Section: New Results

Computation and Dynamical Systems

In [12] , we analyzed the power of dynamical system that are robust to infinitesimal perturbations. While previous works on this question were limited to very specific kinds of systems such as piecewise constant derivative systems, we obtained results for a quite general class of systems: the main hypothesis being smoothness (which is already a prerequisite in systems that perform analog computation). We show that if a system is robust, then the language it recognizes is computable, and the converse: all computable languages can be recognized by a robust smooth system. Those results are true for discrete-time as well as continuous-time dynamical systems on bounded or unbounded domains.

We investigated in [23] , [15] , [33] the isomorphism (conjugacy) problem for dynamical systems. While the decidability in the one-dimensional case is a long-standing open problem, we characterize its exact complexity [23] in higher dimensions. Our result suggest that the isomorphism problem is easier than the factoring and embedding problem (decide if one dynamical system is a subsystem of another). A traditional approach to prove two dynamical systems are not isomorphic is to prove that they have different dynamical invariants. We characterised in terms of complexity and computability classes different well known dynamic invariants (periodic points, Turing degrees) in [23] , [33] .

While Turing machines are usually used for computing, it is an interesting model of dynamical systems, which looks very much like two-dimensional piecewise-affine maps. We investigated dynmacial invariants (entropy and Lyapunov exponents) for Turing machines, and proved quite surprisingly that they are computable. Essentially this means that Turing machines that do interesting computations must do it so slowly that this cannot be seen in their dynamics. This work will be presented in STACS 2014