Poincaré theory for compact abelian one-dimensional solenoidal groups
Abstract
This article presents a generalization of the notion of \emph{Poincaré rotation set} to homeomorphisms of the adèle 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 Poincaré. 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}
- Publication:
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arXiv e-prints
- Pub Date:
- August 2013
- DOI:
- 10.48550/arXiv.1308.1853
- arXiv:
- arXiv:1308.1853
- Bibcode:
- 2013arXiv1308.1853C
- Keywords:
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- Mathematics - Dynamical Systems
- E-Print:
- 15 pages