Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Product Anosov diffeomorphisms and the two-sided limit shadowing property


Author: Bernardo Carvalho
Journal: Proc. Amer. Math. Soc. 146 (2018), 1151-1164
MSC (2010): Primary 37D20; Secondary 37C20
DOI: https://doi.org/10.1090/proc/13790
Published electronically: September 13, 2017
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)

  • [1] D. V. Anosov, Geodesic flows on closed Riemannian manifolds of negative curvature, Trudy Mat. Inst. Steklov. 90 (1967), 209 (Russian). MR 0224110
  • [2] 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, https://doi.org/10.1007/s00574-013-0005-y
  • [3] T. Barbot, Geometrie transverse des flots d'Anosov, Thesis E. N. Sup. Lyon (1992).
  • [4] Bernardo Carvalho, Hyperbolicity, transitivity and the two-sided limit shadowing property, Proc. Amer. Math. Soc. 143 (2015), no. 2, 657-666. MR 3283652, https://doi.org/10.1090/S0002-9939-2014-12250-7
  • [5] B. Carvalho and W. Cordeiro, $ n$-expansive homeomorphisms with the shadowing property, J. Differential Equations 261 (2016), no. 6, 3734-3755. MR 3527644, https://doi.org/10.1016/j.jde.2016.06.003
  • [6] 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, https://doi.org/10.1016/j.jmaa.2014.06.011
  • [7] Sérgio R. Fenley, Anosov flows in $ 3$-manifolds, Ann. of Math. (2) 139 (1994), no. 1, 79-115. MR 1259365, https://doi.org/10.2307/2946628
  • [8] 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
  • [9] 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, https://doi.org/10.2969/jmsj/04130357
  • [10] 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, https://doi.org/10.1017/etds.2012.195
  • [11] Jean Leray and Jules Schauder, Topologie et équations fonctionnelles, Ann. Sci. École Norm. Sup. (3) 51 (1934), 45-78 (French). MR 1509338
  • [12] Sergei Yu. Pilyugin, Shadowing in dynamical systems, Lecture Notes in Mathematics, vol. 1706, Springer-Verlag, Berlin, 1999. MR 1727170
  • [13] S. Smale, Differentiable dynamical systems, Bull. Amer. Math. Soc. 73 (1967), 747-817. MR 0228014, https://doi.org/10.1090/S0002-9904-1967-11798-1

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 37D20, 37C20

Retrieve articles in all journals with MSC (2010): 37D20, 37C20


Additional Information

Bernardo Carvalho
Affiliation: Departamento de Matematica, Universidade Federal de Minas Gerais - UFMG, Belo Horizonte MG, Brazil
Email: bmcarvalho06@gmail.com

DOI: https://doi.org/10.1090/proc/13790
Keywords: Anosov, hyperbolicity, transitivity.
Received by editor(s): May 1, 2016
Received by editor(s) in revised form: February 27, 2017, and April 18, 2017
Published electronically: September 13, 2017
Communicated by: Yingfei Yi
Article copyright: © Copyright 2017 American Mathematical Society

American Mathematical Society