Rocky Mountain Journal of Mathematics

An abelian group associated with topological dynamics

Kazuhiro Kawamura

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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

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

First available in Project Euclid: 26 January 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

Topological transitivity weighted composition operator Banach space of continuous functions


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.

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