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In this work, we study eigenvalue distribution results of a class of highlynon-normal matrix-sequences, which can be viewed as a low rank perturbationdepending on a parameter $\beta>1$ of the basic Toeplitz matrix-sequence$\{T_n(e^{\mathbf{i}\theta})\}_{n\in\N}$, $\mathbf{i}^2=-1$. The latter hasobviously all eigenvalues equal to zero for any matrix order $n$, while for thematrix-sequence under consideration we will show a strong clustering at therange of the generating function $e^{\mathbf{i}\theta}$ i.e. at the complexunit circle. For $\beta\ge 2$ no outliers show up, while for $\beta \in (1,2)$only two outliers are present, which are both real, positive and have finitelimits equal to $\beta-1$ and $(\beta-1)^{-1}$, respectively. The problem looksmathematically innocent, but indeed it is quite challenging since all thesophisticated machinery for deducing the eigenvalue clustering is not easy toapply in the current setting and at most we may hope for weak clusteringresults. In the derivations, we resort to a trick already used for the spectralanalysis of the Google matrix plus several tools from complex analysis. We onlymention that the problem is not an academical curiosity and in fact it stemsfrom problems in dynamical systems and number theory. Numerical experiments inhigh precision are provided and more results are sketched for limit case of$\beta=1$.

The $\beta$ maps and the strong clustering at the complex unit circle