Shearer’s inequality and infimum rule for Shannon entropy and topological entropy (original) (raw)

Packing topological entropy for amenable group actions

Ergodic Theory and Dynamical Systems, 2021

Packing topological entropy is a dynamical analogy of the packing dimension, which can be viewed as a counterpart of Bowen topological entropy. In the present paper we give a systematic study of the packing topological entropy for a continuous G-action dynamical system (X,G)(X,G)(X,G) , where X is a compact metric space and G is a countable infinite discrete amenable group. We first prove a variational principle for amenable packing topological entropy: for any Borel subset Z of X, the packing topological entropy of Z equals the supremum of upper local entropy over all Borel probability measures for which the subset Z has full measure. Then we obtain an entropy inequality concerning amenable packing entropy. Finally, we show that the packing topological entropy of the set of generic points for any invariant Borel probability measure mu\mu mu coincides with the metric entropy if either mu\mu mu is ergodic or the system satisfies a kind of specification property.

Invariant measures of action of amenable groups and their entropy

Measure Algebra and Application, 2024

this article, with the help of the concept of diagonal measure, we define an information function for the amenable group action on a compact metric space and then obtain the entropy of the group action from it. In other words, we show that the integral of the defined information function will be equal to the entropy of the amenable group action.

D S ] 1 9 D ec 2 01 2 The entropy of co-compact open covers

2012

Co-compact entropy is introduced as an invariant of topological conjugation for perfect mappings defined on any Hausdorff space (compactness and metrizability not necessarily required). This is achieved through the consideration of co-compact covers of the space. The advantages of co-compact entropy include: 1) it does not require the space to be compact, and thus generalizes Adler, Konheim and McAndrew’s topological entropy of continuous mappings on compact dynamical systems, and 2) it is an invariant of topological conjugation, compared to Bowen’s entropy that is metric-dependent. Other properties of co-compact entropy are investigated, e.g., the co-compact entropy of a subsystem does not exceed that of the whole system. For the linear system (R, f) defined by f(x) = 2x, the co-compact entropy is zero, while Bowen’s entropy for this system is at least log 2. More general, it is found that co-compact entropy is a lower bound of Bowen’s entropies, and the proof of this result genera...

Entropy in Topological Groups, Part 1

2017

Entropy in Topological Groups Entropy of generalized shifts and combinatorial entropy Let X be a set and λ : X → X a selfmap. (a) For a finite subset D of X the (covariant) combinatorial entropy of λ with respect to D is h c (λ, D) = lim n→∞ |Tn(λ,D)| n ≤ |D|. (b) The number h c (λ) = sup {h c (λ, D) : D ∈ [X ] <ω } is the (covariant) combinatorial entropy of λ. If λ : X → X is finitely many-to-one, the (contravariant) combinatorial entropy h * c (λ) of λ can be defined similarly, by making use of T * n (λ, D) in place of T n (λ, D). Example (Generalized shifts) Let K be a finite group (set) and λ : X → X be a selfmap, X = ∅. Define the generalized shift σ λ : K X → K X by σ λ (g) = g • λ for g : X → K. (a) h top (σ λ) = h c (λ) log |K | (this remains true also for compositions ψ • σ λ or σ λ • ψ, where ψ = (ψ i) ∈ Sym(K) I). (b) if λ : X → X is finitely many-to-one, then the direct sum X K is σ λ-invariant in K X and h alg (σ λ X K) = h * c (λ) log |K |. Entropy in Topological Groups Entropy of generalized shifts and combinatorial entropy Let X be a set and λ : X → X a selfmap. (a) For a finite subset D of X the (covariant) combinatorial entropy of λ with respect to D is h c (λ, D) = lim n→∞ |Tn(λ,D)| n ≤ |D|. (b) The number h c (λ) = sup {h c (λ, D) : D ∈ [X ] <ω } is the (covariant) combinatorial entropy of λ. If λ : X → X is finitely many-to-one, the (contravariant) combinatorial entropy h * c (λ) of λ can be defined similarly, by making use of T * n (λ, D) in place of T n (λ, D). Example (Generalized shifts) Let K be a finite group (set) and λ : X → X be a selfmap, X = ∅. Define the generalized shift σ λ : K X → K X by σ λ (g) = g • λ for g : X → K. (a) h top (σ λ) = h c (λ) log |K | (this remains true also for compositions ψ • σ λ or σ λ • ψ, where ψ = (ψ i) ∈ Sym(K) I). (b) if λ : X → X is finitely many-to-one, then the direct sum X K is σ λ-invariant in K X and h alg (σ λ X K) = h * c (λ) log |K |. Entropy in Topological Groups Entropy of generalized shifts and combinatorial entropy Let X be a set and λ : X → X a selfmap. (a) For a finite subset D of X the (covariant) combinatorial entropy of λ with respect to D is h c (λ, D) = lim n→∞ |Tn(λ,D)| n ≤ |D|. (b) The number h c (λ) = sup {h c (λ, D) : D ∈ [X ] <ω } is the (covariant) combinatorial entropy of λ. If λ : X → X is finitely many-to-one, the (contravariant) combinatorial entropy h * c (λ) of λ can be defined similarly, by making use of T * n (λ, D) in place of T n (λ, D). Example (Generalized shifts) Let K be a finite group (set) and λ : X → X be a selfmap, X = ∅. Define the generalized shift σ λ : K X → K X by σ λ (g) = g • λ for g : X → K. (a) h top (σ λ) = h c (λ) log |K | (this remains true also for compositions ψ • σ λ or σ λ • ψ, where ψ = (ψ i) ∈ Sym(K) I). (b) if λ : X → X is finitely many-to-one, then the direct sum X K is σ λ-invariant in K X and h alg (σ λ X K) = h * c (λ) log |K |. Entropy in Topological Groups Entropy of generalized shifts and combinatorial entropy Let X be a set and λ : X → X a selfmap. (a) For a finite subset D of X the (covariant) combinatorial entropy of λ with respect to D is h c (λ, D) = lim n→∞ |Tn(λ,D)| n

