Joinings in Ergodic Theory

Glossary

Disjoint Measure-Preserving Systems

The two measure-preserving dynamical systems $\left(X,\mathcal{A},\mu, T\right)$ and $\left(Y,\mathcal{B},\nu, S\right)$ are said to be disjoint if their only joining is the product measure μ ⊗ ν.

Joining

Let I be a finite or countable set, and for each i ∈ I, let $\left({X}_i,{\mathcal{A}}_i,{\mu}_i,{T}_i\right)$ be a measure-preserving dynamical system. A joining of these systems is a probability measure on the Cartesian product ∏_i∈IX_i, which has the μ_is as marginals and which is invariant under the product transformation ⊗_i∈IT_i.

Marginal of a Probability Measure on a Product Space

Let λ be a probability measure on the Cartesian product of a finite or countable collection of measurable spaces $\left({\prod}_{i\in I}{X}_i,{\otimes}_{i\in I}{\mathcal{A}}_i\right)$, and let J = j₁, …, j_k be a finite subset of I. The k-fold marginal of λ on ${X}_{j_1},\dots, {X}_{j_k}$ is the probability measure μ defined by:

$$ \forall...