Undecidability and chaos in word-coded symbolic dynamics

The existence of deterministic chaos in dynamical systems is related with Gödelian undecidability of a symbolic dynamical system defined in terms of Markov maps associated with the principal congruence subgroups of the modular group (corresponding to flows on manifold with overall negative curvature). The main ingredients are the group-orbit word-coding of the symbolic dynamics, and the Dehnʹs word problem for the mapping class group of the manifold.