Abstract:

A point x is said to 'escape to infinity' if the sequence of iterates f^k(x) tends to infinity. The functions f we consider are either analytic on the complex plane or quasiregular on n-dimensional Euclidean space; the latter being the natural higher-dimensional generalisation of analytic functions.

For a complex polynomial f, all points that escape to infinity do so at roughly the same speed. For a transcendental (non-polynomial) entire function, Rippon and Stallard showed that there are always points that escape to infinity arbitrarily slowly. We'll see that this result can be generalised to the quasiregular setting. Unexpectedly, part of the method used to prove this 'slow escape' generalisation also allows us to say something about points that escape to infinity very fast. In particular, we can find points for which the escape rate is related to the fastest possible speed of escape. This last result appears to be new even for entire functions.