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2015 An abelian group associated with topological dynamics
Kazuhiro Kawamura
Rocky Mountain J. Math. 45(5): 1457-1470 (2015). DOI: 10.1216/RMJ-2015-45-5-1457


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.


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Kazuhiro Kawamura. "An abelian group associated with topological dynamics." Rocky Mountain J. Math. 45 (5) 1457 - 1470, 2015.


Published: 2015
First available in Project Euclid: 26 January 2016

zbMATH: 1356.37015
MathSciNet: MR3452223
Digital Object Identifier: 10.1216/RMJ-2015-45-5-1457

Primary: ‎37B05‎
Secondary: ‎46E15

Keywords: Banach space of continuous functions , topological transitivity , Weighted composition operator

Rights: Copyright © 2015 Rocky Mountain Mathematics Consortium


Vol.45 • No. 5 • 2015
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