  Metric mean dimension is a geometric invariant of dynamical systems with
infinite topological entropy. We relate this concept with the fractal structure
of the phase space and the H\"older regularity of the map. Afterwards we
improve our general estimates in a family of interval maps by computing the
metric mean dimension in a way similar to the Misiurewicz formula for the
entropy, which in particular shows that our bounds are sharp. Of independent
interest, we develop a dynamical analogue of Minkowski-Bouligand dimension for
maps acting on Ahlfors regular spaces.
