Entropy and Orbit Equivalence in Measure Preserving Dynamics

We present a link between two fundamental notions in the study of measure preserving transformations of a probability space. The study of such transformations, or groups of such transformations, is central to ergodic theory and as dynamical systems often possess natural invariant measures, it is central to dynamics generally.

The first notion is the Kolmogorov-Sinai entropy, a numerical invariant that, vaguely speaking, measures the exponential growth rate of the number epsilon-distinct orbits up to a set of small measure. This is a natural measure of the complexity of the system.

The second idea comes from a fundamental result by H. Dye that says any two free and ergodic measure preserving transformations of a standard probability space are orbit equivalent. That is to say, by measurably rearranging the order of points on the orbit of one ergodic transformation one can modify it to be conjugate to any other.

The link we wish to discuss between these two is that one can place on such rearrangements of orbits a notion of the complexity of the rearrangement intimately related to the notion of entropy. The proof of Dye's theorem is constructed by making a sequence of local rearrangements of the orbits. Each one separately produces a system still conjugate to the original one but in the limit one obtains the dynamics of the other system. One can calculate the complexity of the terms in this sequence of local rearrangements and consider its asymptotic behavior. The core result we will discuss is that two ergodic transformations have the same entropy iff one can be modified into the other by the rearranging of orbits with zero complexity asymptotically.

Refreshments will be served at 2:45 p.m. in the Faculty Lounge, Math Annex (Room 1115).