The Taylor Coefficients Of A Class Of Cauchy Transform Of Hausdorff Measure

In this thesis, we consider the self-similar contractions defined bywith (?), 0 <ρ<ρ_q , q≥4. whereρ_q∈(0,1) was given in [16]. It is known by [16] that {S_j}(?) satisfy the open set condition. Let K be the attractor of {S_j}(?), andμbe the Hausdorff measure with support K. The Cauchy transform ofμis defined byNotice that the compact set K does not containω= 0( for q =4, we need to assume 0 <ρ<ρ_q), so there exists the largest R_q > 0 such that F(z) is analytic in |z| < R_q. The paper [19] has proved that if q=2m, the analytic radius R_q = 1 - 2ρ, and if q = 2m + 1, the analytic radius R_q = |a| where a = S₀(?)S_m(ε_m+1). Since F(z) has Taylor expansion in |z| < R_q, it is easy to prove accoding to the rotation invariance ofμ. that the expansion has the following form:This thesis is the contituation of paper [19]. Our main goal is to study further the asymptotic properties of {(qk)^αR_q^qkb_qk-1}(?). We discuss them in the following two cases: q = 2m(m≥2) and q = 2m + 1(m≥2). It is shown that {(qk)^αR_q^qkb_qk-1}(?) is dense in some bounded domain, whereα= logq/| logρ| is the Hausdorff dimension of K. We also portray further that the accumulation set domain is a non-degenerate bounded closed interval, or the union of some non-degenerate bounded closed intervals. The intervals can always be expressed in terms of a multiplicative periodic functionΦ(z)(Φ(ρz)=Φ(z)).