This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5).
Plane continua X (and maps on them) are often studied by considering their complements in the sphere.The standard approach is to use a Riemann map from the unit disk to a complementary component U of X in the sphere. One of the reasons this approach is useful is that the conformal map encodes the geometry of the boundary of U. We propose to use an alternative to this analytic approach which makes it easier to apply such techniques in the topological setting and certain limit cases where the analytic techniques do not work. It can be shown that conformal external rays can be replaced by metrically defined external rays which leads to the same prime end theory as in the conformal case. Moreover, the conformal map can be replaced by a metrically defined partition of U into disjoint convex sets, which also captures the geometry of the boundary of U. This approach has already been used to obtain additional information about a possible minimal counter example to the plane fixed point problem and to show that all positively oriented maps of the plane, which include all holomorphic maps, must have a fixed point in any non-separating invariant sub-continuum. This approach was also critical in showing that every isotopy of a plane continuum,starting at the identity, can be extended to an isotopy of the entire plane. (Standard Caratheodory kernel convergence is insufficient to establish this result.) We plan to use these results to attack the following problems: does every map of the plane fix a point in any non-separating invariant sub-continuum and is every homogeneous tree-like plane continuum a pseudo arc? Our approach makes extensive use of the notion of a geometric lamination of the unit disk.
Complicated dynamical systems are often studied by analyzing the behavior of a system near periodic points. The study of the existence of such points naturally leads to the notion of a fixed point. The study of the existence of fixed points is an old and well established branch of mathematics. In this project we propose to study the existence of fixed points under maps of the plane onto itself.
