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Mathematics of Complexity and Dynamical Systems pp 205–224Cite as

Entropy in Ergodic Theory

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Article Outline

Glossary

Definition of the Subject

Entropy example: How many questions?

Distribution Entropy

A Gander at Shannon's Noisy Channel Theorem

The Information Function

Entropy of a Process

Entropy of a Transformation

Determinism and Zero‐Entropy

The Pinsker–Field and K‑Automorphisms

Ornstein Theory

Topological Entropy

Three Recent Results

Exodos

Bibliography

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Notes

  1. 1.

    This is the Greek spelling.

  2. 2.

    For instance, see [15,18,24,25].

  3. 3.

    In this paper, unmarked logs will be to base-2. In entropy theory, it does not matter much what base is used, but base-2 is convenient for computing entropy for messages described in bits. When using the natural logarithm, some people refer to the unit of information as a nat. In this paper, I have picked bits rather than nats.

  4. 4.

    This is holds when each probability p is a reciprocal power of two. For general probabilities, the “expected number of questions” interpretation holds in a weaker sense: Throw N darts independently at N copies of the dartboard. Efficiently ask Yes/No questions to determine where all N darts landed. Dividing by N, then sending \({N \to\infty}\), will be the \({p \log\big(\frac{1}{p}\big)}\) sum of Eq. (1).

  5. 5.

    There does not seem to be a standard name for this function. I use \({\boldsymbol{\eta}}\), since an uppercase \({\boldsymbol{\eta}}\) looks like an H, which is the letter that Shannon used to denote what I am calling distribution‐entropy.

  6. 6.

    Curiosity: Just in this paragraph we compute distropy in nats, that is, using natural logarithm. Given a small probability \({p\in[0,1]}\) and setting \({x := 1/p}\), note that \({\boldsymbol{\eta}(p) = \log(x)/x \approx 1/\pi(x)}\), where \({\pi(x)}\) denotes the number of prime numbers less‐equal x. (This approximation is a weak form of the Prime Number Theorem.) Is there any actual connection between the ‘approximate distropy’ function \({\mathcal{H}_\pi(\vec{\mathbf{p}}) := \sum_{p\in \vec{\mathbf{p}}} 1 / \pi (1/p)}\) and Number Theory, other than a coincidence of growth rate?

  7. 7.

    The noise‐process is assumed to be independent of the signal‐process. In contrast, when the perturbation is highly dependent on the signal, then it is sometimes called distortion.

  8. 8.

    I am now at liberty to reveal that our X has always been a Lebesgue space, that is, measure‐isomorphic to an interval of ℝ together with countably many point-atoms (points with positive mass). The equivalence of generating and separating is a technical theorem, due to Rokhlin.

    Assuming μ to be Lebesgue is not much of a limitation. For instance, if μ is a finite measure on any Polish space, then μ extends to a Lebesgue measure on the μ‑completion of the Borel sets. To not mince words: All spaces are Lebesgue spaces unless you are actively looking for trouble.

  9. 9.

    This is sometimes called measure(-theoretic) entropy or (perhaps unfortunately) metric entropy, to distinguish it from topological entropy. Tools known prior to entropy, such as spectral properties, did not distinguish the two Bernoulli‐shifts; see Spectral Theory of Dynamical Systems for the definitions.

  10. 10.

    It is an easier result, undoubtedly known much earlier, that every ergodic T has a countable generating partition – possibly of ∞‑distropy.

  11. 11.

    On the set of ordered K‑set partitions (with K fixed) this convergence is the same as: \({{\textsf{Q}}^{(L)} \to {\textsf{Q}}}\) when \({{\mu\left(Fat\left(\textsf{Q}^{(L)},\textsf{Q}\right)\right)} \to 1}\).

    An alternative approach is the Rokhlin metric, \( \operatorname{Dist}(\textsf{P},\textsf{Q}) := {\mathcal{H}(\textsf{P}|\textsf{Q})} + {\mathcal{H}(\textsf{Q}|\textsf{P})}\), which has the advantage of working for unordered partitions.

  12. 12.

    i. e, \({{{S_n} \to T}}\)  IFF  \({{\forall A \in\mathcal{X}}: {\mu({{S_n}^{-1}(A)} \triangle {{T}^{-1}(A)})} \to 0}\); this is a metric‐topology, since our probability space is countably generated. This can be restated in terms of the unitary operator \({{U_T}}\) on \({\mathbb{L}^{2}(\mu)}\), where \({{U_T(f)} := {f\circ T}}\). Namely, \({{S_n}\to T}\) in the coarse topology  IFF  \({{U_{S_n}}\to{U_T}}\) in the strong operator topology.

  13. 13.

