The connection between topological and algebraic entropy

Topology and its Applications

2019

Self Cite

In analogy to the topological entropy for continuous endomorphisms of totally disconnected locally compact groups, we introduce a notion of topological entropy for continuous endomorphisms of locally linearly compact vector spaces. We study the fundamental properties of this entropy and we prove the Addition Theorem, showing that the topological entropy is additive with respect to short exact sequences. By means of Lefschetz Duality, we connect the topological entropy to the algebraic entropy in a Bridge Theorem. 2010 MSC: Primary: 15A03; 15A04; 22B05. Secondary: 20K30; 37A35.

Section: Introductionmentioning

confidence: 55%

“…Indeed, a Uniqueness Theorem for the topological entropy in the category of compact groups and continuous homomorphisms was proved by Stojanov in [32]. The same result requires a shorter list of axioms restricting to compact abelian groups (see [5,Corollary 3.3]).…”

Section: Introductionmentioning

confidence: 99%

Topological entropy for locally linearly compact vector spaces

Castellano

Topology and its Applications

2019

Self Cite

“…This theorem was proved when G is a torsion abelian group (i.e., K is a totally disconnected compact abelian group) by Weiss [111]; later Peters [90] obtained a proof for G countable and φ an automorphism (i.e., K metrizable and ψ a topological automorphism). The theorem in this general form was recently proved by the authors in [37].…”

Section: Bridge Theoremmentioning

confidence: 84%

Entropy on normed semigroups (towards a unifying approach to entropy)

Dikranjan

2019

Dissertationes Math.

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We present a unifying approach to the study of entropies in Mathematics, such as measure entropy, various forms of topological entropy, several notions of algebraic entropy, and two forms set-theoretic entropy. We take into account only discrete dynamical systems, that is, pairs (X, T ), where X is the underlying space (e.g., a probability space, a compact topological space, a group, a set) and T : X → X is a transformation of X (e.g., a measure preserving transformation, a continuous selfmap, a group homomorphism, a selfmap). We see entropies as functions h : X → R + , associating to each flow (X, φ) of a category X either a non negative real number or ∞.First, we introduce the notion of semigroup entropy h S : S → R + , which is a numerical invariant attached to endomorphisms of the category S of normed semigroups. Then, for a functor F : X → S from any specific category X to S, we define the functorial entropy h F :Clearly, this entropy h F inherits many of the properties of h S , depending also on the properties of the functor F . Motivated by this aspect, we study in detail the properties of h S . Such general scheme, using elementary category theory, permits to obtain many relevant known entropies as functorial entropies h F , for appropriately chosen categories X and functors F : X → S. For example, most of the above mentioned measure entropy, topological entropies, algebraic entropies and set-theoretic entropies are functorial entropies. Furthermore, we exploit our scheme to elaborate a common approach to establish the properties shared by those entropies that we find as functorial entropies. In this way we point out their common nature.Finally, we discuss and deeply analyze through the looking glass of our unifying approach the relations between pairs of entropies. To this end we first formalize the notion of Bridge Theorem between two entropies h 1 : X 1 → R + and h 2 : X 2 → R + with respect to a functor ε : X 1 → X 2 , taking inspiration from the theorem relating the topological and the algebraic entropy by means of Pontryagin duality. Then, for pairs of functorial entropies we use the above scheme to introduce the notion and the related scheme of Strong Bridge Theorem. It allows us to put under the same umbrella various relations between pairs of entropies (e.g., the above mentioned connection of the topological and the algebraic entropy and their relation with the set-theoretic entropy).

“…Moreover, the same connection was shown by Peters in [17] between h alg for topological automorphisms of countable abelian groups and h top for topological automorphisms of metrizable compact abelian groups. These results, known as Bridge Theorems, were recently extended to endomorphisms of abelian groups in [4], to continuous endomorphisms of locally compact abelian groups with totally disconnected Pontryagin dual in [6], and to topological automorphisms of locally compact abelian groups in [22] (in the latter two cases on the Potryagin dual one considers an extension of h top to locally compact groups based on a notion of entropy introduced by Hood in [13] as a generalization of Bowen's entropy from [2] -see also [12]).…”

Section: Introductionmentioning

confidence: 99%

Algebraic Entropy in Locally Linearly Compact Vector Spaces

Castellano

Rings, Polynomials, and Modules

2017

We introduce the algebraic entropy for continuous endomorphisms of locally linearly compact vector spaces over a discrete field, as the natural extension of the algebraic entropy for endomorphisms of discrete vector spaces studied in [10]. We show that the main properties of entropy continue to hold in the general context of locally linearly compact vector spaces, in particular we extend the Addition Theorem.