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Isomorphism Theory in ErgodicTheory

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Encyclopedia of Complexity and Systems Science

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Definition of the Subject

Our main goal in this article is to consider when two measure preserving transformations are in some sense different presentations of the same underlying object. To make this precise we say two measure preserving maps \( { (\mathbf{X},T) } \) and \( { (\mathbf{Y},S) } \) are isomorphic if there exists a measurable map \( { \phi \colon X \to Y } \) such that

  1. (1)

    ϕ is measure preserving,

  2. (2)

    ϕ is invertible almost everywhere and

  3. (3)

    \( { \phi(T(x))=S(\phi(x)) } \) for almost every x.

The main goal of the subject is to construct a collection of invariants of a transformation such that a necessary condition for two transformations to be isomorphic is that the invariant be the same for both transformations. Another goal of the subject is to solve the much more difficult problem of constructing invariants such that the invariants being the same for two transformations is a sufficient condition for the transformation to be isomorphic. Finally we apply these...

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Abbreviations

Almost everywhere:

A property is said to hold almost everywhere (a.e.) if the set on which the property does not hold has measure 0.

Bernoulli shift:

A Bernoulli shift is a stochastic process such that all outputs of the process are independent.

Conditional measure:

For any measure space \( { (X,\mathcal{B},\mu) } \) and σ‑algebra \( { \mathcal{C} \subset \mathcal{B} } \) the conditional measure is a \( { \mathcal{C} } \)‐measurable function g such that \( { \mu(C)=\int_C g \, \mathrm{d}\mu } \) for all \( { C \in \mathcal{C} } \).

Coupling of two measure spaces:

A coupling of two measure spaces \( { (X,\mu,\mathcal{B}) } \) and \( { (Y,\nu,\mathcal{C}) } \) is a measure γ on \( { X \times Y } \) such that \( { \gamma(B \times Y) = \mu(B) } \) for all \( { B \in \mathcal{B} } \) and \( { \gamma(X \times C) = \nu(C) } \) for all \( { C \in \mathcal{C} } \).

Ergodic measure preserving transformation:

A measure preserving transformation is ergodic if the only invariant sets \( { (\mu(A \triangle T^{-1}(A)) = 0) } \) have measure 0 or 1.

Ergodic theorem:

The pointwise ergodic theorem says that for any measure preserving transformation \( { (X,\mathcal{B},\mu) } \) and T and any L 1 function f the time average

$$ \lim_{n \to \infty} \frac{1}{n} \, \sum_{i=1}^{n}f(T^{i}(x)) $$

converges a.e. If the transformation is ergodic then the limit is the space average, \( { \int f \, \mathrm{d}\mu } \) a.e.

Geodesic:

A geodesic on a Riemannian manifold is a distance minimizing path between points.

Horocycle:

A horocycle is a circle in the hyperbolic disk which intersects the boundary of the disk in exactly one point.

Invariant measure:

Likewise a measure μ is said to be invariant with respect to \( { (\mathbf{X},T) } \) provided that \( { \mu(T^{-1}(A))=\mu(A) } \) for all measurable \( { A \in \mathcal{B} } \).

Joining of two measure preserving transformations:

A joining of two measure preserving transformations \( { (\mathbf{X},T) } \) and \( { (\mathbf{Y},S) } \) is a coupling of X and Y which is invariant under \( { T \times S } \).

Markov shift:

A Markov shift is a stochastic process such that the conditional distribution of the future outputs (\( { \{x_n\}_{n \mathchar"313E 0} } \)) of the process conditioned on the last output (x 0) is the same as the distribution conditioned on all of the past outputs of the process (\( { \{x_n\}_{n\leq 0} } \)).

Measure preserving transformation:

A measure preserving transformation consists of a probability space \( { (\mathbf{X},T) } \) and a measurable function \( { T \colon X \to X } \) such that \( { \mu(T^{-1}(A))=\mu(A) } \) for all \( { A \in \mathcal{B} } \).

Measure theoretic entropy:

A numerical invariant of measure preserving transformations that measures the growth in complexity of measurable partitions refined under the iteration of the transformation.

Probability space:

A probability space \( { \mathbf{X}=(X,\mu,\mathcal{B}) } \) is a measure space such that \( { \mu(B)=1 } \).

Rational function, rational map:

A rational function \( { f(z)=g(z)/h(z) } \) is the quotient of two polynomials. The degree of f(z) is the maximum of the degrees of g(z) and h(z). The corresponding rational maps \( { T_f \colon z \to f(z) } \) on the Riemann sphere ℂ are a main object of study in complex dynamics.

Stochastic process:

A stochastic process is a sequence of measurable functions \( { \{x_n\}_{n \in \mathbb{Z}} } \) (or outputs) defined on the same measure space, X. We refer to the value of the functions as outputs.

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Authors and Affiliations

  1. Department of Mathematics, University of Washington, Seattle, USA

    Christopher Hoffman

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  1. Christopher Hoffman
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  1. RAMTECH LIMITED, 122 Escalle Lane, Larkspur, CA, 94939, USA

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

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Hoffman, C. (2009). Isomorphism Theory in ErgodicTheory . In: Meyers, R. (eds) Encyclopedia of Complexity and Systems Science. Springer, New York, NY. https://doi.org/10.1007/978-0-387-30440-3_297

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