A fixed point theorem for branched covering maps of the plane
by
Lex Oversteegen
UAB
Coauthors: A. Blokh

Fixed points of homeomorphisms of the plane have been extensively studied. It is known that any homeomorphism of the plane, which keeps a non-separating continuum invariant, has a fixed point in this continuum. Much less is known about branched covering maps of the plane. Even though all complex polynomials have fixed points, simple examples show that a branched covering map of the plane could have no periodic points at all. Like homeomorphisms, branched covering maps of the plane are either orientation preserving or reversing. Bell announced that all holomorphic maps of the plane have a fixed point in any invariant non-separating subcontinuum. This result was recently generalized to all orientation preserving branched covering maps of the plane. In this talk we will show that if there exists an orientation reversing branched covering map f of the plane of degree -2, which keeps a non-separating continuum invariant and is fixed point free on this continuum. Then there exists a branched covering map g of the plane which has a fully invariant minimal indecomposable subcontinuum X such that g has no fixed point in T(X) (the union of X and all of its bounded complementary domains). It follows that g induces a covering map G of the circle of prime ends for T(X) with exactly three fixed prime ends such that the map g keeps the corresponding external rays invariant. Moreover, g moves points on one of these rays out towards infinity and points on the other two towards X (i.e., X has exactly one outchannel and two inchannels).

Date received: February 28, 2008