Some aspects of rotation theory on compact abelian groups
Volume 161 / 2020
Colloquium Mathematicum 161 (2020), 131-155
MSC: Primary 22-XX, 37-XX; Secondary 22Cxx, 37Axx, 37Bxx.
DOI: 10.4064/cm7593-12-2018
Published online: 24 March 2020
Abstract
We present a generalization of Poincaré’s rotation theory of homeomorphisms of the circle to the case of one-dimensional compact abelian groups which are solenoidal groups, i.e., groups which fiber over the circle with fiber a Cantor abelian group. We define rotation elements à la Poincaré and discuss the dynamical properties of translations on these solenoidal groups. We also study the semiconjugation problem when the rotation element generates a dense subgroup of the solenoidal group. Finally, we comment on the relation between rotation theory and entropy for these homeomorphisms, since unlike the case of the circle, for the solenoids considered here there are homeomorphisms (not homotopic to the identity) with positive entropy.
