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The Banach contraction mapping principle and cohomology

Ludvík Janoš (2000)

Commentationes Mathematicae Universitatis Carolinae

By a dynamical system ( X , T ) we mean the action of the semigroup ( + , + ) on a metrizable topological space X induced by a continuous selfmap T : X X . Let M ( X ) denote the set of all compatible metrics on the space X . Our main objective is to show that a selfmap T of a compact space X is a Banach contraction relative to some d 1 M ( X ) if and only if there exists some d 2 M ( X ) which, regarded as a 1 -cocycle of the system ( X , T ) × ( X , T ) , is a coboundary.

The geometry of laminations

Robbert Fokkink, Lex Oversteegen (1996)

Fundamenta Mathematicae

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.

The structure of disjoint iteration groups on the circle

Krzysztof Ciepliński (2004)

Czechoslovak Mathematical Journal

The aim of the paper is to investigate the structure of disjoint iteration groups on the unit circle 𝕊 1 , that is, families = { F v 𝕊 1 𝕊 1 v V } of homeomorphisms such that F v 1 F v 2 = F v 1 + v 2 , v 1 , v 2 V , and each F v either is the identity mapping or has no fixed point ( ( V , + ) is an arbitrary 2 -divisible nontrivial (i.e., c a r d V > 1 ) abelian group).

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