Hyperbolicity and Cycles

descriptionPublicationkeyboard_double_arrow_right Article 01 Jan 1978Publisher:JSTORJournal:Transactions of the American Mathematical Society, volume 245, page 251 (issn: 0002-9947,

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Authors: Franke, J. E.; Selgrade, J. F.;

doi: 10.2307/1998865 , 10.1090/s0002-9947-1978-0511408-0

Hyperbolicity and Cycles

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Abstract

In this paper cycle points are defined without the assumption of Axiom A. The closure of the set of cycle points C \mathcal {C} being quasi-hyperbolic is shown to be equivalent to Axiom A plus no cycles. Also we give a sufficient condition for C \mathcal {C} to equal the chain recurrent set. In proving these theorems, a spectral decomposition for quasi-hyperbolic invariant sets is used.

Keywords

Axiom A, Dynamical systems with hyperbolic behavior, Local and nonlocal bifurcation theory for dynamical systems, Dynamics induced by flows and semiflows, Cycle Points, Quasihyperbolic Invariant Sets, Spectral Decomposition

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