Omega-limit sets for the Stein-Ulam Spiral map

K. Baranski and M. Misiurewicz

Abstract

In the late 1950's, using computers in the Los Alamos National Laboratory, Stanislaw Ulam and Paul Stein performed a comprehensive research on a class of quadratic maps of the 2-dimensional simplex D to itself. Those maps arise in the theory of population genetics. One of them has the behavior much different than the 96 other ones. We call it the Stein-Ulam Spiral map. In 1972, S. Vallander asked whether the omega-limit set of any interior point of D, except its center, is equal to the boundary of D. We prove that this is the case for the points from a residual subset of D. On the other hand, we show that for any closed invariant subset E of the boundary of D intersecting all three sides of D, the set of points having E as the omega-limit set is relatively large.