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June  2009, 2(2): 393-416. doi: 10.3934/dcdss.2009.2.393

Obstruction argument for transition chains of tori interspersed with gaps

Marian Gidea 1, and  Clark Robinson 2,

1. 

Department of Mathematics, Northeastern Illinois University, Chicago, IL 60625
2. 

Department of Mathematics, Northwestern University, Evanston IL 60208

Received  April 2008 Revised  September 2008 Published  April 2009

We consider a dynamical system whose phase space contains a two-dimensional normally hyperbolic invariant manifold diffeomorphic to an annulus. We assume that the dynamics restricted to the annulus is given by an area preserving monotone twist map. We assume that in the annulus there exist finite sequences of primary invariant Lipschitz tori of dimension $1$, with the property that the unstable manifold of each torus has a topologically crossing intersection with the stable manifold of the next torus in the sequence. We assume that the dynamics along these tori is topologically transitive. We assume that the tori in these sequences, possibly with the exception of the tori at the ends of the sequences, can be $C^0$-approximated from both sides by other primary invariant tori in the annulus. We assume that the region in the annulus between two successive sequences of tori is a Birkhoff zone of instability. We prove the existence of orbits that follow the sequences of invariant tori and cross the Birkhoff zones of instability.
Keywords: Arnold diffusion, correctly aligned windows, shadowing..
Mathematics Subject Classification: Primary, 37J40; 37C50; 37C29; Secondary, 37B3.
Citation: Marian Gidea, Clark Robinson. Obstruction argument for transition chains of tori interspersed with gaps. Discrete and Continuous Dynamical Systems - S, 2009, 2 (2) : 393-416. doi: 10.3934/dcdss.2009.2.393
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