In this paper, we study the spectral property of self-affine measures μ M , D generated by an expanding integer matrix M ∈ M n ( Z ) and a finite digit set D ⊂ Z n , i.e. investigate whether the function in L 2 ( μ M , D ) has a Fourier expansion. Let Z D n = { x ∈ [ 0 , 1 ) n : ∑ d ∈ D e 2 π i ⟨ d , x ⟩ = 0 } and E q n = ( q − 1 Z n ∩ [ 0 , 1 ) n ) ∖ { 0 } for some integer q ⩾ 2 . Through the discussion of the relationships between Z D n and E q n , we establish some criteria to determine whether μ M , D is spectral or non-spectral. Indeed, under those suitable assumptions, if μ M , D is a spectral measure, we find its spectrum; otherwise, we give the maximum number of orthogonal exponential functions in L 2 ( μ M , D ) . As an application, our results can contain some well-known conclusions.

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].

Suppose that q = 2m+ 1 >= 5. Let F be the Cauchy transform of the self-similar measure mu = 1/q Sigma(q-1)(j=0) mu o S-j where S-j (z) = e(2j pi i/q) +rho(z-e(2j pi i/q)) with rho is an element of (0,1). Let K be the attractor of {Sj}(j=0)(q-1) and R-q = dist (0, K). The Laurent coefficients of F in vertical bar z vertical bar > 1 was studied in [18] and [2]. In this paper, we study the asymptotic formula of the Taylor coefficients {b(qk-1)}(k=1)(infinity) of F in vertical bar z vertical bar < R-q. We prove that the supremum of beta satisfying {(qk)(beta) R(q)(qk)b(qk-1)}(k=1)(infinity) is an element of(infinity)(l) is the Hausdorff dimension a of K, and that the accumulation points set of {(qk)(alpha) R(q)(qk)b(qk-1)}(k=1)(infinity) is a union of some non-degenerate line segments. (C) 2020 Elsevier Inc. All rights reserved.

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 $$\varvec{u}=(\eta -2\beta ,\; 2\alpha -\gamma )^T$$ .

Suppose that 0 < vertical bar rho vertical bar < 1 and m >= 2 is an integer. Let mu(rho,m) be the self-similar measure defined by mu(p,m)(.) = 1/m Sigma(m-1)(j=0)mu(rho,m)(rho(-1)(.) - j). Assume that rho = +/-(q/p)(1/r) for some p, q, r is an element of 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 (mu(rho,m)) and m is the best possible. If (q, m) > 1, then there are any number of orthogonal exponential functions in L-2 (mu(rho,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.

Abstract 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.

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].

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 $$\widehat{\mathbb {C}} {\setminus } K$$ .

Let n, b >= 2 be two positive integers. For D = {0, 1, ... ,b-1}, let the self-similar measure mu(n)(b), D be defined by mu(n)(b),(D) = 1/b Sigma(d is an element of D) mu(n)(b),(D)(b(n)x - d). It is known [18] that mu(n)(b),(D) is a spectral measure with a spectrum A(b(n), C) = {Sigma(j=0) (finite) a(j)b(nj) : a(j) is an element of C}, where C = b(n-1) {0, 1, ... , b-1}. In this paper, we give some conditions on tau is an element of Z under which the scaling set tau A(b(n), C) is also a spectrum of mu(n)(b),(D).

Let Dn={0,an,bn}={0,1,2}(mod3),pn∈3Z+, n≥1, satisfy supn≥1⁡max⁡{|an|,|bn|}pn<∞. It is well-known that there exists a unique Borel probability measure μ{pn},{Dn} generated by the following infinite convolution productμ{pn},{Dn}=δp1−1D1⁎δ(p1p2)−1D2⁎⋯ 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 {e2πiλx}λ∈Λ forms an orthonormal basis for L2(μ{pn},{Dn}).

The open mapping theorem, the maximum modulus theorem and the uniqueness theorem are generalized from analytic functions to complex valued functions with C1 condition.Furthermore, the same results in high-dimensional space are established, the applications of the multiplicity of root, index and degree are discussed.

Letting Ωand G be two simply connected domains with locally connected boundary , letting f be a proper holomorphic mapping from Ωonto G, and liftingΩand G onto unit discs, in this work, we have obtained the equality relationship between component numbers of the point on ?G and those of its different inverse image points on ?G.The inequality relationship between the topological degree of f and the number of the different inverse image points is also obtained and discussed .

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 teaching complex analysis,one always faces difficulties in explaining the multiple-value functions.In his textbook,Professor Zhong Yuquan is quite successful in handling this problem.He presents the methods of dividing multiple-valued functions into single-valued branch and calculate the values of the functions through some examples.He summarizes these methods,and concludes that: once an original value is given,one can only get the values at other points through continuous change of variables.This is completely different from the traditional substitution method.However,after the summary he still uses the substitution method to calculate Arcsin2.We think that this defeats his own method just established,and confuses the readers. In this paper,we would like to share with the readers an alternative elementary method to calculate Arcsin2,through a discussion on dividing Arcsinz into single-valued branch.

The separation properties of countably infinite IFS is discussed.First,if the conformal IFS {Si}∞i=0 on a nonempty compact subset X satisfies bounded distortion property and weak separation condition,then |Si(X)|→0.Second,two special classes infinite IFS of similitudes are investigated.The results include that the attractor of the infinite IFS {Si}∞i=0 is unique and compact,and the Hausdorff dimension of other attractor is computed in terms of the spectral radius of certain weighted incidence matrix.Finally,some examples which satisfy the corresponding separation properties are given.

Let D be the open unit disc and let partial derivative D be the boundary of D. For f(z) analytic in D and continuous on (D) over bar, it follows from the open mapping theorem that partial derivative f(D) subset of f(partial derivative 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 derivative 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.

Let A (D) be the space of analytic functions on the open disk D and continuous on (D) over bar. Let partial derivative D be the boundary of D. we are interested in the class of f is an element of A (D) such that the image integral(partial derivative 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) and the mean growth rate of vertical bar f' (z)vertical bar near the boundary. For the specific cases we early 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. (C) 2012 Elsevier Inc. All rights reserved.

Let {Sj}nmj=1 be an iterated function system (IFS) of contractive similitudes on R3 defined by Sj(a)=aj+λ(a-aj),j=1,2,…,nm, where nm is the vertex number of the regular m-polyhedron in R3, aj is the vertex of the regular m-polyhedron, and 0λ1. We give a condition on λ so that IFS {Sj}nmj=1 satisfies the contact condition when λ=ρm. Also, the Hausdroff dimension of the attractor Km of {Sj}nmj=1 is given for 0λ≤ρm.