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A generalisation of Mahler measure and its application in algebraic dynamical systems

Manfred Einsiedler (1999)

Acta Arithmetica

We prove a generalisation of the entropy formula for certain algebraic d -actions given in [2] and [4]. This formula expresses the entropy as the logarithm of the Mahler measure of a Laurent polynomial in d variables with integral coefficients. We replace the rational integers by the integers in a number field and examine the entropy of the corresponding dynamical system.

A remark on the topological entropies of covers and partitions

Pierre-Paul Romagnoli (2007)

Studia Mathematica

We study if the combinatorial entropy of a finite cover can be computed using finite partitions finer than the cover. This relates to an unsolved question in [R] for open covers. We explicitly compute the topological entropy of a fixed clopen cover showing that it is smaller than the infimum of the topological entropy of all finer clopen partitions.

About the Algebraic Yuzvinski Formula

Anna Giordano Bruno, Simone Virili (2015)

Topological Algebra and its Applications

The Algebraic Yuzvinski Formula expresses the algebraic entropy of an endomorphism of a finitedimensional rational vector space as the Mahler measure of its characteristic polynomial. In a recent paper, we have proved this formula, independently fromits counterpart – the Yuzvinski Formula – for the topological entropy proved by Yuzvinski in 1968. In this paper we first compare the proof of the Algebraic Yuzvinski Formula with a proof of the Yuzvinski Formula given by Lind and Ward in 1988, underlying...

An analogue of the Variational Principle for group and pseudogroup actions

Andrzej Biś (2013)

Annales de l’institut Fourier

We generalize to the case of finitely generated groups of homeomorphisms the notion of a local measure entropy introduced by Brin and Katok [7] for a single map. We apply the theory of dimensional type characteristics of a dynamical system elaborated by Pesin [25] to obtain a relationship between the topological entropy of a pseudogroup and a group of homeomorphisms of a metric space, defined by Ghys, Langevin and Walczak in [12], and its local measure entropies. We prove an analogue of the Variational...

Borel isomorphism of SPR Markov shifts

Mike Boyle, Jérôme Buzzi, Ricardo Gómez (2014)

Colloquium Mathematicae

We show that strongly positively recurrent Markov shifts (including shifts of finite type) are classified up to Borel conjugacy by their entropy, period and their numbers of periodic points.

Collective geodesic flows

Léo T. Butler, Gabriel P. Paternain (2003)

Annales de l’institut Fourier

We show that most compact semi-simple Lie groups carry many left invariant metrics with positive topological entropy. We also show that many homogeneous spaces admit collective Riemannian metrics arbitrarily close to the bi-invariant metric and whose geodesic flow has positive topological entropy. Other properties of collective geodesic flows are also discussed.

Commutativity and non-commutativity of topological sequence entropy

Francisco Balibrea, Jose Salvador Cánovas Peña, Víctor Jiménez López (1999)

Annales de l'institut Fourier

In this paper we study the commutativity property for topological sequence entropy. We prove that if X is a compact metric space and f , g : X X are continuous maps then h A ( f g ) = h A ( g f ) for every increasing sequence A if X = [ 0 , 1 ] , and construct a counterexample for the general case. In the interim, we also show that the equality h A ( f ) = h A ( f | n 0 f n ( X ) ) is true if X = [ 0 , 1 ] but does not necessarily hold if X is an arbitrary compact metric space.

Computing explicitly topological sequence entropy: the unimodal case

Victor Jiménez López, Jose Salvador Cánovas Peña (2002)

Annales de l’institut Fourier

Let W ( I ) denote the family of continuous maps f from an interval I = [ a , b ] into itself such that (1) f ( a ) = f ( b ) { a , b } ; (2) they consist of two monotone pieces; and (3) they have periodic points of periods exactly all powers of 2 . The main aim of this paper is to compute explicitly the topological sequence entropy h D ( f ) of any map f W ( I ) respect to the sequence D = ( 2 m - 1 ) m = 1 .

Distributional chaos on tree maps: the star case

Jose S. Cánovas (2001)

Commentationes Mathematicae Universitatis Carolinae

Let 𝕏 = { z : z n [ 0 , 1 ] } , n , and let f : 𝕏 𝕏 be a continuous map having the branching point fixed. We prove that f is distributionally chaotic iff the topological entropy of f is positive.

Dynamical systems arising from elliptic curves

P. D'Ambros, G. Everest, R. Miles, T. Ward (2000)

Colloquium Mathematicae

We exhibit a family of dynamical systems arising from rational points on elliptic curves in an attempt to mimic the familiar toral automorphisms. At the non-archimedean primes, a continuous map is constructed on the local elliptic curve whose topological entropy is given by the local canonical height. Also, a precise formula for the periodic points is given. There follows a discussion of how these local results may be glued together to give a map on the adelic curve. We are able to give a map whose...

Dynamics of commuting homeomorphisms of chainable continua

Christopher Mouron (2010)

Colloquium Mathematicae

A chainable continuum, X, and homeomorphisms, p,q: X → X, are constructed with the following properties: (1) p ∘ q = q ∘ p, (2) p, q have simple dynamics, (3) p ∘ q is a positively continuum-wise fully expansive homeomorphism that has positive entropy and is chaotic in the sense of Devaney and in the sense of Li and Yorke.

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