Given a bi-Lipschitz measure-preserving homeomorphism of a ﬁnite dimensional compact metric measure space, consider the sequence of the
Lipschitz norms of its iterations. We obtain lower bounds on the growth rate of
this sequence assuming that our homeomorphism mixes a Lipschitz function.
In particular, we get a universal lower bound which depends on the dimension of the space but not on the rate of mixing. Furthermore, we get a lower
bound on the growth rate in the case of rapid mixing. The latter turns out to
be sharp: the corresponding example is given by a symbolic dynamical system
associated to the Rudin–Shapiro sequence
Mathematics Subject Classification: 37A05, 37A25, 37C05.