Riffle Shuffles and Their Associated Dynamical Systems

It is shown that for every stationary sequence of random riie permutations there is a natural associated dynamical system consisting of random orbits in the space of sequences from a nite alphabet. For many interesting models of card-shuuing, the associated dynamical systems have simple descriptions in terms of random or deterministic measure-preserving maps of the unit interval. It is shown that the rate of mixing for a card-shuuing process is constrained by the ber entropy h of this map: at least (log N)=h repititions of the shuue are needed to randomize a deck of size N , when N is large.