3:00 p.m., Friday (October 29, 2004)
Math Annex 1100
Department of Mathematics, University of Maryland
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).