Homeomorphisms of bounded length distortion

Jussi Väisälä
1987 Annales Academiae Scientiarum Fennicae Series A I Mathematica  
a homeo' morphism. We let /(a) denote the length of a path a.lf L=l and if (1.2) t(u)lL=t(fa)=Lt(a) for all paths a in D, we say that / is of L-bounded length distortion, abbreviated Z-BLD. In a joint article [MV] of O. Martio and the author, we consider more general BLD maps: discrete open maps of domains of -P into Å' satisfying (1.2). For homeomorphisms and, more generally, for immersions, (1.2) is equivalent to the following condition: Every point in D has a neighborhood U such that /lU is
more » ... U such that /lU is Z-bilipschitz, that is, lx-yllL s lf@)-f(y)l = Llx*yl for all x,y€(J. For this reason, the BLD immersions are often called locally bi lipschitz maps or local quasi-isometries or just quasi-isometries pol, [Ge]. The BLD property can also be defined in terms of upper and lower derivatives. Let L1@) and f (x) be the upper and lower limits of lf (x+h)-.f (x)lllhl as å*0. Then a homeomorphism / is Z-BLD if and only if lr(x)=llL and Lr(x)=L for all x(D. In particular, if / is differentiable aL x, this means (1.4) for all heRz. Every ,L-BLD bomeomorphism is "L2-quasiconformal, but a quasiconformal map is BLD only if its derivative is a.e. bounded away from 0 and -. The purpose of this paper is to identify the domains DcRz which are BLD homeomorphic to a disk or to a half plane. The corresponding problem for bilipschitz maps was solved in the early eighties by Tukia [Tu1], [Tul, Jerison-Kenig FKI and Latfullin [La]; see also [Ge]. Their results can be stated as follows: A bounded domain D is bilipschitz homeomorphic to a disk if and only if its boundary åD is a rectifiable Jordan curve satisfying the chord-arc condition: There is c>l such that (1.5) ( 1.3) o(x, y) = clxyl
doi:10.5186/aasfm.1987.1233 fatcat:5xqfyh3u5fhn5f5kuwlbcnrpbq