On the support of harmonic measure for sets of Cantor type
Lennart Carleson · Annales Academiae Scientiarum Fennicae Series A I Mathematica · 1985
CARLESON l.Introduction.In a recent paper [2] Kaufman and Wu have shown that the support of harmonic measure of the von Koch snowflake domain has dimension strictly smaller than the Hausdorffdimension (log4llog3) of the boundary curve.This result supports a very strong conjecture of Oksendal [4] to the effect that the support of any harmonic measure in the plane has dimension =1.We shall here prove that the Oksendal conjecture is true for any "fractal" curve such as the snowflake.It is also true for any two-dimensional Cantor set with con- stant ratio in the very strong sense that the djmension for these supports is strictly 0, there is no(e) so thatfor n>no(e), we canfind distinct chains C,r, Cnz, ..., C, of lmgth n with rn 1-e'It is clear that this is a description of the support of p.Our scheme is therefore to relate harmonic measure to a stationary p for which we can actually compute the entropy.We also obtain a connection to the geometry of the situation.Since the sup-