Self-affine Measure Research Articles

The cardinality of orthogonal exponentials of planar self-affine measures with two-element digit set

Abstract Let μ M , D {{\mu_{M,D}}} be the planar self-affine measure determined by an expanding integer matrix M ∈ M 2 ⁢ ( ℤ ) {M\in M_{2}(\mathbb{Z})} and a two-element digit set D ⊂ ℤ 2 {D\subset\mathbb{Z}^{2}} . It has been shown that the spectral or non-spectral problem on μ M , D {\mu_{M,D}} is only related to trace ⁡ ( M ) := r 1 {\operatorname{trace}(M):=r_{1}} and det ⁡ ( M ) := r 2 {\det(M):=r_{2}} . In the case when M ∈ M 2 ⁢ ( ℤ ) {M\in M_{2}(\mathbb{Z})} is an expanding matrix and r 1 2 = 3 ⁢ r 2 {r_{1}^{2}=3{r_{2}}} , r 2 ∈ 2 ⁢ ℤ + 1 ∖ { ± 1 } {{r_{2}}\in 2\mathbb{Z}+1\setminus\{\pm 1\}} , the Hilbert space L 2 ⁢ ( μ M , D ) {L^{2}(\mu_{M,D})} contains at most a finite number of orthogonal exponentials, and μ M , D {{\mu_{M,D}}} is a non-spectral measure. The remaining problem in this case is to determine the best upper bound on the cardinality of orthogonal exponentials in the Hilbert space L 2 ⁢ ( μ M , D ) {L^{2}(\mu_{M,D})} . In the present paper, we further the above research to show that there are at most 16 mutually orthogonal exponentials in the corresponding Hilbert space, and the number 16 is the best upper bound. This completes the non-spectrality of self-affine measures in the above case.

Arbitrarily Sparse Spectra for Self-Affine Spectral Measures

Given an expansive matrix R ∈ Md(ℤ) and a finite set of digit B taken from ℤd/R(ℤd). It was shown previously that if we can find an L such that (R, B, L) forms a Hadamard triple, then the associated fractal self-affine measure generated by (R, B) admits an exponential orthonormal basis of certain frequency set Λ, and hence it is termed as a spectral measure. In this paper, we show that if #B < ∣det(R)∣, not only it is spectral, we can also construct arbitrarily sparse spectrum Λ in the sense that its Beurling dimension is zero.

The Spectrality of Self-affine Measure Under the Similar Transformation of $$GL_n(p)$$

The Spectrality of Self-affine Measure Under the Similar Transformation of $$GL_n(p)$$

A converse statement to Hutchinson’s theorem and a dimension gap for self-affine measures

A well-known theorem of J. E. Hutchinson states that if an iterated function system consists of similarity transformations and satisfies the open set condition then its attractor supports a self-similar measure with Hausdorff dimension equal to the similarity dimension. In this article we prove the following result which may be regarded as a form of partial converse: if an iterated function system consists of invertible affine transformations whose linear parts do not preserve a common invariant subspace, and its attractor supports a self-affine measure with Hausdorff dimension equal to the affinity dimension, then the system necessarily consists of similarity transformations. We obtain this result by showing that the equilibrium measures of an affine iterated function system are never Bernoulli measures unless the system either is reducible or consists of similarity transformations. The proof builds on earlier results in the thermodynamic formalism of affine iterated function systems due to Bochi, Feng, Käenmäki, Shmerkin and the first named author and also relies on the work of Benoist on the spectral properties of Zariski-dense subsemigroups of reductive linear groups.

The Assouad dimension of self-affine measures on sponges

AbstractWe derive upper and lower bounds for the Assouad and lower dimensions of self-affine measures in $\mathbb {R}^d$ generated by diagonal matrices and satisfying suitable separation conditions. The upper and lower bounds always coincide for $d=2,3$ , yielding precise explicit formulae for those dimensions. Moreover, there are easy-to-check conditions guaranteeing that the bounds coincide for $d \geqslant 4$ . An interesting consequence of our results is that there can be a ‘dimension gap’ for such self-affine constructions, even in the plane. That is, we show that for some self-affine carpets of ‘Barański type’ the Assouad dimension of all associated self-affine measures strictly exceeds the Assouad dimension of the carpet by some fixed $\delta&gt;0$ depending only on the carpet. We also provide examples of self-affine carpets of ‘Barański type’ where there is no dimension gap and in fact the Assouad dimension of the carpet is equal to the Assouad dimension of a carefully chosen self-affine measure.

