Self-affine Sets Research Articles

Self-affine $2$-attractors and tiles

We study two-digit attractors (2-attractors) in $\mathbb{R}^d$, which are self-affine compact sets defined by two affine contractions with the same linear part. They have widely been studied in the literature under various names (integer self-affine 2-tiles, twindragons, two-digit tiles, 2-reptiles and so on) due to many applications in approximation theory, in the construction of multivariate Haar systems and other wavelet bases, in discrete geometry and in number theory. We obtain a complete classification of isotropic 2-attractors in $\mathbb{R}^d$ and show that all of them are pairwise homeomorphic but not diffeomorphic. In the general, nonisotropic, case we prove that a 2-attractor is uniquely defined by the spectrum of the dilation matrix up to affine similarity. We estimate the number of different 2-attractors in $\mathbb{R}^d$ by analysing integer unitary expanding polynomials with free coefficient $\pm2$. The total number of such polynomials is estimated using the Mahler measure. We present several infinite series of such polynomials. For some 2-attractors their Hölder exponents are found. Some of our results are extended to attractors with an arbitrary number of digits. Bibliography: 63 titles.

Dimension drop of connected part of slicing self-affine sponges

The connected part of a metric space E is defined to be the union of non-trivial connected components of E. We proved that for a class of self-affine sets called slicing self-affine sponges, the connected part of E either coincides with E, or is essentially contained in the attractor of a proper sub-IFS of an iteration of the original IFS. This generalizes an early result of Huang and Rao (2021) [9] on a class of self-similar sets called fractal cubes. Moreover, we show that the result is no longer valid if the slicing property is removed. Consequently, for a Barański carpet E possessing trivial points, the Hausdorff dimension and the box dimension of the connected part of E are strictly less than that of E, respectively. For slicing self-affine sponges in Rd with d≥3, whether the attractor of a sub-IFS has strictly smaller dimensions is an open problem.

Continuity of sub-additive topological pressure with matrix cocycles * *The first author is partially supported by NSFC (11901305), the second author is partially supported by NSFC (11771317, 11790274), the third author is partially supported by NSFC (11871361, 11790274) and Qinglan project from Jiangsu Province.

Let A = {A 1, A 2, …, A k } be a finite collection of contracting affine maps, the corresponding pressure function P(A, s) plays the fundamental role in the study of dimension of self-affine sets. The zero of the pressure function always give the upper bound of the dimension of a self-affine set, and is exactly the dimension of ‘typical’ self-affine sets. In this paper, we consider an expanding base dynamical system, and establish the continuity of the pressure with the singular value function of a Hölder continuous matrix cocycle. This extends Feng and Shmerkin’s result in (Feng and Shmerkin 2014 Geom. Funct. Anal. 24 1101–1128) to a general setting.

On the group of isometries of planar IFS fractals* *The work is supported by NSFC (No. 11971109). The first author is partially supported by the Program for Probability and Statistics: Theory and Application (No. IRTL1704), and the program for innovative research team in science and technology in Fujian province university (No. IRTSTFJ).

For any iterated function system (IFS) on , let K be the attractor. Consider the group of all isometries on K. If K is a self-similar or self-affine set, it is proven that the group must be finite. If K is a bi-Lipschitz IFS fractal, the necessary and sufficient conditions for the infiniteness (or finiteness) of the group are given. For the finite case, the computation of the size of the group is also discussed.

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.

BEURLING DIMENSION AND SELF-AFFINE MEASURES

In this paper, we continue the study in our previous work Beurling dimenison and self-similar measures [J. Funct. Anal. 274 (2018) 2245–2264]. We analyze the Beurling dimension of Bessel sequences and frame spectra of self-affine measures on [Formula: see text]. Unlike the case for the self-similar measures, it is very difficult to obtain a general expression for the dimension of self-affine sets. This implies that the Hausdorff dimension may be not a good candidate that controls the Beurling dimension in general. Instead, we find that the pseudo Hausdorff dimension proposed by He and Lau [Math. Nachr. 281 (2008) 1142–1158] can be a good candidate that can control the Beurling dimension. With the help of the pseudo Hausdorff dimension, we obtain the upper bounds of the Beurling dimension of Bessel sequences. Under suitable condition, we give the lower bound of the Beurling dimension of frame spectra of self-affine measures on [Formula: see text]. Some examples are given to illustrate our theory.

SELF-AFFINE SETS: THE RELATION BETWEEN POSITIVE LEBESGUE MEASURE AND NON-EMPTY INTERIOR

For the self-affine IFS [Formula: see text] satisfying the weak separation condition (WSC), let [Formula: see text] be the corresponding self-affine set. It is proven in this paper that [Formula: see text] has positive Lebesgue measure only if the interior of [Formula: see text] is non-empty.

Properties of quasi-Assouad dimension

The connections between quasi-Assouad dimension and tangents are studied. We apply these results to the calculation of the quasi-Assouad dimension for a class of planar self-affine sets. We also show that sets with decreasing gaps have quasi-Assouad dimension 0 or 1 and exhibit an example of a set in the plane whose quasi-Assouad dimension is smaller than that of its projection onto the \(x\)-axis, showing that quasi-Assouad dimension may increase under Lipschitz mappings. Moreover, for closed sets, we show that the Hausdorff dimension is an upper bound for the quasi-lower Assouad dimension.

Spectrality of a class of self-affine measures and related digit sets

Spectrality of a class of self-affine measures and related digit sets

The box dimensions of exceptional self-affine sets in [formula omitted

We study the box dimensions of self-affine sets in R3 which are generated by a finite collection of generalised permutation matrices. We obtain bounds for the dimensions which hold with very minimal assumptions and give rise to sharp results in many cases. There are many issues in extending the well-established planar theory to R3 including that the principal planar projections are (affine distortions of) self-affine sets with overlaps (rather than self-similar sets) and that the natural modified singular value function fails to be sub-multiplicative in general. We introduce several new techniques to deal with these issues and hopefully provide some insight into the challenges in extending the theory further.

Visible part of dominated self-affine sets in the plane

The dimension of the visible part of self-affine sets, that satisfy domination and a projection condition, is being studied. The main result is that the Assouad dimension of the visible part equals to 1 for all directions outside the set of limit directions of the cylinders of the self-affine set. The result holds regardless of the overlap of the cylinders. The sharpness of the result is also being discussed.

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.

On Symmetric Compactly Supported Wavelets with Vanishing Moments Associated to $$E_d^{(2)}(\mathbb {Z})$$ Dilations

Let A be an expansive linear map on $${{\mathbb {R}}}^d$$ preserving the integer lattice and with $$| \det A|=2$$ . We prove that if there exists a self-affine tile set associated to A, there exists a compactly supported wavelet with any desired number of vanishing moments and some symmetry. We put emphasis on construction of wavelets associated to a linear map A on $${{\mathbb {R}}}^2$$ and to the Quincunx dilation on $${{\mathbb {R}}}^3$$ because we can remove the hypothesis of the existence of the self-affine tile set. Our construction is based on low pass filters by Han in dimension one with the dyadic dilation and multiresolution theory. Finally, for some particular dilation matrices, we realize that unidimensional Daubechies low pass filers can be adapted to obtain compactly supported wavelets with any desired degree of regularity and any fix number of vanishing moments.

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.

Domination, almost additivity, and thermodynamic formalism for planar matrix cocycles

In topics such as the thermodynamic formalism of linear cocycles, the dimension theory of self-affine sets, and the theory of random matrix products, it has often been found useful to assume positivity of the matrix entries in order to simplify or make feasible certain types of calculation. It is natural to ask how positivity may be relaxed or generalised in a way which enables similar calculations to be made in more general contexts. On the one hand one may generalise by considering almost additive or asymptotically additive potentials which mimic the properties enjoyed by the logarithm of the norm of a positive matrix cocycle; on the other hand one may consider matrix cocycles which are dominated, a condition which includes positive matrix cocycles but is more general. In this article we explore the relationship between almost additivity and domination for planar cocycles. We show in particular that a locally constant linear cocycle in the plane is almost additive if and only if it is either conjugate to a cocycle of isometries, or satisfies a property slightly weaker than domination which is introduced in this paper. Applications to matrix thermodynamic formalism are presented.

