Cauchy transforms of self-similar measures: Starlikeness and univalence

Abstract

For the contractive iterated function system S k z = e 2 π i k / m + ρ ( z − e 2 π i k / m ) S_kz=e^{2\pi ik/m}+{\rho (z-e^{2\pi ik/m})} with 0 > ρ > 1 , k = 0 , ⋯ , m − 1 0>\rho >1, k=0,\cdots , m-1 , we let K ⊂ C K\subset \mathbb {C} be the attractor, and let μ \mu be a self-similar measure defined by μ = 1 m ∑ k = 0 m − 1 μ ∘ S k − 1 \mu =\frac 1m\sum _{k=0}^{m-1}\mu \circ S_k^{-1} . We consider the Cauchy transform F F of μ \mu . It is known that the image of F F at a small neighborhood of the boundary of K K has very rich fractal structure, which is coined the Cantor boundary behavior. In this paper, we investigate the behavior of F F away from K K ; it has nice geometry and analytic properties, such as univalence, starlikeness and convexity. We give a detailed investigation for those properties in the general situation as well as certain classical cases of self-similar measures.

Bergman Space Properties of Fractional Derivatives of the Cauchy Transform of a Certain Self-Similar Measure

“…In [10], Stricharz et al studied the Cauchy transform of a self-similar measure µ with compact support K. From numerical data and computer graphics, they considered the Hölder continuity and analyticity of F(z) intuitively. In [11][12][13][14][15], Dong and Lau intensively studied the geometric and analytic properties of F. The precise growth rates of the Laurent coefficients of such F were obtained, and the asymptotic behavior of the coefficients was also discussed in [11]. The geometric properties of F away from K, such as univalence, starlikeness and convexity, were investigated in [14].…”

“…In [11][12][13][14][15], Dong and Lau intensively studied the geometric and analytic properties of F. The precise growth rates of the Laurent coefficients of such F were obtained, and the asymptotic behavior of the coefficients was also discussed in [11]. The geometric properties of F away from K, such as univalence, starlikeness and convexity, were investigated in [14]. Since F is analytic at zero, F has a Taylor expression near zero.…”

On zeros and spectral property of self-affine measures

“…Another amazing fact is that Strichartz proved that the Fourier series related to such spectral measures have much better convergence properties than their classical counterparts on the unit interval [34]. The spectral property, Fourier transform [26,35,36] and Cauchy transform [10][11][12]33] of fractal measure form the main topics in the analysis on fractals.…”

Starlikeness and Convexity of Cauchy Transforms on Regular Polygons

“…, n − 1, induces a self-similar measure µ n,ρ and an attractor K n,ρ . The chaotic behaviour of the Cauchy transform F µ n,ρ near K n,ρ was studied by Dong et al in [2,3,5,6,12]. In particular, K 4,1/2 is the square with vertices {1, i, −1, −i} and µ 4,1/2 is the normalised Lebesgue measure on K 4,1/2 .…”

“…In particular, K 4,1/2 is the square with vertices {1, i, −1, −i} and µ 4,1/2 is the normalised Lebesgue measure on K 4,1/2 . It is shown in [6] that the Cauchy transform F µ 4,1/2 is univalent outside the square K 4,1/2 . Indeed, the authors proved in [7] that F µ 4,1/2 is starlike but not convex in C \ K 4,1/2 .…”