On periodic open disks for diffeomorphisms of surfaces
Salvador Zanata (IME-USP)
In this work we consider periodic open disks for certain diffeomorphisms f : S → S isotopic to the identity on closed orientable surfaces S of genus larger or equal to 1. In case of the torus (genus=1), we assume that the rotation set of f has interior and when genus is larger than 1, we assume a more technical hypothesis, which implies full homotopical complexity of orbits for f. Under these hypotheses, there exists a constant M = M(f) > 0 such that a periodic open disk O either has diameter bounded by M, or it is unbounded. In case the disk is unbounded, we show that there is a partition of O for which every element of this partition is wandering, with the exception of one. And if we consider the maximal periodic open disk which contains O, denoted Omax (it is defined in a very natural way), then when the prime ends rotation number on the boundary of Omax is rational, we prove, under some Cr -generic conditions (for every r ≥ 1), that Omax is a basin of some attractor or of some repeller contained in Omax. We do not know precisely what happens in the irrational case. This is joint work with Andres Koropecki.