A fixed point theorem for plane continua

Hagopian, Charles L.

doi:10.1090/s0002-9904-1971-12690-3

Cited by 25 publications

(10 citation statements)

References 5 publications

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“…Then Bell announced in 1982 (see also Akis [Aki99]) that the Cartwright-Littlewood Theorem can be extended to the class of all holomorphic maps of the plane. For other partial results in this direction see, e.g., [Ham51,Hag71,Bel79,Min90,Hag96,Min99].…”

Section: Prefacementioning

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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Section: Prefacementioning

confidence: 99%

Fixed Point Theorems for Plane Continua with Applications

et al. 2012

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“…Uniquely arcwise connected contractible continua have the fixed point property by Theorem 5 of [43]. Arcwise connected nonseparating plane continua have the fixed point property by Theorem 3 of [15].…”

Section: Theoremmentioning

confidence: 98%

Arc-Smooth Continua

Fugate¹,

Jr.²,

Lum³

1981

Transactions of the American Mathematical Society

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Abstract.Continua admitting arc-structures and arc-smooth continua are introduced as higher dimensional analogues of dendroids and smooth dendroids, respectively. These continua include such spaces as: cones over compacta, convex continua in l2, strongly convex metric continua, injectively metrizable continua, as well as various topological semigroups, partially ordered spaces, and hyperspaces. The arc-smooth continua are shown to coincide with the freely contractible continua and with the metric /T-spaces of Stadtlander. Known characterizations of smoothness in dendroids involving closed partial orders, the set function T, radially convex metrics, continuous selections, and order preserving mappings are extended to the setting of continua with arc-structures. Various consequences of the special contractibility properties of arc-smooth continua are also obtained.Introduction. The purpose of this paper is to introduce and study a well-behaved class of arc-wise connected metric continua called arc-smooth continua) The class of arc-smooth continua, which may be considered as a higher dimensional analogue of the smooth dendroids

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“…Let D be a complementary domain of X such that IczBd D, and let X f = S -D. By Theorem 1 of [1], every subcontinuum of X', and hence every subcontinuum of X, which contains a nonempty open subset of I contains I.…”

Section: If X Is a Continuum In A 2-sphere S And I Is An Indecomposabmentioning

confidence: 99%

Planar images of decomposable continua

Hagopian¹

1972

Pacific J. Math.

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A fixed point theorem for plane continua

Cited by 25 publications

References 5 publications

Fixed Point Theorems for Plane Continua with Applications

Fixed Point Theorems for Plane Continua with Applications

Arc-Smooth Continua

Planar images of decomposable continua

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