Non-trivial collective behavior in coupled map lattices: A transfer operator perspective

We discuss the link between the presence of non-trivial collective behavior in coupled map lattices, and the spectral properties of their transfer (or Perron-Frobenius) operator. In particular, it is argued that non-trivial collective behavior corresponds to a Perron-Frobenius operator possessing a cyclical spectral decomposition known as asymptotic periodicity. We also discuss to what extent changes in the spectral properties of the Perron-Frobenius operator are related to the phase transitions observed between two types of non-trivial collective behavior.