Kronecker Flow on the Infinite Torus

TL;DR

This paper studies Kronecker flows on the infinite torus, extending properties known in finite dimensions to infinite dimensions under certain conditions, and explores their applications to PDEs like the Benjamin-Ono equation.

Contribution

It extends the understanding of Kronecker flows to infinite-dimensional tori, characterizes their properties, and connects them to solutions of PDEs and classification problems.

Findings

Properties like topological transitivity and minimality hold in infinite dimensions under free abelian group conditions.

Constructs orbits with closures homeomorphic to a product of a ball and a Cantor set.

Shows the Benjamin-Ono equation admits solutions with these orbit structures.

Abstract

This article is concerned with Kronecker flows on the infinite torus. The work is partly motivated by the fact that many Hamiltonian PDEs and systems on infinite lattices admit invariant tori, of possibly infinite dimension, on which the dynamics is linearizable. Finite-dimensional Kronecker flows are well understood: the dynamics can be reduced to a non-resonant flow on a subtorus, which is equivalent to being topologically transitive, to minimality, and to unique ergodicity in the projection. We prove that these properties still hold when the dimension of the torus is infinite if and only if the integer (finite) linear combinations of the frequencies form a free abelian group. Next, we construct a class of orbits whose closure is locally homeomorphic to the product of a ball and a Cantor set, extending a recent result by Sakbaev and Volovich. We also show that the Benjamin-Ono equation admits this type of solutions. Finally, we prove the equivalence between a classification problem for Kronecker flows and that for countable abelian groups without torsion.