Dynamics of a piecewise affine homeomorphism of the torus
The simplest forms of chaos in dynamical systems are those which are equivalent to finite-state automata. Yet it is not always obvious when this can be done, nor what are the probabilistic consequences.
Cerbelli and Giona introduced an interesting example of a piecewise affine area-preserving homeomorphism of the 2-torus whose dynamics they proved to have several chaotic properties, e.g. it is mixing and has positive Lyapunov exponent. The chaos is non-uniform, however: the invariant manifolds fold on themselves infinitely often in opposite directions and fill out the torus in a singular way, so they thought this was different from standard types of chaos. Nevertheless, I show it is "pseudo-Anosov", which allows one to deduce several more properties and to quantify the singular behaviour. This is achieved by reduction to a finite-state automaton.