The free quasiworld. Freely quasiconformal and related maps in Banach spaces

Quasimöbius maps, weakly quasimöbius maps and uniform perfectness in quasi-metric spaces

“…[28,Theorem 6.22]), we get from Lemma 2.5 that f is weakly (h 2 , H 2 )-quasimöbius. Furthermore, u•q 1 (W 1 ) and u•q 2 (W 2 ) are two uniformly perfect subsets of R N , since uniform perfectness is preserved by quasimöbius maps (cf.…”

Broken Tubes in Hilbert Spaces

“…There is i such that x, y ∈ G i = int (U i ∪ U i+1 ). As a convex bounded domain, G i is b 1 -uniform with a universal constant b 1 ; see, for example, [Vä5,10.4.2]. Hence there is a b 1 -uniform arc γ : x y in G i .…”

“…Using tower maps (see [Vä5,8.13]) one can find quasihyperbolic maps of a ball with an empty cluster set at each point of an infinite set. However, the following question remains open: 3.6.…”

“…By a result of M. Bonk, J. Heinonen and P. Koskela [BHK,7.12], such domains do no exist in the n-space R n . Broken tubes have been mentioned in the author's papers [Vä3,4.12], [Vä4,3.15], [Vä5,8.15] and [Vä7,4.3] but without an explicit construction. In this article we give a rigorous construction of a broken tube W and a quasihyperbolic homeomorphism F of a straight tube U onto W .…”

Local properties of quasihyperbolic mappings in metric spaces

“…In 1990, based on the idea of quasisymmetry, Väisälä developed a "dimension-free" theory of quasiconformal mappings in infinite-dimensional Banach spaces and obtained many beautiful results. See also [15,16,17,18,19]. In 1998, Heinonen and Koskela [3] showed that these concepts, quasiconformality and quasisymmetry, are quantitatively equivalent in a large class of metric spaces, which includes Euclidean spaces.…”