Wandering domains and invariant conformal structures for mappings of the 2-torus

A C¹ circle diffeomorphism with irrational rotation number need not have any dense orbits. However, any C² circle diffeomorphism with irrational rotation number must in fact be topologically conjugate to an irrational rotation. This paper addresses the analogous matter for the 2-torus. We say that a diffeomorphism f of T², isotopic to the identity, has Denjoy type if hf = Rh, where R is some minimal translation of the torus, and h is a continuous torus mapping homotopic to the identity such that {cursive Greek chi ∈ T² : cardinality (h^-1(cursive Greek chi)) > 1} is nonempty and countable. If f has Denjoy type, the interior of any fiber h^-1(cursive Greek chi), if nonempty, is a wandering domain for f. It is known that there are C² diffeomorphisms of Denjoy type, but not known whether they can be C³. Our main results imply the following Theorem. Let f ∈ Diff₁(T²) have Denjoy type, with minimal set Γ ≠ T². (i) If f preserves a measurable, essentially bounded conformal structure on Γ, then the collection {fⁿ} (considered as mappings of the ideal boundaries of the wandering domains) has unbounded quasisymmetric distortion, and (ii) if f preserves a C^1+Z conformal structure on Γ, then f cannot be C^2+Z. A simple corollary of (ii) is that no C³ diffeomorphism of Denjoy type exists with, for example, circular wandering domains.