In this paper, we study the space of translational limits T (M) of a surface M properly embedded in R3 with nonzero constant
mean curvature and bounded second fundamental form. There is a natural map T which assigns to any surface Σ ∈ T (M) the set
T (Σ) ⊂ T(M). Among various dynamics type results we prove that surfaces in minimal T -invariant sets of T (M) are chord-arc.
We also show that if M has an infinite number of ends, then there exists a nonempty minimal T -invariant set in T (M) consisting
entirely of surfaces with planes of Alexandrov symmetry. Finally, when M has a plane of Alexandrov symmetry, we prove the following
characterization theorem: M has finite topology if and only if M has a finite number of ends greater than one.