Marian Gidea, Clark Robinson
Obstruction argument for transition chains of tori interspersed
with gaps
(447K, pdf)
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.