Algorithmic Information Theory and Cellular Automata Dynamics

We study the ability of discrete dynamical systems to transform/generate randomness in cellular spaces. Thus, we endow the space of bi-infinite sequences by a metric inspired by information distance (defined in the context of Kolmogorov complexity or algorithmic information theory). We prove structural properties of this space (non-separability, completeness, perfectness and infinite topological dimension), which turn out to be useful to understand the transformation of information performed by dynamical systems evolving on it. Finally, we focus on cellular automata and prove a dichotomy theorem: continuous cellular automata are either equivalent to the identity or to a constant one. This means that they cannot produce any amount of randomness.