Vol 212, No 12 (2021)

The homeomorphism class of the isoenergy surface of a billiard book, of low complexity and not necessarily integrable, is determined using methods of low-dimensional topology. In particular, a series of billiard books is constructed that realize isoenergy 3-surfaces homeomorphic to the connected sum of a number of lens spaces and direct products $S^1\times S^2$.The Fomenko-Zieschang invariants, which classify Liouville foliations on isoenergy surfaces up to fibrewise homeomorphisms – that is, up to Liouville equivalence of the corresponding integrable Hamiltonian systems – are calculated for several integrable billiards of this type.Bibliography: 14 titles.

Given a tuple of $m+1$ germs of arbitrary analytic functions at a fixed point, we introduce the polynomial Hermite-Pade $m$-system, which includes the Hermite-Pade polynomials of types I and II. In the generic case we find the weak asymptotics of the polynomials of the Hermite-Pade $m$-system constructed from the tuple of germs of functions $1, f_1,…,f_m$ that are meromorphic on an $(m+1)$-sheeted compact Riemann surface $\mathfrak R$. We show that if $f_j = f^j$ for some meromorphic function $f$ on $\mathfrak R$, then with the help of the ratios of polynomials of the Hermite-Pade $m$-system we recover the values of $f$ on all sheets of the Nuttall partition of $\mathfrak R$, apart from the last sheet. Bibliography: 18 titles.

Let $R$ be a $p$-adically complete ring equipped with a $\delta$-structure. We construct a functorial group homomorphism from the Milnor $K$-group $K^{M}_{n}(R)$ to the quotient of the $p$-adic completion of the module of differential forms $\widehat{\Omega}^{n-1}_{R}/d\widehat{\Omega}^{n-2}_{R}$. This homomorphism is a $p$-adic analogue of the Bloch map defined for the relative Milnor $K$-groups of nilpotent extensions of rings of nilpotency degree $N$ for which the number $N!$ is invertible. Bibliography: 12 titles.