(Non-)expansivity in functional envelopes

We study expansivity in functional envelopes of dynamical systems. Our main result is that if the phase space of the original system contains an arc or if it contains a free infinite zero dimensional set, then the expansivity in the functional envelope with Hausdorff metric is impossible. This is in contrast with the fact that, when considering the uniform metric in functional envelopes, the expansivity of a system is equivalent with the expansivity of its functional envelope.