Vol 212, No 5 (2021)

An arrow matrix is a matrix with zeros outside the main diagonal, the first row and the first column. We consider the space $M_{\operatorname{St}_n,\lambda}$ of Hermitian arrow $(n+1)\times (n+1)$-matrices with fixed simple spectrum $\lambda$. We prove that this space is a smooth $2n$-manifold with a locally standard torus action: we describe the topology and combinatorics of its orbit space. If $n\geqslant 3$, the orbit space $M_{\operatorname{St}_n,\lambda}/T^n$ is not a polytope, hence $M_{\operatorname{St}_n,\lambda}$ is not a quasitoric manifold. However, there is an action of a semidirect product $T^n\rtimes\Sigma_n$ on $M_{\operatorname{St}_n,\lambda}$, and the orbit space of this action is a certain simple polytope $\mathscr{B}^n$ obtained from the cube by cutting off codimension-2 faces. In the case $n=3$, the space $M_{\operatorname{St}_3,\lambda}/T^3$ is a solid torus with boundary subdivided into hexagons in a regular way. This description allows us to compute the cohomology ring and equivariant cohomology ring of the 6-dimensional manifold $M_{\operatorname{St}_3,\lambda}$ and another manifold, its twin. Bibliography: 32 titles.

Interpolation sequences of the form $\{\pm\lambda_n\}$ $(\lambda_n > 0)$ are investigated, and also the problem of when the system of exponentials $\{e^{\pm\lambda_n z}\}$ is nonspanning on the family of arbitrary rectifiable curves in the uniform norm.In terms of the interpolation nodes (or equivalently, the exponents of the system of exponentials) a criterion for the interpolation problem to be solvable is established and the strong nonspanning property of $\{e^{\pm\lambda_n z}\}$ is proved. This significantly improves some known results, in particular, results due to Korevaar, Dixon and Berndtsson.Bibliography: 23 titles.

It is proved that in each homotopy class of continuous mappings of the two-dimensional torus to itself that induce a hyperbolic action on the fundamental group, as long as it is free of expanding mappings, there exists an $A$-endomorphism $f$ whose nonwandering set consists of an attracting hyperbolic sink and a nontrivial one-dimensional collapsing repeller, which is a one-dimensional orientable lamination, locally homeomorphic to the direct product of a Cantor set and a line segment. Moreover, the unstable $Df$-invariant subbundle of the tangent space to the repeller has the property of uniqueness. Bibliography: 23 titles.