Aliasghar Sarizadeh
In the present work, we study the attractors of iterated function systems (IFSs) on connected and compact metric spaces. We prove that the whole of the phase space of a forward minimal IFS, for which some map admits an attracting fixed point, is an attractor.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
We provide a new algorithm (called the grid algorithm) designed to generate the image of the attractor of a generalized iterated function system on a finite dimensional space and we compare it with the deterministic algorithm regarding generalized iterated function systems presented by P. Jaros, L. Maslanka and F. Strobin in [Algorithms generating images of attractors of generalized iterated function systems, Numer. Algorithms, 73 (2016), 477-499].
Generalized Iterated Function System and Common Attractors of Generalized Hutchinson Operators in Dislocated Metric Spaces
针对错位度量空间，本文提出广义迭代函数系统以构造广义压缩映射的公共分形，并证明有理压缩映射吸引子问题的适定性及广义拼贴定理。
If $F$ and $G$ are iterated function systems, then any infinite word $W$ in the symbols $F$ and $G$ induces a limit set. It is natural to ask whether this Cantor set can also be realized as the limit set of a single $C^{1 + α}$ iterated function system $H$. We prove that under certain assumptions on $F$, $G$ and $H$, the answer is no. This problem is motivated by the spectral theory of one-dimensional quasicrystals.
This paper contains four main results associated with an attractor of a projective iterated function system (IFS). The first theorem characterizes when a projective IFS has an attractor which avoids a hyperplane. The second theorem establishes that a projective IFS has at most one attractor. In the third theorem the classical duality between points and hyperplanes in projective space leads to connections between attractors that avoid hyperplanes and repellers that avoid points as well as hyperplane attractors that avoid points and repellers that avoid hyperplanes. Finally, an index is defined …
The fast basin of an attractor of an iterated function system (IFS) is the set of points in the domain of the IFS whose orbits under the associated semigroup intersect the attractor. Fast basins can have non-integer dimension and comprise a class of deterministic fractal sets. The relationship between the basin and the fast basin of a point-fibred attractor is analyzed. To better understand the topology and geometry of fast basins, and because of analogies with analytic continuation, branched fractal manifolds are introduced. A branched fractal manifold is a metric space constructed from the e…
Iterated function systems (IFSs) and their attractors have been central to the theory of fractal geometry almost from its inception. And contractivity of the functions in the IFS has been central to the theory of iterated functions systems. If the functions in the IFS are contractions, then the IFS is guaranteed to have a unique attractor. Recently, however, there has been an interest in what occurs to the attractor at the boundary between contractvity and expansion of the IFS. That is the subject of this paper. For a family $F_t$ of IFSs depending on a real parameter $t>0$, the existence and …
Guided by classical concepts, we define the notion of \emph{ends} of an iterated function system and prove that the number of ends is an upper bound for the number of nondegenerate components of its attractor. The remaining isolated points are then linked to idempotent maps. A commutative diagram illustrates the natural relationships between the infinite walks in a semigroup and components of an attractor in more detail. We show in particular that, if an iterated function system is one-ended, the associated attractor is connected, and ask whether every connected attractor (fractal) conversely …
The paper is devoted to searching algorithms which will allow to generate images of attractors of \emph{generalized iterated function systems} (GIFS in short), which are certain generalization of classical iterated function systems, defined by Mihail and Miculescu in 2008, and then intensively investigated in the last years (the idea is that instead of selfmaps of a metric space $X$, we consider mappings form the Cartesian product $X\times...\times X$ to $X$). Two presented algorithms are counterparts of classical \emph{deterministic algorithm} and so-called \emph{chaos game}. The third and fo…
We prove that almost every finite collection of matrices in $GL_d(\mathbb{R})$ and $SL_d(\mathbb{R})$ with positive entries is Diophantine. Next we restrict ourselves to the case $d=2$. A finite set of $SL_2(\mathbb{R})$ matrices induces a (generalized) iterated function system on the projective line $\mathbb{RP}^1$. Assuming uniform hyperbolicity and the Diophantine property, we show that the dimension of the attractor equals the minimum of 1 and the critical exponent.
