| Abstract: | This article concerns the iteration of quasiregular mappings on (mathbb {R}^d) and entire functions on (mathbb {C}). It is shown that there are always points at which the iterates of a quasiregular map tend to infinity at a controlled rate. Moreover, an asymptotic rate of escape result is proved that is new even for transcendental entire functions. Let (f:mathbb {R}^drightarrow mathbb {R}^d) be quasiregular of transcendental type. Using novel methods of proof, we generalise results of Rippon and Stallard in complex dynamics to show that the Julia set of f contains points at which the iterates (f^n) tend to infinity arbitrarily slowly. We also prove that, for any large R, there is a point x with modulus approximately R such that the growth of (|f^n(x)|) is asymptotic to the iterated maximum modulus (M^{n}(R,f)). |
