# American Institute of Mathematical Sciences

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 & Continuous Dynamical Systems - A, 2006, 16 (4) : 871-882. doi: 10.3934/dcds.2006.16.871
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