Eva Miranda

Renzo Bruera, Robert Cardona, Eva Miranda et al.

We explore the relationship between Turing completeness and topological entropy of dynamical systems. We first prove that a natural class of Turing machines that we call "branching Turing machines" (which includes most of the known examples of universal Turing machines) has positive topological entropy. Motivated by the recent construction of Turing complete Euler flows, we deduce that any Turing complete dynamics with a continuous encoding that simulates a universal branching machine is chaotic. On the other hand, we show that, unexpectedly, universal Turing machines with zero topological entropy (and even zero speed) can be constructed, unveiling the independence of chaos and universality at the symbolic level.

Eva Miranda, Isaac Ramos

We show that two-dimensional billiard systems are Turing complete, in the sense that the halting of any Turing machine with a given input is equivalent to a certain bounded trajectory in this system entering a specified open set. Billiards serve as idealized models of particle motion with elastic reflections and arise naturally as limits of smooth Hamiltonian systems under steep confining potentials. Our results establish the existence of undecidable trajectories in physically natural billiard-type models, including billiard-type models arising in hard-sphere gases and in collision-chain limits of celestial mechanics.