  We study systems of globally coupled interval maps, where the identical
individual maps have two expanding, fractional linear, onto branches, and where
the coupling is introduced via a parameter - common to all individual maps -
that depends in an analytic way on the mean field of the system. We show: 1)
For the range of coupling parameters we consider, finite-size coupled systems
always have a unique invariant probability density which is strictly positive
and analytic, and all finite-size systems exhibit exponential decay of
correlations. 2) For the same range of parameters, the self-consistent
Perron-Frobenius operator which captures essential aspects of the corresponding
infinite-size system (arising as the limit of the above when the system size
tends to infinity), undergoes a supercritical pitchfork bifurcation from a
unique stable equilibrium to the coexistence of two stable and one unstable
equilibrium.
