Mathematics

Rotation sets of $\mathrm{SO(3)}$-extensions of quasiperiodic flows

A. N. Saharov

Nizhny Novgorod State Agrarian and Technological University named after L.Ya. Florentyev

Abstract: In this paper, we construct a class of special flows on a multidimensional torus and a topological invariant of such flows, i.e. a rotation set. Such flows arise while reducing linear systems of differential equations with quasiperiodic coefficients to a triangular form. In the process of such a reduction, we obtain a system of nonlinear differential equations on a multidimensional torus, which generates a projective flow induced by the original linear system. In this paper, we use known results from the theory of matrix groups and Lie algebras and construct an algorithm for $\mathrm{SO(n)}$-extension of a quasiperiodic linear system. The resulting system of equations admits a reduction in order, which allows us to write the right-hand sides as trigonometric polynomials in Euler angles on a sphere. The case $n=3$ is considered separately. The equations defining the projective flow are written explicitly. The projective flow is defined on a torus of dimension $m+2$, where $m$ is the dimension of the original torus. The structure of this flow is determined by topological invariants of the flow. For example, a non-singular flow on a two-dimensional torus has a topological invariant – the rotation number (A. Poincare). Using M. Herman's method, it is possible to prove the existence and uniqueness of the rotation vector $(\rho_1,\rho_2)$ for the projective flow on $\mathbb{T}^{m+2}$. Using S. Schwartzman's theory defining the rotation set for flows on compact metric spaces, it is shown that the component $\rho_2=0$. Here, the fact is used that the dimension of the maximal toric subalgebra of the algebra $\mathrm{so}(3)$ is equal to one.

Keywords: linear extensions, group extension, projective extension, toric subalgebra, rotation vector, asymptotic cycles

UDC: 517.93

MSC: 34D20

Received: 15.02.2025
Accepted: 28.05.2025

DOI: 10.15507/2079-6900.27.202502.171-184