The Banach contraction mapping principle and cohomology
Janoš, Ludvík
Commentationes Mathematicae Universitatis Carolinae, Tome 41 (2000), p. 605-610 / Harvested from Czech Digital Mathematics Library
Access to full text
Full (PDF)

By a dynamical system $(X,T)$ we mean the action of the semigroup $(\Bbb Z^+,+)$ 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.

Publié le : 2000-01-01
Classification:  37B25,  37B99,  47H10,  54H15,  54H20,  54H25 
@article{119193,
     author = {Ludv\'\i k Jano\v s},
     title = {The Banach contraction mapping principle and cohomology},
     journal = {Commentationes Mathematicae Universitatis Carolinae},
     volume = {41},
     year = {2000},
     pages = {605-610},
     zbl = {1087.37502},
     mrnumber = {1795089},
     language = {en},
     url = {http://dml.mathdoc.fr/item/119193}
}

Janoš, Ludvík. The Banach contraction mapping principle and cohomology. Commentationes Mathematicae Universitatis Carolinae, Tome 41 (2000) pp. 605-610. http://gdmltest.u-ga.fr/item/119193/

Edelstein M. On fixed and periodic points under contractive mappings, J. London Math. Soc. 37 (1962), 74-79. (1962) | MR 0133102 | Zbl 0113.16503

Huissi M. Sur les solutions globales de l'equation des cocycles, Aequationes Math. 45 (1993), 195-206. (1993) | MR 1212385

Iwanik A. Ergodicity for piecewise smooth cocycles over toral rotations, Fund. Math. 157 (1998), 235-244. (1998) | MR 1636890

Janoš L. A converse of Banach's contraction theorem, Proc. Amer. Math. Soc. 18 (1967), 287-289. (1967) | MR 0208589 | Zbl 0148.43001

Janoš L.; Ko H.M.; Tau K.K. Edelstein's contractivity and attractors, Proc. Amer. Math. Soc. 76 (1979), 339-344. (1979) | MR 0537101

Meyers P.R. A converse to Banach's contraction theorem, J. Res. Nat. Bureau of Standard 71B (1967), 73-76. (1967) | MR 0221469 | Zbl 0161.19803

Moore C.; Schmidt K. Coboundaries and homomorphisms for nonsingular actions and a problem of H. Helson, Proc. London Math. Soc. 40 (1980), 443-475. (1980) | MR 0572015

Nussbaum R. Some asymptotic fixed point theorem, Trans. Amer. Math. Soc. 171 (1972), 349-375. (1972) | MR 0310719

Opoitsev V.J. A converse to the principle of contracting maps, Russian Math. Surveys 31 (1976), 175-204. (1976) | Zbl 0351.54025

Parry W.; Tuncel S. Classification Problems in Ergodic Theory, London Math. Soc. Lecture Note Series 67, Cambridge University Press, Cambridge, 1982. | MR 0666871 | Zbl 0487.28014

Rus I.A. Weakly Picard mappings, Comment. Math. Univ. Carolinae 34 (1993), 769-773. (1993) | MR 1263804 | Zbl 0787.54045

Volný D. Coboundaries over irrational rotations, Studia Math. 126 (1997), 253-271. (1997) | MR 1475922