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A Conley-type decomposition of the strong chain recurrent set

Published online by Cambridge University Press:  07 September 2017

OLGA BERNARDI  and
ANNA FLORIO
OLGA BERNARDI
Affiliation:
Dipartimento di Matematica ‘Tullio Levi-Civita’, Università di Padova, Via Trieste 63, 35121 Padova, Italy email obern@math.unipd.it
ANNA FLORIO
Affiliation:
Laboratoire de Mathématiques d’Avignon, Avignon Université, 84018 Avignon, France
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Abstract

For a continuous flow on a compact metric space, the aim of this paper is to prove a Conley-type decomposition of the strong chain recurrent set. We first discuss in detail the main properties of strong chain recurrent sets. We then introduce the notion of strongly stable set as a closed invariant set which is the intersection of the $\unicode[STIX]{x1D714}$-limits of a specific family of nested and definitively invariant neighborhoods of itself. This notion strengthens that of stable set; moreover, any attractor turns out to be strongly stable. We then show that strongly stable sets play the role of attractors in the decomposition of the strong chain recurrent set; indeed, we prove that the strong chain recurrent set coincides with the intersection of all strongly stable sets and their complementary.

Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 39 , Issue 5 , May 2019 , pp. 1261 - 1274
Copyright
© Cambridge University Press, 2017 

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