The Banach contraction mapping principle and cohomology

Ludvík Janoš

Commentationes Mathematicae Universitatis Carolinae (2000)

  • Volume: 41, Issue: 3, page 605-610
  • ISSN: 0010-2628

Abstract

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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.

How to cite

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Janoš, Ludvík. "The Banach contraction mapping principle and cohomology." Commentationes Mathematicae Universitatis Carolinae 41.3 (2000): 605-610. <http://eudml.org/doc/22513>.

@article{Janoš2000,
abstract = {By a dynamical system $(X,T)$ we mean the action of the semigroup $(\mathbb \{Z\}^+,+)$ on a metrizable topological space $X$ induced by a continuous selfmap $T:X\rightarrow X$. 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 $d_1\in M(X)$ if and only if there exists some $d_2\in M(X)$ which, regarded as a $1$-cocycle of the system $(X,T)\times (X,T)$, is a coboundary.},
author = {Janoš, Ludvík},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {$B$-system; $E$-system},
language = {eng},
number = {3},
pages = {605-610},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {The Banach contraction mapping principle and cohomology},
url = {http://eudml.org/doc/22513},
volume = {41},
year = {2000},
}

TY - JOUR
AU - Janoš, Ludvík
TI - The Banach contraction mapping principle and cohomology
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2000
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 41
IS - 3
SP - 605
EP - 610
AB - By a dynamical system $(X,T)$ we mean the action of the semigroup $(\mathbb {Z}^+,+)$ on a metrizable topological space $X$ induced by a continuous selfmap $T:X\rightarrow X$. 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 $d_1\in M(X)$ if and only if there exists some $d_2\in M(X)$ which, regarded as a $1$-cocycle of the system $(X,T)\times (X,T)$, is a coboundary.
LA - eng
KW - $B$-system; $E$-system
UR - http://eudml.org/doc/22513
ER -

References

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  8. Nussbaum R., Some asymptotic fixed point theorem, Trans. Amer. Math. Soc. 171 (1972), 349-375. (1972) Zbl0256.47040MR0310719
  9. Opoitsev V.J., A converse to the principle of contracting maps, Russian Math. Surveys 31 (1976), 175-204. (1976) Zbl0351.54025
  10. Parry W., Tuncel S., Classification Problems in Ergodic Theory, London Math. Soc. Lecture Note Series 67, Cambridge University Press, Cambridge, 1982. Zbl0487.28014MR0666871
  11. Rus I.A., Weakly Picard mappings, Comment. Math. Univ. Carolinae 34 (1993), 769-773. (1993) Zbl0787.54045MR1263804
  12. Volný D., Coboundaries over irrational rotations, Studia Math. 126 (1997), 253-271. (1997) Zbl0890.28011MR1475922

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