# American Institute of Mathematical Sciences

March  2003, 9(2): 287-308. doi: 10.3934/dcds.2003.9.287

## Orbital and weak shadowing properties

 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, Japan

Received  August 2001 Revised  June 2002 Published  December 2002

We study weak and orbital shadowing properties of dynamical systems related to the following approach: we look for exact trajectories lying in small neighborhoods of approximate ones (or containing approximate ones in their small neighborhoods) or for exact trajectories such that the Hausdorff distances between their closures and closures of approximate trajectories are small.
These properties are characterized for linear diffeomorphisms. We also study some $C^1$-open sets of diffeomorphisms defined in terms of these properties. It is shown that the $C^1$-interior of the set of diffeomorphisms having the orbital shadowing property coincides with the set of structurally stable diffeomorphisms.
Citation: S. Yu. Pilyugin, A. A. Rodionova, Kazuhiro Sakai. Orbital and weak shadowing properties. Discrete & Continuous Dynamical Systems, 2003, 9 (2) : 287-308. doi: 10.3934/dcds.2003.9.287
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