Self-affine Measure Research Articles

A class of spectral self-affine measures with four-element digit sets

The self-affine measure μM,D associated with an expanding matrix M∈Mn(Z) and a finite digit set D⊂Zn is uniquely determined by the self-affine identity with equal weight. In this paper we construct a class of self-affine measures μM,D with four-element digit sets in the higher dimensions (n≥3) such that the Hilbert space L2(μM,D) possesses an orthogonal exponential basis. That is, μM,D is spectral. Such a spectral measure cannot be obtained from the condition of compatible pair. This extends the corresponding result in the plane.

Extensions of Laba-Wang's condition for spectral pairs

Let be a self-affine measure associated with an expanding matrix and a finite digit set . We consider in this paper the spectrality of . In the case when is a compatible pair for some , a necessary condition is obtained for the spectral pair . This condition is shown to be equivalent to the known necessary conditions for the same spectral pair. Moreover, we prove that all these necessary conditions are not sufficient in the higher dimensions, but they are sufficient in the dimension one. This extends Laba-Wang's condition for spectral pairs.

Spectral self-affine measures with prime determinant

The self-affine measure \(\mu _{M,D}\) relating to an expanding matrix \(M\in M_{n}(\mathbb Z )\) and a finite digit set \(D\subset \mathbb Z ^n\) is a unique probability measure satisfying the self-affine identity with equal weight. In the present paper, we shall study the spectrality of \(\mu _{M,D}\) in the case when \(|\det (M)|=p\) is a prime. The main result shows that under certain mild conditions, if there are two points \(s_{1}, s_{2}\in \mathbb R ^{n}, s_{1}-s_{2}\in \mathbb Z ^{n}\) such that the exponential functions \(e_{s_{1}}(x), e_{s_{2}}(x)\) are orthogonal in \(L^{2}(\mu _{M,D})\), then the self-affine measure \(\mu _{M,D}\) is a spectral measure with lattice spectrum. This gives some sufficient conditions for a self-affine measure to be a lattice spectral measure.

Spectrality of self-affine measures on the three-dimensional Sierpinski gasket

AbstractThe self-affine measure μM, D corresponding to M = diag[p1, p2, p3] (pj ∈ ℤ \ {0, ± 1}, j = 1, 2, 3) and D = {0, e1, e2, e3} in the space ℝ3 is supported on the three-dimensional Sierpinski gasket T(M, D), where e1, e2, e3 are the standard basis of unit column vectors in ℝ3. We shall determine the spectrality and non-spectrality of μM, D, and show that if pj ∈ 2ℤ \ {0, 2} for j = 1, 2, 3, then μM, D is a spectral measure, and if pj ∈ (2ℤ + 1) \ {±1} for j = 1, 2, 3, then μM, D is a non-spectral measure and there exist at most 4 mutually orthogonal exponential functions in L2(μM, D), where the number 4 is the best possible. This generalizes the known results on the spectrality of self-affine measures.

Spectrality of a class of self-affine measures with decomposable digit sets

The self-affine measure associated with an expanding matrix and a finite digit set is uniquely determined by the self-affine identity with equal weight. The spectral and non-spectral problems on the selfaffine measures have some surprising connections with a number of areas in mathematics, and have been received much attention in recent years. In the present paper, we shall determine the spectrality and non-spectrality of a class of self-affine measures with decomposable digit sets. We present a method to deal with such case, and clarify the spectrality and non-spectrality of a class of self-affine measures by applying this method.

Spectra of a class of self-affine measures

In the present paper we will study the spectral property of a class of self-affine measures under the condition of compatible pair. We first answer a question of Dutkay and Jorgensen concerning the relationship between spectral self-affine measure and compatible pair. We then consider the spectra of Bernoulli convolutions and obtain a sharp result which extends the corresponding result of Jorgensen, Kornelson and Shuman. Finally, we provide a structural property for the integer spectrum of a spectral self-affine measure.

