Jörg Neunhäuserer
We describe a class of fractal attractors induced by β-shifts. We use a coding by these shifts to show that the systems are mixing with topological entropy log βand have an ergodic measure of full entropy. Moreover we determine the Hausdorff dimension of the attractor.
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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…
We provide a new algorithm to generate images of the generalized fuzzy fractal attractors described in Oliveira-2017. We also provide some important results on the approximation of fractal operators to discrete subspaces with application to discrete versions of the deterministic algorithm for fractal image generation in the cases of IFS recovering the classical images from Barnsley et al., Fuzzy IFS from Cabrelli-1992 and GIFS's from Jaros-2016.
Study of attractors and fractal functions on the product spaces and Dimensional aspects
本文研究了产品空间上的吸引子与分形函数，探讨了产品Hausdorff度量与吸引子维度的关系，建立了同构性和维度界。
Fractal dimension for Inhomogeneous graph-directed attractors
该研究针对非齐次有向图迭代函数系统，定义了对应的分离条件，估计了吸引子的上、下盒维数，并初步探讨了其维数的连续性，丰富了分形几何中相关吸引子维数的研究。
Let $0< λ< μ<1$ and $λ+μ>1$. In this note we prove that for the vast majority of such parameters the top of the attractor $A_{λ,μ}$ of the IFS $\{(λx,μy), (μx+1-μ, λy+1-λ)\}$ is the graph of a continuous, strictly increasing function. Despite this, for most parameters, $A_{λ, μ}$ has a box dimension strictly greater than 1, showing that the upper boundary is not representative of the complexity of the fractal. Finally, we prove that if $λμ\ge 2^{-1/6}$, then $A_{λ,μ}$ has a non-empty interior.
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 …
Exponential attractors with explicit fractal dimensions for retarded functional differential equations
本文提出一种结合挤压性质与Banach空间有限子空间覆盖的新方法，构造具有显式分形维数的无穷维动力系统指数吸引子，推广了Hilbert空间的相关方法，并应用于两类滞后方程。
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.
Valuation theory, generalized IFS attractors and fractals
本文以赋值环和赋值域为例，探讨拓扑IFS吸引子与分形空间概念的推广方式，以覆盖更一般的数学设定。
Iterated Function Systems (IFSs) have been at the heart of fractal geometry almost from its origin, and several generalizations for the notion of IFS have been suggested. Subdivision schemes are widely used in computer graphics and attempts have been made to link fractals generated by IFSs to limits generated by subdivision schemes. With an eye towards establishing connection between non-stationary subdivision schemes and fractals, this paper introduces the notion of "trajectories of maps defined by function systems" which may be considered as a new generalization of the traditional IFS. The s…
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.
In this paper, we deal with the part of Fractal Theory related to finite families of (weak) contractions, called iterated function systems (IFS, herein). An attractor is a compact set which remains invariant for such a family. Thus, we consider spaces homeomorphic to attractors of either IFS or weak IFS, as well, which we will refer to as Banach and topological fractals, respectively. We present a collection of counterexamples in order to show that all the presented definitions are essential, though they are not equivalent in general.
Box dimension of fractal functions on attractors
本文在一般吸引子定义域上统一研究分形插值函数，在插值点非等距、尺度向量可变且映射非仿射的广义条件下给出盒维数上下界，并求得m维立方体与谢尔宾斯基地毯上非仿射分形函数的精确盒维数。
A fractal function is a function whose graph is the attractor of an iterated function system. This paper generalizes analytic continuation of an analytic function to continuation of a fractal function.
Cabrelli, Forte, Molter and Vrscay in 1992 considered a {fuzzy} version of the theory of iterated function systems (IFSs in short) and their fractals%The idea was to extend the classical Hutchinson-Barnsley operator to selfmaps of a metric space to appropriate selfmaps of space of fuzzy , which now is quite rich and important part of the fractals theory. On the other hand, Miculescu and Mihail in 2008 introduced another generalization of the IFSs' theory - instead of selfmaps of a metric space $X$, they considered mappings defined on the finite Cartesian product $X^m$. %It turns out that many …
We prove an estimation of the Kolmogorov $ε$-entropy in H of the unitary ball in the space V, where H is a Hilbert space and V is a Sobolev-like subspace of H. Then, by means of Zelik's result [5], an estimate of the fractal dimension of the attractors of some nonlinear parabolic equations is established.
We consider finite systems of contractive homeomorphisms of a complete metric space, which are non-redundant on every level. In general this separation condition is weaker than the strong open set condition and is not equivalent to the weak separation property. We prove that this separation condition is equivalent to the strong Markov property (see definition below). We also show that the set of $N$-tuples of contractive homeomorphisms, which are non-redundant on every level, is a $G_δ$ set in the topology of pointwise convergence of every component mapping with an additional requirement that …
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。
Finite Fractal Dimension of uniform attractors for non-autonomous dynamical systems with infinite dimensional symbol space
本文研究了非自治动力系统一致吸引子的盒计数维数上界，在符号空间无限维的情况下，通过拉回吸引子的半连续性条件得出有限维上界，并应用于反应扩散方程验证。
This work is aimed by the spirit of 1946 Moran's Theorem, which ensures that both the box and the Hausdorff dimensions for any attractor could be calculated as the solution of an equation involving only its similarity factors. To achieve such result, the open set condition (OSC, herein) is required to be satisfied by the pieces of the attractor in order to guarantee that they do not overlap too much. In this paper, we generalize the classical Moran's through a fractal dimension model which allows to obtain the fractal dimension of any IFS-attractor equipped with its natural fractal structure w…
Topological dimensions of global attractors for a delayed reaction-diffusion equation on an unbounded domain
研究无界域上时滞反应扩散方程全局吸引子的Hausdorff与分形维数估计，通过辅助Hilbert空间重述方程并利用扰动半群方法证明全局解与吸引子存在性，最终获得维数上界。
Let $\{S_i\}_{i=1}^{l}$ be an iterated function system(IFS) on $\mathbb{R}^d$ with attractor K. Let $π$ be the canonical projection. In this paper we define a new concept called "projection pressure" $P_π(ϕ)$ for $ϕ\in C(\mathbb{R}^d)$ under certain affine IFS, and show the variational principle about the projection pressure. Furthermore we check that the unique zero root of "projection pressure" still satisfies Bowen's equation when each $S_i$ is the similar map with the same compression ratio. Using the root of Bowen's equation, we can get the Hausdorff dimension of the attractor $K$.
