On Minkowski dimension of Jordan curves

Abstract

This paper has its origin in a question raised by McMullen (McM08): Under what general circumstances does a smooth family of conformal mapst: D → C with �0 = id satisfy (a) d 2 dt 2 H.dim(�t(@D)) � � t=0 = lim r→1 1 4�|log(1 − r)| ˆ |z|=r | u � '(z)| 2 |dz|? McMullen has shown that (a) is true for some families (�t) arising from some dynamical systems. In order to answer this question, we consider a general analytic 1-parameter family (�t), t ∈ U, a neighborhood of 0, conformal maps with �0 = id andt(0) = 0, ∀t ∈ U defined ast(z) = ´ z 0 e tb(u) du, b ∈ B, where B is the Bloch space. By using a probability argument, we first describe a relatively large class of functions in B for which (�t)t∈U satisfies (a), where Hausdorff dimension is replaced by Minkowski dimension. This class is defined in terms of the square function of the associated dyadic martingale of Re(b). The second principal result of this paper is a counter- example which is reminiscent of Kahane and Piranian's construction of non-Smirnov domain. We have constructed a singular Bloch function b such that if we consider the associated family (�t) as above, thent(@D) is rectifiable for t 0 such that M.dim(�t(@D)) ≥ 1 + ct 2 (t > 0 small), thus contradicting (a), where the Hausdorff dimension replaced by the Minkowski dimension.