Title: Abstract dynamical systems
Joint work with: Sina Greenwood, Steve Watson, Robin Knight and Dave
McIntyre
Abstract: Let $X$ be any (infinte) set and let $T:X\to X$ be a function. A
natural and intriguing question to ask is whether there is a topology on
$X$ with respect to which $T$ is continuous. Clearly any selfmap is
continuous with respect to the discrete and indiscrete topologies, so we
are really looking for `nice' topologies that make $T$ continuous. This
specific question was asked by Ellis in 1953, but it is closely related to
Bessaga's converse to the Banach Fixed Point Theorem and work of Neumann,
Mekler and Truss on automorphisms of the set of rationals.
It turns out that there are surprisingly simple answers to this question
when we take `nice' to mean compact, Hausdorff or separable metric. In the
first case, the answer is in terms of the orbit structure of $T$; in the
second, it is in terms of the cardinality of $X$. On the other hand, there
are good reasons why the question is hard if we ask for a compact metric
topology.
In the talk we will discuss these and related results and look at how one
might start to attack them.