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Joinings in Ergodic Theory

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Mathematics of Complexity and Dynamical Systems

Article Outline

Glossary

Definition of the Subject

Introduction

Joinings of Two or More Dynamical Systems

Self-Joinings

Some Applications and Future Directions

Bibliography

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Notes

  1. 1.

    An introduction to finite‐rank transformations can be found e. g. in [32]; we also refer the reader to the quite complete survey [12].

Abbreviations

Disjoint measure‐preserving systems:

The two measure‐preserving dynamical systems \({(X,\mathcal{A},\mu,T)}\) and \( (Y,\mathcal{B},\nu,S) \) are said to be disjoint if their only joining is the product measure \({\mu\otimes\nu}\).

Joining:

Let I be a finite or countable set, and for each \({i\in I}\), 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 \({\prod_{i\in I} X_i}\), which has the μ i 's as marginals, and which is invariant under the product transformation \({\bigotimes_{i\in I} T_i}\).

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 \({\left(\prod_{i\in I}X_i,\bigotimes_{i\in I}\mathcal{A}_i\right)}\), and let \({J= \{ j_1, \dots ,j_k \}}\) 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:

$$ \begin{aligned} &\forall A_1\in \mathcal{A}_{j_1}, \dots , A_k\in \mathcal{A}_{j_k}\:,\\ &\kern-1pt\mu(A_1\times\cdots\times A_k) \mathrel{\mathop:}= \lambda\left(A_1\times\cdots\times A_k\times\prod_{i\in I\setminus J}X_i \right)\:. \end{aligned}$$
Markov intertwining:

Let \({(X,\mathcal{A},\mu,T)}\) and \({(Y,\mathcal{B},\nu,S)}\) be two measure‐preserving dynamical systems. We call Markov intertwining of T and S any operator \({P\colon\ L^2(X,\mu)\to L^2(Y,\nu)}\) enjoying the following properties:

  • \({P U_T = U_S P}\), where U T and U S are the unitary operators on \({L^2(X,\mu)}\) and \({L^2(Y,\nu)}\) associated respectively to T and S (i. e. \({U_Tf(x)=f(Tx)}\), and \({U_Sg(y)=g(Sy)}\)).

  • \({P \mathbb{1}_X=\mathbb{1}_{Y}}\),

  • \({f\ge0}\) implies \({P f\ge0}\), and \({g\ge0}\) implies \({P^* g\ge0}\), where \({P^*}\) is the adjoint operator of P.

Minimal self‐joinings:

Let \({k\ge2}\) be an integer. The ergodic measure‐preserving dynamical system T has k‑fold minimal self‐joinings if, for any ergodic joining λ of k copies of T, we can partition the set \({\{1, \dots ,k\}}\) of coordinates into subsets \({J_1, \dots ,J_\ell}\) such that

  1. 1.

    For j 1 and j 2 belonging to the same J i , the marginal of λ on the coordinates j 1 and j 2 is supported on the graph of T n for some integer n (depending on j 1 and j 2);

  2. 2.

    For \({j_1\in J_1, \dots ,j_\ell\in J_\ell}\), the coordinates \({j_1, \dots ,j_\ell}\) are independent.

We say that T has minimal self‐joinings if T has k‑fold minimal self‐joinings for every \({k\ge2}\).

Off‐diagonal self‐joinings:

Let \({(X,\mathcal{A},\mu,T)}\) be a measure‐preserving dynamical system, and S be an invertible measure‐preserving transformation of \({(X,\mathcal{A},\mu)}\) commuting with T. Then the probability measure \( \Delta \) S defined on \({X\times X}\) by

$$ \Delta_S(A\times B) \mathrel{\mathop:}= \mu(A\cap S^{-1}B) $$
(1)

is a 2‑fold self‐joining of T supported on the graph of S. We call it an off‐diagonal self‐joining of T.

Process in a measure‐preserving dynamical systems:

Let \({(X,\mathcal{A},\mu,T)}\) be a measure‐preserving dynamical system, and let \({(E,\mathcal{B}(E))}\) be a measurable space (which may be a finite or countable set, or \({\mathbb{R}^d}\), or \({\mathbb{C}^d}\)…). For any E‑valued random variable ξ defined on the probability space \({(X,\mathcal{A},\mu)}\), we can consider the stochastic process \({(\xi_i)_{i\in\mathbb{Z}}}\) defined by

$$ \xi_i\mathrel{\mathop:}= \xi\circ T^i\:. $$

Since T preserves the probability measure μ, \({(\xi_i)_{i\in\mathbb{Z}}}\) is a stationary process: For any ℓ and n, the distribution of \({(\xi_0, \dots ,\xi_\ell)}\) is the same as the probability distribution of \({(\xi_n, \dots ,\xi_{n+\ell})}\).

Self‐joining:

Let T be a measure‐preserving dynamical system. A self‐joining of T is a joining of a family \({\left(X_i,\mathcal{A}_i,\mu_i,T_i\right)_{i\in I}}\) of systems where each T i is a copy of T. If I is finite and has cardinal k, we speak of a k‑fold self‐joining of T.

Simplicity:

For \({k\ge2}\), we say that the ergodic measure‐preserving dynamical system T is k‑fold simple if, for any ergodic joining λ of k copies of T, we can partition the set \({\{1, \dots ,k\}}\) of coordinates into subsets \({J_1, \dots ,J_\ell}\) such that

  1. 1.

    for j 1 and j 2 belonging to the same J i , the marginal of λ on the coordinates j 1 and j 2 is supported on the graph of some \({S\in C(T)}\) (depending on j 1 and j 2);

  2. 2.

    for \({j_1\in J_1, \dots ,j_\ell\in J_\ell}\), the coordinates \({j_1, \dots ,j_\ell}\) are independent.

We say that T is simple if T is k‑fold simple for every \({k\ge2}\).

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Authors and Affiliations

  1. Laboratoire de Mathématiques Raphaël Salem, CNRS – Université de Rouen, Saint Étienne du Rouvray, France

    Thierry de la Rue

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    Robert A. Meyers Ph. D. (Editor-in-Chief) (Editor-in-Chief)

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de la Rue, T. (2012). Joinings in Ergodic Theory. In: Meyers, R. (eds) Mathematics of Complexity and Dynamical Systems. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-1806-1_49

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