Open Set Condition Research Articles

A self-similar set with non-locally connected components

Luo, Rao and Xiong [Topol. Appl. 322 (2022), 108271] conjectured that if a planar self-similar iterated function system with the open set condition does not involve rotations or reflections, then every connected component of the attractor is locally connected. We create a homogeneous counterexample of Lalley–Gatzouras type, which disproves this conjecture.

Shrinking target problems in d-decaying Gauss-like iterated function systems

Let Φ={ϕn}n≥1 be an infinite iterated function system (IFS) on [0,1] satisfying the open set condition with the open set (0,1) and let J be its attractor. For any x∈J, there is a unique integer sequence {ωn(x)}n≥1, called the digit sequence of x, such thatx=limn→∞⁡ϕω1(x)∘⋯∘ϕωn(x)(1) except at countably many points. In this paper, we investigate the shrinking target problem in the infinite iterated function system with a polynomial decay of the derivative. More precisely, we consider the size of the setW(T,x0)={x∈J:Tn(x)∈Itn(x0) for infinitely many n∈N}, where T:J⟼J is the expanding map induced by the left shift and Itn(x0) denotes the tn-th cylinder of x0∈J. As a continuation of one result of Li et al. (2014) [11], we obtain the exact Hausdorff dimension of the set W(T,x0) in some general infinite function systems.

Rethinking group activity recognition under the open set condition

Rethinking group activity recognition under the open set condition

Transversal family of non-autonomous conformal iterated function systems

We study Non-autonomous Iterated Function Systems (NIFSs) with overlaps. A NIFS on a compact subset X\subset\mathbb{R}^{m} is a sequence \Phi=(\{\phi^{(j)}_{i}\}_{i\in I^{(j)}})_{j=1}^{\infty} of collections of uniformly contracting maps \phi^{(j)}_{i}\colon X\rightarrow X , where I^{(j)} is a finite set. In comparison to usual iterated function systems, we allow the contractions \phi^{(j)}_{i} applied at each step j to depend on j . In this paper, we focus on a family of parameterized NIFSs on \mathbb{R}^{m} . Here, we do not assume the open set condition. We show that if a d -parameter family of such systems satisfies the transversality condition, then for almost every parameter value the Hausdorff dimension of the limit set is the minimum of m and the Bowen dimension. Moreover, we give an example of a family \{\Phi_{t}\}_{t\in U} of parameterized NIFSs such that \{\Phi_{t}\}_{t\in U} satisfies the transversality condition but \Phi_{t} does not satisfy the open set condition for any t\in U .

A dimensional mass transference principle from ball to rectangles for projections of Gibbs measures and applications

Mass transference principle are tools designed to provide estimates for the Hausdorff dimension of points approximable at a certain rate by a specific sequence of points (for instance rationals). Usually, such results are established provided that ambiant space is equipped with an Ahlfors regular measure (see [7,28] for instance). In the case of balls, Barral and Seuret established that mass transference principles still holds provided that the ambiant measure is a Gibbs measure or a self-similar measure satisfying the open set condition ([3]). In this article, we establish a mass transference principle “from ball to rectangle” assuming that the ambiant measure is the projection of Gibbs measure on the dyadic grid. Such measures are not Ahlfors regular in general and this result extends in particular the mass transference principle from ball to rectangle in the case of the Lebesgue measure, established in [28], and partially the result of Barral and Seuret [3]. As an application, given b∈N, we estimate the dimension of weighted approximation of points of [0,1]2 approximable by points with rational coordinates whose expansion in base b has prescribed frequencies. These results are new and surprising in the sens that, although the Lebesgue measure “scales” naturally rectangles, this needs not be the case for projection of Gibbs measures.

Intersections of the pieces of self-similar dendrites in the plane

We prove that each self-similar dendrite in the plane has the weak separation property and that the ramification orders of its points are bounded. We show that there are self-similar dendrites in the plane that satisfy the Open Set Condition, such that several pieces may intersect in the same non-trivial subdendrite.

ON SELF-SIMILARITY OF QUADRANGLE

Wen [Some problems in fractal geometry, in Report in Chinese Conference on Fractal Geometry and Dynamical Systems, 2023] posed the following questions: when a quadrangle is a self-similar set with the open set condition (OSC). It is well known that any convex quadrangle is a self-similar set. For the concave quadrangles, this case is more complicated. Loosely speaking, some concave polygons may be not self-similar. In this paper, we prove two results. First, we give a necessary and sufficient condition such that a concave quadrangle is a self-similar set. Moreover, we also give a necessary condition for a concave quadrangle to be a self-similar set with the OSC. For the convex quadrangle, we investigate the trapezoid. We show under some condition that the trapezoid is a self-similar set with the OSC. In particular, for any trapezoid, if the ratio of the lengths of two bases is rational, then the trapezoid is a self-similar set with the OSC.

