Leírás
W e study mean field coupled map systems of uniformly expanding circle
maps. We first consider N globally coupled doubling maps of the circle
with diffusive coupling. Reconsidering and extending the results of
Fernandez we prove ergodicity breaking for N=3 and N=4 and showcase
some synchronization phenomena for various values of N in case of
strong coupling. We then introduce the continuum limit of the system,
where we generalize the doubling map to a smooth uniformly expanding
circle map T. Now the state of the system is described by a density
function and the evolution of an initial density with respect to the
transfer operator of the coupled dynamics is studied. We show that for
weak enough coupling, a unique, asymptotically stable invariant
density exists in a suitable function space. Furthermore, we show that
this invariant density depends Lipschitz continuously on the coupling
parameter. For sufficiently strong coupling, we prove convergence to
a point mass which can be interpreted as chaotic synchronization. To
conclude, we provide some outlook on the case of discontinuous T.