Description

• Title:
  Rank-two solenoidal endomorphisms
• Creator:
  Ha, Ku Yong;
  Lee, Jong Bum
• Description:
  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.
• Object availability:
  free access
• Rights: https://creativecommons.org/licenses/by-nd/4.0 ;
  Prawa autorskie (c) 2023 Ku Yong Ha, Jong Bum Lee
• Date:
  2023.02.26
• Type:
  info:eu-repo/semantics/article;
  info:eu-repo/semantics/publishedVersion;
  Recenzowany artykuł
• Language:
  English
• Source:
  Topological Methods in Nonlinear Analysis; Vol 61, No 1 (March 2023); 291 - 329;
  1230-3429
• Publisher:
  Nicolaus Copernicus University in Toruń
• Relation: https://apcz.umk.pl/TMNA/article/view/42825/34992 ;https://apcz.umk.pl/TMNA/article/view/42825/34993
• Subject:
  $p$-adic absolute value;
  Pontryagin dual;
  Reidemeister number;
  solenoid;
  solenoidal endomorphism;
  subgroup index;
  torsion-free abelian group;
  Primary: 37C25;
  Secondary: 55M20
• Identifier: https://apcz.umk.pl/TMNA/article/view/42825
• Data provider:
  Akademicka Platforma Czasopism
• Can I use it?:
  seek permission
• Type:
  other