Numerical evaluation of singular integrals on non-disjoint self-similar fractal sets

We consider the numerical evaluation of a class of double integrals with respect to a pair of self-similar measures over a self-similar fractal set (the attractor of an iterated function system), with a weakly singular integrand of logarithmic or algebraic type. In a recent paper (Gibbs et al. Numer. Algorithms 92, 2071–2124 2023), it was shown that when the fractal set is “disjoint” in a certain sense (an example being the Cantor set), the self-similarity of the measures, combined with the homogeneity properties of the integrand, can be exploited to express the singular integral exactly in terms of regular integrals, which can be readily approximated numerically. In this paper, we present a methodology for extending these results to cases where the fractal is non-disjoint but non-overlapping (in the sense that the open set condition holds). Our approach applies to many well-known examples including the Sierpinski triangle, the Vicsek fractal, the Sierpinski carpet, and the Koch snowflake.

A dichotomy on the self-similarity of graph-directed attractors

This paper seeks conditions that ensure that the attractor of a graph directed iterated function system (GD-IFS) cannot be realised as the attractor of a standard iterated function system (IFS). For a strongly connected directed graph, it is known that, if all directed circuits go through a particular vertex, then for any GD-IFS of similarities on \mathbb{R} based on the graph and satisfying the convex open set condition (COSC), its attractor associated with that vertex is also the attractor of a (COSC) standard IFS. In this paper we show the following complementary result. If there exists a directed circuit which does not go through a certain vertex, then there exists a GD-IFS based on the graph such that the attractor associated with that vertex is not the attractor of any standard IFS of similarities. Indeed, we give algebraic conditions for such GD-IFS attractors not to be attractors of standard IFSs, and thus show that ‘almost-all’ COSC GD-IFSs based on the graph have attractors associated with this vertex that are not the attractors of any COSC standard IFS.

Cross-corpus open set bird species recognition by vocalization

In the wild, bird vocalizations of the same species across different populations may be different (e.g., so called dialect). Besides, the number of species is unknown in advance. These two facts make the task of bird species recognition based on vocalization a challenging one. This study treats this task as an open set recognition (OSR) cross-corpus scenario. We propose Instance Frequency Normalization (IFN) to remove instance-specific differences across different corpora. Furthermore, an x-vector feature extraction model integrated Time Delay Neural Network (TDNN) and Long Short-Term Memory (LSTM) are designed to better capture sequence information. Finally, the threshold-based Probabilistic Linear Discriminant Analysis (PLDA) is introduced to discriminate the extracted x-vector features to discover the unknown classes. When compared to the best results of the existing method, the average ACCs for the single-corpus and cross-corpus experiments are improved, implying that our method can provide a potential solution and improve performance for cross-corpus bird species recognition based on vocalization in open set condition.

Mean Lipschitz–Killing curvatures for homogeneous random fractals

Homogeneous random fractals form a probabilistic extension of self-similar sets with more dependencies than in random recursive constructions. For such random fractals we consider mean values of the Lipschitz–Killing curvatures of their parallel sets for small parallel radii. Under the uniform strong open set condition and some further geometric assumptions, we show that rescaled limits of these mean values exist as the parallel radius tends to 0 . Moreover, integral representations are derived for these limits which extend those known in the deterministic case.

Distribution of δ-connected components of self-affine sponges of Lalley-Gatzouras type

Let (E,ρ) be a metric space and let hE(δ) be the cardinality of the set of δ-connected components of E. In literature, in case of that E is a self-conformal set satisfying the open set condition or E is a self-affine Sierpiński sponge, necessary and sufficient condition is given for the validity of the relationhE(δ)≍δ−dimB⁡E, when δ→0. In this paper, we generalize the above result to self-affine sponges of Lalley-Gatzouras type; actually in this case, we show that there exists a Bernoulli measure μ such that for any cylinder W, it holds thathW(δ)≍μ(W)δ−dimB⁡E, when δ→0.

LOCAL GEOMETRY OF SELF-SIMILAR SETS: TYPICAL BALLS, TANGENT MEASURES AND ASYMPTOTIC SPECTRA

We analyze the local geometric structure of self-similar sets with open set condition through the study of the properties of a distinguished family of spherical neighborhoods, the typical balls. We quantify the complexity of the local geometry of self-similar sets, showing that there are uncountably many classes of spherical neighborhoods that are not equivalent under similitudes. We show that at a tangent level, the uniformity of the Euclidean space is recuperated in the sense that any typical ball is a tangent measure of the measure [Formula: see text] at [Formula: see text]-a.e. point, where [Formula: see text] is any self-similar measure. We characterize the spectrum of asymptotic densities of metric measures in terms of the packing and centered Hausdorff measures. As an example, we compute the spectrum of asymptotic densities of the Sierpiński gasket.

