Topological entropy

Adler, Roy L.; Konheim, Alan G.; McAndrew, M. H.

doi:10.1090/s0002-9947-1965-0175106-9

Cited by 1,141 publications

(998 citation statements)

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“…For a chaotic system, channel capacity is equivalent to the topological entropy because it de"nes the rate at which information is generated by the system [117]. To give a concrete example, consider symbol sequences consisting of a string of n symbols generated by the dynamics.…”

Section: Encoding Digital Messages Using Chaos Controlmentioning

confidence: 99%

The control of chaos: theory and applications

et al. 2000

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Section: Encoding Digital Messages Using Chaos Controlmentioning

confidence: 99%

The control of chaos: theory and applications

et al. 2000

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“…We consider first the case m = 2. Let v (1) = (v (1) n ) n be defined by (2) ), that concludes the proof for the case m = 2. It is possible to extend in an easy way this argument for an arbitrary m > 0.…”

Section: Proof the Canonical Isomorphism Of Vector Spacesmentioning

confidence: 67%

“…The attempt to give a precise answer to the above questions, not in the setting of vector spaces, but in the less elementary setting of Abelian groups, originated the theory of the algebraic entropy (see [1,6,14,20]). Nowadays, this theory is extended to R-modules over arbitrary rings R (see [2,15,16,18,21]), and also to topological groups (see [9]).…”

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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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“…Various forms of entropy exist in the literature. For instance, Adler, Konheim, and McAndrew introduced the notion of topological entropy in [1] for continuous maps of compact topological spaces. Measure-theoretic entropy was introduced by Kolmogorov in [28] and later improved by Sinaȋ in [47] for measure-preserving morphisms of probability spaces, and in [4] Bellon and Viallet introduced a notion of algebraic entropy for dominant rational self-maps of projective space.…”

Section: Definitionmentioning

confidence: 99%

Entropy and flatness in local algebraic dynamics

Majidi-Zolbanin¹,

Miasnikov²,

Szpiro³

2013

Publicacions Matemàtiques

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For a local endomorphism of a noetherian local ring we introduce a notion of entropy, along with two other asymptotic invariants. We use this notion of entropy to extend numerical conditions in Kunz' regularity criterion to every contracting endomorphism of a noetherian local ring, and to give a characteristic-free interpretation of the definition of Hilbert-Kunz multiplicity. We also show that every finite endomorphism of a complete noetherian local ring of equal characteristic can be lifted to a finite endomorphism of a complete regular local ring. The local ring of an algebraic or analytic variety at a point fixed by a finite self-morphism inherits a local endomorphism whose entropy is well-defined. This situation arises at the vertex of the affine cone over a projective variety with a polarized self-morphism, where we compare entropy with degree.

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Topological entropy

Cited by 1,141 publications

References 1 publication

The control of chaos: theory and applications

The control of chaos: theory and applications

A soft introduction to algebraic entropy

Entropy and flatness in local algebraic dynamics

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