scholarly journals Aperiodicity at the boundary of chaos

2017 ◽  
Vol 38 (7) ◽  
pp. 2683-2728
Author(s):  
STEVEN HURDER ◽  
ANA RECHTMAN
Keyword(s):  
Topological Entropy ◽  
Periodic Orbits ◽  
Parameter Family ◽  
Positive Topological Entropy ◽  
Dynamical Properties ◽  
Entropy Zero

We consider the dynamical properties of $C^{\infty }$-variations of the flow on an aperiodic Kuperberg plug $\mathbb{K}$. Our main result is that there exists a smooth one-parameter family of plugs $\mathbb{K}_{\unicode[STIX]{x1D716}}$ for $\unicode[STIX]{x1D716}\in (-a,a)$ and $a<1$, such that: (1) the plug $\mathbb{K}_{0}=\mathbb{K}$ is a generic Kuperberg plug; (2) for $\unicode[STIX]{x1D716}<0$, the flow in the plug $\mathbb{K}_{\unicode[STIX]{x1D716}}$ has two periodic orbits that bound an invariant cylinder, all other orbits of the flow are wandering, and the flow has topological entropy zero; (3) for $\unicode[STIX]{x1D716}>0$, the flow in the plug $\mathbb{K}_{\unicode[STIX]{x1D716}}$ has positive topological entropy, and an abundance of periodic orbits.

SMOOTH TRIANGULAR MAPS OF TYPE 2∞ WITH POSITIVE TOPOLOGICAL ENTROPY

1995 ◽  
Vol 05 (05) ◽  
pp. 1319-1324 ◽  
Author(s):  
FRANCISCO BALIBREA ◽  
FRANCISCO ESQUEMBRE ◽  
ANTONIO LINERO
Keyword(s):  
Topological Entropy ◽  
Periodic Orbits ◽  
Positive Topological Entropy ◽  
Period 2 ◽  
Triangular Maps

We explicitly construct for any k in ℕ a [Formula: see text]-differentiable triangular map in the square I2 with the following properties: (a) it has periodic orbits of period 2n for any n and no other periodic orbits, (b) the topological entropy is positive, and (c) the set of recurrent points contains properly the set of uniformly recurrent points.

scholarly journals Forward triplets and topological entropy on trees

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Lluís Alsedà ◽  
David Juher ◽  
Francesc Mañosas
Keyword(s):  
Topological Entropy ◽  
Real Root ◽  
Positive Entropy ◽  
Minimum Entropy ◽  
Periodic Patterns ◽  
Simple Criterion ◽  
Positive Topological Entropy ◽  
Entropy Zero ◽  
New Criterion ◽  
Tree Map

