Given a bi-Lipschitz measure-preserving homeomorphism of a compact metric measure space of finite dimension, consider the sequence formed by the Lipschitz norms of its iterations. I shall discuss the growth of this sequence for some classes of homeomorphisms. For homeomorphisms which mix Lipschitz functions I shall present some universal lower estimations on the growth and lower bounds which depend on the rate of mixing. My talk will based on a joint paper with Leonid Polterovich.