scholarly journals A Geometric Approach to Voiculescu-Brown Entropy

2004 ◽  
Vol 47 (4) ◽  
pp. 553-565
Author(s):  
David Kerr
Keyword(s):  
Topological Entropy ◽  
Basic Problem ◽  
Geometric Approach ◽  
Topological Dynamics ◽  
Positive Entropy ◽  
Algebraic Dynamics

AbstractA basic problem in dynamics is to identify systems with positive entropy, i.e., systems which are “chaotic.” While there is a vast collection of results addressing this issue in topological dynamics, the phenomenon of positive entropy remains by and large a mystery within the broader noncommutative domain of C*-algebraic dynamics. To shed some light on the noncommutative situation we propose a geometric perspective inspired by work of Glasner and Weiss on topological entropy. This is a written version of the author’s talk at theWinter 2002Meeting of the CanadianMathematical Society in Ottawa, Ontario.

scholarly journals A topological lens for a measure-preserving system

2010 ◽  
Vol 31 (1) ◽  
pp. 49-75 ◽  
Author(s):  
E. GLASNER ◽  
M. LEMAŃCZYK ◽  
B. WEISS
Keyword(s):  
Topological Entropy ◽  
Periodic Points ◽  
Positive Entropy ◽  
Topological Properties ◽  
Topologically Transitive ◽  
Topological System ◽  
Zero Entropy ◽  
Weakly Mixing ◽  
Dynamical Properties

AbstractWe introduce a functor which associates to every measure-preserving system (X,ℬ,μ,T) a topological system $(C_2(\mu ),\tilde {T})$ defined on the space of twofold couplings of μ, called the topological lens of T. We show that often the topological lens ‘magnifies’ the basic measure dynamical properties of T in terms of the corresponding topological properties of $\tilde {T}$. Some of our main results are as follows: (i) T is weakly mixing if and only if $\tilde {T}$ is topologically transitive (if and only if it is topologically weakly mixing); (ii) T has zero entropy if and only if $\tilde {T}$ has zero topological entropy, and T has positive entropy if and only if $\tilde {T}$ has infinite topological entropy; (iii) for T a K-system, the topological lens is a P-system (i.e. it is topologically transitive and the set of periodic points is dense; such systems are also called chaotic in the sense of Devaney).

scholarly journals ℤd-actions with prescribed topological and ergodic properties

2011 ◽  
Vol 32 (1) ◽  
pp. 191-209 ◽  
Author(s):  
YURI LIMA
Keyword(s):  
Topological Entropy ◽  
Topological Dynamics ◽  
Ergodic Transformations ◽  
Positive Topological Entropy ◽  
Ergodic Properties ◽  
Uniquely Ergodic

AbstractWe extend constructions of Hahn and Katznelson [On the entropy of uniquely ergodic transformations. Trans. Amer. Math. Soc.126 (1967), 335–360] and Pavlov [Some counterexamples in topological dynamics. Ergod. Th. & Dynam. Sys.28 (2008), 1291–1322] to ℤd-actions on symbolic dynamical spaces with prescribed topological and ergodic properties. More specifically, we describe a method to build ℤd-actions which are (totally) minimal, (totally) strictly ergodic and have positive topological entropy.

A geometric approach to semi-dispersing billiards (Survey)

1998 ◽  
Vol 18 (2) ◽  
pp. 303-319 ◽  
Author(s):  
D. BURAGO ◽  
S. FERLEGER ◽  
A. KONONENKO
Keyword(s):  
Topological Entropy ◽  
Riemannian Geometry ◽  
Geometric Approach ◽  
Open Problems ◽  
Periodic Trajectories ◽  
Uniform Estimates ◽  
Dispersing Billiards

We summarize the results of several recent papers, together with a few new results, which rely on a connection between semi-dispersing billiards and non-regular Riemannian geometry. We use this connection to solve several open problems about the existence of uniform estimates on the number of collisions, topological entropy and periodic trajectories of such billiards.

TOPOLOGICAL ENTROPY FOR MULTIDIMENSIONAL PERTURBATIONS OF ONE-DIMENSIONAL MAPS

2001 ◽  
Vol 11 (05) ◽  
pp. 1443-1446 ◽  
Author(s):  
MICHAŁ MISIUREWICZ ◽  
PIOTR ZGLICZYŃSKI
Keyword(s):  
Topological Entropy ◽  
Positive Entropy ◽  
One Dimensional ◽  
Interval Map ◽  
One Dimensional Maps

We prove that if an interval map of positive entropy is perturbed to a compact multidimensional map then the topological entropy cannot drop down considerably if the perturbation is small.