Dimension of the repeller for a piecewise expanding affine map

In this paper, we study the dimension theory of a class of piecewise affine systems in euclidean spaces suggested by Michael Barnsley, with some applications to the fractal image compression. It is a more general version of the class considered in the work of Keane, Simon and Solomyak [The dimension of graph directed attractors with overlaps on the line, with an application to a problem in fractal image recognition. {\it Fund. Math.}, {\bf 180}(3):279-292, 2003] and can be considered as the continuation of the works [On the dimension of self-affine sets and measures with overlaps. {\it Proc. Amer. Math. Soc.}, {\bf 144}(10):4427-4440, 2016], [On the dimension of triangular self-affine sets. {\it Erg. Th. \& Dynam. Sys.}, to appear.] by the authors. We also present some applications of our results for the generalized Takagi functions and fractal interpolation functions.

CONNECTEDNESS OF A CLASS OF SELF-AFFINE CARPETS

Let [Formula: see text], where [Formula: see text] are integers and [Formula: see text] be a digit set. Then the pair [Formula: see text] generates a fractal set [Formula: see text] satisfying [Formula: see text] which is a unit square. However, if we remove one digit from [Formula: see text], then the structure of [Formula: see text] will become very interesting. A well-known example is the Sierpinski carpet. In this paper, we study the resulting self-affine sets of moving a digit in [Formula: see text] to a different place. That is, we consider a digit set [Formula: see text], where [Formula: see text]. We give a complete characterization for the connectedness of self-affine carpet [Formula: see text] in terms of the domains of [Formula: see text] and [Formula: see text].

Open set condition and pseudo Hausdorff measure of self-affine IFSs

Let A be an n × n real expanding matrix and be a finite subset of with . The family of maps is called a self-affine iterated function system (self-affine IFS). The self-affine set is the unique compact set determined by satisfying the set-valued equation . The number with q = |det(A)|, is the so-called pseudo similarity dimension of K. As shown by He and Lau, one can associate with A and any number s ⩾ 0 a natural pseudo Hausdorff measure denoted by . In this paper, we show that, if s is chosen to be the pseudo similarity dimension of K, then the condition holds if and only if the IFS satisfies the open set condition (OSC). This extends the well-known result for the self-similar case that the OSC is equivalent to K having positive Hausdorff measure for a suitable s. Furthermore, we relate the exact value of pseudo Hausdorff measure to a notion of upper s-density with respect to the pseudo norm w(x) associated with A for the measure in the case that .

Analyticity of the affinity dimension for planar iterated function systems with matrices which preserve a cone

The sub-additive pressure function for an affine iterated function system (IFS) and the affinity dimension, defined as the unique solution s0 to P(s0) = 1, were introduced by Falconer in his seminal 1988 paper on self-affine fractals. The affinity dimension prescribes a value for the Hausdorff dimension of a self-affine set which is known to be correct in generic cases and in an increasing range of explicit cases. It was shown by Feng and Shmerkin in 2014 that the affinity dimension depends continuously on the IFS. In this article we prove that when the linear parts of the affinities which define the IFS are matrices which strictly preserve a common cone, the sub-additive pressure is locally real analytic as a function of the matrix coefficients of the linear parts of the affinities. In this setting we also show that the sub-additive pressure is piecewise real analytic in s, implying that the affinity dimension is locally analytic in the matrix coefficients. Combining this with a recent result of Bárány et al we obtain results concerning the analyticity of the Hausdorff dimension for certain families of planar self-affine sets.

Birkhoff and Lyapunov Spectra on Planar Self-affine Sets

Abstract Working on strongly irreducible planar self-affine sets satisfying the strong open set condition, we calculate the Birkhoff spectrum of continuous potentials and the Lyapunov spectrum.