Alex Kruckman, University of California, Berkeley
“Properly Ergodic Structures”
One natural notion of ”random L-structure” is a probability measure on the space of L-structures with domain ω which is invariant and ergodic for the natural action of S∞ on this space. We call such measures ”ergodicstructures.” Ergodic structures arise as limits of sequences of finite structures which are convergent in the appropriate sense, generalizing the graph limits (or ”graphons”) of Lovsz and Szegedy. In this talk, I will address the properly ergodic case, in which no isomorphism class of countable structures is given measure 1. In joint work with Ackerman, Freer, and Patel, we give a characterization of those theories (in any countable fragment of Lω1,ω) which admit properly ergodic models. The main tools are a Morley-Scott analysis of an ergodic structure, the Aldous-Hoover representation theorem, and an ”AFP construction” - a method of producing ergodic structures via inverse limits of discrete probability measures on finite structures.
M3-3103