Convolution semigroups for automorphism dynamics

TL;DR

This paper introduces a new convolution operation for invariant types and measures in first-order theories, connecting automorphism group actions, Ellis semigroups, and definable groups through a generalized product.

Contribution

It develops a novel convolution operation for invariant types and measures, generalizing existing products and establishing new correspondence theorems in model theory.

Findings

Established homeomorphisms between Ellis semigroups and collections of types and measures

Defined a new convolution operation encoding standard definable convolution over groups

Proved correspondence theorems linking idempotent measures to automorphism subgroups

Abstract

Initially motivated by Hrushovski's paper on definability patterns, we obtain homeomorphisms between Ellis semigroups related to natural actions of the automorphism groups of first order structures and certain collections of types and Keisler measures. Thus, we can transfer the semigroup operation from these Ellis semigroups to the corresponding collections of types and Keisler measures. By generalizing this transferred product, we obtain a new convolution operation for invariant types and measures in arbitrary first-order theories. We develop its general theory and prove several correspondence theorems between idempotent measures and closed subgroups of the automorphism group of a sufficiently large (so-called monster) model with respect to the relatively definable topology. Via the affine sort construction, we demonstrate that this new notion of convolution encodes the standard definable convolution operation over definable groups.