On covering mappings on solenoids

Abstract

Abstract. Covering mappings on the dyadic solenoid are studied. Some results stated by Zhou Youcheng (2000) are discussed in a more detailed way by indicating certain inaccuracies in their proofs. These are either corrected or supplemented, or else suitable counterexamples are constructed. Some open questions are asked and connections with related results are considered.In this paper a space always means a metric space, and a continuum is defined as a compact connected space. A mapping means a continuous function. We denote by N the set of all positive integers, by C the space of all complex numbers (equipped with the natural topology), and by S 1 the unit circle, i.e., S 1 = {z ∈ C : |z| = 1}. As usual, i means the imaginary unit, i.e., i 2 = −1, and exp stands for the exponential function, i.e., exp(iα) = cos α + i sin α.

Two families of Exel–Larsen crossed products

“…The P -adic solenoid is the inverse limit Σ P := lim ← − (T, z → z p i ), which is a compact abelian group. We will use results of Charatonik-Covarrubias [9] and Gumerov [17] to study C(Σ P ) ⋊ α,L N × .…”