    Fix an \({\varepsilon > 0}\) and an \({N > 1/\varepsilon}\). Points \({x,T(x),\dots, T^N(x)}\) have some two at distance less than \({\frac{1}{N}}\); say, \({\operatorname{Dist}({T^i(x),T^j(x)}) < \varepsilon}\), for some \({0\le i<j\le N}\). Since T is an isometry, \( \varepsilon > \operatorname{Dist}({x,T^k(x)}) > 0 \), where \({k:=j-i}\). So the T k‐orbit of x is ε‐dense.

  14. 14.

    In engineering circles, this is called the Almost‐everywhere equi‐partition theorem.

  15. 15.

    Traditionally, this called the Pinsker algebra where, in this context, “algebra” is understood to mean “σ‑algebra”.

  16. 16.

    Because we only work on a compact space, we can omit “finite”. Some generalizations of topological entropy to non‐compact spaces require that only finite open‐covers be used [37].

  17. 17.

    When X is not compact, the definitions need not coincide; e. g. [37]. And topologically‐equivalent metrics, but which are not uniformly equivalent, may give the same T different entropies (see p. 171 in [23]).

  18. 18.

    Perhaps the ∅-bad‐length, 574, is shorter than the 1-bad‐length because, say, ∅s take less tape‐space than 1s and so – being written more densely – cause ambiguity sooner.

  19. 19.

    A popular computer‐algebra‐system was not, at least under my inexpert tutelage, able to simplify this. However, once top-ent gave the correct answer, the software was able to detect the equality.

  20. 20.

    The ergodic measures are the extreme points of \({{\mathcal{M} (T)}}\); call them \({{\mathcal{M}_{\text{Erg}} (T)}}\). This \({{\mathcal{M} (T)}}\) is the set of barycenters obtained from Borel probability measures on \({{\mathcal{M}_{\text{Erg}} (T)}}\) (see Krein-Milman_theorem, Choquet_theory in [60]). In this instance, what explains the failure to have an ergodic maximal‐entropy measure? Let \({{\mu_{k}}}\) be an invariant ergodic measure on \({{Y_{k}}}\). These measures do converge to the one-point (ergodic) probability measure \({{\mu_{\infty}}}\) on \({{Y_{\infty}}}\). But the map \({\mu \mapsto \mathcal{E}_{\mu}(T)}\) is not continuous at \({{\mu_{\infty}}}\).

  21. 21.

    Measures \({{\alpha_{L}}\to{\mu}}\)  IFF  \({{\int{f}{{\text{d}} {\alpha_{L}}}}\to{\int{f}{{\text{d}} {\mu}}}}\), for each continuous \({f\colon X \to \mathbb{R}}\). This metrizable topology makes \({\mathcal{M}}\) compact. Always, \({{\mathcal{M}(T)}}\) is a non-void compact subset (see Measure Preserving Systems.)

  22. 22.

    For any two sets \({B,{B^{\prime}}\subset X}\), the union \({{\partial B} \cup {\partial{B^{\prime}}}}\) is a superset of the three boundaries \({\partial{(B\cup{B^{\prime}})}, {\partial{(B\cap{B^{\prime}})}}, {\partial{(B\smallsetminus{B^{\prime}})}}}\).

Abbreviations

Measure space:

A measure space \({(X,\mathcal{X},\mu)}\) is a set X, a field (that is, a σ‑algebra) \({\mathcal{X}}\) of subsets of X, and a countably‐additive measure \({\mu\colon \mathcal{X} \to [0,\infty]}\). (We often just write \({(X,\mu)}\), with the field implicit.) For a collection \({\mathcal{C} \subset \mathcal{X}}\), use \({\operatorname{Fld}(\mathcal{C}) \supset \mathcal{C}}\) for the smallest field including \({\mathcal{C}}\). The number \({\mu(B)}\) is the “μ-mass of  B ”.

Measure‐preserving map:

A measure‐preserving map \({\psi\colon (X,\mathcal{X},\mu) \to (Y,\mathcal{Y},\nu)}\) is a map \({\psi\colon X \to Y}\) such that the inverse image of each \({B \in \mathcal{Y}}\) is in \({\mathcal{X}}\), and \({\mu({\psi}^{-1} (B)) = \nu(B)}\). A (measure‐preserving) transformation is a measure‐preserving map \( T\colon (X,\mathcal{X},\mu) \to (X,\mathcal{X},\mu) \). Condense this notation to \({(T\colon X,\mathcal{X},\mu)}\) or \({(T\colon X,\mu)}\).

Probability space:

A probability space is a measure space \({(X,\mu)}\) with \({\mu(X)=1}\); this μ is a probability measure. All our maps/transformations in this article are on probability spaces.

