Generalized Specification Property and Distributional Chaos

2003 ◽  
Vol 13 (07) ◽  
pp. 1683-1694 ◽  
Author(s):  
F. Balibrea ◽  
B. Schweizer ◽  
A. Sklar ◽  
J. Smítal
Keyword(s):  
Compact Interval ◽  
Limit Set ◽  
Distributional Chaos ◽  
Specification Property ◽  
Continuous Map ◽  
Basic Set ◽  
Made In

Let f be a continuous map from a compact interval into itself. Continuing the work begun by Schweizer and Smítal [1994], we prove that the restriction of f to any basic set (i.e. any nonsolenoidal, infinite, maximal ω-limit set) satisfies a generalization of the specification property. We apply this generalization to establish several conjectures made in the abovementioned paper, e.g. the fact that distributional chaos is stable.

scholarly journals ω-limit sets for maps of the interval

1986 ◽  
Vol 6 (3) ◽  
pp. 335-344 ◽  
Author(s):  
Louis Block ◽  
Ethan M. Coven
Keyword(s):  
Periodic Points ◽  
Compact Interval ◽  
Minimal Set ◽  
Limit Sets ◽  
Piecewise Monotone ◽  
Continuous Map ◽  
Limit Points

AbstractLet f denote a continuous map of a compact interval to itself, P(f) the set of periodic points of f and Λ(f) the set of ω-limit points of f. Sarkovskǐi has shown that Λ(f) is closed, and hence ⊆Λ(f), and Nitecki has shown that if f is piecewise monotone, then Λ(f)=. We prove that if x∈Λ(f)−, then the set of ω-limit points of x is an infinite minimal set. This result provides the inspiration for the construction of a map f for which Λ(f)≠.

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.

Extreme Chaos and Transitivity

2003 ◽  
Vol 13 (07) ◽  
pp. 1695-1700 ◽  
Author(s):  
Marta Babilonová-Štefánková
Keyword(s):  
Topological Entropy ◽  
Distributional Chaos ◽  
Positive Topological Entropy ◽  
Continuous Map ◽  
Almost Everywhere ◽  
Provided Examples

In the eighties, Misiurewicz, Bruckner and Hu provided examples of functions chaotic in the sense of Li and Yorke almost everywhere. In this paper we show that similar results are true for distributional chaos, introduced in [Schweizer & Smítal, 1994]. In fact, we show that any bitransitive continuous map of the interval is conjugate to a map uniformly distributionally chaotic almost everywhere. Using a result of A. M. Blokh we get as a consequence that for any map f with positive topological entropy there is a k such that fk is semiconjugate to a continuous map uniformly distributionally chaotic almost everywhere, and consequently, chaotic in the sense of Li and Yorke almost everywhere.

Asymptotic average shadowing property, almost specification property and distributional chaos

2016 ◽  
Vol 30 (03) ◽  
pp. 1650001 ◽  
Author(s):  
Lidong Wang ◽  
Xiang Wang ◽  
Fengchun Lei ◽  
Heng Liu
Keyword(s):  
Dynamical System ◽  
Orbit Closure ◽  
Shadowing Property ◽  
Distributional Chaos ◽  
Specification Property ◽  
Asymptotic Average Shadowing

It is proved that a nontrivial compact dynamical system with asymptotic average shadowing property (AASP) displays uniformly distributional chaos or distributional chaos in a sequence. Moreover, distributional chaos in a system with AASP can be uniform and dense in the measure center, that is, there is an uncountable uniformly distributionally scrambled set consisting of such points that the orbit closure of every point contains the measure center. As a corollary, the similar results hold for the system with almost specification property.

scholarly journals Work Passion e Identidade: reflexões de uma meta-síntese

2021 ◽  
Vol 20 (3) ◽  
pp. 131-151
Author(s):  
Samantha Frohlich ◽  
Adriana Roseli Wunsch Takahashi
Keyword(s):  
Web Of Science ◽  
Future Studies ◽  
Basic Set ◽  
The Subject ◽  
Study Case ◽  
Synthesis Methodology ◽  
Time And Energy ◽  
Subject Making ◽  
Made In ◽  
Work Passion

O presente estudo tem como objetivo analisar como a Work Passion afeta na Identidade dos indivíduos em suas atividades, aos quais foram analisados estudos de caso a partir da metodologia de meta-síntese proposta por Hoon (2013). Considerando esses dois conceitos, ressalta-se que para Vallarand et al., (2003) a paixão pelo trabalho existe quando as pessoas julgam importantes ou investem tempo e energia na atividade. Nesse sentido, as atividades de valor passam a ser internalizadas na identidade da pessoa.  Para alcançar os resultados foram feitas buscas em quatro bases de dados: Web of Science, Scielo, Spell e Scopus, em que os termos utilizados para busca foram “Work Passion*” AND “Identity” AND “Study Case”. A síntese realizada neste estudo contribuiu para o avanço da literatura em Work Passion e Identidade ao propor um conjunto básico de características presentes nos artigos analisados aos quais chama-se atenção para o tema e suas variáveis, avançando principalmente no estudo da literatura do tema tornando-se útil para futuros estudos.  ABSTRACTThe present study aims to analyze how Work Passion affects the Identity of individuals in their activities, to which case studies were analyzed from the meta-synthesis methodology proposed by Hoon (2013). Considering these two concepts, it is noteworthy that for Vallarand et al. (2003) the passion for work exists when people consider it important or invest time and energy in the activity. In this sense, the valuable activities become internalized in the person's identity.  To reach the results, searches were made in four databases: Web of Science, Scielo, Spell and Scopus, in which the terms used for the search were "Work Passion*" AND "Identity" AND "Study Case". The synthesis performed in this study contributed to the advancement of the literature on Work Passion and Identity by proposing a basic set of characteristics present in the analyzed articles to which attention is drawn to the theme and its variables, advancing mainly the study of the literature on the subject, making it useful for future studies.

