Erratum to: Transitivity of generic semigroups of area-preserving surface diffeomorphisms

Abstract

Theorem 12 in [2] claims that for a Moser generic diffeomorphism f and an invariant residual domain U , there are no periodic points in the boundary of U . This is claimed to be a direct consequence of [3]. However, Mather’s theorem is weaker than that; specifically, it says that the rotation number of the prime ends compactification on any end of U is irrational. On the sphere, Mather’s result implies what is claimed in Theorem 12, and the proof (which can be found in [1]) uses the existence of homoclinic intersections for periodic points of f guaranteed by a theorem of Pixton. This can also be done in T2 using [4]. However, for higher genus, the claim remains open. We will prove a weaker version of Theorem 12 of [2]. To do this, we define for each f a Cr -residual set G f , and we prove that the statement of Theorem 12 of [2] holds if we additionally require that the residual domain U be g-invariant for some g ∈ G f . This is enough for our proofs. The modifications needed in the statements and proofs of [2] are as follows: