Differential Equations and Control Processes
(Differencialnie Uravnenia i Protsesy Upravlenia)

The Mane Theorem and the Shadowing Theory

Author(s):

Sergei Yurievich Piliygin

Faculty of Mathematics and Mechanics,
St.Petersburg State University
Universitetsky prospekt, 28,
198504, Peterhof, St. Petersburg, Russia
Professor
Doctor in physical and mathematical science, professor

sp@sp1196.spb.edu

Abstract:

In many papers devoted to connections between the theory of shadowing of pseudotrajectories of dynamical systems and the theory of structural stability, the authors apply the Mane theorem on characterization of structural stability in terms of the analytic strong transversality condition. The original proof of this theorem contains a proof of the implication "the analytic strong transversality condition implies the density of periodic points in the nonwandering set", which is based on a nontrivial theory of hyperbolic limit sets. In this short note, we show that the proof of this implication can be essentially simplified in the case of a diffeomorphism having the shadowing property. Bibl. 17 titles

References:

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