Superattracting fixed points of quasiregular mappings

Abstract

Abstract. We investigate the rate of convergence of the iterates of an n-dimensional quasiregular mapping within the basin of attraction of a fixed point of high local index. A key tool is a refinement of a result that gives bounds on the distortion of the image of a small spherical shell. This result also has applications to the rate of growth of quasiregular mappings of polynomial type, and to the rate at which the iterates of such maps can escape to infinity.

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

“…It was shown in [11] that if f is such a map satisfying deg f > K I (f ), then I(f ) = ∅ and ∂I(f ) is perfect. Moreover, I(f ) = A(f ) for such maps [13]. The inclusion J(f ) ⊂ ∂I(f ) also holds for such maps, and again it may be strict; cf.…”

On the mean radius of quasiconformal mappings

“…Theorem 2.1 (Theorem 1.1, [2]). Let U ⊂ R n be a domain for n ≥ 2 and let f : U → R n be a non-constant quasiregular map.…”

The orbits of generalized derivatives

“…Moreover, [4,Theorem 1.5] shows that O(x) lies in a ring {x ∈ R n : 1/C ′ ≤ |x| ≤ C ′ } for some constant C ′ ≥ 1 depending only on n, K O (f ) and i(x 0 , f ).…”