Role of unstable periodic orbits in phase transitions of coupled map lattices

The thermodynamic formalism for dynamical systems with many degrees of freedom is extended to deal with time averages and fluctuations of some macroscopic quantity along typical orbits, and applied to coupled map lattices exhibiting phase transitions. Thereby, it turns out that a seed of phase transition is embedded as an anomalous distribution of unstable periodic orbits, which appears as a so-called q-phase transition in the spatiotemporal configuration space. This intimate relation between phase transitions and q-phase transitions leads to one natural way of defining transitions and their order in extended chaotic systems. Furthermore, a basis is obtained on which we can treat locally introduced control parameters as macroscopic “temperature” in some cases involved with phase transitions.