Cantor boundary behavior of analytic functions

Let A(D)A(D) be the space of analytic functions on the open disk DD and continuous on D¯. Let ∂D∂D be the boundary of DD, we are interested in the class of f∈A(D)f∈A(D) such that the image f(∂D)f(∂D) is a curve that forms loops everywhere. This fractal behavior was first raised by Lund et al. (1998) [21] in the study of the Cauchy transform of the Hausdorff measure on the Sierpinski gasket. We formulate the property as the Cantor boundary behavior   (CBB) and establish two sufficient conditions through the distribution of zeros of f′(z)f′(z) and the mean growth rate of |f′(z)||f′(z)| near the boundary. For the specific cases we carry out a detailed investigation on the gap series and the complex Weierstrass functions; the CBB for the Cauchy transform on the Sierpinski gasket will appear elsewhere.