Factor map:

A factor map

$$ \psi\colon (T\colon X,\mathcal{X},\mu) \to (S\colon Y,\mathcal{Y},\nu) $$

is a measure‐preserving map \({\psi\colon X \to Y}\) which intertwines the transformations, \({\psi\circ T = S \circ\psi}\). And ψ is an isomorphism if – after deleting a nullset (a mass-zero set) in each space – this ψ is a bijection and \({{\psi}^{-1}}\) is also a factor map.

Almost everywhere (a.e.):

A measure‐theoretic statement holds almost everywhere, abbreviated a.e., if it holds off of a nullset. (Eugene Gutkin once remarked to me that the problem with Measure Theory is … that you have to say “almost everywhere”, almost everywhere.) For example, \({B \smash{\overset{a.e.}{\supset} A }}\) means that \({\mu(B \smallsetminus A)}\) is zero. The a.e. will usually be implicit.

Probability vector:

A probability vector \({\vec{v} = (v_{1},v_{2},\dots)}\) is a list of non‐negative reals whose sum is 1. We generally assume that probability vectors and partitions (see below) have finitely many components. Write “countable probability vector/partition”, when finitely or denumerably many components are considered.

Partition:

A partition \({\textsf{P}=(A_{1},A_{2},\dots)}\) splits X into pairwise disjoint subsets \({A_{i}\in\mathcal{X}}\) so that the disjoint union \({\bigsqcup_{i} A_{i}}\) is all of X. Each A i is an atom of \({\textsf{P}}\). Use \({|\textsf{P}|}\) or \({^{\#}\textsf{P}}\) for the number of atoms. When \({\textsf{P}}\) partitions a probability space, then it yields a probability vector \({\vec{v}}\), where \({v_{j} := \mu(A_{j})}\). Lastly, use \({\textsf{P} \langle x \rangle}\) to denote the \({\textsf{P}}\)‐atom that owns x.

Fonts:

We use the font \({\mathcal{H},\mathcal{E},\mathcal{I}}\) for distribution‐entropy, entropy and the information function. In contrast, the script font \({\mathcal{ABC}\dots}\) will be used for collections of sets; usually subfields of \({\mathcal{X}}\). Use \({\mathbb{E}(\cdot)}\) for the (conditional) expectation operator.

Notation:

\({\mathbb{Z} = \text{integers}}\), \({\mathbb{Z}_{+} = \text{positive integers}}\), and \( \mathbb{N} = \text{natural numbers} = {0,1,2,\dots}\). (Some well‐meaning folk use ℕ for \({\mathbb{Z}_{+}}\), saying ‘Nothing could be more natural than the positive integers’. And this is why \({0\in\mathbb{N}}\). Use \({\lceil \cdot \rceil}\) and \({\lfloor \cdot \rfloor}\) for the ceiling and floor functions; \({\lfloor \cdot \rfloor}\) is also called the “greatest‐integer function”. For an interval \({J := [a,b) \subset [-\infty,+\infty]}\), let \({[a\dots b)}\) denote the interval of integers \({J \cap\mathbb{Z}}\) (with a similar convention for closed and open intervals). E. g., \({({\text{e}} \dots \pi] = ({\text{e}} \dots \pi) = \{3\}}\).

For subsets A and B of the same space, Ω, use \({A\subset B}\) for inclusion and \({A \varsubsetneqq B}\) for proper inclusion. The difference set \({B \smallsetminus A}\) is \({\{\omega \in B \:|\: \omega \notin A\}}\). Employ A c for the complement \({\Omega\smallsetminus A}\). Since we work in a probability space, if we let \({x := \mu(A)}\), then a convenient convention is to have

$$ x^{c} \quad\text{denote}\quad 1 - x\:, $$

since then \({\mu(A^{c})}\) equals x c.

Use \({A \triangle B}\) for the symmetric difference \( [A \smallsetminus B] \cup [B \smallsetminus A] \). For a collection \({\mathcal{C}=\{{E_{j}}\}_{j}}\) of sets in Ω, let the disjoint union \({\bigsqcup_{j}{E_{j}}}\) or \({\bigsqcup(\mathcal{C})}\) represent the union \({\bigcup_{j}{E_{j}}}\) and also assert that the sets are pairwise disjoint.

Use “\({\forall_{\mathrm{large}} n}\)” to mean: “\({\exists n_0}\) such that \({\forall n > {n_0}}\)”. To refer to left hand side of an Eq. (20), use LhS(20); do analogously for RhS(20), the right hand side.

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  1. University of Florida, Gainesville, USA

    Jonathan L. F. King

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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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King, J.L.F. (2012). Entropy 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_14

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