Equicontinuity of maps on a dendrite with finite branch points

Sun, Tai Xiang; Su, Guang Wang; Xi, Hong Jian; Kong, Xiangming

doi:10.1007/s10114-017-6289-x

Cited by 10 publications

(6 citation statements)

References 22 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…A few words regarding the equivalence between (a) and (d) are in order. By [5,Theorem 4.12] together with [14,Lemma 2.6], the implication (a)⇒(d) still holds if X is merely a dendrite with finitely many branching points. The reverse implication holds for an arbitrary dendrite and a surjective map by [10,Theorem 5.2].…”

Section: Examples and Open Problems This Section Contains Examplesmentioning

confidence: 99%

Equicontinuous mappings on finite trees

Acosta

Fernández-Bretón

2021

Fund. Math.

View full text Add to dashboard Cite

If X is a finite tree and f : X → X is a map, in the Main Theorem of this paper (Theorem 1.8), we find eight conditions, each of which is equivalent to f being equicontinuous. To name just a few of the results obtained: the equicontinuity of f is equivalent to there being no arc A ⊆ X satisfying A f n [A] for some n ∈ N. It is also equivalent to the statement that for some nonprincipal ultrafilter u, the function f u : X → X is continuous (in other words, failure of equicontinuity of f is equivalent to the failure of continuity of every element of the Ellis remainder g ∈ E(X, f) *). One of the tools used in the proofs is the Ramsey-theoretic result known as Hindman's theorem. Our results generalize the ones shown by Vidal-Escobar and García-Ferreira (2019), and complement those of Bruckner and Ceder (1992), Mai (2003) and Camargo, Rincón and Uzcátegui (2019).

show abstract

Section: Examples and Open Problems This Section Contains Examplesmentioning

confidence: 99%

Equicontinuous mappings on finite trees

Acosta

Fernández-Bretón

2021

Fund. Math.

View full text Add to dashboard Cite

show abstract

“…f n (X) is a periodic homeomorphism, or is topologically conjugate to the irrational rotation of the unit circle S 1 . In [11,22,23], the authors studied equicontinuous dendrite maps. For a dendrite X with countable set of endpoints, in [22,Theorem 2.8] it is shown that f is equicontinuous if and only if Ω(x, f n ) = ω(x, f n ) for any x ∈ X and each n ∈ N. For a dendrite X with finite branch points, in [23,Theorem 28] it is proved that the following statements are equivalent:…”

Section: Introductionmentioning

confidence: 99%

Equicontinuous local dendrite maps

2021

View full text Add to dashboard Cite

Let X be a local dendrite, and f : X → X be a map. Denote by E(X) the set of endpoints of X. We show that if E(X) is countable, then the following are equivalent:(1) f is equicontinuous;(2) <img src="data:image/png;base64,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" alt="" /> fn (X) = R(f);(3) f| <img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAACoAAAA/CAYAAACW2FpzAAAE5ElEQVRoBe2af0zUZRzH4Y671FhMMtIRGhE4rc2to1JnKbOGLn8ABRTJmL8yXdPYUKkQRFpoiKekoiOSmZqbmhWIDih3MozgD8RIcKQSU0QXP3bI5Pb93l4N5eC48+783n2XWt/bvvs+v+55Xs/783zuefZ5zotH5OP1iHCigMptqf+jon1cPr2PnPRPSN+6H8OV21aiinTU/0B+djrpWwqp+NO6zqqZk6RMihqp+jKJxRsKOPbTUXavnUuIfyhv59XSLVyl5LN44tcXcab5Jp1tdRz5IoUdlUYnWPZVsoDuz9zGedNQ5+a2UpJ1fozRzSIyqZCmvqE6EGne/TlFbWbrQpdpWUDzco/TazOUcGEz00dqePbdb7kkDK/sLd7GV+dtCoc3scvJAlqZk85Ra4WMtexIiCa1IIeoCX5MfL+Q33sGxu5tIH91FgaJy1QWULp+o+DTZNalZ5GxdgWJiz5mb00X/cYVWkrYGDOFZ4LDmbNwAXPnLyWvustOMVcF8oAOjCIab9DebbVYrUYXe9ppvW5EtCqTkpQVVMrAUtsqoFIVc9VeUdSVQlLr5VXUdJWqA5tJfm82L4UFEeA3Au2o0YwLnsy0+cvYsKuEP7ql7UiWCckE2suF71KIDPFF5e2Ft9qXwBemEjFnAVHz3uA1XShPar3x8vJGExBOYq6Bdom8noOKLRxf/TL+am80T+lIyDxMTZv9tiN2NVORv2ZgMo8RvHA7NRLOJZ6Bmjv4ZZ0OX5WasRFplLbex/5trKcw6UV8VT4ERhXQdO/9wWLxwbdHoF2lH/Ccj4rRr2dRLUEdxFaOLA5Dq3qCmbmN97VbuQ8q1JGhG4HKLwJ9oxsbY3cZq0I1qALiOHTD9YJ1G9RUmUyYj5qg5SftjniD9nKaMNOSF8Eo78eJ3NN25wDjrLmboCJN2VPRqANZesL2JOpsuOF15hY9s7Qq/BOOuZysm6B9lH84HrVmBjmX3DC7hbfvJMsD1Whn6mlxYX03QW9TsmQcKu1sdlofmC0A9/s2GVgd4oNmajYXXczXY9Bd111I4QzaZGDN8wpov0RDplcUtVkyyhq1EeTeWcWZBnVRnGlQCtuE4ky2itwzrzjToCyKMw1KYZv4rztTHxWrJqAe+RaFHbZzl5A3VbN+sg/aWXr+cnFadFNREK/9yo8/N3FLApd9UzPttcWcargb9LWvHypxG3Soi38npYDKrfNDp6gjn3rAoGaulW0nLS2Nxa+OYVLKWRyFoh4wqGWB9McJgpksC6j5b6oP6tHvO8NVoZfm8iJ2bt2M/nAdnY7sZeFw+ZYR1Hgqi43FF8mPfJrwuFR2GtowiVfIjQjjo9OWy06Bc3uW805MNNHRjp+YxC0YLDd5dyYhG6jI5YoyGm5VkTIpkLhDA0EtoZ4MXQgryy2gLqVz0EA20Lv9i81bmDF+CcUDcTGxMZvpQYl8Pyw2KiIKAoKTR7QL38gKaubG1/MYG1XE3e1doGHTKwQvO4Hx2gGy9zT233xybu8K4uNiiY11/MQl5TgwfZUcXt9H+cqJzN5+eSBCLHJB/ybTFmWycf0OTne451HdZ79hU8Y6Yqf4ExCeQGrGJg7W2f9ISfp5Enu66RnGI2C82Yn91YKDpehBsSRQD8bx+KsKqMcS2nSgKGojiMfZh1jR4X/4+geRrDru1x/giQAAAABJRU5ErkJggg==" alt="" /> fn (X) is equicontinuous;(4) f| <img src="data:image/png;base64,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" alt="" />fn (X) is a pointwise periodic homeomorphism or is topologically conjugate to an irrational rotation of S 1 ;(5) ω(x, f) = Ω(x, f) for all x ∈ X.This result generalizes [17, Theorem 5.2], [24, Theorem 2] and [11, Theorem 2.8].

