Stable Homeomorphisms of the Pseudo-Arc

Lewis, Wayne

doi:10.4153/cjm-1979-041-1

Cited by 20 publications

(4 citation statements)

References 4 publications

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“…This lemma follows immediately from Theorem 8 in [11], stating that if p and q are distinct points of P \ U , where U is open in P , such that the subcontinuum M irreducible between p and q does not intersect cl(U ), then there is a homeomorphism h : P → P with h(p) = q and h|U = 1 U (cf. also [8,Theorem] …”

Section: Lemma Let U Be An Open Subset Of the Pseudo-arc P Such Thatmentioning

confidence: 97%

Hereditarily indecomposable continua with exactly n autohomeomorphisms

Pol

2002

Colloq. Math.

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Abstract. The main goal of this paper is to construct, for every n, m ∈ N, a hereditarily indecomposable continuum X nm of dimension m which has exactly n autohomeomorphisms.1. Introduction. All spaces considered are assumed to be metrizable separable. Our terminology follows [6] and [10]. A continuum X is hereditarily indecomposable, abbreviated HI, if for any two intersecting subcon-denotes the group of all homeomorphisms of X onto X. A continuum X is rigid if the identity 1 X is the only homeomorphism of X onto X, i.e., G(X) = {1 X }. In [5] H. Cook gave an example of a rigid, 1-dimensional, HI continuum. Recently M. Reńska [18] constructed, for every m ∈ N, an HI rigid m-dimensional Cantor manifold. The main goal of this paper is to prove the following theorem.

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Section: Lemma Let U Be An Open Subset Of the Pseudo-arc P Such Thatmentioning

confidence: 97%

Hereditarily indecomposable continua with exactly n autohomeomorphisms

Pol

2002

Colloq. Math.

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“…3, at least one of those pieces would not be connected. We refer the interested reader to [24] for computer generated pictures of continua that correspond to inverse limit spaces of different spaces.…”

Section: Next Consider the Sequences Inmentioning

confidence: 99%

An Echo State Network Imparts a Curve Fitting

Manjunath

2022

IEEE Trans. Neural Netw. Learning Syst.

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Recurrent neural networks are successfully employed in processing information from temporal data. Approaches to training such networks are varied, and reservoir computing based attainments such as the echo state network provides great ease in training. Akin to many machine learning algorithms rendering an interpolation function or fitting a curve, we observe that a driven system such as a recurrent neural network renders a continuous curve fitting if and only if it satisfies the echo state property. The domain of the learnt curve is an abstract space of left-infinite sequence of inputs and the codomain is the space of readout values. When the input originates from discrete-time dynamical systems, we find theoretical conditions under which a topological conjugacy between the input and reservoir dynamics can exist, and present some numerical results relating the linearity in the reservoir to the forecasting abilities of the echo state networks.

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“…. , n [27]; and 2) given two points x and y and an open subset U , if there is a subcontinuum of the pseudo-arc containing x and y which is disjoint from U , then there is a homeomorphism h of the pseudo-arc to itself such that h(x) = y and h is the identity on U [33]. These properties should be compared with similar properties enjoyed by the circle S 1 : 1 ) given two sets of n points x 1 , .…”

Section: Introductionmentioning

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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Stable Homeomorphisms of the Pseudo-Arc

Cited by 20 publications

References 4 publications

Hereditarily indecomposable continua with exactly n autohomeomorphisms

Hereditarily indecomposable continua with exactly n autohomeomorphisms

An Echo State Network Imparts a Curve Fitting

A complete classification of homogeneous plane continua

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