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In this paper, we study the spectrality of Sierpinski-Moran measure defined as an infinite convolution measure: $$\begin{aligned} \mu _{\{M_j\},\{{\mathcal {D}}_j\}}=\delta _{M_1^{-1}{\mathcal {D}}_1}*\delta _{(M_1M_2)^{-1}{\mathcal {D}}_2}*\cdots , \end{aligned}$$ where $${\mathcal {D}}_n=\{(0,0)^t,(a_n,0)^t,(0,b_n)^t\}\subset {\mathbb {Z}}^{2}$$ and $$\;M_n=\text {diag}(s_n,t_n)\in M_2(\mathbb Z)$$ are $$2\times 2$$ expanding diagonal matrices. Our goal is to investigate the existence of Fourier orthonormal basis for $$L^2(\mu _{\{M_n\},\{{\mathcal {D}}_n\}})$$ , i.e., find an exponential function system $$\{e^{2\pi i\langle \lambda ,x \rangle }\}_{\lambda \in \Lambda }$$ forming an orthonormal basis for $$L^2(\mu _{\{M_n\},\{{\mathcal {D}}_n\}})$$ . Some sufficient conditions for this aim are given, and some spectra $$\Lambda $$ are found.

Let $$\mu $$ be a Borel probability measure on $${\mathbb {R}}^n$$ . We call $$\mu $$ a spectral measure if there exists a countable set $$\Lambda \subset {\mathbb {R}}^n$$ such that $$E_\Lambda :=\{e^{2\pi i }:\lambda \in \Lambda \}$$ forms an orthogonal basis for the Hilbert space $$L^2(\mu )$$ . Let the measure $$\mu _{\{M,{\mathcal {D}}_n\}}$$ be defined by the following expression $$\mu _{\{M,{\mathcal {D}}_n\}}=\delta _{M^{-1}{\mathcal {D}}_1}*\delta _{M^{-2}{\mathcal {D}}_2}*\cdots $$ , where $$M=\text {diag}(\rho ^{-1},\rho ^{-1})$$ with $$|\rho

Let $$\mu _{M, D}$$ be the self-affine measure generated by an expanding integer matrix $$M\in M_{2}(\mathbb {Z})$$ and an integer three-element digit set $$D=\{(0,0)^T, (\alpha ,\beta )^T,(\gamma ,\eta )^T\}$$ . In this paper, we show that if $$3\mid \det (M)$$ and $$3\not \mid \alpha \eta -\beta \gamma $$ , then $$L^2(\mu _{M,D})$$ has an orthogonal basis of exponential functions if and only if $$M^*\varvec{u}\in 3\mathbb {Z}^2$$ , where $$\varvec{u}=(\eta -2\beta ,\; 2\alpha -\gamma )^T$$ .

For $n\geq 3$ , let $Q_n\subset \mathbb {C}$ be an arbitrary regular n-sided polygon. We prove that the Cauchy transform $F_{Q_n}$ of the normalised two-dimensional Lebesgue measure on $Q_n$ is univalent and starlike but not convex in $\widehat {\mathbb {C}}\setminus Q_n$ .

Let the 2 × 2 {2\times 2} expanding matrix R k {R_{k}} be an integer Jordan matrix, i.e., R k = diag ⁡ ( r k , s k ) {R_{k}=\operatorname{diag}(r_{k},s_{k})} or R k = J ⁢ ( p k ) {R_{k}=J(p_{k})} , and let D k = { 0 , 1 , … , q k - 1 } ⁢ v {D_{k}=\{0,1,\ldots,q_{k}-1\}v} with v = ( 1 , 1 ) T {v=(1,1)^{T}} and 2 ≤ q k ≤ p k , r k , s k {2\leq q_{k}\leq p_{k},r_{k},s_{k}} for each natural number k. We show that the sequence of Hadamard triples { ( R k , D k , C k ) } {\{(R_{k},D_{k},C_{k})\}} admits a spectrum of the associated Moran measure provided that lim inf k → ∞ ⁡ 2 ⁢ q k ⁢ ∥ R k - 1 ∥ < 1 {\liminf_{k\to\infty}2q_{k}\lVert R_{k}^{-1}\rVert .

