A , can then be associated with A . The Reduction Theorem allows us to relate rotational invariant sets under s n to those with the same rotation number under s n - 1 . We will discuss s 3 as a concrete example. This is a report on joint work with Bell, Fokkink, Mayer and Tymchatyn. We will give a brief history of the plane fixed point problem including all recent results.
Holomorphic maps are prototypes of positively oriented maps.
Moreover, any positively oriented map has a fixed point in a non-separating invariant subcontinuum. This generalizes an old result for orientation preserving homeomorphisms of the plane. In order to generalize this result to negatively oriented maps of the plane, we will show that any minimal counter example to the plane fixed point problem for a negatively oriented map must have at least one dense generalized in-channel. The arc property of Kelley is a property similar to the property of Kelley but stronger. The continua guaranteed by the definition are additionally required to be arcwise connected.