Arnol'd cat map lattices.

We construct Arnol'd cat map lattice field theories in phase space and configuration space. In phase space we impose that the evolution operator of the linearly coupled maps be an element of the symplectic group, in direct generalization of the case of one map. To this end we exploit the correspondence between the cat map and the Fibonacci sequence. The chaotic properties of these systems also can be understood from the equations of motion in configuration space. These describe inverted harmonic oscillators, where the runaway behavior of the potential competes with the toroidal compactification of the phase space. We highlight the spatiotemporal chaotic properties of these systems using standard benchmarks for probing deterministic chaos of dynamical systems, namely, the complete dense set of unstable periodic orbits, which, for long periods, lead to ergodicity and mixing. The spectrum of the periods exhibits a strong dependence on the strength and the range of the interaction.