Twist points of the von Koch snowflake

Abstract

It is known that the set of twist points in the boundary of the von Koch snowflake domain has full harmonic measure. We provide a new, simple proof, based on the doubling property of the harmonic measure, and on the existence of an equivalent measure, invariant and ergodic with respect to the shift. The von Koch snowflake domain D is the union of an increasing sequence of open polygons Dn, where Do is an equilateral triangle, and D+1 is obtained from Dn by replacing the middle third of each side of Dn by the two other sides of the equilateral triangle based on the removed segment and lying outside Dn [vKO6]. A point w E bD is called a twist point if liminf arg[z w] = oo, lim sup arg[z w] = +oo z-*w,zEa z-*wzEa for each curve a in D ending at w [P92, p.141]. Let vp be the harmonic measure on bD for D with respect to the center p of Do. We show that the following known theorem admits a new, simple proof, based on the doubling property of vap and on the existence of a measure equivalent to vip, invariant and ergodic with respect to the shift. Theorem. The set of twist points in the boundary of the von Koch snowflake has full harmonic measure. Let Tn be the collection of the sides of Dn. For n = 1, 2,... we now define a labeling ?: Tn -+ A, where A = {1, r7, cl Cr }. For each x E Tn (n = 0, 1, 2, ... ), the left third of x is an element of Tn+, and is labeled by 1; the right third of x (also an element of Tn+1) is labeled by r; the equilateral triangle based on the middle third of x has two other sides (again, elements of Tn+i): the one adjacent to the left (right) third is labeled by cl (Cr). The subset of Tn+, just described is denoted by A+(x); its elements are called direct descendants of x; each y E Tn+, is a direct descendant of a unique element of Tn, denoted by y-. Let T = Un=0 Tn. For x E T we write lxi = n if x E Tn. Let C be the collection of the vertices of all the polygons Dn; since C is countable, its harmonic measure is zero [HK76, p.247]. For each w E bD C there is a Received by the editors November 1, 1996. 1991 Mathematics Subject Classification. Primary 31A15, 30C35.