Analys- och sannolikhetsseminarium

Abstrakt finns enbart på engelska: Two probability measure-preserving bijections of a standard probability space are orbit equivalent if they have the same orbits up to conjugacy. However, Dye proved that this relation cannot distinguish between ergodic bijections. Quantitative orbit equivalence proposes to strengthen the definition of orbit equivalence, and aims at bridging the gap between the well-studied but very complicated relation of conjugacy, and the trivial relation of orbit equivalence.

In recent years, odometers have been a central class of systems for explicit constructions of orbit equivalences, using their combinatorial structure. In this talk we introduce a construction of orbit equivalence between odometers and new systems that we call odomutants. This richer class, favourable to counter-examples, provides three flexibility results for quantitative orbit equivalence. Here is an example.

It follows from work of Kerr and Li that if the cocycles of an orbit equivalence are log-integrable, the entropy is preserved. Our construction of odomutants shows that their result is optimal, namely we find odomutants of positive entropy orbit equivalent to an odometer, with almost log-integrable cocycles.