THE HAUSI}ORFF DIMEI\SIOiT OF THE BRAI\CH SET OF A QUÄSIREGUTAR MAPPII\G

Leb G be a domain in the z-dimensional euclidean space-8 " , n 2 2. Consider & non-constant quasiregular mapping f: G--> R ". Let Bf denote the branch set of /. By 16l m(B) : m(fB1) : 0, where m is the z-dimensional Lebesgue me&surein R". Thenalso H*(By):H'(fB):0,where Hn, &)0,is the a-dimensional Hausdorff outer measure in R*. On the other hand, in [3] it is shown by an example that dimr By atd. dirll., fB, the Hausdorff dimensions of B, and fBr, can be arbitrarily close to z. In this paper we prove the following results. Let i'(r, /) denote the local topological index of f at r. If / is as above, then (l.l) dimrfB, I c ' I n, where the constant c ' depends only on z and the maximal dilatation K(f) of f. If, in addition, i(f) : sup{t(r,/) | reBr} < oo, then (1.2) dimrB, I c <n, where the constant c depends only ort n, K(f) and, i'(f). Ib remains an open question whether c actually depends on i(f). If it does not, then always dim'B, qn, too. We shall prove (f.I) and (f.2) using a similar method to Reöetnjak'