On the [formula omitted]-orthogonal exponentials

Let μ M , D be a self-affine measure corresponding to a given affine iterated function system { ϕ d ( x ) = M − 1 ( x + d ) } d ∈ D . In the present paper we will study the problem of how to determine the L 2 ( μ M , D ) -space has finite or infinite orthogonal exponentials. Such research is motivated by a conjecture on the non-spectrality of μ M , D . We first obtain a non-spectral criterion which extends the result of Dutkay and Jorgensen. In opposition to the condition of this criterion, we then obtain some conditions which imply the infinite orthogonal systems in L 2 ( μ M , D ) . These are necessary for further investigation on the spectrality of μ M , D . As an application, we completely settle the corresponding problem for the generalized planar Sierpinski family.

Non-spectral self-affine measure problem on the plane domain

The self-affine measure μ M , D corresponding to an expanding integer matrix M = [ a b c d ] and D = { ( 0 0 ) , ( 1 0 ) , ( 0 1 ) } is supported on the attractor (or invariant set) of the iterated function system { ϕ d ( x ) = M − 1 ( x + d ) } d ∈ D . In the present paper we show that if ( a + d ) 2 = 4 ( a d − b c ) and a d − b c is not a multiple of 3, then there exist at most 3 mutually orthogonal exponential functions in L 2 ( μ M , D ) , and the number 3 is the best. This extends several known results on the non-spectral self-affine measure problem. The proof of such result depends on the characterization of the zero set of the Fourier transform μ ˆ M , D , and provides a way of dealing with the non-spectral problem.

The quantization dimension of the self-affine measures on general Sierpiński carpets

Let μ be a self-affine measure on a general Sierpinski carpet E. We give a characterization for the upper and lower quantization dimension of μ in terms of revised cylinder sets. Using this characterization, we prove that the quantization dimension D r (μ) of μ exists for all r > 0 under an additional condition. We establish an explicit formula for D r (μ) and show that it increases to the box-counting dimension \({dim_B^* \mu}\) of μ as r tends to infinity. For a class of Sierpinski carpets E and the uniform measures μ on E, we show that the quantization dimension of μ coincides with its box-counting dimension and that the D r (μ)-dimensional upper and lower quantization coefficient of μ are both positive and finite.

Self-similar and self-affine sets: measure of the intersection of two copies

AbstractLetK⊂ℝdbe a self-similar or self-affine set and letμbe a self-similar or self-affine measure on it. Let 𝒢 be the group of affine maps, similitudes, isometries or translations of ℝd. Under various assumptions (such as separation conditions, or the assumption that the transformations are small perturbations, or thatKis a so-called Sierpiński sponge) we prove theorems of the following types, which are closely related to each other.•(Non-stability)There exists a constantc<1 such that for everyg∈𝒢 we have eitherμ(K∩g(K))<c⋅μ(K) orK⊂g(K).•(Measure and topology)For everyg∈𝒢 we haveμ(K∩g(K))>0⟺∫K(K∩g(K))≠0̸ (where ∫Kis interior relative toK).•(Extension)The measureμhas a 𝒢-invariant extension to ℝd.Moreover, in many situations we characterize thosegfor whichμ(K∩g(K))>0. We also obtain results about thosegfor whichg(K)⊂Korg(K)⊃K.

Non-spectrality of planar self-affine measures with three-elements digit set

The self-affine measure μ M , D associated with an affine iterated function system { ϕ d ( x ) = M − 1 ( x + d ) } d ∈ D is uniquely determined. The problems of determining the spectrality or non-spectrality of a measure μ M , D have been received much attention in recent years. One of the non-spectral problem on μ M , D is to estimate the number of orthogonal exponentials in L 2 ( μ M , D ) and to find them. In the present paper we show that for an expanding integer matrix M ∈ M 2 ( Z ) and the three-elements digit set D given by M = [ a b d c ] and D = { ( 0 0 ) , ( 1 0 ) , ( 0 1 ) } , if a c − b d ∉ 3 Z , then there exist at most 3 mutually orthogonal exponentials in L 2 ( μ M , D ) , and the number 3 is the best. This confirms the three-elements digit set conjecture on the non-spectrality of self-affine measures in the plane

Covariance matrices of self-affine measures

In this paper we derive a formula for the covariance matrix of any self-affine measure, i.e. a probability measure μ satisfying μ = ∑ k = 1 l p k μ ∘ S k − 1 , where { S k ( x ) = A k x + b k } 1 ≤ k ≤ l is a family of affine contractive maps and { p k } 1 ≤ k ≤ l is a set of probability weights. In particular if for every k , A k = A then the formula has the following form D 2 X = [ I ⊗ I − A ⊗ A ] − 1 D 2 B , where D 2 X denotes the covariance matrix of the measure μ and D 2 B denotes a covariance matrix of a discrete random variable B with values b k , and corresponding probabilities p k .