This paper is in the form of an essay. It defines fractal tops and code space structures associated with set-attractors of hyperbolic iterated function systems (IFSs). The fractal top of an IFS is associated with a certain shift invariant subspace of code space, whence the entropy of the IFS, and of its set-attractor, may be defined. Given any ordered pair of hyperbolic IFSs, each with the same number of maps, there is a natural transformation, constructed with the aid of fractal tops, whose domain is the attractor A of the first IFS and whose range is contained in the attractor B of the secon…
The paper concerns fractal homeomorphism between the attractors of two bi-affine iterated function systems. After a general discussion of bi-affine functions, conditions are provided under which a bi-affine iterated function system is contractive, thus guaranteeing an attractor. After a general discussion of fractal homeomorphism, fractal homeomorphisms are constructed for a specific type of bi-affine iterated function system.
We study weakly hyperbolic iterated function systems on compact spaces, as defined by Edalat, but in the more general setting of a compact parameter space. We prove the existence of attractors, both in the topological and measure theoretical viewpoint, and prove that the measure theoretical attractor is ergodic. We also define weakly hyperbolic iterated functions systems for complete spaces and compact parameter space, and prove that this definition extends the one given by Edalat. Furthermore, we study the question of existence of the attractors in this setting. Finally, we prove a version of…
In this paper are investigated strictly self-similar fractals that are composed of an infinite number of regular star-polygons, also known as Sierpinski $n$-gons, $n$-flakes or polyflakes. Construction scheme for Sierpinsky $n$-gon and $n$-flake is presented where the dimensions of the Sierpinsky $\infty$-gon and the $\infty$-flake are computed to be 1 and 2, respectively. These fractals are put in a general context and Iterated Function Systems are applied for the visualisation of the geometric iterations of the initial polygons, as well as the visualisation of sets of points that lie on the …
This paper presents a sufficient condition for a continuum in $R^n$ to be embeddable in $R^n$ in such a way that its image is not an attractor of any iterated function system. An example of a continuum in $R^2$ that is not an attractor of any weakly contracting iterated function system is also given.
This paper examines thresholds for certain properties of the attractor of a general one-parameter affine family of iterated functions systems. As the parameter increases, the iterated function system becomes less contractive, and the attractor evolves. Thresholds are studied for the following properties: the existence of an attractor, the connectivity of the attractor, and the existence of non-empty interior of the attractor. Also discussed are transition phenomena between existence and non-existence of an attractor.
We consider iterated function systems on the real line that consist of continuous, piecewise linear functions. Under a mild separation condition, we show that the Hausdorff and box dimensions of the attractor are equal to the minimum of 1 and the exponent which comes from the most natural system of covers of the attractor.
Separation Conditions for Iterated Function Systems with a Common Attractor
本文证明满足开集条件但不满足强分离条件的迭代函数系统的吸引子，不能由满足强分离条件的有限相似迭代函数系统生成，去除了齐次性假设并否定回答了相关民间问题。
Upper box dimension of inhomogeneous attractors of $C^1$ iterated function systems
本文证明$C^1$迭代函数系统非齐次吸引子的上盒维数受限于系统奇异维数与凝聚集上盒维数的最大值，推广了Burrell和Fraser的结果，并扩展至轨道集。
We study the attractor of Iterated Function Systems composed of infinitely many affine, homogeneous maps. In the special case of second generation IFS, defined herein, we conjecture that the attractor consists of a finite number of non-overlapping intervals. Numerical techniques are described to test this conjecture, and a partial rigorous result in this direction is proven.
Every quasi-attractor of an iterated function system (IFS) of continuous functions on a first-countable Hausdorff topological space is renderable by the probabilistic chaos game. By contrast, we prove that the backward minimality is a necessary condition to get the deterministic chaos game. As a consequence, we obtain that an IFS of homeomorphisms of the circle is renderable by the deterministic chaos game if and only if it is forward and backward minimal. This result provides examples of attractors (a forward but no backward minimal IFS on the circle) that are not renderable by the determinis…
Miculescu and Mihail in 2008 introduced the concept of a \emph{generalized iterated function system} (GIFS in~short), a particular extension of the classical IFS. The idea is that, instead of families of selfmaps of a metric space~$X$, GIFSs consist of maps defined on a finite Cartesian {$m$-th power} $X^m$ with values in $X$ (in such a case we say that a GIFS is \emph{of order} $m$). It turned out that a great part of the classical {Hutchinson theory} has natural counterpart in this GIFSs' framework. On the other hand, there are known only few examples of~fractal sets which are generated by G…
We investigate the topological and metric properties of attractors of an iterated function system (IFS) whose functions may not be contractive. We focus, in particular, on invertible IFSs of finitely many maps on a compact metric space. We rely on ideas Kieninger and McGehee and Wiandt, restricted to what is, in many ways, a simpler setting, but focused on a special type of attractor, namely point-fibred minimal (locally) invariant sets. This allows us to give short proofs of some of the key ideas.
