scholarly journals Periodic points and smooth rays

2021 ◽  
Vol 25 (8) ◽  
pp. 170-178
Author(s):  
Carsten Petersen ◽  
Saeed Zakeri
Keyword(s):  
Periodic Point ◽  
Julia Set ◽  
Periodic Points ◽  
Connected Component ◽  
Landing Point ◽  
Content Type ◽  
Polynomial Map ◽  
External Ray

Let P : C → C P: \mathbb {C} \to \mathbb {C} be a polynomial map with disconnected filled Julia set K P K_P and let z 0 z_0 be a repelling or parabolic periodic point of P P . We show that if the connected component of K P K_P containing z 0 z_0 is non-degenerate, then z 0 z_0 is the landing point of at least one smooth external ray. The statement is optimal in the sense that all but one cycle of rays landing at z 0 z_0 may be broken.

scholarly journals The periodic points of ε-contractive maps in fuzzy metric spaces

2021 ◽  
Vol 22 (2) ◽  
pp. 311
Author(s):  
Taixiang Sun ◽  
Caihong Han ◽  
Guangwang Su ◽  
Bin Qin ◽  
Lue Li
Keyword(s):  
Periodic Point ◽  
Metric Space ◽  
Metric Spaces ◽  
Periodic Points ◽  
Fuzzy Metric Space ◽  
Connected Component ◽  
Fuzzy Metric Spaces ◽  
Fuzzy Metric ◽  
Contractive Maps

<p>In this paper, we introduce the notion of ε-contractive maps in fuzzy metric space (X, M, ∗) and study the periodicities of ε-contractive maps. We show that if (X, M, ∗) is compact and f : X −→ X is ε-contractive, then P(f) = ∩ ∞n=1f n (X) and each connected component of X contains at most one periodic point of f, where P(f) is the set of periodic points of f. Furthermore, we present two examples to illustrate the applicability of the obtained results.</p>

scholarly journals A landing theorem for entire functions with bounded post-singular sets

2020 ◽  
Vol 30 (6) ◽  
pp. 1465-1530
Author(s):  
Anna Miriam Benini ◽  
Lasse Rempe
Keyword(s):  
Entire Function ◽  
Periodic Point ◽  
Infinite Order ◽  
Entire Functions ◽  
Complex Polynomial ◽  
Landing Point ◽  
External Rays ◽  
Singular Sets ◽  
External Ray ◽  
Polynomial Dynamics

AbstractThe Douady-Hubbard landing theorem for periodic external rays is one of the cornerstones of the study of polynomial dynamics. It states that, for a complex polynomial f with bounded postcritical set, every periodic external ray lands at a repelling or parabolic periodic point, and conversely every repelling or parabolic point is the landing point of at least one periodic external ray. We prove an analogue of this theorem for an entire function f with bounded postsingular set. If f has finite order of growth, then it is known that the escaping set I(f) contains certain curves called periodic hairs; we show that every periodic hair lands at a repelling or parabolic periodic point, and conversely every repelling or parabolic periodic point is the landing point of at least one periodic hair. For a postsingularly bounded entire function f of infinite order, such hairs may not exist. Therefore we introduce certain dynamically natural connected subsets of I(f), called dreadlocks. We show that every periodic dreadlock lands at a repelling or parabolic periodic point, and conversely every repelling or parabolic periodic point is the landing point of at least one periodic dreadlock. More generally, we prove that every point of a hyperbolic set is the landing point of a dreadlock.

scholarly journals Periodic points for amenable group actions on uniquely arcwise connected continua

2020 ◽  
pp. 1-12
Author(s):  
ENHUI SHI ◽  
XIANGDONG YE
Keyword(s):  
Periodic Point ◽  
Amenable Group ◽  
Group Actions ◽  
Periodic Points ◽  
Arcwise Connected

Abstract We show that any action of a countable amenable group on a uniquely arcwise connected continuum has a periodic point of order $\leq 2$ .

DYNAMICS OF MONOTONE GRAPH, DENDRITE AND DENDROID MAPS

2011 ◽  
Vol 21 (11) ◽  
pp. 3205-3215 ◽  
Author(s):  
ISSAM NAGHMOUCHI
Keyword(s):  
Periodic Point ◽  
Periodic Points ◽  
Nonempty Interior ◽  
Limit Set ◽  
Limit Sets ◽  
Dendrite Map ◽  
Graph Map ◽  
Monotone Graph

We show that, for monotone graph map f, all the ω-limit sets are finite whenever f has periodic point and for monotone dendrite map, any infinite ω-limit set does not contain periodic points. As a consequence, monotone graph and dendrite maps have no Li–Yorke pairs. However, we built a homeomorphism on a dendroid with a scrambled set having nonempty interior.