Orthogonal exponentials on the generalized three-dimensional Sierpinski gasket

The self-affine measure μ M , D corresponding to the expanding integer matrix M = [ p 0 m 0 p 0 0 0 p ] and D = { ( 0 0 0 ) , ( 1 0 0 ) , ( 0 1 0 ) , ( 0 0 1 ) } is supported on the generalized three-dimensional Sierpinski gasket T ( M , D ) , where p is odd. In the present paper we show that there exist at most 7 mutually orthogonal exponential functions in L 2 ( μ M , D ) . This generalizes the result of Dutkay and Jorgensen [D.E. Dutkay, P.E.T. Jorgensen, Analysis of orthogonality and of orbits in affine iterated function systems, Math. Z. 256 (2007) 801–823] on the non-spectral self-affine measure problem. By using the same method, we also obtain that for self-affine measure μ M , D corresponding to the expanding integer matrix M = [ p 0 0 p ] and D = { ( 0 0 ) , ( 1 0 ) , ( 0 1 ) , ( 1 1 ) } , where p is odd, there exist at most 5 mutually orthogonal exponential functions in L 2 ( μ M , D ) .

Singularity of certain self-affine measures

The self-affine measure associated with an iterated function system and a weight is uniquely determined. The problem of determining whether a self-affine measure is absolutely continuous or singular has been studied extensively in recent years. In the present paper we consider the singularity of certain self-affine measures. We obtain a sufficient condition for such measures being singular. Two applications of this result are given, which extend several known results in a simple manner.

Analysis of μ R,D -Orthogonality in Affine Iterated Function Systems

Let μR,D be a self-affine measure associated with an expanding integer matrix R∈Mn(ℤ) and a finite subset D⊆ℤn. In the present paper we study the μR,D-orthogonality and compatible pair conditions. We also show that any set of μR,D-orthogonal exponentials contains at most 3 elements on the generalized plane Sierpinski gasket and the number 3 is the best.

Non-spectral problem for a class of planar self-affine measures

The self-affine measure μ M , D corresponding to an expanding matrix M ∈ M n ( R ) and a finite subset D ⊂ R n is supported on the attractor (or invariant set) of the iterated function system { ϕ d ( x ) = M −1 ( x + d ) } d ∈ D . The spectral and non-spectral problems on μ M , D , including the spectrum-tiling problem implied in them, have received much attention in recent years. One of the non-spectral problem on μ M , D is to estimate the number of orthogonal exponentials in L 2 ( μ M , D ) and to find them. In the present paper we show that if a , b , c ∈ Z , | a | > 1 , | c | > 1 and a c ∈ Z ∖ ( 3 Z ) , M = [ a b 0 c ] and D = { ( 0 0 ) , ( 1 0 ) , ( 0 1 ) } , then there exist at most 3 mutually orthogonal exponentials in L 2 ( μ M , D ) , and the number 3 is the best. This extends several known conclusions. The proof of such result depends on the characterization of the zero set of the Fourier transform μ ˆ M , D , and provides a way of dealing with the non-spectral problem.

Orthogonal exponentials on the generalized plane Sierpinski gasket

The self-affine measure μ M p , D corresponding to M p = 2 p 0 2 ( p ∈ Z ) and D = 0 0 , 1 0 , 0 1 is supported on the the generalized plane Sierpinski gasket T ( M p , D ) . In the present paper we show that there exist at most 3 mutually orthogonal exponential functions in L 2 ( μ M p , D ) , and the number 3 is the best. This generalizes several known results on the non-spectral self-affine measure problem.

Formula omitted]-Orthogonality and compatible pair

Let μ M , D be a self-affine measure associated with an expanding integer matrix M ∈ M n ( Z ) and a finite subset D ⊂ Z n . In the present paper we study the μ M , D -orthogonality and compatible pair conditions as well as relations between them. The research here is based on the structure of vanishing sums of roots of unity, and is closely related to the problem of spectral self-affine measure.