Rank-two solenoidal endomorphisms

Abstract

Let $G$ be a torsion-free abelian group of rank two and let $\phi$ be an endomorphism of $G$, called a rank-two \emph{solenoidal endomorphism}. Then it is represented by a $2\times 2$-matrix $M_\phi$ with rational entries. The purpose of this article is to prove the following: The group, $\mathrm{coker}(\phi)$, of the cokernut of $\phi$ is finite if and only if $M_\phi$ is nonsingular, and if it is so, then we give an explicit formula for the order of $\mathrm{coker}(\phi)$, $[G:\mathrm{im}(\phi)]$, in terms of $p$-adic absolute values of the determinant of $M_\phi$. Since $G$ is abelian, the Reidemeister number of $\phi$ is equal to the order of the cokernut of $\mathrm{id}-\phi$ and, when it is finite, it is equal to the number of fixed points of the Pontryagin dual $\widehat\phi$ of $\phi$. Thereby, we solve completely the problem raised in \cite{Miles} of finding the possible sequences of periodic point counts for \emph{all} endomorphisms of the rank-two solenoids.