ИСТИНА

Let $V\cong\mathbb R^k$ be a $k$-dimensional real vector space, and let $\Gamma=\{\gamma_1,\ldots,\gamma_m\}$ be a sequence (a \emph{configuration}) of $m$ vectors in the dual space~$V^*$. We consider the action of $V$ on the complex space $\mathbb C^m$ given by \begin{equation}\label{ract} \begin{aligned} V\times\mathbb C^m&\longrightarrow\mathbb C^m\\ (\mb v,\mb z)&\mapsto \bigl(z_1e^{\langle\gamma_1,\mb v\rangle},\ldots,z_me^{\langle\gamma_m,\mb v\rangle}\bigr). \end{aligned} \end{equation} This is a very classical dynamical system taking its origin in the works of Poincar\'e. There is a well-known relationship between linear properties of the vector configuration $\Gamma$ and the topology of the foliation of $\mathbb C^m$ by the orbits of~\eqref{ract}. We systematise the existing knowledge on this relationship and proceed by analysing the topology of the nondegenerate leaf space using some recent constructions of toric topology.