Rank-two solenoidal endomorphisms - Ha, Ku Yong; Lee, Jong Bum (Topological Methods in Nonlinear Analysis, 2023)
Let G be a torsion-free abelian group of rank two and let φ be an endomorphism of G, called a rank-two solenoidal endomorphism. Then it is represented by a 2× 2-matrix M_φ with rational entries. The purpose of this article is to prove the following: The group, coker(φ), of the cokernut of φ is finite if and only if M_φ is nonsingular, and if it is so, then we give an explicit formula for the order of coker(φ), [G:im(φ)], in terms of p-adic absolute values of the determinant of M_φ. Since G is abelian, the Reidemeister number of φ is equal to the order of the cokernut of id-φ and, when it is finite, it is equal to the number of fixed points of the Pontryagin dual φ of φ. Thereby, we solve completely the problem raised in Miles of finding the possible sequences of periodic point counts for all endomorphisms of the rank-two solenoids.