December  2006, 16(4): 871-882. doi: 10.3934/dcds.2006.16.871

Transversality properties and $C^1$-open sets of diffeomorphisms with weak shadowing

1. 

Faculty of Mathematics and Mechanics, St. Petersburg State University, University av., 28, 198504, St. Petersburg, Russian Federation, Russian Federation

2. 

Department of Mathematics, Utsunomiya University, Utsunomiya 321-8505

Received  December 2004 Revised  May 2006 Published  September 2006

Let Int$^1WS(M)$ be the $C^1$-interior of the set of diffeomorphisms of a smooth closed manifold $M$ having the weak shadowing property. The second author has shown that if $\dim M = 2$ and all of the sources and sinks of a diffeomorphism $f \in$ Int$^1WS(M)$ are trivial, then $f$ is structurally stable. In this paper, we show that there exist diffeomorphisms $f \in$ Int$^1WS(M)$, $\dim M = 2$, such that $(i)$ $f$ belongs to the $C^1$-interior of diffeomorphisms for which the $C^0$-transversality condition is not satisfied, $(ii)$ $f$ has a saddle connection. These results are based on the following theorem: if the phase diagram of an $\Omega$-stable diffeomorphism $f$ of a manifold $M$ of arbitrary dimension does not contain chains of length $m > 3$, then $f$ has the weak shadowing property.
Citation: S. Yu. Pilyugin, Kazuhiro Sakai, O. A. Tarakanov. Transversality properties and $C^1$-open sets of diffeomorphisms with weak shadowing. Discrete and Continuous Dynamical Systems, 2006, 16 (4) : 871-882. doi: 10.3934/dcds.2006.16.871
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