## Rocky Mountain Journal of Mathematics

### An abelian group associated with topological dynamics

Kazuhiro Kawamura

#### 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

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

#### 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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