The Entropy of Co-Compact Open Covers

Entropy, 2013

Co-compact entropy is introduced as an invariant of topological conjugation for perfect mappings defined on any Hausdorff space (compactness and metrizability are not necessarily required). This is achieved through the consideration of co-compact covers of the space. The advantages of co-compact entropy include: (1) it does not require the space to be compact and, thus, generalizes Adler, Konheim and McAndrew's topological entropy of continuous mappings on compact dynamical systems; and (2) it is an invariant of topological conjugation, compared to Bowen's entropy, which is metric-dependent. Other properties of co-compact entropy are investigated, e.g., the co-compact entropy of a subsystem does not exceed that of the whole system. For the linear system, (R, f), defined by f (x) = 2x, the co-compact entropy is zero, while Bowen's entropy for this system is at least log 2. More generally, it is found that co-compact entropy is a lower bound of Bowen's entropies, and the proof of this result also generates the Lebesgue Covering Theorem to co-compact open covers of non-compact metric spaces.

Topological Entropy and Algebraic Entropy for group endomorphisms

The notion of entropy appears in many fields and this paper is a survey about entropies in several branches of Mathematics. We are mainly concerned with the topological and the algebraic entropy in the context of continuous endomorphisms of locally compact groups, paying special attention to the case of compact and discrete groups respectively. The basic properties of these entropies, as well as many examples, are recalled. Also new entropy functions are proposed, as well as generalizations of several known definitions and results. Furthermore we give some connections with other topics in Mathematics as Mahler measure and Lehmer Problem from Number Theory, and the growth rate of groups and Milnor Problem from Geometric Group Theory. Most of the results are covered by complete proofs or references to appropriate sources.

Topological entropy, upper Carath

Colloquium Mathematicum, 2020

We study dynamical systems given by the action T : G × X → X of a finitely generated semigroup G with identity 1 on a compact metric space X by continuous selfmaps and with T (1, −) = idX. For any finite generating set G1 of G containing 1, the receptive topological entropy of G1 (in the sense of Ghys et al. (1988) and Hofmann and Stoyanov (1995)) is shown to coincide with the limit of upper capacities of dynamically defined Carathéodory structures on X depending on G1, and a similar result holds true for the classical topological entropy when G is amenable. Moreover, the receptive topological entropy and the topological entropy of G1 are lower bounded by respective generalizations of Katok's δ-measure entropy, for δ ∈ (0, 1). In the case when T (g, −) is a locally expanding selfmap of X for every g ∈ G \ {1}, we show that the receptive topological entropy of G1 dominates the Hausdorff dimension of X modulo a factor log λ determined by the expanding coefficients of the elements of {T (g, −) : g ∈ G1 \ {1}}.

Algebraic entropy for amenable semigroup actions

Journal of Algebra, 2020

We introduce two notions of algebraic entropy for actions of cancellative right amenable semigroups S on discrete abelian groups A by endomorphisms; these extend the classical algebraic entropy for endomorphisms of abelian groups, corresponding to the case S = N. We investigate the fundamental properties of the algebraic entropy and compute it in several examples, paying special attention to the case when S is an amenable group. For actions of cancellative right amenable monoids on torsion abelian groups, we prove the so called Addition Theorem. In the same setting, we see that a Bridge Theorem connects the algebraic entropy with the topological entropy of the dual action by means of the Pontryagin duality, so that we derive an Addition Theorem for the topological entropy of actions of cancellative left amenable monoids on totally disconnected compact abelian groups.

Finiteness of Topological Entropy for Locally Compact Abelian Groups

Glasgow Mathematical Journal, 2020

We study the locally compact abelian groups in the class mathfrakEltinfty{\mathfrak E_{ \lt \infty }}mathfrakEltinfty , that is, having only continuous endomorphisms of finite topological entropy, and in its subclass mathfrakE0\mathfrak E_0mathfrakE0 , that is, having all continuous endomorphisms with vanishing topological entropy. We discuss the reduction of the problem to the case of periodic locally compact abelian groups, and then to locally compact abelian p-groups. We show that locally compact abelian p-groups of finite rank belong to mathfrakEltinfty{\mathfrak E_{ \lt \infty }}mathfrakEltinfty , and that those of them that belong to mathfrakE_0\mathfrak E_0mathfrakE_0 are precisely the ones with discrete maximal divisible subgroup. Furthermore, the topological entropy of endomorphisms of locally compact abelian p-groups of finite rank coincides with the logarithm of their scale. The backbone of the paper is the Addition Theorem for continuous endomorphisms of locally compact abelian groups. Various versions of the Addition Theorem are established in the paper and used in ...