Manuscript submitted to Website: http://AIMsciences.org AIMS’ Journals Volume X, Number 0X, XX 200X pp. X–XX OBSTRUCTION ARGUMENT FOR TRANSITION CHAINS OF TORI INTERSPERSED WITH GAPS Marian Gidea Department of Mathematics, Northeastern Illinois University Chicago, IL 60625, USA Clark Robinson Department of Mathematics, Northwestern University, Evanston, IL 60208 Evanston, IL 60208 (Communicated by the associate editor name) Abstract. 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, 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. 1. Introduction. This paper is a continuation to [23], in which we describe a topological method for proving the existence of orbits that shadow transition chains of primary invariant tori interspersed with Birkhoff zones of instability. In [23] the boundaries of the Birkhoff zones of instabilities were assumed to be smooth. In this paper we consider the general case when all the primary invariant tori in the transition chains and at the boundaries of the Birkhoff zones of instabilities are only Lipschitz. Here by a primary torus in an annulus we mean a 1-dimensional torus that cannot be homotopically deformed into a point in the annulus. Given a normally hyperbolic invariant manifold diffeomorphic to an annulus, by a transition chain of primary invariant tori in the annulus we mean a finite or countable sequence of primary invariant C 1 -smooth (Lipschitz) tori with the following properties: (i) the unstable manifold of each torus intersects transversally (topologically crossing) the stable manifold of the subsequent torus in the sequence, (ii) the motion on each torus is topologically transitive. The definition of topological crossing is given in Section 5. 2000 Mathematics Subject Classification. Primary, 37J40; 37C50; 37C29; Secondary, 37B30. Key words and phrases. Arnold diffusion, correctly aligned windows, shadowing. The first author is supported by NSF grant DMS 0601016. 1