Article

 Title: The Banach contraction mapping principle and cohomology (English) Author: Janoš, Ludvík Language: English Journal: Commentationes Mathematicae Universitatis Carolinae ISSN: 0010-2628 (print) ISSN: 1213-7243 (online) Volume: 41 Issue: 3 Year: 2000 Pages: 605-610 . Category: math . Summary: 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. (English) Keyword: $B$-system Keyword: $E$-system MSC: 37B25 MSC: 37B99 MSC: 47H10 MSC: 54H15 MSC: 54H20 MSC: 54H25 idZBL: Zbl 1087.37502 idMR: MR1795089 . Date available: 2009-01-08T19:05:33Z Last updated: 2012-04-30 Stable URL: http://hdl.handle.net/10338.dmlcz/119193 . Reference: [1] Edelstein M.: On fixed and periodic points under contractive mappings.J. London Math. Soc. 37 (1962), 74-79. Zbl 0113.16503, MR 0133102 Reference: [2] Huissi M.: Sur les solutions globales de l'equation des cocycles.Aequationes Math. 45 (1993), 195-206. MR 1212385 Reference: [3] Iwanik A.: Ergodicity for piecewise smooth cocycles over toral rotations.Fund. Math. 157 (1998), 235-244. MR 1636890 Reference: [4] Janoš L.: A converse of Banach's contraction theorem.Proc. Amer. Math. Soc. 18 (1967), 287-289. Zbl 0148.43001, MR 0208589 Reference: [5] Janoš L., Ko H.M., Tau K.K.: Edelstein's contractivity and attractors.Proc. Amer. Math. Soc. 76 (1979), 339-344. MR 0537101 Reference: [6] Meyers P.R.: A converse to Banach's contraction theorem.J. Res. Nat. Bureau of Standard 71B (1967), 73-76. Zbl 0161.19803, MR 0221469 Reference: [7] Moore C., Schmidt K.: Coboundaries and homomorphisms for nonsingular actions and a problem of H. Helson.Proc. London Math. Soc. 40 (1980), 443-475. MR 0572015 Reference: [8] Nussbaum R.: Some asymptotic fixed point theorem.Trans. Amer. Math. Soc. 171 (1972), 349-375. MR 0310719 Reference: [9] Opoitsev V.J.: A converse to the principle of contracting maps.Russian Math. Surveys 31 (1976), 175-204. Zbl 0351.54025 Reference: [10] Parry W., Tuncel S.: Classification Problems in Ergodic Theory.London Math. Soc. Lecture Note Series 67, Cambridge University Press, Cambridge, 1982. Zbl 0487.28014, MR 0666871 Reference: [11] Rus I.A.: Weakly Picard mappings.Comment. Math. Univ. Carolinae 34 (1993), 769-773. Zbl 0787.54045, MR 1263804 Reference: [12] Volný D.: Coboundaries over irrational rotations.Studia Math. 126 (1997), 253-271. MR 1475922 .

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