show abstract

“…These results have been partially extended to dendrites, that is, locally connected continua without simple closed curves [6,9,10,11]. For instance, Sun et al [9,Theorem 2.8] showed that if X is a dendrite such that it has less that 2 ℵ 0 end points and f : X → X is a continuous map, then f is equicontinuous if, and only if, ω(x, f n ) = Ω(x, f n ) for all x ∈ X and n ∈ N. Also, in [10,Theorem 2.8], it is shown that if X is a dendrite with finite branch points and f : X → X is a continuous map, then equicontinuity is equivalent to ω(x, f ) = Ω(x, f ) for each x ∈ X. The main result in the present paper is Theorem 4.12, where we show the following: Let X be a dendrite, f : X → X be a continuous map and Per(f ) be the collection of periodic points.…”

Section: Introductionmentioning

confidence: 99%

“…There has been a lot of interest in the study of dynamical system defined on dendrites [6,9,10,11]. A particular interesting issue is to determine when a continuous map f : X → X is equicontinuous, i.e.…”

Section: Introductionmentioning

confidence: 99%

Equicontinuity of maps on dendrites

Camargo

Rincón

Uzcátegui

2019

Chaos, Solitons & Fractals

View full text Add to dashboard Cite

Given a dendrite X and a continuous map f : X → X, we show the following are equivalent: (i) ω f is continuous and Per(f ) = n∈N f n (X); (ii) ω(x, f ) = Ω(x, f ) for each x ∈ X; and (iii) f is equicontinuous. Furthermore, we present some examples illustrating our results.2010 Mathematics Subject Classification. Primary: 54H20. Secondary: 37B45. 1 Definitions and preliminariesLet Z be a metric space, then given A ⊆ Z and ǫ > 0, the open ball about A of radius ǫ is denoted by V(A, ǫ). The interior, clousure, boundary and cardinality of A are denoted by A • , A, Bd(A) and |A|, respectively. A map is a continuous function. Given a compact metric space X, we denote by 2 X the set of all nonempty closed subsets of X, topologized by the Hausdorff metric which is defined as follows: for A, B ∈ 2 X , we setA continuum is a nonempty, compact, connected metric space. We say that a continuum X is a simple closed curve if X is homeomorphic to S 1 = {z ∈ C : |z| = 1}. A continuum X is a dendrite provided that X is locally connected and does not contain a simple closed curve. It is clear that dendrites are uniquely arcwise connected continua. So, if X is a dendrite and x, y ∈ X, we denote by [x, y] the unique arc joining x and y; also (x, y) will denote [x, y] \ {x, y}.Let X be a continuum. A point p ∈ X is called an end point of X provided that whenever U is open and p ∈ U, there exists an open set V ⊆ U such that x ∈ V and |Bd(V )| = 1. We denote by End(X) the collection of end points of X. Furthermore, p is a cut point of X, if X \ {p} is disconnected.

show abstract

Equicontinuity of maps on a dendrite with finite branch points

Abstract: Let (T, d) be a dendrite with finite branch points and f be a continuous map from T to T .

Cited by 10 publications

References 22 publications

Equicontinuous mappings on finite trees

Equicontinuous mappings on finite trees

Equicontinuous local dendrite maps

Equicontinuity of maps on dendrites

Contact Info

Product

Resources

About