<p style='text-indent:20px;'>We provide a new and very simple criterion of positive topological entropy for tree maps. We prove that a tree map <inline-formula><tex-math id="M1">\begin{document}$ f $\end{document}</tex-math></inline-formula> has positive entropy if and only if some iterate <inline-formula><tex-math id="M2">\begin{document}$ f^k $\end{document}</tex-math></inline-formula> has a periodic orbit with three aligned points consecutive in time, that is, a triplet <inline-formula><tex-math id="M3">\begin{document}$ (a,b,c) $\end{document}</tex-math></inline-formula> such that <inline-formula><tex-math id="M4">\begin{document}$ f^k(a) = b $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M5">\begin{document}$ f^k(b) = c $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M6">\begin{document}$ b $\end{document}</tex-math></inline-formula> belongs to the interior of the unique interval connecting <inline-formula><tex-math id="M7">\begin{document}$ a $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M8">\begin{document}$ c $\end{document}</tex-math></inline-formula> (a <i>forward triplet</i> of <inline-formula><tex-math id="M9">\begin{document}$ f^k $\end{document}</tex-math></inline-formula>). We also prove a new criterion of entropy zero for simplicial <inline-formula><tex-math id="M10">\begin{document}$ n $\end{document}</tex-math></inline-formula>-periodic patterns <inline-formula><tex-math id="M11">\begin{document}$ P $\end{document}</tex-math></inline-formula> based on the non existence of forward triplets of <inline-formula><tex-math id="M12">\begin{document}$ f^k $\end{document}</tex-math></inline-formula> for any <inline-formula><tex-math id="M13">\begin{document}$ 1\le k&lt;n $\end{document}</tex-math></inline-formula> inside <inline-formula><tex-math id="M14">\begin{document}$ P $\end{document}</tex-math></inline-formula>. Finally, we study the set <inline-formula><tex-math id="M15">\begin{document}$ \mathcal{X}_n $\end{document}</tex-math></inline-formula> of all <inline-formula><tex-math id="M16">\begin{document}$ n $\end{document}</tex-math></inline-formula>-periodic patterns <inline-formula><tex-math id="M17">\begin{document}$ P $\end{document}</tex-math></inline-formula> that have a forward triplet inside <inline-formula><tex-math id="M18">\begin{document}$ P $\end{document}</tex-math></inline-formula>. For any <inline-formula><tex-math id="M19">\begin{document}$ n $\end{document}</tex-math></inline-formula>, we define a pattern that attains the minimum entropy in <inline-formula><tex-math id="M20">\begin{document}$ \mathcal{X}_n $\end{document}</tex-math></inline-formula> and prove that this entropy is the unique real root in <inline-formula><tex-math id="M21">\begin{document}$ (1,\infty) $\end{document}</tex-math></inline-formula> of the polynomial <inline-formula><tex-math id="M22">\begin{document}$ x^n-2x-1 $\end{document}</tex-math></inline-formula>.</p>

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.

scholarly journals Ivanov’s Theorem for Admissible Pairs Applicable to Impulsive Differential Equations and Inclusions on Tori

Mathematics ◽  
2020 ◽  
Vol 8 (9) ◽  
pp. 1602
Author(s):  
Jan Andres ◽  
Jerzy Jezierski
Keyword(s):  
Differential Equations ◽  
Topological Entropy ◽  
Lower Estimate ◽  
Deterministic Chaos ◽  
Impulsive Differential Equations ◽  
Nielsen Numbers ◽  
Positive Topological Entropy ◽  
Admissible Pairs ◽  
Impulsive Dynamics ◽  
Differential Equations And Inclusions

The main aim of this article is two-fold: (i) to generalize into a multivalued setting the classical Ivanov theorem about the lower estimate of a topological entropy in terms of the asymptotic Nielsen numbers, and (ii) to apply the related inequality for admissible pairs to impulsive differential equations and inclusions on tori. In case of a positive topological entropy, the obtained result can be regarded as a nontrivial contribution to deterministic chaos for multivalued impulsive dynamics.

Minimal self-joinings and positive topological entropy

1995 ◽  
Vol 120 (3-4) ◽  
pp. 205-222 ◽  
Author(s):  
F. Blanchard ◽  
E. Glasner ◽  
J. Kwiatkowski
Keyword(s):  
Topological Entropy ◽  
Minimal Self ◽  
Positive Topological Entropy

scholarly journals On the class of square Petrie matrices induced by cyclic permutations

2004 ◽  
Vol 2004 (31) ◽  
pp. 1617-1622
Author(s):  
Bau-Sen Du
Keyword(s):  
Finite Field ◽  
Zeta Function ◽  
Topological Entropy ◽  
Characteristic Polynomial ◽  
Characteristic Polynomials ◽  
Dynamical Properties

Letn≥2be an integer and letP={1,2,…,n,n+1}. LetZpdenote the finite field{0,1,2,…,p−1}, wherep≥2is a prime. Then every mapσonPdetermines a realn×nPetrie matrixAσwhich is known to contain information on the dynamical properties such as topological entropy and the Artin-Mazur zeta function of the linearization ofσ. In this paper, we show that ifσis acyclicpermutation onP, then all such matricesAσare similar to one another overZ2(but not overZpfor any primep≥3) and their characteristic polynomials overZ2are all equal to∑k=0nxk. As a consequence, we obtain that ifσis acyclicpermutation onP, then the coefficients of the characteristic polynomial ofAσare all odd integers and hence nonzero.

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