Vol 216, No 11 (2025)

For groups $G\subseteq\operatorname{Homeo}+({\mathbb R})$ of orientation-preserving line homeomorphisms with a nonempty minimal set a new criterion is obtained for the existence of a projectively invariant Borel measure finite on compact sets. It is shown that the existence of a projectively invariant Borel measure finite on compact sets is equivalent to the metabelianity of the canonical quotient group $ G/HG$, where the normal subgroup $HG$ consists of the homeomorphisms in $G$ that fix all points in the minimal set. It is shown that for groups $G\subseteq\operatorname{Homeo}+({\mathbb R})$ of orientation-preserving line homeomorphisms with a nonempty minimal set, in the space of quotient groups $G/HG$ the class of metabelian groups coincides with the class of groups with finite normal series whose quotients contain no free subsemigroups with two generators, and the class of Abelian groups coincides with the class of groups not containing free subsemigroups with two generators. On this basis, for the class of solvable groups $G\subseteq\operatorname{Homeo}+({\mathbb R})$ of orientation-preserving line homeomorphisms with a nonempty minimal set, a nontrivial quotient group and without a freely acting homeomorphism it is shown that it is combinatorially complex: such a group is not a group with finite normal series the quotients of whose terms contain no free subsemigroups with two generators.

We study the asymptotic behaviour of small deviation probabilities for a critical Galton–Watson process with infinite variance of the offspring sizes of particles and apply the result obtained to investigate the structure of a reduced critical Galton-Watson process.

For flows satisfying Smale's Axiom A on closed manifolds of dimension $n\ge3$ the structure of codimension-one basis sets is described, which are either extending attractors or contracting repellers. For such nonmixing basis sets special trapping neighbourhoods with boundary components homeomorphic to ${\mathbb S}^{n-2}\times{\mathbb S}^1$ are constructed. This makes it possible to construct a compactification (support) of the basin of a basis set. which is a locally trivial fibre bundle over the circle, and the extension of the original flow to the support is a dynamical suspension and a structurally stable flow of attractor-repeller type.

We study diagonal arrangement complements $D(\mathcal K)$ in $\mathbb C^m$. We consider the class of simplicial complexes $\mathcal K$ in which any two missing faces have a common vertex, and we prove that the coordinate arrangement complement $U(\mathcal K)$ is the double suspension of the diagonal arrangement complement $D(\mathcal K)$. In the case of subspace arrangements in $\mathbb R^m$ the coordinate arrangement complement $U_{\mathbb R}(\mathcal K)$ is the single suspension of $D_{\mathbb R}(\mathcal K)$.