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The physical Church thesis and the sensitivity to initial conditions

The physical Church thesis is a thesis about nature that expresses that all that can be computed by a physical system — a machine — is computable in the sense of computability theory. At a first look, this thesis seems contradictory with the existence, in nature, of chaotic dynamical systems, that is systems whose evolution cannot be “computed” because of their sensitivity to initial conditions. The goal of this note is to show that there exist dynamical systems that are both computable and chaotic, and thus that the existence in nature of chaotic dynamical system is not, per se, a refutation of the physical Church thesis. Thus, chaos seems to be compatible with computability, in the same way as it is compatible with determinism.