scholarly journals Examples of expanding maps with some special properties

1987 ◽  
Vol 36 (3) ◽  
pp. 469-474 ◽  
Author(s):  
Bau-Sen Du
Keyword(s):  
Fixed Point ◽  
Positive Integer ◽  
Periodic Point ◽  
Topological Entropy ◽  
Real Line ◽  
Periodic Points ◽  
Unit Interval ◽  
Expanding Maps ◽  
The Real ◽  
Piecewise Monotonic

Let I be the unit interval [0, 1] of the real line. For integers k ≥ 1 and n ≥ 2, we construct simple piecewise monotonic expanding maps Fk, n in C0 (I, I) with the following three properties: (1) The positive integer n is an expanding constant for Fk, n for all k; (2) The topological entropy of Fk, n is greater than or equal to log n for all k; (3) Fk, n has periodic points of least period 2k · 3, but no periodic point of least period 2k−1 (2m+1) for any positive integer m. This is in contrast to the fact that there are expanding (but not piecewise monotonic) maps in C0(I, I) with very large expanding constants which have exactly one fixed point, say, at x = 1, but no other periodic point.

Periodic Point at Real Axis for a Generalized 3x+1 Function

2011 ◽  
Vol 55-57 ◽  
pp. 1670-1674 ◽  
Author(s):  
Shuai Liu ◽  
Zheng Xuan Wang
Keyword(s):  
Periodic Point ◽  
Real Axis ◽  
Exponential Function ◽  
Periodic Points ◽  
Divergence Points ◽  
Fractal Character ◽  
Convergence And Divergence ◽  
Complex Exponential

In order to study the fractal character of representative complex exponential function just as generalized 3x+1 function T(x). In this essay, we proved that T(x) has periodic points of every period in bound (n, n+1) when n>1 in real axis. Then, we found the distribution of 2-periods points of T(x) in real axis. We put forward the bottom bound of 2-periodic point’s number and proved it. Moreover, we found the number of T(x)’s 2-periodic points in different bounds to validate our conclusion. Then, we extended the conclusion to i-periods points and find similar conclusion. Finally, we proved there exist endless convergence and divergence points of T(x) in real axis.

On Devaney chaotic generalized shift dynamical systems

2013 ◽  
Vol 50 (4) ◽  
pp. 509-522 ◽  
Author(s):  
Fatemah Shirazi ◽  
Javad Sarkooh ◽  
Bahman Taherkhani
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Periodic Point ◽  
Periodic Points ◽  
List Type ◽  
Topologically Transitive ◽  
Generalized Shift ◽  
One To One ◽  
Shift Dynamical System ◽  
Infinite Sequences

In the following text we prove that in a generalized shift dynamical system (XГ, σφ) for infinite countable Г and discrete X with at least two elements the following statements are equivalent: the dynamical system (XГ, σφ) is chaotic in the sense of Devaneythe dynamical system (XГ, σφ) is topologically transitivethe map φ: Г → Г is one to one without any periodic point.Also for infinite countable Г and finite discrete X with at least two elements (XГ, σφ) is exact Devaney chaotic, if and only if φ: Г → Г is one to one and φ: Г → Г has niether periodic points nor φ-backwarding infinite sequences.

scholarly journals On common fixed and periodic points of commuting functions

1998 ◽  
Vol 21 (2) ◽  
pp. 269-276 ◽  
Author(s):  
Aliasghar Alikhani-Koopaei
Keyword(s):  
Fixed Point ◽  
Common Fixed Point ◽  
Periodic Point ◽  
Periodic Points ◽  
Continuous Functions ◽  
Contractive Condition ◽  
Unique Common Fixed Point

It is known that two commuting continuous functions on an interval need not have a common fixed point. It is not known if such two functions have a common periodic point. In this paper we first give some results in this direction. We then define a new contractive condition, under which two continuous functions must have a unique common fixed point.

Polynomial shift-like maps in ℂk

2019 ◽  
Vol 30 (12) ◽  
pp. 1950066
Author(s):  
Sayani Bera
Keyword(s):  
Periodic Points ◽  
Basins Of Attraction ◽  
Hyperbolic Polynomial ◽  
Polynomial Map ◽  
Fatou Components

The purpose of this paper is to explore a few properties of polynomial shift-like automorphisms of [Formula: see text] We first prove that a [Formula: see text]-shift-like polynomial map (say [Formula: see text]) degenerates essentially to a polynomial map in [Formula: see text]-dimensions as [Formula: see text] Second, we show that a shift-like map obtained by perturbing a hyperbolic polynomial (i.e. [Formula: see text], where [Formula: see text] is sufficiently small) has finitely many Fatou components, consisting of basins of attraction of periodic points and the component at infinity.

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