Homeomorphisms of the real line with singularities.

Given a real number a ≠ 0, we consider the set of homeomorphisms f : R \ {0} → R \ {a} where {(x, y) : x = 0} is a vertical asymtote, {(x, y) : y = a} is a horizontal asymtote and f is strictly increasing in each connected component (-∞, 0) and (0,+∞). In this context, similar to circle homeomorphisms, all possible dynamics are shown. It is established the relationship between existence of periodic orbits and the limit sets. Also, whenever f-n(0) ≠ a for all n ∈ N, then the non-existence of periodic orbits leads to a non-trivial limit set, which is either the whole line R or perfect and nowhere dense. It is shown a notion of separation of points that leads to transitivity. [ABSTRACT FROM AUTHOR]