Vol 213, No 2 (2022)

We study billiards on compact connected domains in $\mathbb{R}^3$ bounded by a finite number of confocal quadrics meeting in dihedral angles equal to ${\pi}/{2}$. Billiards in such domains are integrable due to having three first integrals in involution inside the domain. We introduce two equivalence relations: combinatorial equivalence of billiard domains determined by the structure of their boundaries, and weak equivalence of the corresponding billiard systems on them. Billiard domains in $\mathbb{R}^3$ are classified with respect to combinatorial equivalence, resulting in 35 pairwise nonequivalent classes. For each of these obtained classes, we look for the homeomorphism class of the nonsingular isoenergy 5-manifold, and we show this to be one of three types: either $S^5$, or $S^1\times S^4$, or $S^2\times S^3$. We obtain 24 classes of pairwise nonequivalent (with respect to weak equivalence) Liouville foliations of billiards on these domains restricted to a nonsingular energy level. We also define bifurcation atoms of three-dimensional tori corresponding to the arcs of the bifurcation diagram. Bibliography: 59 titles.

On an infinite-dimensional torus $\mathbb{T}^{\infty} = E/2\pi\mathbb{Z}^{\infty}$, where $E$ is an infinite-dimensional real Banach space and $\mathbb{Z}^{\infty}$ is an abstract integer lattice, a special class of diffeomorphisms $\operatorname{Diff}(\mathbb{T}^{\infty})$ is considered. It consists of the maps $G\colon \mathbb{T}^{\infty}\to\mathbb{T}^{\infty}$ equal to sums of invertible bounded linear operators preserving $\mathbb{Z}^{\infty}$ and $C^1$-smooth periodic additives. Necessary and sufficient conditions ensuring that such maps are hyperbolic (that is, are Anosov diffeomorphisms) are obtained. Bibliography: 15 titles.

The Chmutov-Lando theorem claims that the value of a weight system (a function on the chord diagrams that satisfies the four-term Vassiliev relations) corresponding to the Lie algebra $\mathfrak{sl}_2$ depends only on the intersection graph of the chord diagram. We compute the values of the $\mathfrak{sl}_2$ weight system at the graphs in several infinite series, which are the joins of a graph with a small number of vertices and a discrete graph. In particular, we calculate these values for a series in which the initial graph is the cycle on five vertices; the graphs in this series, apart from the initial one, are not intersection graphs. We also derive a formula for the generating functions of the projections of graphs equal to the joins of an arbitrary graph and a discrete graph to the subspace of primitive elements of the Hopf algebra of graphs. Using the formula thus obtained, we calculate the values of the $\mathfrak{sl}_2$ weight system at projections of the graphs of the indicated form onto the subspace of primitive elements. Our calculations confirm Lando's conjecture concerning the values of the $\mathfrak{sl}_2$ weight system at projections onto the subspace of primitives. Bibliography: 17 titles.