In this paper, we study cut sets of attractors of iteration function systems (IFS) in $\mathbb{R}^d$. Under natural conditions, we show that all irreducible cut sets of these attractors are perfect sets or single points. This leads to a criterion for the existence of cut points of IFS attractors. If the IFS attractors are self-affine tiles, our results become algorithmically checkable and can be used to exhibit cut points with the help of Hata graphs. This enables us to construct cut points of some self-affine tiles studied in the literature.
This paper investigates the dimension theory of some families of continuous piecewise linear iterated function systems. For one family, we show that the Hausdorff dimension of the attractor is equal to the exponential growth rate obtained from the most natural covering system. We also prove that for Lebesgue typical parameters, the 1-dimensional Lebesgue measure of the underlying attractor is positive, if this number is bigger than 1 and all the contraction ratios are positive.
Blending attractors of Iterated Function Systems
本文提出一种基于编码映射混合分形吸引子的新方法，引入混合系数度量相似性，并给出离散逼近算法及误差估计。
We give a systematic account of iterated function systems (IFS) of weak contractions of different types (Browder, Rakotch, topological). We show that the existence of attractors and asymptotically stable invariant measures, and the validity of the random iteration algorithm ("chaos game"), can be obtained rather easily for weakly contractive systems. We show that the class of attractors of weakly contractive IFSs is essentially wider than the class of classical IFSs' fractals. On the other hand, we show that, in reasonable spaces, a typical compact set is not an attractor of any weakly contrac…
We consider iterated function systems $\mathrm{IFS}(T_1,\dots,T_k)$ consisting of continuous self maps of a compact metric space $X$. We introduce the subset $S_{\mathrm{t}}$ of {\emph{weakly hyperbolic sequences}} $ξ=ξ_0\ldotsξ_n \ldots \in Σ_k^+$ having the property that $\bigcap_n T_{ξ_{0}}\circ\cdots\circ T_{ξ_{n}}(X)$ is a point $\{π(ξ)\}$. The target set $π(S_{\mathrm{t}})$ plays a role similar to the semifractal introduced by Lasota-Myjak. Assuming that $S_{\mathrm{t}}\ne \emptyset$ (the only hyperbolic-like condition we assume) we prove that the IFS has at most one strict attractor and…
We study the topological properties of attractors of Iterated Function Systems (I.F.S.) on the real line, consisting of affine maps of homogeneous contraction ratio. These maps define what we call a second generation I.F.S.: they are uncountably many and the set of their fixed points is a Cantor set. We prove that when this latter either is the attractor of a finite, non-singular, hyperbolic, I.F.S. (of first generation), or it possesses a particular dissection property, the attractor of the second generation I.F.S. consists of finitely many closed intervals.
This paper discusses, certain algebraic, analytic, and topological results on partial iterated function systems($IFS_p$'s). Also, the article proves the Collage theorem for partial iterated function systems. Further, it provides a method to address the points in the attractor of a partial iterated function system and obtain results related to the address of points in the attractor. The completeness of the partial metric space of contractions with a fixed contractivity factor is proved, under suitable conditions. Also, it demonstrates the continuity of the map that associates each contraction i…
Iterated relation systems on Riemannian manifolds
针对黎曼流形上多值或非收缩关系定义的分形，本文提出迭代关系系统，证明其吸引子可与图导向迭代函数系统等同，并在图开集或有限型条件下获得维数公式，改进了已有方法。
In 2013 Balka and Máthé showed that in uncountable polish spaces the typical compact set is not a fractal of any IFS. In 2008 Miculescu and Mihail introduced a concept of a generalized iterated function system (GIFS in short), a particular extension of classical IFS, in which they considered families of mappings defined on finite Cartesian product $X^m$ with values in $X$. Recently, Secelean extended these considerations to mappings defined on the space $\ell_\infty(X)$ of all bounded sequences of elements of $X$ endowed with supremum metric. In the paper we show that in Euclidean spaces a typ…
