This paper considers transition tori of Arnold which have transverse
heteroclinic
intersections. Using a sequence of correctly aligned windows, we prove the
existence of an orbit which comes near an arbitrary infinite sequence of
these transition tori. By our proof, we aim to clarify the reason that
it is possible to take an infinite or bi-infinite chain of such tori.
This fact is implicit in the treatment of obstructing sets used by Arnold,
but does not seem transparent.
Our treatment of the transition tori is based
on an earlier article of Easton about this topic, but removes some of the
undesirable assumptions which he made.
The explanation is given in terms of
correctly aligned windows introduced by Easton. Recently, these ideas have
been developed using the Conley index by Carbinatto, Kwapisz, and
Mischaikow and several others. We use the statement given by M. Gidea.
Besides seeking to clarify the situation for transition tori, we hope to
make people in Celestial Mechanics aware of the results on topological
horseshoes without the necessity of proving the differentiably transverse
homoclinic intersection.