Decidability of Chaos for Some Families of Dynamical Systems

We show that existence of positive Lyapounov exponents and/or SRB measures are undecidable (in the algorithmic sense) properties within some parametrized families of interesting dynamical systems: quadratic family and Hénon maps. Because the existence of positive exponents (or SRB measures) is, in a natural way, a manifestation of “chaos”, these results may be understood as saying that the chaotic character of a dynamical system is undecidable. Our investigation is directly motivated by questions asked by Carleson and Smale in this direction.