On the support of harmonic measure for sets of Cantor type
1985; Finnish Academy of Science and Letters; Volume: 10; Linguagem: Inglês
10.5186/aasfm.1985.1014
ISSN0066-1953
Autores Tópico(s)Advanced Mathematical Modeling in Engineering
ResumoCARLESONl. 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 < 1 even when the Hausdorff dimension of the Cantor set is close to 2.If one thinks of harmonic measure as hitting probability it may not be so sur- prising that the results are consequences of results on stationary processes.More precisely, they belong to the field of information theory and we shall here review what is needed (see e.g.[]).After this manuscript was finished I was informed that A. Manning has a similar result for invariant measures of a polynomial map. (See [3].)Let x:{""}:be a double infinite sequence of symbols xo taken from a finite set, "alphabet".Let p be a probability measure on the set of x's.We assume p to be stationary and ergodic with respect to the shift Z: Tx:{xna1\7-.Denote by C a "chain" C:(xt, xz, .. ., x,), of length, n.The entropy H of p is defined as fI : lim Hnsup rr' n-fl nn where Ho --Z ", p(C") log tt(C).McMillan's basic theorem now describes how many Cnthat actually occur.Theorem.Gioen e>0, there is no(e) so thatfor n>no(e), we canfind distinct chains C,r, Cnz, ..., C, of lmgth n with rn<sn(Hte) so that Zi='P(c,) > 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-
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