Product Anosov diffeomorphisms and the two-sided limit shadowing property

Carvalho, Bernardo

doi:10.1090/proc/13790

Product Anosov diffeomorphisms and the two-sided limit shadowing property
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Proc. Amer. Math. Soc. 146 (2018), 1151-1164 Request permission

Abstract:

We characterize product Anosov diffeomorphisms in terms of the two-sided limit shadowing property. It is proved that an Anosov diffeomorphism is a product Anosov diffeomorphism if and only if any lift to the universal covering has the unique two-sided limit shadowing property. Then we introduce two maps in a suitable Banach space such that fixed points of these maps are related with shadowing orbits on the universal covering.

References

D. V. Anosov, Geodesic flows on closed Riemannian manifolds of negative curvature, Trudy Mat. Inst. Steklov. 90 (1967), 209 (Russian). MR 0224110
A. Arbieto, B. Carvalho, W. Cordeiro, and D. J. Obata, On bi-Lyapunov stable homoclinic classes, Bull. Braz. Math. Soc. (N.S.) 44 (2013), no. 1, 105–127. MR 3077636, DOI 10.1007/s00574-013-0005-y
T. Barbot, Geometrie transverse des flots d’Anosov, Thesis E. N. Sup. Lyon (1992).
Bernardo Carvalho, Hyperbolicity, transitivity and the two-sided limit shadowing property, Proc. Amer. Math. Soc. 143 (2015), no. 2, 657–666. MR 3283652, DOI 10.1090/S0002-9939-2014-12250-7
B. Carvalho and W. Cordeiro, $n$-expansive homeomorphisms with the shadowing property, J. Differential Equations 261 (2016), no. 6, 3734–3755. MR 3527644, DOI 10.1016/j.jde.2016.06.003
Bernardo Carvalho and Dominik Kwietniak, On homeomorphisms with the two-sided limit shadowing property, J. Math. Anal. Appl. 420 (2014), no. 1, 801–813. MR 3229854, DOI 10.1016/j.jmaa.2014.06.011
Sérgio R. Fenley, Anosov flows in $3$-manifolds, Ann. of Math. (2) 139 (1994), no. 1, 79–115. MR 1259365, DOI 10.2307/2946628
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
Koichi Hiraide, Expansive homeomorphisms with the pseudo-orbit tracing property of $n$-tori, J. Math. Soc. Japan 41 (1989), no. 3, 357–389. MR 999503, DOI 10.2969/jmsj/04130357
Andres Koropecki and Enrique R. Pujals, Some consequences of the shadowing property in low dimensions, Ergodic Theory Dynam. Systems 34 (2014), no. 4, 1273–1309. MR 3227156, DOI 10.1017/etds.2012.195
Jean Leray and Jules Schauder, Topologie et équations fonctionnelles, Ann. Sci. École Norm. Sup. (3) 51 (1934), 45–78 (French). MR 1509338
Sergei Yu. Pilyugin, Shadowing in dynamical systems, Lecture Notes in Mathematics, vol. 1706, Springer-Verlag, Berlin, 1999. MR 1727170
S. Smale, Differentiable dynamical systems, Bull. Amer. Math. Soc. 73 (1967), 747–817. MR 228014, DOI 10.1090/S0002-9904-1967-11798-1