Fourier decay for homogeneous self-affine measures

We show that for Lebesgue almost all d -tuples (\theta_1,\ldots,\theta_d) , with |\theta_j|&gt;1 , any self-affine measure for a homogeneous non-degenerate iterated function system \{Ax+a_j\}_{j=1}^m in \mathbb{R}^d , where A^{-1} is a diagonal matrix with the entries (\theta_1,\ldots,\theta_d) , has power Fourier decay at infinity.

Spectrality of a Class of Self-Affine Measures with Prime Determinant

We study the spectrality of a class of self-affine measures with prime determinant. Spectral measures are connected with fractal geometry that shows some kind of geometrical self-similarity under magnification. To make the self-affine measure becomes a spectral measure with lattice spectrum, we provide two new sufficient conditions related to the elements of digit set and zero set, respectively. The two sufficient conditions are more precise and easier to be verified as compared with the previous research. Moreover, these conditions offer a fresh perspective on a conjecture of Lagarias and Wang.

Hausdorff dimension of planar self-affine sets and measures with overlaps

We prove that if \mu is a self-affine measure in the plane whose defining IFS acts totally irreducibly on \mathbb{RP}^1 and satisfies an exponential separation condition, then its dimension is equal to its Lyapunov dimension.We also treat a class of reducible systems. This extends our previous work on the subject with Bárány to the overlapping case.

Spectrality of generalized Sierpinski-type self-affine measures

In this work, we study the spectral property of generalized Sierpinski-type self-affine measures μ M , D on R 2 generated by an expanding integer matrix M ∈ M 2 ( Z ) with det ⁡ ( M ) ∈ 3 Z and a non-collinear integer digit set D = { ( 0 , 0 ) t , ( α 1 , α 2 ) t , ( β 1 , β 2 ) t } with α 1 β 2 − α 2 β 1 ∈ 3 Z . We give the sufficient and necessary conditions for μ M , D to be a spectral measure, i.e., there exists a countable subset Λ ⊂ R 2 such that E ( Λ ) = { e 2 π i 〈 λ , x 〉 : λ ∈ Λ } forms an orthonormal basis for L 2 ( μ M , D ) . This completely settles the spectrality of the self-affine measure μ M , D .

INFINITE ORTHOGONAL EXPONENTIALS OF A CLASS OF SELF-AFFINE 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].

The spectral property of a class of self-affine measures on [formula omitted

Let 0<|ρ|<1 and I2 be the 2×2 identical matrix. For an expanding real matrix M=ρ−1I2∈M2(R) and a four-element digit set D={(0,0)t,(1,0)t,(0,1)t,(−1,−1)t}⊂Z2 with cardinality |D|, let μM,D be the self-affine measure defined by μM,D(⋅)=1|D|∑d∈DμM,D(M(⋅)−d). In this paper, we prove that L2(μM,D) has an exponential orthonormal basis if and only if ρ−1∈2Z∖{0}.

Uniformity of spectral self-affine measures

For an arbitrary self-affine measure defined by a self-affine iterated function system and a family of probability weights, it is proven in this article that, if a self-affine measure is a spectral measure, then the probability weights must be equal and measure non-overlap holds in a weaker sense. In particular, all spectral integral self-affine measures satisfy the OSC.

On the spectrality of self-affine measures with four digits on ℝ2

In this paper, we study the spectral property of the self-affine measure [Formula: see text] generated by an expanding real matrix [Formula: see text] and the four-element digit set [Formula: see text]. We show that [Formula: see text] is a spectral measure, i.e. there exists a discrete set [Formula: see text] such that the collection of exponential functions [Formula: see text] forms an orthonormal basis for [Formula: see text], if and only if [Formula: see text] for some [Formula: see text]. A similar characterization for Bernoulli convolution is provided by Dai [X.-R. Dai, When does a Bernoulli convolution admit a spectrum? Adv. Math. 231(3) (2012) 1681–1693], over which [Formula: see text]. Furthermore, we provide an equivalent characterization for the maximal bi-zero set of [Formula: see text] by extending the concept of tree-mapping in [X.-R. Dai, X.-G. He and C. K. Lai, Spectral property of Cantor measures with consecutive digits, Adv. Math. 242 (2013) 187–208]. We also extend these results to the more general self-affine measures.

