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 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.

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.

Minimum entropy deconvolution and simplicity: A noniterative algorithm

Geophysics ◽  
1985 ◽  
Vol 50 (3) ◽  
pp. 394-413 ◽  
Author(s):  
Carlos A. Cabrelli
Keyword(s):  
Synthetic Data ◽  
Computational Method ◽  
Minimum Entropy ◽  
Autocorrelation Matrix ◽  
Numerical Tests ◽  
Convolution Model ◽  
Single Input ◽  
Definition Of ◽  
Single Matrix ◽  
New Criterion

Minimum entropy deconvolution (MED) is a technique developed by Wiggins (1978) with the purpose of separating the components of a signal, as the convolution model of a smooth wavelet with a series of impulses. The advantage of this method, as compared with traditional methods, is that it obviates strong hypotheses over the components, which require only the simplicity of the output. The degree of simplicity is measured with the Varimax norm for factor analysis. An iterative algorithm for computation of the filter is derived from this norm, having as an outstanding characteristic its stability in presence of noise. Geometrical analysis of the Varimax norm suggests the definition of a new criterion for simplicity: the D norm. In case of multiple inputs, the D norm is obtained through modification of the kurtosis norm. One of the most outstanding characteristics of the new criterion, by comparison with the Varimax norm, is that a noniterative algorithm for computation of the deconvolution filter can be derived from the D norm. This is significant because the standard MED algorithm frequently requires in each iteration the inversion of an autocorrelation matrix whose order is the length of the filter, while the new algorithm derived from the D norm requires the inversion of a single matrix. On the other hand, results of numerical tests, performed jointly with Graciela A. Canziani, show that the new algorithm produces outputs of greater simplicity than those produced by the traditional MED algorithm. These considerations imply that the D criterion yields a new computational method for minimum entropy deconvolution. A section of numerical examples is included, where the results of an extensive simulation study with synthetic data are analyzed. The numerical computations show in all cases a remarkable improvement resulting from use of the D norm. The properties of stability in the presence of noise are preserved as shown in the examples. In the case of a single input, the relation between the D norm and the spiking filter is analyzed (Appendix B).

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.

scholarly journals Number-theoretic positive entropy shifts with small centralizer and large normalizer

2020 ◽  
pp. 1-26
Author(s):  
M. BAAKE ◽  
Á. BUSTOS ◽  
C. HUCK ◽  
M. LEMAŃCZYK ◽  
A. NICKEL
Keyword(s):  
Dynamical System ◽  
Number Fields ◽  
Lattice Points ◽  
Topological Invariants ◽  
Positive Entropy ◽  
Quadratic Number ◽  
Positive Topological Entropy ◽  
Higher Dimensional ◽  
Real Quadratic ◽  
Shift Dynamical System

Abstract Higher-dimensional binary shifts of number-theoretic origin with positive topological entropy are considered. We are particularly interested in analysing their symmetries and extended symmetries. They form groups, known as the topological centralizer and normalizer of the shift dynamical system, which are natural topological invariants. Here, our focus is on shift spaces with trivial centralizers, but large normalizers. In particular, we discuss several systems where the normalizer is an infinite extension of the centralizer, including the visible lattice points and the k-free integers in some real quadratic number fields.

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 Chain recurrent points of a tree map

1999 ◽  
Vol 59 (2) ◽  
pp. 181-186 ◽  
Author(s):  
Tao Li ◽  
Xiangdong Ye
Keyword(s):  
Topological Entropy ◽  
Periodic Points ◽  
Limit Set ◽  
Recurrent Point ◽  
Continuous Map ◽  
Tree Map

We generalise a result of Hosaka and Kato by proving that if the set of periodic points of a continuous map of a tree is closed then each chain recurrent point is a periodic one. We also show that the topological entropy of a tree map is zero if and only if thew-limit set of each chain recurrent point (which is not periodic) contains no periodic points.

Stability of the maximal measure for piecewise monotonic interval maps

1997 ◽  
Vol 17 (6) ◽  
pp. 1419-1436 ◽  
Author(s):  
PETER RAITH
Keyword(s):  
Topological Entropy ◽  
Stability Condition ◽  
Finite Union ◽  
Small Perturbations ◽  
Interval Maps ◽  
Positive Topological Entropy ◽  
Maximal Measure ◽  
Closed Intervals ◽  
Piecewise Monotonic

Let $T:X\to{\Bbb R}$ be a piecewise monotonic map, where $X$ is a finite union of closed intervals. Define $R(T)=\bigcap_{n=0}^{\infty} \overline{T^{-n}X}$, and suppose that $(R(T),T)$ has a unique maximal measure $\mu$. The influence of small perturbations of $T$ on the maximal measure is investigated. If $(R(T),T)$ has positive topological entropy, and if a certain stability condition is satisfied, then every piecewise monotonic map $\tilde{T}$, which is contained in a sufficiently small neighbourhood of $T$, has a unique maximal measure $\tilde{\mu}$, and the map $\tilde{T}\mapsto\tilde{\mu}$ is continuous at $T$.

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