Invariant measures for Cartesian powers of Chacon infinite transformation

TL;DR

This paper classifies all invariant measures for Cartesian powers of the infinite measure-preserving Chacon transformation, revealing a richer structure than in the finite case and deriving properties like trivial centralizer.

Contribution

It characterizes all invariant measures for Cartesian powers of the infinite Chacon transformation, including new types of measures not supported on graphs, and establishes key dynamical properties.

Findings

All ergodic invariant measures are products of diagonal measures.

The class of diagonal measures includes some singular measures not supported on graphs.

The infinite Chacon transformation has trivial centralizer and no nontrivial factors.

Abstract

We describe all boundedly finite measures which are invariant by Cartesian powers of an infinite measure preserving version of Chacon transformation. All such ergodic measures are products of so-called diagonal measures, which are measures generalizing in some way the measures supported on a graph. Unlike what happens in the finite-measure case, this class of diagonal measures is not reduced to measures supported on a graph arising from powers of the transformation: it also contains some weird invariant measures, whose marginals are singular with respect to the measure invariant by the transformation. We derive from these results that the infinite Chacon transformation has trivial centralizer, and has no nontrivial factor. At the end of the paper, we prove a result of independent interest, providing sufficient conditions for an infinite measure preserving dynamical system defined on a Cartesian product to decompose into a direct product of two dynamical systems.