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卷 216, 编号 11 (2025)
On the metabelianity of the canonical quotient groups of orientation-preserving line homeomorphisms
摘要
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.
Matematicheskii Sbornik. 2025;216(11):3-40
3-40
Support of the measure in an integral representation for a Nevanlinna function defined by a limit periodic continued fraction
摘要
In terms of a continued fraction with limit-periodic parameters, which is similar to a Schur continued fraction and converges to the Nevanlinna function, a representation for the support of the measure associated with the limiting Nevanlinna function is obtained.
Matematicheskii Sbornik. 2025;216(11):41-61
41-61
Probabilities of small deviations of a critical Galton–Watson process with infinite variance of the number of the direct descendants of particles
摘要
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.
Matematicheskii Sbornik. 2025;216(11):62-89
62-89
Some lower bounds for optimal sampling recovery of functions with mixed smoothness
摘要
Recently there was a substantial progress in the problem of sampling recovery on function classes with mixed smoothness. It was mostly done by proving new and sometimes optimal upper bounds for both linear sampling recovery and nonlinear sampling recovery. In this paper we address the problem of lower bounds for the optimal rates of nonlinear sampling recovery. In the case of linear recovery one can use the well-developed theory of estimating the Kolmogorov and linear widths to establish some lower bounds for the optimal rates. In the case of nonlinear recovery we cannot use the above approach. It seems like the only technique which is available now is based on some simple observations. We demonstrate how these observations can be used.
Matematicheskii Sbornik. 2025;216(11):90-107
90-107
A-flows with codimension one basic sets
摘要
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.
Matematicheskii Sbornik. 2025;216(11):108-134
108-134
Unique expansions in number systems via refinement equations
摘要
Using the subdivision scheme theory we develop a criterion to check if each natural number has at most one representation in the $n$ -ary number system with a set of nonnegative integer digits $A=\{a_1, a_2,…, a_n\}$ that contains zero. This uniqueness property is shown to be equivalent to a certain restriction on the zeros of the trigonometric polynomial $\sum_{k=1}^n e^{-2\pi i a_k t}$ . From this criterion, under a natural condition of irreducibility for $A$ , we deduce that in the case of a prime number $n$ uniqueness holds if and only if the digits of $A$ are distinct modulo $n$ , whereas for any composite $n$ we show that the latter condition is not necessary. We also establish a connection of this uniqueness with the problem of semigroup freeness for affine integer functions of equal integer slope; in combination with the two criteria, this allows us to fill the gap in a work of Klarner on the question of Erdős about the densities of affine integer orbits and to establish a simple algorithm to check freeness and the positivity of density in the case when the slope is a prime number.
Matematicheskii Sbornik. 2025;216(11):135-149
135-149
Connection between coordinate and diagonal arrangement complements
摘要
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 $D_{\mathbb R}(\mathcal K)$ .
Matematicheskii Sbornik. 2025;216(11):150-166
150-166