scholarly journals Non-wandering points and the depth of a graph map

2000 ◽  
Vol 69 (2) ◽  
pp. 143-152 ◽  
Author(s):  
Xiangdong Ye
Keyword(s):  
Topological Entropy ◽  
Topological Structure ◽  
Piecewise Monotone ◽  
Continuous Map ◽  
Derived Set ◽  
Limit Points ◽  
Graph Map

AbstractLef: G → G be a continuous map of a graph and let d(A) denote the derived set (or limit points) of A ⊂ G. We prove that d(Ω(f)) ⊂ λ (f) and the depth of f is at most three. We also prove that if f is piecewise monotone or has zero topological entropy, then the depth of f is at most two. Furthermore, we obtain some results on the topological structure of Ω(f).

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 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.

ENTROPY OF MAPS WITH HORIZONTAL GAPS

2004 ◽  
Vol 14 (04) ◽  
pp. 1489-1492 ◽  
Author(s):  
MICHAŁ MISIUREWICZ
Keyword(s):  
Topological Entropy ◽  
Entropy Function ◽  
Devil's Staircase ◽  
Piecewise Monotone ◽  
Interval Maps ◽  
Continuous Map ◽  
Devil’S Staircase ◽  
Staircase Structure

We study the behavior of topological entropy in one-parameter families of interval maps obtained from a continuous map f by truncating it at the level depending on the parameter. When f is piecewise monotone, the entropy function has the devil's staircase structure.

scholarly journals On the topological entropy of transitive maps of the interval

1991 ◽  
Vol 44 (2) ◽  
pp. 207-213 ◽  
Author(s):  
Ethan M. Coven ◽  
Melissa C. Hidalgo
Keyword(s):  
Topological Entropy ◽  
Piecewise Linear ◽  
Turning Points ◽  
Invariant Set ◽  
Invariant Sets ◽  
Piecewise Monotone ◽  
Continuous Map ◽  
Linear Maps ◽  
Piecewise Linear Maps

The topological entropy of a continuous map of the interval is the supremum of the topological entropies of the piecewise linear maps associated to its finite invariant sets. We show that for transitive maps, this supremum is attained at some finite invariant set if and only if the map is piecewise monotone and the set contains the endpoints of the interval and the turning points of the map.

scholarly journals On Some Properties of the Limit Points of (z(n)/n)n

Mathematics ◽  
2021 ◽  
Vol 9 (16) ◽  
pp. 1931
Author(s):  
Eva Trojovská ◽  
Kandasamy Venkatachalam
Keyword(s):  
Recent Result ◽  
Real Number ◽  
Topological Structure ◽  
Positive Real Number ◽  
Fibonacci Numbers ◽  
Positive Real ◽  
Derived Set ◽  
Limit Points ◽  
Interior Points ◽  
Theoretical Results

Let (Fn)n≥0 be the sequence of Fibonacci numbers. The order of appearance of an integer n≥1 is defined as z(n):=min{k≥1:n∣Fk}. Let Z′ be the set of all limit points of {z(n)/n:n≥1}. By some theoretical results on the growth of the sequence (z(n)/n)n≥1, we gain a better understanding of the topological structure of the derived set Z′. For instance, {0,1,32,2}⊆Z′⊆[0,2] and Z′ does not have any interior points. A recent result of Trojovská implies the existence of a positive real number t<2 such that Z′∩(t,2) is the empty set. In this paper, we improve this result by proving that (127,2) is the largest subinterval of [0,2] which does not intersect Z′. In addition, we show a connection between the sequence (xn)n, for which z(xn)/xn tends to r>0 (as n→∞), and the number of preimages of r under the map m↦z(m)/m.

scholarly journals Topological Entropy and Specialα-Limit Points of Graph Maps

2011 ◽  
Vol 2011 ◽  
pp. 1-7
Author(s):  
Taixiang Sun ◽  
Guangwang Su ◽  
Hailan Liang ◽  
Qiuli He
Keyword(s):  
Topological Entropy ◽  
Continuous Map ◽  
Limit Points

LetGa graph andf:G→Gbe a continuous map. Denote byh(f),R(f), andSA(f)the topological entropy, the set of recurrent points, and the set of specialα-limit points off, respectively. In this paper, we show thath(f)>0if and only ifSA(f)-R(f)≠∅.

scholarly journals The depth and the attracting centre for a continuous map on a fuzzy metric interval

2020 ◽  
Vol 21 (2) ◽  
pp. 285
Author(s):  
Taixiang Sun ◽  
Lue Li ◽  
Guangwang Su ◽  
Caihong Han ◽  
Guoen Xia
Keyword(s):  
Positive Integer ◽  
Fuzzy Metric ◽  
Continuous Map ◽  
Limit Points

<p>Let I be a fuzzy metric interval and f be a continuous map from I to I. Denote by R(f), Ω(f) and ω(x, f) the set of recurrent points of f, the set of non-wandering points of f and the set of ω- limit points of x under f, respectively. Write ω(f) = ∪x∈Iω(x, f), ωn+1(f) = ∪x∈ωn(f)ω(x, f) and Ωn+1(f) = Ω(f|Ωn(f)) for any positive integer n. In this paper, we show that Ω2(f) = R(f) and the depth of f is at most 2, and ω3(f) = ω2(f) and the depth of the attracting centre of f is at most 2.</p>

ESSENTIAL ENTROPY-CARRYING HORSESHOES AS SET LIMITS

2012 ◽  
Vol 22 (08) ◽  
pp. 1250195 ◽  
Author(s):  
STEVEN M. PEDERSON
Keyword(s):  
Topological Entropy ◽  
Convergent Sequence ◽  
Invariant Set ◽  
Invariant Sets ◽  
Piecewise Monotone ◽  
Interval Maps ◽  
Monotone Map ◽  
Monotone Maps

This paper studies the set limit of a sequence of invariant sets corresponding to a convergent sequence of piecewise monotone interval maps. To do this, the notion of essential entropy-carrying set is introduced. A piecewise monotone map f with an essential entropy-carrying horseshoe S(f) and a sequence of piecewise monotone maps [Formula: see text] converging to f is considered. It is proven that if each gi has an invariant set T(gi) with at least as much topological entropy as f, then the set limit of [Formula: see text] contains S(f).

scholarly journals Topological conjugacies of piecewise monotone interval maps

2001 ◽  
Vol 25 (2) ◽  
pp. 119-127 ◽  
Author(s):  
Nikos A. Fotiades ◽  
Moses A. Boudourides
Keyword(s):  
Topological Entropy ◽  
Piecewise Linear ◽  
Topological Conjugacy ◽  
Markov Condition ◽  
Piecewise Monotone ◽  
Interval Maps ◽  
Linear Maps ◽  
Piecewise Linear Maps ◽  
Topological Conjugacies

Our aim is to establish the topological conjugacy between piecewise monotone expansive interval maps and piecewise linear maps. First, we are concerned with maps satisfying a Markov condition and next with those admitting a certain countable partition. Finally, we compute the topological entropy in the Markov case.

Topological Structure of Non–wandering Set of a Graph Map

2005 ◽  
Vol 21 (4) ◽  
pp. 873-880
Author(s):  
Rong Bao Gu ◽  
Tai Xiang Sun ◽  
Ting Ting Zheng
Keyword(s):  
Topological Structure ◽  
Graph Map
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