GENERALIZED CANTOR-INTEGERS AND INTERVAL DENSITY OF HOMOGENEOUS CANTOR SETS

In this paper, we study the generalized Cantor-integers [Formula: see text] with the base conversion function [Formula: see text] being strictly increasing and satisfying [Formula: see text] and [Formula: see text]. We show that the sequence [Formula: see text] with [Formula: see text] is dense in the closed interval with the endpoints being its inferior and superior, respectively. Moreover, every homogeneous Cantor set [Formula: see text] satisfying open set condition can be induced by some generalized Cantor-integers, we get the exact point which attains the maximal interval density of the form [Formula: see text] with respect to the self-similar probability measure supported on [Formula: see text]. This result partially confirms a conjecture of E. Ayer and R. S. Strichartz [Exact Hausdorff measure and intervals of maximum density for Cantor sets, Trans. Am. Math. Soc. 351(9) (1999) 3725–3741].

A DICHOTOMY OF STRONG OPEN SET CONDITION OF SELF-CONFORMAL SETS

Let [Formula: see text] be a [Formula: see text] self-conformal set. It is well known that [Formula: see text] satisfies the strong open set condition provided it satisfies the open set condition. A point [Formula: see text] is called a trivial point of [Formula: see text] if [Formula: see text] is a connected component of [Formula: see text]. Let [Formula: see text] be a strong open set for [Formula: see text]. In this paper, we show that whether [Formula: see text] contains a trivial point of [Formula: see text] determines several metrical and topological properties of [Formula: see text], including the maximal power law property and the dimension drop of the connected part of [Formula: see text].

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.

Hausdorff and Fourier dimension of graph of continuous additive processes

An additive process is a stochastic process with independent increments and that is continuous in probability. In this paper, we study the almost sure Hausdorff and Fourier dimension of the graph of continuous additive processes with zero mean. Such processes can be represented as Xt=BV(t) where B is Brownian motion and V is a continuous increasing function. We show that these dimensions depend on the local uniform Hölder indices. In particular, if V is locally uniformly bi-Lipschitz, then the Hausdorff dimension of the graph will be 3/2. We also show that the Fourier dimension almost surely is positive if V admits at least one point with positive lower Hölder regularity.It is also possible to estimate the Hausdorff dimension of the graph through the Lq spectrum of V. We will show that if V is generated by a self-similar measure on R1 with convex open set condition, the Hausdorff dimension of the graph can be precisely computed by its Lq spectrum. An illustrating example of the Cantor Devil Staircase function, the Hausdorff dimension of the graph is 1+12⋅log2log3. Moreover, we will show that the graph of the Brownian staircase surprisingly has Fourier dimension zero almost surely.

On the connected components of IFS fractals

Let K be the attractor of a self-conformal IFS, μ be the corresponding invariant measure for some probability weights and Λ(K) be the union of all trivial connected components of K. Assume that the IFS satisfies the bounded distortion property and the strong open set condition with an open set V whose closure V‾ is a union of finitely many connected components. It is proven that the four statements dimH⁡(K∖Λ(K))<dimH⁡(K), Hs(V∩Λ(K))>0(wheres=dimH⁡(K)), V∩Λ(K)≠∅ and μ(Λ(K))=1 are equivalent. Some weaker conclusions are also given for attractors generated by bi-Lipschitz IFSs.

On affine iterated function systems which robustly admit an invariant affine subspace

In this note we give a simple sufficient condition for an affine iterated function system to admit an invariant affine subspace persistently with respect to changes in the translation parameters. This yields further examples of tuples of contracting linear maps which do not satisfy the conclusions of Falconer’s theorem on the Hausdorff dimension of almost every self-affine set. We also obtain new examples of iterated function systems of similarity transformations which cannot satisfy the open set condition for any choice of translation parameters, and resolve a related question of Peres and Solomyak.

Numerical quadrature for singular integrals on fractals

We present and analyse numerical quadrature rules for evaluating regular and singular integrals on self-similar fractal sets. The integration domain Gamma subset mathbb {R}^n is assumed to be the compact attractor of an iterated function system of contracting similarities satisfying the open set condition. Integration is with respect to any “invariant” (also known as “balanced” or “self-similar”) measure supported on Gamma, including in particular the Hausdorff measure mathcal {H}^d restricted to Gamma, where d is the Hausdorff dimension of Gamma. Both single and double integrals are considered. Our focus is on composite quadrature rules in which integrals over Gamma are decomposed into sums of integrals over suitable partitions of Gamma into self-similar subsets. For certain singular integrands of logarithmic or algebraic type, we show how in the context of such a partitioning the invariance property of the measure can be exploited to express the singular integral exactly in terms of regular integrals. For the evaluation of these regular integrals, we adopt a composite barycentre rule, which for sufficiently regular integrands exhibits second-order convergence with respect to the maximum diameter of the subsets. As an application we show how this approach, combined with a singularity-subtraction technique, can be used to accurately evaluate the singular double integrals that arise in Hausdorff-measure Galerkin boundary element methods for acoustic wave scattering by fractal screens.