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April  2007, 17(2): 223-245. doi: 10.3934/dcds.2007.17.223

Pseudo-orbit shadowing in the $C^1$ topology

Flavio Abdenur 1, and  Lorenzo J. Díaz 1,

1. 

Dep. Matemática PUC-Rio, Marquês de São Vicente 225 22453-900, Rio de Janeiro, Brazil, Brazil

Received  December 2005 Revised  September 2006 Published  November 2006

We prove that the shadowing property does not hold for diffeomorphisms in an open and dense subset of the set of $C^1$-robustly non-hyperbolic transitive diffeomorphisms (i.e., diffeomorphisms with a $C^1$-neighborhood consisting of non-hyperbolic transitive diffeomorphisms).
Keywords: Heterodimensional cycle, Periodic point, Transitivity., Index, Pseudo-orbit Shadowing, Homoclinic class.
Mathematics Subject Classification: Primary: 37C20, 37C25; Secondary: 37D3.
Citation: Flavio Abdenur, Lorenzo J. Díaz. Pseudo-orbit shadowing in the $C^1$ topology. Discrete & Continuous Dynamical Systems, 2007, 17 (2) : 223-245. doi: 10.3934/dcds.2007.17.223
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