Write S = {z ∈ C : |z| = 1}. A character of a locally compact abelian group G is a continuous group homomorphism G → S. We denote by Ĝ the set of characters of G, where for φ1, φ2 ∈ Ĝ and x ∈ G, we define (φ1φ2)(x) = φ1(x)φ2(x). We assign Ĝ the final topology for the family of functions {φ 7→ φ(x) : x ∈ G}, i.e., the coarsest topology on Ĝ so that for each x ∈ G, the function φ 7→ φ(x) is continuous Ĝ→ S. With this topology, it is a fact that Ĝ is itself a locally compact abelian group, called… Expand

It is known that the Collatz Conjecture (and the study of similar maps, here called "Hydra maps") can be stated in terms of solution sets of functional equations; or, equivalently, the fixed points… Expand