Compositions of confluent mappings and some other classes of functions

Lelek, A.; Read, D. R.

doi:10.4064/cm-29-1-101-112

Cited by 27 publications

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“…We also give some characterizations of these mappings and some examples. Theorems 3.8 and 3.9 generalize some earlier results in [10], and Theorems 3.17 and 3.18 are analogous to Theorem 2 in [16]. Composition, product and union properties are proved also.…”

Section: Introductionsupporting

confidence: 72%

Confluent and Related Mappings Defined by Means of Quasi-Components

Grispolakis

1978

Can. j. math.

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Section: Introductionsupporting

confidence: 72%

Confluent and Related Mappings Defined by Means of Quasi-Components

Grispolakis

1978

Can. j. math.

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“…It is known (see e.g. [32] and [40]) that a hereditarily indecomposable and weakly chainable continuum must be tree-like.…”

Section: Discussion and Questionsmentioning

confidence: 99%

A complete classification of homogeneous plane continua

Hoehn

Oversteegen

2016

Acta Math.

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We show that the only compact and connected subsets (i.e. continua) X of the plane R 2 which contain more than one point and are homogeneous, in the sense that the group of homeomorphisms of X acts transitively on X, are, up to homeomorphism, the circle S 1 , the pseudo-arc, and the circle of pseudo-arcs. These latter two spaces are fractal-like objects which do not contain any arcs. It follows that any compact and homogeneous space in the plane has the form X × Z, where X is either a point or one of the three homogeneous continua above, and Z is either a finite set or the Cantor set.The main technical result in this paper is a new characterization of the pseudo-arc. Following Lelek, we say that a continuum X has span zero provided for every continuum C and every pair of maps f, g : C → X such that f (C) ⊂ g(C) there exists c 0 ∈ C so that f (c 0 ) = g(c 0 ). We show that a continuum is homeomorphic to the pseudo-arc if and only if it is hereditarily indecomposable (i.e., every subcontinuum is indecomposable) and has span zero.

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“…By Moore's Plane Decomposition Theorem [Dav86], X is homeomorphic to C. Since f is confluent, it is easy to see that g is confluent. By a theorem of Lelek and Read [LR74] g is open since it is confluent and light (also see [Nad92, Theorem 13.26]). Since h and g factor the perfect map f through a Hausdorff space C, both h and g are perfect [Eng89, 3.7.5].…”

Section: Proofmentioning

confidence: 99%

Fixed Point Theorems for Plane Continua with Applications

et al. 2012

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Part 1. Basic Theory Chapter 2. Preliminaries and outline of Part 1 2.1. Index 2.2. Variation 2.3. Classes of maps 2.4. Partitioning domains Chapter 3. Tools 3.1. Stability of Index 3.2. Index and variation for finite partitions 3.3. Locating arcs of negative variation 3.4. Crosscuts and bumping arcs 3.5. Index and Variation for Carathéodory Loops 3.6. Prime Ends 3.7. Oriented maps 3.8. Induced maps of prime ends Chapter 4. Partitions of domains in the sphere 4.1. Kulkarni-Pinkall Partitions 4.2. Hyperbolic foliation of simply connected domains 4.3. Schoenflies Theorem 4.4. Prime ends Part 2. Applications of basic theory Chapter 5. Description of main results of Part 2 5.1. Outchannels 5.2. Fixed points in invariant continua 5.3. Fixed points in non-invariant continua -the case of dendrites 5.4. Fixed points in non-invariant continua -the planar case 5.5. The polynomial case Chapter 6. Outchannels and their properties 6.1. Outchannels v vi CONTENTS 6.2. Uniqueness of the Outchannel Chapter 7. Fixed points 7.1. Fixed points in invariant continua 7.2. Dendrites 7.3. Non-invariant continua and positively oriented maps of the plane 7.4. Maps with isolated fixed points 7.5. Applications to complex dynamics Bibliography Index

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Compositions of confluent mappings and some other classes of functions

Cited by 27 publications

References 0 publications

Confluent and Related Mappings Defined by Means of Quasi-Components

Confluent and Related Mappings Defined by Means of Quasi-Components

A complete classification of homogeneous plane continua

Fixed Point Theorems for Plane Continua with Applications

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