Inducing Fixed Points in the Stone-čech Compactification

If f is an autohomeomorphism of some space X, then βf denotes its Stone-Čech extension to βX. For each n ≤ ω, we give an example of a first countable, strongly zerodimensional, subparacompact X and a map f such that every point of X has an orbit of size n under f and βf has a fixed point. We give an example of a normal, zero-dimensional X such that f is fixed-point-free but βf is not. We note that it is impossible for every point of X to have an orbit of size 3 and βX to have a point with orbit of size 2. For every Tychonoff space X there is a unique compact, Hausdorff space βX, the StoneČech compactification of X, which contains X as a dense subspace and has the property that every autohomeomorphism and every continuous R-valued map f on X can be uniquely extended to one on βX, denoted βf (see, for example, [E]). Even if an autohomeomorphism f has no fixed points, βf may do. Such induced fixed points can be regarded as ideal and, following work by van Douwen and Watson, we describe examples of spaces with fixed-pointfree autohomeomorphisms which nevertheless have ideal fixed points. For each n ≤ ω, we give an example of a first countable, strongly zero-dimensional, subparacompact X and a map f such that every point of X has an orbit of size n under f and βf has a fixed point. Since neither these examples, nor those described by Watson in [W], are normal, we also give an example of a normal, zero-dimensional X such that f is fixed-point-free but βf is not. This example is based on the space described in [D]. Answering a question from [W], we note that it is impossible for every point of X to have an orbit of size 3 and βX to have a point with orbit of size 2. We also show that a set can be topologized so that a fixed-point-free permutation is an autohomeomorphism with an ideal fixed point if and only if the set is uncountable.