In this paper we consider Iterated Function Systems (IFS) on the real line consisting of continuous piecewise linear functions. We assume some bounds on the contraction ratios of the functions, but we do not assume any separation condition. Moreover, we do not require that the functions of the IFS are injective, but we assume that their derivatives are separated from zero. We prove that if we fix all the slopes but perturb all other parameters, then for all parameters outside of an exceptional set of less than full packing dimension, the Hausdorff dimension of the attractor is equal to the exp…
In this paper we propose a new model of random graph directed fractals that extends the current well-known model of random graph directed iterated function systems, $V$-variable attractors, and fractal and Mandelbrot percolation. We study its dimensional properties for similarities with and without overlaps. In particular we show that for the two classes of $1$-variable and $\infty$-variable random graph directed attractors we introduce, the Hausdorff and upper box counting dimension coincide almost surely, irrespective of overlap. Under the additional assumption of the uniform strong separati…
We consider iterated functions systems (IFS) on compact metric spaces and introduce the concept of target sets. Such sets have very rich dynamical properties and play a similar role as semifractals introduced by Lasota and Myjak do for regular IFSs. We study sufficient conditions which guarantee that the closure of the target set is a local attractor for the IFS. As a corollary, we establish necessary and sufficient conditions for the IFS having a global attractor. We give an example of a non-regular IFS whose target set is nonempty, showing that our approach gives rise to a "new class" of sem…
A Closed Graph Theorem for hyperbolic iterated function systems
本研究引入双曲迭代函数系统的态射概念，证明态射图像为迭代函数系统吸引子并得到闭图像定理，给出了研究迭代函数系统拓扑共轭问题的新途径。
On nonlinear iterated function systems with overlaps
本文构造了一个由线性分式变换组成的带重叠非线性迭代函数系统示例，证明在两映射共享不动点时，吸引子维数仍等于共形维数，从而不存在“维数下降”现象。
Fractal Attractors in Random Nonlinear Iterated Function Systems: Existence, Stability, and Dimensional Properties
本研究提出随机非线性迭代函数系统框架，证明其分形吸引子的存在性与稳定性，并通过模拟和盒计数法揭示维数1.4至1.89的复杂结构，拓展了随机分形建模能力。
We prove an extension of M. Hata's theorem [4] for planar Markov Iterated Function Systems satisfying a strong version of the Open Set Condition. More precisely, if the attractor of such a system is connected, then it is locally connected. We construct counterexamples to show that all additional hypothesis are necessary.
Distinct dimensions for attractors of bi-Lipschitz iterated function systems
本文构造了双Lipschitz迭代函数系统，其吸引子具有五种不同维数，并证明了正熵遍历测度前推非精确维数及吸引子内部空但正Lebesgue测度的例子。
The Chain Graph of a general Iterated Function System
研究一般迭代函数系统的链图，关注具有紧致动力学的IFS，引入链图编码其定性动力学特征，主要结果为具有紧致动力学的IFS的链图与其在全局吸引子上限制的链图一致，且链图节点数至多与哈钦森映射图的节点数相同。
Stimulated by recent problems in the theory of iterated function systems, we provide a variant of the Banach converse theorem for multivalued maps. In particular, we show that attractors of continuous multivalued maps in a metric space are stable. Moreover, such attractors in locally compact, complete metric spaces may be obtained by means of the Banach theorem in the hyperspace.
The study of generalized iterated function systems (GIFS) was introduced by Mihail and Miculescu in 2008. We provide a new approach to study those systems as the limit of the Hutchinson-Barnsley setting for infinite iterated function systems (IIFS) which has been developed by many authors in the last years. We show that any attractor of a contractive generalized iterated function system is the limit with respect to Hausdorff-Pompeiu metric of attractors of contractive infinite iterated function systems. We also prove that any Hutchinson measure for a contractive generalized iterated function s…
On the equicontinuity of the Hutchinson operator
本文证明了一类广泛迭代函数系统的Hutchinson算子是等度连续的，特别地，每个具有吸引子的迭代函数系统都有等度连续的Hutchinson算子。
Split square and split carpet as examples of non-metrizable IFS attractors
本文定义分裂正方形与分裂地毯两个不可度量化紧空间，并用分裂区间的乘积构造它们，证明二者均可作为迭代函数系统的吸引子。
We study countable compact spaces as potential attractors of iterated function systems. We give an example of a convergent sequence in the real line which is not an IFS-attractor and for each countable ordinal $δ$ we show that a countable compact space of height $δ+1$ can be embedded in the real line so that it becomes the attractor of an IFS. On the other hand, we show that a scattered compact metric space of limit height is never an IFS-attractor.