scholarly journals The chain recurrent set, attractors, and explosions

1985 ◽  
Vol 5 (3) ◽  
pp. 321-327 ◽  
Author(s):  
Louis Block ◽  
John E. Franke
Keyword(s):  
Compact Space ◽  
Metric Space ◽  
Periodic Points ◽  
Limit Set ◽  
Compact Metric Space ◽  
Continuous Map ◽  
Special Case ◽  
Equivalent Conditions ◽  
Chain Recurrent Set ◽  
Recurrent Set

AbstractCharles Conley has shown that for a flow on a compact metric space, a point x is chain recurrent if and only if any attractor which contains the & ω-limit set of x also contains x. In this paper we show that the same statement holds for a continuous map of a compact metric space to itself, and additional equivalent conditions can be given. A stronger result is obtained if the space is locally connected.It follows, as a special case, that if a map of the circle to itself has no periodic points then every point is chain recurrent. Also, for any homeomorphism of the circle to itself, the chain recurrent set is either the set of periodic points or the entire circle. Finally, we use the equivalent conditions mentioned above to show that for any continuous map f of a compact space to itself, if the non-wandering set equals the chain recurrent set then f does not permit Ω-explosions. The converse holds on manifolds.

Perturbation Theorems for Relative Spectral Problems

1972 ◽  
Vol 24 (1) ◽  
pp. 72-81 ◽  
Author(s):  
Edward Hughes
Keyword(s):  
Hilbert Space ◽  
Boundary Conditions ◽  
Boundary Value Problems ◽  
Eigenvalue Problems ◽  
Broad Class ◽  
Boundary Value ◽  
Compact Interval ◽  
Eigenfunction Expansions ◽  
Made In ◽  
Number Of Authors

Eigenvalue problems of the form Af = λBf, where λ is a complex parameter and A and B are operators on a Hilbert Space, have been considered by a number of authors (e.g., [1; 3; 5; 7; 10]). In this paper, we shall be concerned with the existence and nature of eigenfunction expansions associated with such problems, with no assumptions of self-adjointness. The form of the theorems to be given here is: if the system (A, B) is spectral and complete (definitions below), and F and G are operators satisfying certain “smallness” conditions, then (A + F, B + G) is also spectral and complete. The hypotheses for these theorems are chosen with an eye to applying the results to boundary-value problems on a compact interval. Such applications, together with an examination of circumstances under which the system (Dn, Dm) (D denoting differentiation) is spectral and complete under a broad class of boundary conditions, will be made in a later paper.

scholarly journals The attracting centre of a continuous self-map of the interval

1988 ◽  
Vol 8 (2) ◽  
pp. 205-213 ◽  
Author(s):  
Xiong Jincheng
Keyword(s):  
Periodic Point ◽  
Limit Point ◽  
Periodic Points ◽  
Compact Interval ◽  
Continuous Map ◽  
Limit Points ◽  
Countably Infinite

AbstractLet ƒ denote a continuous map of a compact interval I to itself. A point x ∈ I is called a γ-limit point of ƒ if it is both an ω-limit point and an α-limit point of some point y ∈ I. Let Γ denote the set of γ-limit points. In the present paper, we show that (1) −Γ is either empty or countably infinite, where denotes the closure of the set P of periodic points, (2) x ∈ I is a γ-limit point if and only if there exist y1 and y2 in I such that x is an ω-limit point of y1, and y1 is an ω-limit point of y2, and if and only if there exists a sequence y1, y2,…of points in I such that x is an ω-limit point of y1, and yi is an ω-limit point of yi+1 for every i ≥ 1, and (3) the period of each periodic point of ƒ is a power of 2 if and only if every γ-limit point is recurrent.

A characterization of ω-limit sets of maps of the interval with zero topological entropy

1993 ◽  
Vol 13 (1) ◽  
pp. 7-19 ◽  
Author(s):  
A. M. Bruckner ◽  
J. Smítal
Keyword(s):  
Topological Entropy ◽  
Cantor Set ◽  
Limit Set ◽  
Limit Sets ◽  
Continuous Map

AbstractWe prove that an infiniteW⊂ (0, 1) is an ω-limit set for a continuous map ƒ of [0,1] with zero topological entropy iffW=Q∪PwhereQis a Cantor set, andPis countable, disjoint fromQ, dense inWif non-empty, and such that for any intervalJcontiguous toQ, card (J∩P) ≤ 1 if 0 or 1 is inJ, and card (J∩P) ≤ 2 otherwise. Moreover, we prove a conjecture by A. N. Šarkovskii from 1967 thatPcan contain points from infinitely many orbits, and consequently, that the system of ω-limit sets containingQand contained inW, can be uncountable.

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