Non-Spectral Problem of Self-Affine Measures in R&amp;lt;sup&amp;gt;3&amp;lt;/sup&amp;gt;

The problems of determining the spectrality or non-spectrality of a measure have been received much attention in recent years. One of the non-spectral problems on μM,D is to estimate the number of orthogonal exponentials in L2(μM,D). In the present paper, we establish some relations inside the zero set by the Fourier transform of the self-affine measure μM,D. Based on these facts, we show that 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 extends several known conclusions.

Spectrality of a class of planar self-affine measures with three-element digit sets

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

Assouad dimension of planar self-affine sets

We calculate the Assouad dimension of a planar self-affine set $X$ satisfying the strong separation condition and the projection condition and show that $X$ is minimal for the conformal Assouad dimension. Furthermore, we see that such a self-affine set $X$ adheres to very strong tangential regularity by showing that any two points of $X$, which are generic with respect to a self-affine measure having simple Lyapunov spectrum, share the same collection of tangent sets.

Fourier transform of self-affine measures

Suppose F is a self-affine set on Rd, d≥2, which is not a singleton, associated to affine contractions fj=Aj+bj, Aj∈GL(d,R), bj∈Rd, j∈A, for some finite A. We prove that if the group Γ generated by the matrices Aj, j∈A, forms a proximal and totally irreducible subgroup of GL(d,R), then any self-affine measure μ=∑pjfjμ, ∑pj=1, 0<pj<1, j∈A, on F is a Rajchman measure: the Fourier transform μˆ(ξ)→0 as |ξ|→∞. As an application this shows that self-affine sets with proximal and totally irreducible linear parts are sets of rectangular multiplicity for multiple trigonometric series. Moreover, if the Zariski closure of Γ is connected real split Lie group in the Zariski topology, then μˆ(ξ) has a power decay at infinity. Hence μ is Lp improving for all 1<p<∞ and F has positive Fourier dimension. In dimension d=2,3 the irreducibility of Γ and non-compactness of the image of Γ in PGL(d,R) is enough for power decay of μˆ. The proof is based on quantitative renewal theorems for random walks on the sphere Sd−1.

Orthogonal exponentials of self-affine measures on ℝn

Let the self-affine measure [Formula: see text] be generated by an expanding matrix [Formula: see text] and a finite integer digit set [Formula: see text], where [Formula: see text] with [Formula: see text] and [Formula: see text]. In this paper, we show that if [Formula: see text] for an integer [Formula: see text], then [Formula: see text] admits 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].

ASYMPTOTIC ORDER OF THE GEOMETRIC MEAN ERROR FOR SELF-AFFINE MEASURES ON BEDFORD–MCMULLEN CARPETS

Let [Formula: see text] be a Bedford–McMullen carpet associated with a set of affine mappings [Formula: see text] and let [Formula: see text] be the self-affine measure associated with [Formula: see text] and a probability vector [Formula: see text]. We study the asymptotics of the geometric mean error in the quantization for [Formula: see text]. Let [Formula: see text] be the Hausdorff dimension for [Formula: see text]. Assuming a separation condition for [Formula: see text], we prove that the [Formula: see text]th geometric error for [Formula: see text] is of the same order as [Formula: see text].

ORTHOGONAL EXPONENTIAL FUNCTIONS OF THE PLANAR SELF-AFFINE MEASURES WITH FOUR DIGITS

For the self-affine measure [Formula: see text] generated by an expanding matrix [Formula: see text] and an integer digit set [Formula: see text] with [Formula: see text], Su et al. proved that if [Formula: see text], then [Formula: see text] contains an infinite orthogonal set of exponential functions if and only if [Formula: see text] [J. Su, Y. Liu and J. C. Liu, Non-spectrality of the planar self-affine measures with four-element digit sets, Fractals (2019), https://doi.org/10.1142/S0218348X19501159 ]. In this paper, we show that the above conclusion also holds for [Formula: see text]. So, a complete characterization of [Formula: see text] containing an infinite orthogonal set of exponential functions is given.