Foundations of local iterated function systems
本文研究了局部迭代函数系统的基础，证明了其具有紧吸引子，并展示了非自相似吸引子的构造，同时揭示了其与经典框架的不同之处。
The aim of this paper is to establish some results regarding Infinite Iterated Function Systems with the help of the Tarski-Kantorovitch fixed-point principles for maps on partially ordered sets. To this end we introduce two new classes of Infinite Iterated Function Systems which are well suited for applying the aforementioned principle. We also study some properties of the canonical projection from the shift space of an Infinite Iterated Function System belonging to one of the two introduced classes to its attractor.
Continuity of the natural dimension of piecewise linear iterated function systems
研究分段线性迭代函数系统的自然维数随参数连续变化，并据此证明吸引子具有正勒贝格测度。
Borel complexity of the family of attractors for weak IFSs
本文研究弱迭代函数系统吸引子族的 Borel 复杂度，证明 wIFS$^d$ 是 $G_{δσ}$-难且非 $F_{σδ}$，并在一维情形下证明其为解析子集，从而刻画了其与经典 IFS 吸引子族的复杂度差异。
In this paper we study random iterated function systems. Our main result gives sufficient conditions for an analogue of a well known theorem due to Khintchine from Diophantine approximation to hold almost surely for stochastically self-similar and self-affine random iterated function systems.
We study the minimality of almost every orbital branch of minimal iterated function systems (IFSs). We prove that this kind of minimality holds for forward and backward minimal IFSs generated by orientation-preserving homeomorphisms of the circle. We provide new examples of iterated functions systems where this behavior persists under perturbation of the generators.
A topological approach to fuzzy iterated function systems
本文统一 Hutchinson–Barnsley 理论的拓扑与模糊集推广，证明 Tychonoff 空间上压缩 fuzzy IFS 生成模糊吸引子，并推出 Hausdorff 空间中拓扑压缩情形的较弱吸引子结论。
Moran-Type Iterated Function Systems and Dimensions of Moran Self-Similar Sets
本文提出Moran型迭代函数系统及自相似集的维度理论，通过定义吸引子和测度并引入分离条件，探讨了自相似集的维度计算方法。
In this paper, we focus attention on extending the topological conjugacy of adding machine maps and minimal systems to iterated function systems. We provide necessary and sufficient conditions for an iterated function system to be conjugated to an adding machine map. It is proved that every minimal iterated function system which has some non-periodic regular point is semi-conjugate to an adding machine map. Furthermore, we investigate the topological conjugacy of an infinite family of tent maps, as well as the restriction of a map to its $ω-$limit set with an iterated function system.
For geometric Lorenz attractors (including the classical Lorenz attractor) we obtain a greatly simplified proof of the central limit theorem which applies also to the more general class of codimension two singular hyperbolic attractors. We also obtain the functional central limit theorem and moment estimates, as well as iterated versions of these results. A consequence is deterministic homogenisation (convergence to a stochastic differential equation) for fast-slow dynamical systems whenever the fast dynamics is singularly hyperbolic of codimension two.
We investigate combinatorial issues relating to the use of random orbit approximations to the attractor of an iterated function system with the aim of clarifying the role of the stochastic process during generation the orbit. A Baire category counterpart of almost sure convergence is presented; and a link between topological and probabilistic methods is observed.
Recurrent and gradient dynamics of Iterated Function Systems
本文提出一种用于研究一般迭代函数系统定性动力学的框架，通过二元关系编码点的下游关系，分离周期行为与梯度行为。
We show how geometric methods from the general theory of fractal dimensions and iterated function systems can be deployed to study symbolic dynamics in the zero entropy regime. More precisely, we establish a dimensional characterization of the topological notion of amorphic complexity. For subshifts with discrete spectrum associated to constant length substitutions, this characterization allows us to derive bounds for the amorphic complexity by interpreting the subshift as the attractor of an iterated function system in a suitable quotient space. As a result, we obtain the general finiteness a…
Tilings from Tops of Overlapping Iterated Function Systems
本文研究了如何通过重叠迭代函数系统的顶端扩展自相似铺砌，提出了一种收敛于铺砌的近似超瓷砖序列的方法，并探讨了有限类型系统、非周期性单体铺砌及无限多种相关原型瓷砖的实例。
We prove that any compact manifold without boundary admits a pair of diffeomorphisms that generates $C^1$ robustly minimal dynamics. We apply the results to the construction of blenders and robustly transitive skew product diffeomorphisms.
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。