Let Δ be an equilateral triangle, we define the Cauchy transform of the normalized two-dimensional Lebesgue measure μ on Δ by F ( z ) = ∫ Δ d μ ( w ) z − w . We prove that F is univalent and starli...

Suppose that $0 and $m\geqslant 2$ is an integer. Let $\unicode[STIX]{x1D707}_{\unicode[STIX]{x1D70C},m}$ be the self-similar measure defined by $\unicode[STIX]{x1D707}_{\unicode[STIX]{x1D70C},m}(\cdot )=\frac{1}{m}\sum _{j=0}^{m-1}\unicode[STIX]{x1D707}_{\unicode[STIX]{x1D70C},m}(\unicode[STIX]{x1D70C}^{-1}(\cdot )-j)$. Assume that $\unicode[STIX]{x1D70C}=\pm (q/p)^{1/r}$ for some $p,q,r\in \mathbb{N}^{+}$ with $(p,q)=1$ and $(p,m)=1$. We prove that if $(q,m)=1$, then there are at most $m$ mutually orthogonal exponential functions in $L^{2}(\unicode[STIX]{x1D707}_{\unicode[STIX]{x1D70C},m})$ and $m$ is the best possible. If $(q,m)>1$, then there are any number of orthogonal exponential functions in $L^{2}(\unicode[STIX]{x1D707}_{\unicode[STIX]{x1D70C},m})$.

For an integral self-affine spectral measure, if the zeros of its Fourier transform are all integral vectors, it is proven that any its spectrum has a tree structure. For any subset with such tree structure, a sufficient condition and a necessary condition for the subset to be a spectrum are given, respectively. Applications are given to some known results as special cases.

Let μ M , D {\mu_{M,D}} be a self-affine measure generated by an expanding diagonal matrix M ∈ M 3 ⁢ ( ℝ ) {M\in M_{3}(\mathbb{R})} with entries ρ 1 , ρ 2 , ρ 3 {\rho_{1},\rho_{2},\rho_{3}} and the digit set D = { ( 0 , 0 , 0 ) t , ( 1 , 0 , 0 ) t , ( 0 , 1 , 0 ) t , ( 0 , 0 , 1 ) t } {D=\{(0,0,0)^{t},(1,0,0)^{t},(0,1,0)^{t},(0,0,1)^{t}\}} . In this paper, we prove that for any ρ 1 , ρ 2 , ρ 3 ∈ ( 1 , ∞ ) {\rho_{1},\rho_{2},\rho_{3}\in(1,\infty)} , if ρ 1 , ρ 2 , ρ 3 ∈ { ± x 1 r : x ∈ ℚ + , r ∈ ℤ + } {\rho_{1},\rho_{2},\rho_{3}\in\{\pm x^{\frac{1}{r}}:x\in\mathbb{Q}^{+},r\in% \mathbb{Z}^{+}\}} , then L 2 ⁢ ( μ M , D ) {L^{2}(\mu_{M,D})} contains an infinite orthogonal set of exponential functions if and only if there exist two numbers of ρ 1 , ρ 2 , ρ 3 {\rho_{1},\rho_{2},\rho_{3}} that are in the set { ± ( p q ) 1 r : p ∈ 2 ⁢ ℤ + , q ∈ 2 ⁢ ℤ + - 1 ⁢ and ⁢ r ∈ ℤ + } {\{\pm(\frac{p}{q})^{\frac{1}{r}}:p\in 2\mathbb{Z}^{+},q\in 2\mathbb{Z}^{+}-1% \text{ and }r\in\mathbb{Z}^{+}\}} . In particular, if ρ 1 , ρ 2 , ρ 3 ∈ { p q : p , q ∈ 2 ⁢ ℤ + 1 } {\rho_{1},\rho_{2},\rho_{3}\in\{\frac{p}{q}:p,q\in 2\mathbb{Z}+1\}} , then there exist at most 4 mutually orthogonal exponential functions in L 2 ⁢ ( μ M , D ) {L^{2}(\mu_{M,D})} , and the number 4 is the best possible.

