002-9947-1979-0534124-9
S. E. Newhouse, On codimension one Anosov diffeomorphisms, Amer. J. Math. 92 (1970), 761–770. MR 277004, DOI 10.2307/2373372
J. Palis and S. Smale, Structural stability theorems, Global Analysis (Proc. Sympos. Pure Math., Vol. XIV, Berkeley, Calif., 1968) Amer. Math. Soc., Providence, R.I., 1970, pp. 223–231. MR 0267603
Ken Palmer, Shadowing in dynamical systems, Mathematics and its Applications, vol. 501, Kluwer Academic Publishers, Dordrecht, 2000. Theory and applications. MR 1885537, DOI 10.1007/978-1-4757-3210-8
Sergei Yu. Pilyugin, Shadowing in dynamical systems, Lecture Notes in Mathematics, vol. 1706, Springer-Verlag, Berlin, 1999. MR 1727170
Sergei Yu. Pilyugin, Sets of dynamical systems with various limit shadowing properties, J. Dynam. Differential Equations 19 (2007), no. 3, 747–775. MR 2350246, DOI 10.1007/s10884-007-9073-2
Clark Robinson, Stability theorems and hyperbolicity in dynamical systems, Rocky Mountain J. Math. 7 (1977), no. 3, 425–437. MR 494300, DOI 10.1216/RMJ-1977-7-3-425
Kazuhiro Sakai, Naoya Sumi, and Kenichiro Yamamoto, Diffeomorphisms satisfying the specification property, Proc. Amer. Math. Soc. 138 (2010), no. 1, 315–321. MR 2550197, DOI 10.1090/S0002-9939-09-10085-0
Dmitry Todorov, Generalizations of analogs of theorems of Maizel and Pliss and their application in shadowing theory, Discrete Contin. Dyn. Syst. 33 (2013), no. 9, 4187–4205. MR 3038058, DOI 10.3934/dcds.2013.33.4187
Dawei Yang and Shaobo Gan, Expansive homoclinic classes, Nonlinearity 22 (2009), no. 4, 729–733. MR 2486353, DOI 10.1088/0951-7715/22/4/002