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Non-separating subcontinua of planar continua

100%
Daniel D., Islas C., Leonel R., Tymchatyn E.
Colloquium Mathematicum
|
2015
|
tom 141
|
nr 1
143-148
EN
We revisit an old question of Knaster by demonstrating that each non-degenerate plane hereditarily unicoherent continuum X contains a proper, non-degenerate subcontinuum which does not separate X.
2
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Hereditarily indecomposable inverse limits of graphs

100%
Kawamura K., Tuncali H., Tymchatyn E.
Fundamenta Mathematicae
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2005
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tom 185
|
nr 3
195-210
EN
We prove the following theorem: Let G be a compact connected graph and let f: G → G be a piecewise linear surjection which satisfies the following condition: for each nondegenerate subcontinuum A of G, there is a positive integer n such that fⁿ(A) = G. Then, for each ε > 0, there is a map $f_{ε}: G → G$ which is ε-close to f such that the inverse limit $(G,f_{ε})$ is hereditarily indecomposable.
3
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Topology and measure of buried points in Julia sets

100%
Curry C., Mayer J., Tymchatyn E.
Fundamenta Mathematicae
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2013
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tom 222
|
nr 1
1-17
EN
It is well-known that the set of buried points of a Julia set of a rational function (also called the residual Julia set) is topologically "fat" in the sense that it is a dense $G_{δ}$ if it is non-empty. We show that it is, in many cases, a full-measure subset of the Julia set with respect to conformal measure and the measure of maximal entropy. We also address Hausdorff dimension of buried points in the same cases, and discuss connectivity and topological dimension of the set of buried points. Finally, we present a non-dynamical example of a plane continuum whose set of buried points is a dense and hereditarily disconnected (components are points) $G_{δ}$, but not totally disconnected (not all quasi-components are points).
4
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On continuous extension of uniformly continuous functions and metrics

100%
Banakh T., Brodskiy N., Stasyuk I., Tymchatyn E.
Colloquium Mathematicum
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2009
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tom 116
|
nr 2
191-202
EN
We prove that there exists a continuous regular, positive homogeneous extension operator for the family of all uniformly continuous bounded real-valued functions whose domains are closed subsets of a bounded metric space (X,d). In particular, this operator preserves Lipschitz functions. A similar result is obtained for partial metrics and ultrametrics.
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