For an infinite Blaschke product B B , does there necessarily exist δ > 0 \delta >0 such that each w w satisfying | w | > δ |w|>\delta is assumed infinitely often by B B ? Stephenson raised this question in 1979 and then constructed a counterexample in 1988 to prove that the answer to his problem is negative. In this paper, we find two sufficient conditions under which the answer to the problem is positive.

Let n,b≥2 be two positive integers. For D={0,1,⋯,b−1}, let the self‐similar measure μbn,D be defined by μbn,D=1b∑d∈Dμbn,D(bnx−d). It is known [18] that μbn,D is a spectral measure with a spectrum Λ(bn,C)=∑j=0finiteajbnj:aj∈C,where C=bn−1{0,1,⋯,b−1}. In this paper, we give some conditions on τ∈Z under which the scaling set τΛ(bn,C) is also a spectrum of μbn,D.

Suppose that K is the square with vertexes $$\{1, i, -1, -i\}$$ and $$\mu =\frac{1}{2}{\mathcal {L}}^2$$ is the normalized two-dimensional Lebesgue measure on K, let F(z) be the Cauchy transform of $$\mu $$ . Dong et al. (Trans Am Math Soc 369:4817–4842, 2017) proved that F(z) is univalent in $$\widehat{\mathbb {C}} {\setminus } K$$ . In this paper, we show that F(z) is starlike in $$\widehat{\mathbb {C}} {\setminus } K$$ , but not convex in $$\widehat{\mathbb {C}} {\setminus } K$$ .

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 .

We study the spectral property of a class of Sierpinski-type self-affine measures μ M , D ( ⋅ ) = 1 3 ∑ d ∈ D μ M , D ( M ( ⋅ ) − d ) on R 2 , where M = [ ρ 1 − 1 a 0 ρ 2 − 1 ] is a real upper triangular expanding matrix and D = { ( 0 0 ) , ( d 1 0 ) , ( d 2 d 3 ) } is a three-element real digit set with d 1 d 3 ≠ 0 . A necessary and sufficient condition for μ M , D to be a spectral measure is established.

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 ) .

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 } ) .

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.

In this paper, we study infinite families of orthogonal exponentials of some self-affine measures. The digit set [Formula: see text] and any [Formula: see text] expanding integer matrix [Formula: see text] can generate a self-affine measure [Formula: see text]. Let [Formula: see text] and [Formula: see text] be the transposed conjugate of [Formula: see text], where [Formula: see text] and the elements of [Formula: see text] come from [Formula: see text]. In this paper, we prove the following results. For [Formula: see text], [Formula: see text] is a spectral measure. For [Formula: see text], there are infinite families of orthogonal exponentials, but none of them forms an orthogonal basis in [Formula: see text].

Let [Formula: see text] be an arithmetic digit set for each [Formula: see text], where [Formula: see text], and let [Formula: see text] be a sequence of integers larger than 1. In this paper, we prove that the Moran measure [Formula: see text] generated by infinite convolution of finite atomic measures [Formula: see text] is a spectral measure if [Formula: see text] and [Formula: see text].

Let [Formula: see text] be a prime and [Formula: see text] be a sequence of finite arithmetic digit sets in [Formula: see text] with [Formula: see text] uniformly bounded, and let [Formula: see text] be the discrete probability measure on the finite set [Formula: see text] with equal distribution. For [Formula: see text], the infinite Bernoulli convolution [Formula: see text] converges weakly to a Borel probability measure (Moran measure). In this paper, we study the existence of exponential orthonormal basis for [Formula: see text].

In the paper, we mainly discuss the connectedness of two kinds of self-affine sets. One is generated by matrix A = ( p 0 − a q ) and digit set D = { ( i s , j t ) 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 = ( p 0 − a q ) and digit set D = { ( i s , ( d i + 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.

