On topologically and quasiconformally homogeneous continua
Beverly L. Brechner, Timo Erkama · Annales Academiae Scientiarum Fennicae Series A I Mathematica · 1979
A subset M of the Riemann sphere is called quasiconformally homogeneous if for each pair of points P and Q of M there is a quasiconformal map E defined in a neighborhood of M such that E(M):M and q(P):Q.For information about quasiconformal mappings, see [].Recently the second author showed [3] that a simple closed curve is quasicon- formally homogeneous if and only if it is a quasicircle (i.e., the image of a circle under a quasiconformal map).In this note we prove the following more general result.Theorem l.Euery non-degenerate quasiconformally homogeneous continuum is a quasicircle.Note that a continuum is called non-degenerate if it is an inflnite proper subset of the sphere.