Co-induction in dynamical systems

2011 ◽  
Vol 32 (3) ◽  
pp. 919-940 ◽  
Author(s):  
ANTHONY H. DOOLEY ◽  
GUOHUA ZHANG
Keyword(s):  
Dynamical Systems ◽  
Topological Entropy ◽  
Amenable Group ◽  
Positive Entropy ◽  
Completely Positive ◽  
Infinite Subgroup

AbstractIf a countable amenable group G contains an infinite subgroup Γ, one may define, from a measurable action of Γ, the so-called co-induced measurable action of G. These actions were defined and studied by Dooley, Golodets, Rudolph and Sinelsh’chikov. In this paper, starting from a topological action of Γ, we define the co-induced topological action of G. We establish a number of properties of this construction, notably, that the G-action has the topological entropy of the Γ-action and has uniformly positive entropy (completely positive entropy, respectively) if and only if the Γ-action has uniformly positive entropy (completely positive entropy, respectively). We also study the Pinsker algebra of the co-induced action.

Can one always lower topological entropy?

1991 ◽  
Vol 11 (3) ◽  
pp. 535-546 ◽  
Author(s):  
M. Shub ◽  
B. Weiss
Keyword(s):  
Topological Entropy ◽  
The Other ◽  
Positive Entropy ◽  
Other Hand ◽  
Topological System

AbstractWe consider the problem of when does a positive entropy topological system have a continuous factor with strictly smaller entropy. In many cases it is shown that such small entropy factors exist. On the other hand, classes of examples are given where differentiable factors must preserve some of the original entropy.

scholarly journals Topological Entropy of Cournot-Puu Duopoly

2010 ◽  
Vol 2010 ◽  
pp. 1-12 ◽  
Author(s):  
Jose S. Cánovas ◽  
David López Medina
Keyword(s):  
Topological Entropy ◽  
Positive Entropy ◽  
Dynamical Complexity ◽  
Duopoly Model ◽  
Parameter Values

The aim of this paper is to analyze a classical duopoly model introduced by Tönu Puu in 1991. For that, we compute the topological entropy of the model and characterize those parameter values with positive entropy. Although topological entropy is a measure of the dynamical complexity of the model, we will show that such complexity could not be observed.

scholarly journals Positive-entropy integrable systems and the Toda lattice, II

2010 ◽  
Vol 149 (3) ◽  
pp. 491-538 ◽  
Author(s):  
LEO T. BUTLER
Keyword(s):  
Topological Entropy ◽  
Integrable Systems ◽  
Cotangent Bundle ◽  
Toda Lattice ◽  
Entropy Function ◽  
Topological Conjugacy ◽  
Lax Representation ◽  
Positive Entropy ◽  
Toral Automorphisms

AbstractThis paper constructs completely integrable convex Hamiltonians on the cotangent bundle of certain k bundles over l. A central role is played by the Lax representation of a Bogoyavlenskij–Toda lattice. The classification of these systems, up to iso-energetic topological conjugacy, is related to the classification of abelian groups of Anosov toral automorphisms by their topological entropy function.

scholarly journals The minimum tree for a given zero-entropy period

2005 ◽  
Vol 2005 (19) ◽  
pp. 3025-3033 ◽  
Author(s):  
Esther Barrabés ◽  
David Juher
Keyword(s):  
Periodic Orbit ◽  
Topological Entropy ◽  
Positive Entropy ◽  
Minimum Number ◽  
Zero Entropy

We answer the following question: given anyn∈ℕ, which is the minimum number of endpointsenof a tree admitting a zero-entropy mapfwith a periodic orbit of periodn? We prove thaten=s1s2…sk−∑i=2ksisi+1…sk, wheren=s1s2…skis the decomposition ofninto a product of primes such thatsi≤si+1for1≤i<k. As a corollary, we get a criterion to decide whether a mapfdefined on a tree witheendpoints has positive entropy: iffhas a periodic orbit of periodmwithem>e, then the topological entropy offis positive.

ON TOPOLOGICAL DYNAMICS OF SEQUENCES OF CONTINUOUS MAPS

1995 ◽  
Vol 05 (05) ◽  
pp. 1437-1438 ◽  
Author(s):  
SERGIĬ KOLYADA ◽  
LUBOMÍR SNOHA
Keyword(s):  
Dynamical System ◽  
Topological Space ◽  
Topological Entropy ◽  
Discrete Dynamical System ◽  
Topological Dynamics ◽  
Limit Sets ◽  
Compact Topological Space ◽  
Continuous Maps ◽  
Uniformly Convergent

We define and study ω-limit sets and topological entropy for a nonautonomous discrete dynamical system given by a sequence [Formula: see text] of continuous selfmaps of a compact topological space. A special attention is paid to the case when the space is metric and the sequence [Formula: see text] either forms an equicontinuous family of maps or is uniformly convergent. We also show that for any continuous maps f and g from a compact topological space into itself the topological entropies h(f ◦ g) and h(g ◦ f) are equal.

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