In this paper, we consider the non-spectral problem for the planar self-affine measures $\mu_{M,D}$ generated by an expanding integer matrix $M\in M_2(\mathbb{Z})$ and a finite digit set $D\subset\mathbb{Z}^2$. Let $p\geq2$ be a positive integer, $E_p^2:=\frac{1}{p}\{(i,j)^t:0\leq i,j\leq p-1\}$ and $\mathcal{Z}_{D}^2:=\{x\in[0, 1)^2:\sum_{d\in D}{e^{2\pi i\langle d,x\rangle}}=0\}$. We show that if $\emptyset\neq\mathcal{Z}_{D}^2\subset E_p^2\setminus\{0\}$ and $\gcd(\det(M),p)=1$, then there exist at most $p^2$ mutually orthogonal exponential functions in $L^2(\mu_{M,D})$. In particular, if $p$ is a prime, then the number $p^2$ is the best., Comment: 14 pages

Let [Formula: see text],[Formula: see text][Formula: see text],[Formula: see text][Formula: see text], satisfy [Formula: see text]. Let [Formula: see text] be the infinite convolution of probability measures with finite support and equal distribution. In this paper, we show that if [Formula: see text], then there exists a discrete set [Formula: see text] such that [Formula: see text] is an orthonormal basis for [Formula: see text].

For a positive integer [Formula: see text], let [Formula: see text]. Let the self-affine measure [Formula: see text] be generated by an expanding real matrix [Formula: see text] and a finite digit set [Formula: see text], where [Formula: see text] with [Formula: see text] and [Formula: see text] is the [Formula: see text]th column of the [Formula: see text] identical matrix [Formula: see text], [Formula: see text]. In this paper, we prove that [Formula: see text] contains an infinite orthogonal set of exponential functions if and only if there exists [Formula: see text] such that [Formula: see text] for some [Formula: see text] with [Formula: see text] and [Formula: see text].

For an analytic function f on the open unit disk D and continuous on D ¯ , the Cantor boundary behavior (CBB) is used to describe the curve f ( ∂ D ) 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 f ′ ( z ) near the boundary and the fast mean growth rate of f ′ ( z ) 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.

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

The Cauchy transform of a measure has been used to study the analytic capacity and uniform rectifiability of subsets in C . Recently, Lund et al. (Experiment. Math. 7 (1998) 177) have initiated the study of such transform F of self-similar measure. In this and the forecoming papers (Starlikeness and the Cauchy transform of some self-similar measures, in preparation; The Cauchy transform on the Sierpinski gasket, in preparation), we study the analytic and geometric behavior as well as the fractal behavior of the transform F. The main concentration here is on the Laurent coefficients {an}n=0∞ of F. We give asymptotic formulas for {an}n=0∞ and for F(k)(z) near the support of μ, hence the precise growth rates on |an| and |F(k)| are determined. These formulas are connected with some multiplicative periodic functions, which reflect the self-similarity of μ and K. As a by-product, we also discover new identities of certain infinite products and series.

We generalize Bazilevic’s Theorem and provide an alternative proof of an important theorem of Hamilton.

Let\(\mathbb{D}\) be the open unit disc and let \(\partial \mathbb{D}\) be the boundary of \(\mathbb{D}\). For f(z) analytic in \(\mathbb{D}\) and continuous on \(\overline{\mathbb{D}}\), it follows from the open mapping theorem that \(\partial f(\mathbb{D}) \subset f(\partial \mathbb{D})\). These two sets have very rich and intrigue geometric properties. When f(z) is univalent, then they are equal and there is a large literature to study their boundary behaviors. Our interest is on the class of analytic functions f(z) for which the image curves \(f(\partial \mathbb{D})\) form infinitely many loops everywhere, they are not univalent of course. We formulate this as the Cantor boundary behavior. We give sufficient conditions for such property, making use of the distribution of the zeros of f ′ and the mean growth rate of f ′ . Examples includes the complex Weierstrass functions, and the Cauchy transform of the canonical Hausdorff measure on the Sierpiski gasket.