The Pinsker subgroup of an algebraic flow

Dikranjan, Dikran; Bruno, Anna Giordano

doi:10.1016/j.jpaa.2011.06.018

Cited by 25 publications

(77 citation statements)

References 19 publications

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“…On the other hand, there is a relevant difference with respect to the case of a single endomorphism when the acting semigroup S is cyclic and finite. In fact, if A is an abelian group and φ : A → A is an endomorphism such that φ n = φ m for some distinct n, m ∈ N, then h alg (φ) = 0 (see [23,28]). This is no more true in general for the algebraic entropy defined in this paper, as we see in item (a) of the next example.…”

Section: Definitionsmentioning

confidence: 99%

“…The following is a useful technical consequence of the above lemma. Following [23,28], call an action S α A of a cancellative right amenable monoid S on an abelian group A (a) locally nilpotent if for every a ∈ A there exists s ∈ S such that α(s)(a) = 0;…”

Section: Computing Entropy Using Generatorsmentioning

confidence: 99%

“…The interest in this direction increased after [30], where a rather complete description in the case of torsion abelian groups was obtained, including an Addition Theorem and a Uniqueness Theorem. These were generalized to all abelian groups in [23]. Details and results can be found in [2,4,23,29,38,42,43], in [40,41] for the non-abelian case, in [66,67] for the algebraic entropy for modules; see also the surveys [26,28,33,44,45].…”

Section: Introductionmentioning

confidence: 99%

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Algebraic entropy for amenable semigroup actions

Dikranjan¹,

Fornasiero²,

Bruno³

2020

Journal of Algebra

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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.In Example 2.28 we show that the naïve generalization of this lemma for semidirect products fails, while Theorem 2.27 gives the correct generalization.Recall that a semigroup S is called Ore semigroup (or right reversible semigroup as in [63]) if it satisfies the left Ore condition, that is, for every a, b ∈ S there exist f, g ∈ S such that f a = gb.It is a well-known fact (see [63, Proposition 1.23]) that a right amenable semigroup S satisfies the left Ore condition, and that the group generated by S is of the form S −1 S.We use the following consequence of this property.Corollary 2.10. Let S be a right amenable semigroup, and s 1 , . . . , s k ∈ S. Then there exist t, r 1 , . . . , r k ∈ S such that t = r i s i for every i ∈ {1, . . . , k}.We conclude with the following relation between amenable monoids and amenable groups.

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Section: Definitionsmentioning

confidence: 99%

Section: Computing Entropy Using Generatorsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Algebraic entropy for amenable semigroup actions

Dikranjan¹,

Fornasiero²,

Bruno³

2020

Journal of Algebra

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“…So the results proved in Section 2 for the discrete length function L on Mod.R/ hold true for the discrete length function ent L , but on the subcategory lFin L OEX of Mod.ROEX/. In particular, given a locally [3,4].…”

Section: Uniqueness Of the Entropy Functionmentioning

confidence: 99%

Length functions, multiplicities and algebraic entropy

2013

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We consider algebraic entropy defined using a general discrete length function L and will consider the resulting entropy in the setting of ROEX -modules. Then entropy will be viewed as a function h L on modules over the polynomial ring ROEX extending L. In this framework we obtain the main results of this paper, namely that under some mild conditions the induced entropy is additive, thus entropy becomes an operator from the length functions on R-modules to length functions on ROEX -modules. Furthermore, if one requires that the induced length function h L satisfies two very natural conditions, then this process is uniquely determined. When R is Noetherian, we will deduce that, in this setting, entropy coincides with the multiplicity symbol as conjectured by the second named author. As an application we show that if R is also commutative, the L-entropy of the right Bernoulli shift on the infinite direct product of a module of finite positive length has value 1, generalizing a result proved for Abelian groups by A. Giordano Bruno.

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“…Actually, since the definition of the Peters entropy involves only sums of elements and not their multiplication by scalars, one can consider only the structure of Abelian group of V , disregarding that of K -vector space. We refer to [3,4,14] for many interesting results on Peters entropy in the Abelian groups setting.…”

Section: Example 23mentioning

confidence: 99%

A soft introduction to algebraic entropy

Bruno

Salce

2012

Arab. J. Math.

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The goal of this mainly expository paper is to develop the theory of the algebraic entropy in the basic setting of vector spaces V over a field K . Many complications encountered in more general settings do not appear at this first level. We will prove the basic properties of the algebraic entropy of linear transformations φ : V → V of vector spaces and its characterization as the rank of V viewed as module over the polynomial ring K [X ] through the action of φ. The two main theorems on the algebraic entropy, namely, the Addition Theorem and the Uniqueness Theorem, whose proofs require many efforts in more general settings, are easily deduced from the above characterization. The adjoint algebraic entropy of a linear transformation, its connection with the algebraic entropy of the adjoint map of the dual space and the dichotomy of its behavior are also illustrated. Mathematics Subject Classification

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The Pinsker subgroup of an algebraic flow

Cited by 25 publications

References 19 publications

Algebraic entropy for amenable semigroup actions

Algebraic entropy for amenable semigroup actions

Length functions, multiplicities and algebraic entropy

A soft introduction to algebraic entropy

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