scholarly journals Stirring our way to Sharkovsky's theorem

1997 ◽  
Vol 56 (3) ◽  
pp. 453-458
Author(s):  
Seth Patinkin
Keyword(s):  
Topological Space ◽  
Periodic Point ◽  
Real Line ◽  
Continuous Functions ◽  
Cycle Structure ◽  
Intermediate Value Theorem ◽  
The Real ◽  
Continuous Map ◽  
Sharkovsky's Theorem ◽  
Intermediate Value

The periodic-point or cycle structure of a continuous map of a topological space has been a subject of great interest since A.N. Sharkovsky completely explained the hierarchy of periodic orders of a continuous map f: R → R, where R is the real line. In this paper the topological idea of “stirring” is invoked in an effort to obtain a transparent proof of a generalisation of Sharkovsky's Theorem to continuous functions f: I → I where I is any interval. The stirring approach avoids all graph-theoretical and symbolic abstraction of the problem in favour of a more concrete intermediate-value-theorem-oriented analysis of cycles inside an interval.

scholarly journals A generalization of a theorem of marotto

1981 ◽  
Vol 82 ◽  
pp. 83-97 ◽  
Author(s):  
Kenichi Shiraiwa ◽  
Masahiro Kurata
Keyword(s):  
Euclidean Space ◽  
Periodic Point ◽  
Real Line ◽  
Chaotic Behavior ◽  
Dimensional Euclidean Space ◽  
Homoclinic Point ◽  
Minimal Period ◽  
The Real ◽  
Continuous Map ◽  
Transversal Homoclinic Point

In 1975, Li and Yorke [3] found the following fact. Let f: I→ I be a continuous map of the compact interval I of the real line R into itself. If f has a periodic point of minimal period three, then f exhibits chaotic behavior. The above result is generalized by F.R. Marotto [4] in 1978 for the multi-dimensional case as follows. Let f: Rn → Rn be a differentiate map of the n-dimensional Euclidean space Rn (n ≧ 1) into itself. If f has a snap-back repeller, then f exhibits chaotic behavior.In this paper, we give a generalization of the above theorem of Marotto. Our theorem can also be regarded as a generalization of the Smale’s results on the transversal homoclinic point of a diffeomorphism.

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.

scholarly journals Remarks on uniformly symmetrically continuous functions

2016 ◽  
Vol 09 (03) ◽  
pp. 1650069
Author(s):  
Tammatada Khemaratchatakumthorn ◽  
Prapanpong Pongsriiam
Keyword(s):  
Real Line ◽  
Nonempty Subset ◽  
Continuous Functions ◽  
The Real ◽  
Definition Of ◽  
Symmetrically Continuous Functions ◽  
Uniformly Continuous

We give the definition of uniform symmetric continuity for functions defined on a nonempty subset of the real line. Then we investigate the properties of uniformly symmetrically continuous functions and compare them with those of symmetrically continuous functions and uniformly continuous functions. We obtain some characterizations of uniformly symmetrically continuous functions. Several examples are also given.

Intermediate value theorems for holomorphic maps in complex Banach spaces

1991 ◽  
Vol 109 (3) ◽  
pp. 539-540 ◽  
Author(s):  
Kazimierz Włodarczyk
Keyword(s):  
Banach Spaces ◽  
Bounded Domain ◽  
Mathematical Analysis ◽  
Closed Interval ◽  
Holomorphic Maps ◽  
Intermediate Value Theorem ◽  
Continuous Map ◽  
Analytic Maps ◽  
Intermediate Value

One of the most celebrated theorems of mathematical analysis is the intermediate value theorem of Bolzano which, in a simple case, states that a real-valued continuous map f of a closed interval [a, b], such that f(a) and f(b) have different signs, has a zero in (a, b). Recently, Shih in [5] observed that without loss of generality we may suppose that a 7 < 0 < b and f(a) < 0 < f(b) and, consequently, the condition f(a).f(b) < 0 becomes x.f(x) > 0 for x∈∂Ω where ∂Ω denotes the boundary of the interval Ω = (a, b); then the conclusion is that f has at least one zero in ω. It is a remarkable fact that Shih extends this form of Boizano's theorem to analytic maps in ℂ [5] and, subsequently, in ℂn [6]. He proved that if Ω is a bounded domain in ℂn containing the origin, is continuous in and analytic in Ω and Re for z∈∂Ω, then f has exactly one zero in Ω. In this paper we extend Shih's result to Banach spaces.

Topological properties of the multifunction space L(X) of cusco maps

2008 ◽  
Vol 58 (6) ◽  
Author(s):  
Ľ. Holá ◽  
Tanvi Jain ◽  
R. McCoy
Keyword(s):  
Topological Space ◽  
Real Line ◽  
Vietoris Topology ◽  
Topological Properties ◽  
Connected Subset ◽  
The Real ◽  
Upper Semicontinuous ◽  
Set Valued Mapping ◽  
Nonempty Compact

AbstractA set-valued mapping F from a topological space X to a topological space Y is called a cusco map if F is upper semicontinuous and F(x) is a nonempty, compact and connected subset of Y for each x ∈ X. We denote by L(X), the space of all subsets F of X × ℝ such that F is the graph of a cusco map from the space X to the real line ℝ. In this paper, we study topological properties of L(X) endowed with the Vietoris topology.

Statistical σ approximation to Bögel-type continuous functions

2011 ◽  
Vol 48 (4) ◽  
pp. 475-488 ◽  
Author(s):  
Sevda Karakuş ◽  
Kamil Demirci
Keyword(s):  
Compact Subset ◽  
Real Line ◽  
Statistical Convergence ◽  
Approximation Theorem ◽  
Continuous Functions ◽  
Linear Operators ◽  
Positive Linear Operators ◽  
Approximation Result ◽  
The Real

In this paper, using the concept of statistical σ-convergence which is stronger than the statistical convergence, we obtain a statistical σ-approximation theorem for sequences of positive linear operators defined on the space of all real valued B-continuous functions on a compact subset of the real line. Then, we construct an example such that our new approximation result works but its classical and statistical cases do not work. Also we compute the rate of statistical σ-convergence of sequence of positive linear operators.

Uniformities on a Product

1972 ◽  
Vol 24 (3) ◽  
pp. 379-389 ◽  
Author(s):  
Anthony W. Hager
Keyword(s):  
Topological Space ◽  
Cardinal Number ◽  
Uniform Space ◽  
Continuous Functions ◽  
Topological Spaces ◽  
Infinite Cardinal ◽  
The Real ◽  
Completely Regular

All topological spaces shall be uniformizable (completely regular Hausdorff). A uniformity on X shall be viewed as a collection μ of coverings of X, via the manner of Tukey [20] and Isbell [16], and the associated uniform space denoted μX. Given the uniformizable topological space X, we shall be concerned with compatible uniformities as follows (discussed more carefully in § 1). The fine uniformity α (finest compatible with the topology); the “cardinal reflections“ αm of α (m an infinite cardinal number) ; αc, the weak uniformity generated by the real-valued continuous functions.With μ standing, generically, for one of these uniformities, we consider the question: when is μ(X × Y) = μX × μY For μ = αℵ0 (the finest compatible precompact uniformity), the problem is equivalent to that of whenβ(X × Y) = βX × βY,β denoting Stone-Cech compactification; this is answered by the theorem of Glicksberg [9]. For μ = α, we have Isbell's generalization [16, VI1.32].

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