Erratum to “Generalizations of the Poincaré–Birkhoff theorem”

Abstract

Other results of [1] which rely on Theorem (2.1) are correct as stated and follow from the result above. A disk U in the covering space A is called positively returning for f if f(U)∩U = ∅ and fn(U)∩T k(U) = ∅ for some n, k > 0, where T is the generator of the infinite cyclic group of covering translations. Negatively returning disks are defined similarly. The difference in the statement of the theorem above and the statement of Theorem (2.1) of [1] is the additional requirement in item 3 that the positively and negatively returning disks in the covering space are lifts of disks in the annulus. Equivalently the positively and negatively returning disks in the covering space must be disjoint from their image under T k, k = 0 (standard arguments show it suffices that T (U) ∩ U = ∅). This additional hypothesis is satisfied in other results from [1] which use Theorem (2.1). I do not know whether Theorem (2.1) of [1], as originally stated, is true or not.