Reduction and irreducibility for words and tree-words
On span and weakly chainable continua
The geometry of laminations
A lamination is a continuum which locally is the product of a Cantor set and an arc. We investigate the topological structure and embedding properties of laminations. We prove that a nondegenerate lamination cannot be tree-like and that a planar lamination has at least four complementary domains. Furthermore, a lamination in the plane can be obtained by a lakes of Wada construction.
A hereditarily indecomposable, hereditarily non-chainable planar tree-like continuum
On hereditarily decomposable hereditarily equivalent non-metric continua
On span and chainable continua
On the span of weakly-chainable continua
A fixed point theorem for branched covering maps of the plane
It is known that every homeomorphism of the plane which admits an invariant non-separating continuum has a fixed point in the continuum. In this paper we show that any branched covering map of the plane of degree d, |d| ≤ 2, which has an invariant, non-separating continuum Y, either has a fixed point in Y, or is such that Y contains a minimal (in the sense of inclusion among invariant continua), fully invariant, non-separating subcontinuum X. In the latter case, f has to be of degree -2 and X has...
On homogeneous totally disconnected 1-dimensional spaces
The Cantor set and the set of irrational numbers are examples of 0-dimensional, totally disconnected, homogeneous spaces which admit elegant characterizations and which play a crucial role in analysis and dynamical systems. In this paper we will start the study of 1-dimensional, totally disconnected, homogeneous spaces. We will provide a characterization of such spaces and use it to show that many examples of such spaces which exist in the literature in various fields are all homeomorphic. In particular,...
The homeomorphism group of the hairy arc
On the structure of tranches in continuously irreducible continua
On confluently graph-like compacta
For any class 𝒦 of compacta and any compactum X we say that: (a) X is confluently 𝒦-representable if X is homeomorphic to the inverse limit of an inverse sequence of members of 𝒦 with confluent bonding mappings, and (b) X is confluently 𝒦-like provided that X admits, for every ε >0, a confluent ε-mapping onto a member of 𝒦. The symbol 𝕃ℂ stands for the class of all locally connected compacta. It is proved in this paper that for each compactum X and each family 𝒦 of graphs, X is confluently...
The Menger curve Characterization and extension of homeomorphisms of non-locally-separating closed subsets
CONTENTS1. Introduction.................................................................................................................................................52. Partitioning Peano continua......................................................................................................................103. Peano continua and cross-connectedness...............................................................................................184. The characterization of the Menger curve.................................................................................................285....
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