Analytical Approach for Piecewise Linear Coupled Map Lattices

A simple construction is presented, which generalises piecewise linear one-dimensional Markov maps to an arbitrary number of dimensions. The corresponding coupled map lattice, known as a simplicial mapping in the mathematical literature, allows for an analytical investigation. Especially the spin Hamiltonian, which is generated by the symbolic dynamics is accessible. As an example a formal relation between a globally coupled system and an Ising mean field model is established. The phase transition in the limit of infinite system size is analysed and analytical results are compared with numerical simulations.