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Let D n = { 0 , a n , b n } = { 0 , 1 , 2 } ( m o d 3 ) , p n ∈ 3 Z + , n ≥ 1 , satisfy sup n ≥ 1 ⁡ max ⁡ { | a n | , | b n | } p n < ∞ . It is well-known that there exists a unique Borel probability measure μ { p n } , { D n } generated by the following infinite convolution product μ { p n } , { D n } = δ p 1 − 1 D 1 ⁎ δ ( p 1 p 2 ) − 1 D 2 ⁎ ⋯ in the weak convergence. In this paper, we give some conditions to ensure that there exists a discrete set Λ such that the exponential function system { e 2 π i λ x } λ ∈ Λ forms an orthonormal basis for L 2 ( μ { p n } , { D n } ) . [ABSTRACT FROM AUTHOR]

In the paper, we mainly discuss the connectedness of two kinds of self-affine sets. One is generated by matrix A=(p0-aq) and digit set D={(is,jt)T:i=0,1,...,|q|-1,j=0,1,...,|p|-1}, where s,t≠0 and p,q∈Z with 3≤|p|+1<|q|<2|p|-1. The other is generated by matrix A=(p0-aq) and digit set D={(is,(di+j)t)T:i=0,1,...,|p|-1,j=0,1,...,|q|-1}, where s,t≠0, and p,q,d∈Z with |p|,|q|≥2. The sufficient or necessary conditions for their connectedness are revealed. [ABSTRACT FROM AUTHOR]

Abstract: For an analytic function on the open unit disk and continuous on , the Cantor boundary behavior (CBB) is used to describe the curve that forms infinitely many fractal-look loops everywhere. The class of analytic functions with the CBB was formulated and investigated in Dong et al. [6]. In this note, our main objective is to give further discuss of the criteria of CBB in Dong et al. [6]. We show that the two major criteria, the accumulation of the zeros of near the boundary and the fast mean growth rate of near the boundary, do not imply each other. Also we make an improvement of another criterion, which allows us to have more examples of CBB. [Copyright &y& Elsevier]

Let U(p,k) denote the class of functions f which are areally mean p-valent and have k maximal growth directions in Δ={z: &z.sfnc;z&z.sfnc;<1}. In this paper we give an asymptotic formula of the mean square of the logarithmic derivative f′/f for f∈U(p,k) with some restricted condition. This generalizes the corresponding result in [J. Anal. Math. 36 (1979) 36–43]. Also we construct a function f∈U(p,k) which does not satisfy this asymptotic formula. [Copyright &y& Elsevier]

Suppose that 0 < | ρ | < 1 and m ≥ 2 is an integer. Let μ ρ , m be the self-similar measure defined by μ ρ , m ( ⋅ ) = 1 m ∑ j = 0 m − 1 μ ρ , m ( ρ − 1 ( ⋅ ) − j ) . In this paper, we prove that L 2 ( μ ρ , m ) contains an infinite orthonormal set of exponential functions if and only if ρ = ± ( q / p ) 1 / r for some p , q , r ∈ N + with gcd ( p , q ) = 1 and gcd ( p , m ) > 1 . [ABSTRACT FROM AUTHOR]

For a tangential slit, the behavior of the driving function in the Loewner differential equation is less clear. In this paper, we investigate the tangential slit φ ( Γ ) , where φ is a univalent real analytic function near the origin, and where Γ is a circular arc tangent at the origin. Our main aim is to give an interesting way to prove the asymptotic property of the driving function which generates the tangential slit φ ( Γ ) . [ABSTRACT FROM AUTHOR]