Vol 214, No 2 (2023)

The topology of the Liouville foliations of integrable magnetic topological billiards, systems in which a ball moves on piecewise smooth two-dimensional surfaces in a constant magnetic field, is considered. The Fomenko-Zieschang invariants of Liouville equivalence are calculated for the Hamiltonian systems arising, and the topology of invariant 3-manifolds, isointegral and isoenergy ones, is investigated. The Liouville equivalence of such billiards to some known Hamiltonian systems is discovered, for instance, to the geodesic flows on 2-surfaces and to systems of rigid body dynamics. In particular, peculiar saddle singularities are discovered in which singular circles have different orientations — such systems were also previously encountered in mechanical systems in a magnetic field on surfaces of revolution homeomorphic to a 2-sphere. Bibliography: 13 titles.

For any $\mathbb{A}^1$-invariant cohomology theory that satisfies Nisnevich excision on the category of smooth schemes over a field $k$ it is proved that the Cousin complex on the complement $U-D$ to the strict normal-crossing divisor $D$ in a local essentially smooth scheme $U$ is acyclic. This claim is also proved for the schemes $(X-D)\times(\mathbb{A}^1_k-Z_0)\times…\times(\mathbb{A}^1_k-Z_l)$, where $Z_0,…,Z_l$ is a finite family of closed subschemes in the affine line over $k$. Bibliography: 32 titles.

Conditions on spaces $X$ with generalized distance $\rho_X$ are investigates under which analogues of Banach's and Nadler's fixed-point theorems and Arutyunov's coincidence-point theorem can be obtained for mappings on such spaces. This is shown to hold if each geometric progression with ratio $<1$ (that is, each sequence $\{ x_i\}\subset X$ satisfying $\rho_X(x_{i+1},x_i)\leq \gamma \rho_X(x_i,x_{i-1})$, $ i=1,2,…$, with some $\gamma < 1$) is convergent. Examples of spaces with and without this property are given. In particular, the required property holds in a complete $f$-quasimetric space $X$ if the distance $\rho_X$ in it satisfies $\rho_X(x,z) \leq \rho_X(x,y)+(\rho_X(y,z))^\eta$, $x,y,z \in X$, for some $\eta\in (0,1)$, that is, if the function $f\colon\mathbb{R}_+^{2} \to \mathbb{R}_+$ is given by $f(r_1,r_2)=r_1 + r_2^{\eta}$. Next, for $f(r_1,r_2)=\max\{ r_1^{\eta}, r_2^{\eta} \}$, where $\eta \in (0,2^{-1}]$, it is shown that for any $\gamma > 0$ there exists an $f$-quasimetric space containing a geometric progression with ratio $\gamma$ which is not a Cauchy sequence. The ‘zero-one law’, which means that either each geometric progression with ratio $<1$ is a Cauchy sequence or, for any $\gamma\in (0,1)$, there exists a geometric progression with ratio $\gamma$ that is not Cauchy, is discussed for $f$-quasimetric spaces. Bibliography: 29 titles.