Rocky Mountain Journal of Mathematics

An abelian group associated with topological dynamics

Kazuhiro Kawamura

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Abstract

For a continuous surjection $T:X \to X$ on a compact metric space $X$ and a unimodular continuous weight on $X$, we consider a weighted composition operator $U_{T,w}$ on the Banach space $C(X)$ of complex-valued continuous functions on $X$ with the sup norm. The set ${\mathcal W}_{T}$ of all weights $w$, for which the operator $U_{T,w}$ has an eigenvalue with a unimodular eigenfunction, forms a topological abelian group. The group ${\mathcal W}_{T}$ admits a homomorphism $W_T$ to the first integral \v{C}ech cohomology of the space $X$. The image and the kernel of $W_T$ carry topological and ergodic aspects of the dynamics $T$. A concrete description of $\operatorname{Im}W_{T}$ and $\operatorname{Ker}W_{T}$ is given for positively expansive eventually-onto open maps (under an assumption on the induced homomorphism of the first \v{C}ech cohomology) and minimal rotations on tori.

Article information

Source
Rocky Mountain J. Math., Volume 45, Number 5 (2015), 1457-1470.

Dates
First available in Project Euclid: 26 January 2016

Permanent link to this document
https://projecteuclid.org/euclid.rmjm/1453817249

Digital Object Identifier
doi:10.1216/RMJ-2015-45-5-1457

Mathematical Reviews number (MathSciNet)
MR3452223

Zentralblatt MATH identifier
1356.37015

Subjects
Primary: 37B05: Transformations and group actions with special properties (minimality, distality, proximality, etc.)
Secondary: 46E15: Banach spaces of continuous, differentiable or analytic functions

Keywords
Topological transitivity weighted composition operator Banach space of continuous functions

Citation

Kawamura, Kazuhiro. An abelian group associated with topological dynamics. Rocky Mountain J. Math. 45 (2015), no. 5, 1457--1470. doi:10.1216/RMJ-2015-45-5-1457. https://projecteuclid.org/euclid.rmjm/1453817249


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References

  • N. Aoki, Expanding maps of solenoids, Monatsh. Math. 105 (1988), 1–34.
  • ––––, Topological dynamics, in Topics in general topology, K. Morita and J. Nagata, eds., North-Holland, Amsterdam, 1989.
  • N. Aoki and K. Hiraide, Topological theory of dynamical systemsRecent advances, North-Holland Math. Lib. 52 (1994), North-Holland.
  • L Baggett, On circle-valued cocycles of an ergodic measure-preserving transformation, Israel J. Math. 61 (1988), 29–38.
  • M. Bestvina, Characterizing $k$-dimensional universal Menger compactum, Mem. Amer. Math. Soc. 71, 1988.
  • F.E. Browder, On the iteration of transformations in non-compact minimal dynamical systems, Proc. Amer. Math. Soc. 9 (1958), 773–780.
  • A. Chigogidze, K. Kawamura and E.D. Tymchatyn, Menger manifolds, in Continua with Houston problem book, Marcel Dekker, New York, 1995.
  • E.M. Coven and W.L. Reddy, Positively expansive maps of compact manifolds, in Global Theory of Dynamical Systems, Lecture Notes in Math. 819 (1980), 96-110.
  • W.H. Gottschalk and G.A. Hedlund, Topological dynamics, Amer. Math. Soc. Colloq. Publ. 36, 1955.
  • K. Hiraide, Positively expansive open maps of Peano spaces, Topol. Appl. 37 (1990), 213–220.
  • K.H. Hoffmann and S. Morris, The structure of compact groups, Gruyter Stud. Math. 25, Walter de Gruyter, 1998.
  • A. Katok and B. Hasselblatt, An introduction to modern theory of dynamical systems, Cambridge University Press, Cambridge, 1995.
  • K. Kawamura, A survey on Menger manifold theoryUpdate, Topol. Appl. 101 (2000), 83–91.
  • P. Liardet and D. Voln\' y, Sums of continuous and differentiable functions in dynamical systems, Israel J. Math. 98 (1997), 29–60.
  • A.N. Quas, Rigidity of continuous coboundaries, Bull. Lond. Math. Soc. 29 (1997), 595–600.
  • I. Rosenholtz, Local expansions, derivatives and fixed points, Fund. Math. 91 (1976), 1–4.
  • K. Sakai, Periodic points of positively expansive maps, Proc. Amer. Math. Soc. 94 (1985), 531–534.
  • ––––, Various shadowing properties for positively expansive maps, Topol. Appl. 131 (2003), 15–31.
  • M. Shub, Endomorphisms of compact differentiable manifolds, Amer. J. Math. 91 (1969), 175–199.
  • C.W. Stark, Minimal dynamics on Menger manifolds, Topol. Appl. 90 (1998), 21–30.
  • P. Walters, An introduction to ergodic theory, G.T.M. 79, Springer, Berlin, 1982.