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Propriétés dynamiques des régions d’instabilité. (Dynamical properties of regions of instability). (French) Zbl 0653.58014
Let f be a C 1-diffeomorphism of the annulus \(A=T\) \(1\times R\) which is homotopic to the identity such that f preserves a measure on A and admits a region of instability in the sense that there exist two continuous maps \(\psi\) \(+,\psi\)-: T \(1\to R\) satisfying the following properties: (i) \(\psi\) \(-<\psi\) \(+\), (ii) the graphs \(G^{\pm}=\{(t,\psi^{\pm}(t)| \quad t\in T\quad 1\}\) of \(\psi^{\pm}\) are f-invariant (iii) the region \(D=\{(t,r)\in A| \quad \psi \quad -(t)\leq r\leq \psi \quad +(t)\}\) contains no other graph of a continuous map of T 1 into R, which is f- invariant. The author studies the behavior of the f-orbits in a neighborhood of \(G^{\pm}\) and shows, among other things, that in each neighbourhood of G \(+\) (resp. G -) there exists a point whose \(\alpha\) and \(\omega\)-limit sets are contained in G - (resp. G \(+)\).
Reviewer: A.Morimoto

MSC:
37C75 Stability theory for smooth dynamical systems
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References:
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