scholarly journals A characterization of chaos

1986 ◽  
Vol 34 (2) ◽  
pp. 283-292 ◽  
Author(s):  
K. Janková ◽  
J. Smítal
Keyword(s):  
Topological Entropy ◽  
Real Interval ◽  
Complex Behaviour ◽  
Positive Topological Entropy ◽  
Continuous Map ◽  
Continuous Mappings

Consider the continuous mappings f from a compact real interval to itself. We show that when f has a positive topological entropy (or equivalently, when f has a cycle of order ≠ 2n, n = 0, 1, 2, …) then f has a more complex behaviour than chaoticity in the sense of Li and Yorke: something like strong or uniform chaoticity, distinguishable on a certain level ɛ > 0. Recent results of the second author then imply that any continuous map has exactly one of the following properties: It is either strongly chaotic or every trajectory is approximable by cycles. Also some other conditions characterizing chaos are given.

scholarly journals A CHARACTERIZATION OF THE SET Ω (f)\ ω (f) FOR CONTINUOUS MAPS OF THE INTERVAL WITH ZERO TOPOLOGICAL ENTROPY

1995 ◽  
Vol 05 (05) ◽  
pp. 1433-1435
Author(s):  
F. BALIBREA ◽  
J. SMÍTAL
Keyword(s):  
Topological Entropy ◽  
Minimal Set ◽  
Continuous Map ◽  
Continuous Maps ◽  
Limit Points ◽  
Infinite Set ◽  
Nonwandering Points

We give a characterization of the set of nonwandering points of a continuous map f of the interval with zero topological entropy, attracted to a single (infinite) minimal set Q. We show that such a map f can have a unique infinite minimal set Q and an infinite set B ⊂ Ω (f)\ ω (f) (of nonwandering points that are not ω-limit points) attracted to Q and such that B has infinite intersections with infinitely many disjoint orbits of f.

scholarly journals A remark on positive topological entropy ofN-buffer switched flow networks

2005 ◽  
Vol 2005 (2) ◽  
pp. 93-99 ◽  
Author(s):  
Xiao-Song Yang
Keyword(s):  
Topological Entropy ◽  
Metric Space ◽  
Elementary Proof ◽  
Flow Networks ◽  
Compact Metric Space ◽  
Positive Topological Entropy ◽  
Continuous Map ◽  
Simple Theorem

We present a simpler elementary proof on positive topological entropy of theN-buffer switched flow networks based on a new simple theorem on positive topological entropy of continuous map on compact metric space.

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.

scholarly journals Distribution of Maps with Transversal Homoclinic Orbits in a Continuous Map Space

2011 ◽  
Vol 2011 ◽  
pp. 1-11
Author(s):  
Qiuju Xing ◽  
Yuming Shi
Keyword(s):  
Banach Space ◽  
Topological Entropy ◽  
Homoclinic Orbits ◽  
Positive Topological Entropy ◽  
Continuous Map ◽  
Bounded Set ◽  
Transversal Homoclinic Orbits ◽  
Continuous Maps ◽  
Dense Set

This paper is concerned with distribution of maps with transversal homoclinic orbits in a continuous map space, which consists of continuous maps defined in a closed and bounded set of a Banach space. By the transversal homoclinic theorem, it is shown that the map space contains a dense set of maps that have transversal homoclinic orbits and are chaotic in the sense of both Li-Yorke and Devaney with positive topological entropy.

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.

scholarly journals Keplerian trigonometry

2021 ◽  
Author(s):  
Alessandro Gambini ◽  
Giorgio Nicoletti ◽  
Daniele Ritelli
Keyword(s):  
Elliptic Curve ◽  
Real Interval ◽  
Planar Curve ◽  
Signed Area

AbstractTaking the hint from usual parametrization of circle and hyperbola, and inspired by the pathwork initiated by Cayley and Dixon for the parametrization of the “Fermat” elliptic curve $$x^3+y^3=1$$ x 3 + y 3 = 1 , we develop an axiomatic study of what we call “Keplerian maps”, that is, functions $${{\,\mathrm{{\mathbf {m}}}\,}}(\kappa )$$ m ( κ ) mapping a real interval to a planar curve, whose variable $$\kappa $$ κ measures twice the signed area swept out by the O-ray when moving from 0 to $$\kappa $$ κ . Then, given a characterization of k-curves, the images of such maps, we show how to recover the k-map of a given parametric or algebraic k-curve, by means of suitable differential problems.

SIMPLE AND COMPLEX DYNAMICS FOR ONE-DIMENSIONAL MANIFOLD MAPS

1995 ◽  
Vol 05 (05) ◽  
pp. 1351-1355
Author(s):  
VLADIMIR FEDORENKO
Keyword(s):  
Topological Entropy ◽  
Complex Dynamics ◽  
Periodic Points ◽  
Dimensional Manifold ◽  
One Dimensional ◽  
Circle Maps ◽  
Interval Maps ◽  
Chain Recurrent Set ◽  
Recurrent Set

We give a characterization of complex and simple interval maps and circle maps (in the sense of positive or zero topological entropy respectively), formulated in terms of the description of the dynamics of the map on its chain recurrent set. We also describe the behavior of complex maps on their periodic points.

Perturbation of the fixed point problem for continuous mappings

2020 ◽  
pp. 241-249
Author(s):  
Zukhra T. Zhukovskaya ◽  
Sergey E. Zhukovskiy
Keyword(s):  
Banach Space ◽  
Fixed Point ◽  
Fixed Points ◽  
Continuous Mapping ◽  
Fixed Point Problem ◽  
Point Problem ◽  
Bounded Subset ◽  
Completely Continuous ◽  
Continuous Map ◽  
Continuous Mappings

We consider the problem of a double fixed point of pairs of continuous mappings defined on a convex closed bounded subset of a Banach space. It is shown that if one of the mappings is completely continuous and the other is continuous, then the property of the existence of fixed points is stable under contracting perturbations of the mappings. We obtain estimates for the distance from a given pair of points to double fixed points of perturbed mappings. We consider the problem of a fixed point of a completely continuous mapping on a convex closed bounded subset of a Banach space. It is shown that the property of the existence of a fixed point of a completely continuous map is stable under contracting perturbations. Estimates of the distance from a given point to a fixed point are obtained. As an application of the obtained results, the solvability of a difference equation of a special type is proved.

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.

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