Alastair Fletcher

THE ESCAPING SET OF A QUASIREGULAR MAPPING

We show that if the maximum modulus of a quasiregular mapping f : RN → RN grows sufficiently rapidly, then there exists a nonempty escaping set I(f) consisting of points whose forward orbits under iteration of f tend to infinity. We also construct a quasiregular mapping for which the closure of I(f) has a bounded component. This stands in contrast to the situation for entire functions in the complex plane, for which all components of the closure of I(f) are unbounded and where it is in fact conjectured that all components of I(f) are unbounded.

Quasiregular dynamics on the n-sphere

In this paper, we investigate the boundary of the escaping set I ( f ) for quasiregular mappings on ℝ n , both in the uniformly quasiregular case and in the polynomial type case. The aim is to show that ∂I ( f ) is the Julia set J ( f ) when the latter is defined, and shares properties with the Julia set when J ( f ) is not defined.

The fast escaping set for quasiregular mappings

The fast escaping set of a transcendental entire function is the set of all points which tend to infinity under iteration as fast as possible compatible with the growth of the function. We study the analogous set for quasiregular mappings in higher dimensions and show, among other things, that various equivalent definitions of the fast escaping set for transcendental entire functions in the plane also coincide for quasiregular mappings. We also exhibit a class of quasiregular mappings for which the fast escaping set has the structure of a spider’s web.

Local Rigidity of Infinite-Dimensional Teichmüller Spaces

This paper presents a rigidity theorem for infinite-dimensional Bergman spaces of hyperbolic Riemann surfaces, which states that the Bergman space

Julia sets of uniformly quasiregular mappings are uniformly perfect

A^{1}(M)

Infinite dimensional Teichmüller spaces

, for such a Riemann surface

The Julia Set and the Fast Escaping Set of a Quasiregular Mapping

M

Chaotic dynamics of a quasiregular sine mapping

, is isomorphic to the Banach space of summable sequence,

Iteration of quasiregular tangent functions in three dimensions

l^{1}

On asymptotic Teichmüller space

. This implies that whenever