  We describe a topological mechanism for the existence of diffusing orbits in
a dynamical system satisfying the following assumptions: (i) the phase space
contains a normally hyperbolic invariant manifold diffeomorphic to a
two-dimensional annulus, (ii) the restriction of the dynamics to the annulus is
an area preserving monotone twist map, (iii) the annulus contains sequences of
invariant one-dimensional tori that form transition chains, i.e., the unstable
manifold of each torus has a topologically transverse intersection with the
stable manifold of the next torus in the sequence, (iv) the transition chains
of tori are interspersed with gaps created by resonances, (v) within each gap
there is prescribed a finite collection of Aubry-Mather sets. Under these
assumptions, there exist trajectories that follow the transition chains, cross
over the gaps, and follow the Aubry-Mather sets within each gap, in any
specified order. This mechanism is related to the Arnold diffusion problem in
Hamiltonian systems. In particular, we prove the existence of diffusing
trajectories in the large gap problem of Hamiltonian systems. The argument is
topological and constructive.
