An abelian group associated with topological dynamics

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.