Poincar\'e theory for compact abelian one-dimensional solenoidal groups

TL;DR

This paper extends Poincaré rotation theory to homeomorphisms of one-dimensional solenoidal groups, generalizing classical circle rotation results to more complex compact abelian groups using Pontryagin duality.

Contribution

It introduces a new notion of rotation set for homeomorphisms of solenoidal groups, generalizing classical rotation theory to these groups via Pontryagin duality.

Findings

Rotation sets are closed intervals or single points in the solenoid context.

Homeomorphisms with irrational rotation numbers are semiconjugate to translations.

The theory applies to all one-dimensional compact abelian solenoidal groups.

Abstract

This article presents a generalization of the notion of \emph{Poincar\'e rotation set} to homeomorphisms of the ad\`ele class group $\mathbb{A}/\mathbb{Q}$ of the rational numbers $\mathbb{Q}$, which is a connected compact abelian group which can be identified with the one-dimensional universal solenoid $\mathbf{S}$, \ie the algebraic universal covering of the circle. The definition is first introduced in general for homeomorphisms of $\mathbf{S}$ which are isotopic to a translation, and then specializing in homeomorphisms of $\mathbf{S}$ isotopic to the identity, in which case the rotation set is a closed interval contained in the base leaf (the connected component of the identity). If in the latter case the rotation interval reduces to a single element $\rho$ and $\rho$ is irrational (\ie it is a monothetic generator of $\mathbf{S}$), we show that the homeomorphism is semiconjugate to the translation $z\mapsto\rho{z}$, like in the classical theory of Poincar\'e. This theory is valid for any general compact abelian one dimensional solenoidal group $\mathbf{S}_G$, which are Pontryagin duals of dense subgroups $G$ of the rational numbers with the discrete topology. These solenoidal groups are one-dimensional laminations which are locally homeomorphic to the product of a Cantor set by an interval so they behave very much like a ``diffuse'' version of the circle. Our approach differs from others because we use Pontryagin duality of compact abelian groups to define the rotation sets. \end{abstract}