Asymmetric coupling of nonchaotic Rulkov neurons: Fractal attractors, quasimultistability, and final state sensitivity
本文研究了两个非对称电耦合的非混沌Rulkov神经元系统，发现了由混沌伪吸引子引发的拟多稳态、复杂分形吸引域及极端终态敏感性，对神经生物学应用具有重要意义。
Fractals as Sculptures
本文证明在紧致有界度量空间上，迭代函数系统存在紧致不变子空间，使吸引子可表示为递减紧致空间序列的交集。
We develop the dimension theory for a class of linear solenoids, which have a "fractal" attractor. We will find the dimension of the attractor, proof formulas for the dimension of ergodic measures on this attractor and discuss the question whether there exists a measure of full dimension.
Generalized $q$-dimensions of measures on nonautonomous fractals
本文研究非自治吸引子（经典Moran集和迭代函数系统吸引子的推广）支撑测度的广义$q$维，给出了相似吸引子的维数公式、仿射集的上界及仿射集变体$q\neq1$时的维数公式。
We study the fractal dimension of a class of solenoidal attractors in dimensions greater or equal than 3, proving that if the contraction is sufficiently strong, the expansion is close to conformal and the attractor satisfy a geometrical condition of transversality between its components, then the Hausdorff and box-counting dimension of every stable section of the attractor have the same value, which corresponds to the zero of the topological pressure as in Bowen's formula. We also calculate the dimension of the attractor and prove that it is continuous in this class.
In this paper, we will show that any geometric Lorenz flow in a definite class satisfies the parameter-shifted shadowing property.
The statistical and Milnor attractors of step skew products over the Bernoulli shift are studied. For the case of the fiber a circle we prove that for a topologically generic step skew product the statistical and the Milnor attractor coincide and are Lyapunov stable. For this end we study some properties of the projection of the attractor onto the fiber, which might be of independent interest. For the case of the fiber being a segment we give a description of the Milnor attractor as the closure of the union of graphs of finitely many almost everywhere defined functions from the base of the ske…
In this paper, we characterize a novel separation property for IFS-attractors on complete metric spaces. Such a separation property is weaker than the strong open set condition (SOSC) and becomes necessary to reach the equality between the similarity and the Hausdorff dimensions of strict self-similar sets. We also investigate the size of the overlaps from the viewpoint of that separation property. In addition, we contribute some equivalent conditions to reach the equality between the similarity dimension and a new Hausdorff type dimension for IFS-attractors introduced by the authors in terms …
Chaotic Dynamics and Fractal Geometry in Ring Lattice Systems of Nonchaotic Rulkov Neurons
本文研究60维环晶格中电耦合非混沌Rulkov神经元的动力学，发现混沌放电、同步簇放电及超混沌，并借助Kaplan-Yorke猜想揭示吸引子分形维数可达45维，阐明简单模型交互涌现复杂集体行为的机制。
Under fairly general assumptions, we prove that every compact invariant set $\mathcal I$ of the semiflow generated by the semilinear damped wave equation u_{tt}+αu_t+β(x)u-\Deltau = f(x,u), (t,x)\in[0,+\infty[\timesΩ, u = 0, (t,x)\in[0,+\infty[\times\partialΩin $H^1_0(Ω)\times L^2(Ω) has finite Hausdorff and fractal dimension. Here $Ω$ is a regular, possibly unbounded, domain in $\R^3$ and $f(x,u)$ is a nonlinearity of critical growth. The nonlinearity $f(x,u)$ needs not to satisfy any dissipativeness assumption and the invariant subset $\mathcal I$ needs not to be an attractor. If $f(x,u)$ is…
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…
Finite-dimensionality of attractors for wave equations with degenerate nonlocal damping
本文针对一类具有单点退化非局部阻尼的波动方程，提出单独处理退化点附近维数的新方法，证明了其全局吸引子的有限维性。
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。
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。
We consider the iterated function systems (IFSs) that consist of three general similitudes in the plane with centres at three non-collinear points, and with a common contraction factor $\la\in(0,1)$. As is well known, for $\la=1/2$ the invariant set, $§_\la$, is a fractal called the Sierpiński sieve, and for $\la<1/2$ it is also a fractal. Our goal is to study $§_\la$ for this IFS for $1/2<\la<2/3$, i.e., when there are "overlaps" in $§_\la$ as well as "holes". In this introductory paper we show that despite the overlaps (i.e., the Open Set Condition breaking down completely), the attractor ca…
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