Showing papers on "Iterated function system published in 2018"

The Chain Properties and Average Shadowing Property of Iterated Function Systems

Abstract: This note proves that an iterated function system is chain transitive (resp., chain mixing, transitive) if and only if the step skew product corresponding to the iterated function system is chain transitive (resp., chain mixing, transitive). As an application, it is obtained that an iterated function system with the (asymptotic) average shadowing property is chain mixing, improving the main results in Bahabadi (Georgian Math J 22:179–184, 2015).

Equilibrium states of generalised singular value potentials and applications to affine iterated function systems

Abstract: We completely describe the equilibrium states of a class of potentials over the full shift which includes Falconer's singular value function for affine iterated function systems with invertible affinities. We show that the number of distinct ergodic equilibrium states of such a potential is bounded by a number depending only on the dimension, answering a question of A. Kaenmaki. We prove that all such equilibrium states are fully supported and satisfy a Gibbs inequality with respect to a suitable subadditive potential. We apply these results to demonstrate that the affinity dimension of an iterated function system with invertible affinities is always strictly reduced when any one of the maps is removed, resolving a folklore open problem in the dimension theory of self-affine fractals. We deduce a natural criterion under which the Hausdorff dimension of the attractor has the same strict reduction property.

Spectral Asymptotics of One-Dimensional Fractal Laplacians in the Absence of Second-Order Identities

Abstract: We observe that some self-similar measures defined by finite or infinite iterated function systems with overlaps are in certain sense essentially of finite type, which allows us to extract useful measure-theoretic properties of iterates of the measure. We develop a technique to obtain a closed formula for the spectral dimension of the Laplacian defined by a self-similar measure satisfying this condition. For Laplacians defined by fractal measures with overlaps, spectral dimension has been obtained earlier only for a small class of one-dimensional self-similar measures satisfying Strichartz second-order self-similar identities. The main technique we use relies on the vector-valued renewal theorem proved by Lau, Wang and Chu[ 24 ].

Weak θ -contractions and some fixed point results with applications to fractal theory

Abstract: In this paper, we define weak θ-contractions on a metric space into itself by extending θ-contractions introduced by Jleli and Samet (J. Inequal. Appl. 2014:38, 2014) and utilize the same to prove some fixed point results besides proving some relation-theoretic fixed point results in generalized metric spaces. Moreover, we give some applications to fractal theory improving the classical Hutchinson–Barnsley′s theory of iterated function systems. We also give illustrative examples to exhibit the utility of our results.

Box dimension of bilinear fractal interpolation surfaces

Abstract: Bilinear fractal interpolation surfaces were introduced by Ruan and Xu in 2015. In this paper, we present the formula for their box dimension under certain constraint conditions.

Iterated function systems of logistic maps: Synchronization and intermittency

Abstract: We discuss iterated function systems generated by finitely many logistic maps, with a focus on synchronization and intermittency. We provide sufficient conditions for synchronization, involving negative Lyapunov exponents and minimal dynamics. A number of results that clarify the scope of these conditions are included. We analyze a mechanism for intermittency that involves the full map x → 4x(1 - x) as one of the generators of the iterated function system. For iterated function systems generated by x → 2x(1 - x) and x → 4x(1 - x) we prove the existence of a σ-finite stationary measure.

Turbulent flow over urban-like fractals: prognostic roughness model for unresolved generations

Abstract: Large-eddy simulation (LES) has been used to model turbulent channel flow over urban-like, fractal topographies, constructed via iterated function system (IFS). By using the IFS approach, t...

A hybrid of fractal image coding and fractal dimension for an efficient retrieval method

Abstract: Fractal image coding (FIC) based on the inverse problem of an iterated function system plays an essential role in several areas of computer graphics and in many other interesting applications. Through FIC, an image can be transformed to compressed representative parameters and be expressed in a simple geometric way. Dealing with digital images requires storing a large number of images in databases, where searching such databases is time consuming. Therefore, finding a new technique that facilitates this task is a challenge that has received increasing attention from many researchers. In this study, a new method that combines fractal dimension (FD) which is an indicator of image complexity with the FIC scheme is proposed. Classifying images in databases according to their texture by using FD helps reduce the retrieval time of query images. The validity of the proposed method is evaluated using geosciences images. Result shows that the method is computationally attractive.

Dimension of self-affine sets for fixed translation vectors

Abstract: An affine iterated function system is a finite collection of affine invertible contractions and the invariant set associated to the mappings is called self-affine. In 1988, Falconer proved that, for given matrices, the Hausdorff dimension of the self-affine set is the affinity dimension for Lebesgue almost every translation vectors. Similar statement was proven by Jordan, Pollicott, and Simon in 2007 for the dimension of self-affine measures. In this article, we have an orthogonal approach. We introduce a class of self-affine systems in which, given translation vectors, we get the same results for Lebesgue almost all matrices. The proofs rely on Ledrappier-Young theory that was recently verified for affine iterated function systems by Barany and Kaenmaki, and a new transversality condition, and in particular they do not depend on properties of the Furstenberg measure. This allows our results to hold for self-affine sets and measures in any Euclidean space.

On intrinsic and extrinsic rational approximation to Cantor sets

Abstract: We establish various new results on a problem proposed by K. Mahler in 1984 concerning rational approximation to fractal sets by rational numbers inside and outside the set in question, respectively. Some of them provide a natural continuation and improvement of recent results of Broderick, Fishman and Reich and Fishman and Simmons. A key feature is that many of our new results apply to more general, multi-dimensional fractal sets and require only mild assumptions on the iterated function system. Moreover we provide a non-trivial lower bound for the distance of a rational number $p/q$ outside the Cantor middle third set $C$ to the set $C$, in terms of the denominator $q$. We further discuss patterns of rational numbers in fractal sets. We want to highlight two of them: Firstly, an upper bound for the number of rational (algebraic) numbers in a fractal set up to a given height (and degree) for a wide class of fractal sets. Secondly we find properties of the denominator structure of rational points in ''missing digit'' Cantor sets, generalizing claims of Nagy and Bloshchitsyn.

A New Approach to Numerical Computation of Hausdorff Dimension of Iterated Function Systems: Applications to Complex Continued Fractions

Abstract: In a previous paper (Falk and Nussbaum, in $$C^m$$ Eigenfunctions of Perron–Frobenius operators and a new approach to numerical computation of hausdorff dimension: applications in $$\mathbb {R}^1$$ , 2016. ArXiv e-prints arXiv:1612.00870 ), the authors developed a new approach to the computation of the Hausdorff dimension of the invariant set of an iterated function system or IFS and studied some applications in one dimension. The key idea, which has been known in varying degrees of generality for many years, is to associate to the IFS a parametrized family of positive, linear, Perron-Frobenius operators $$L_s$$ . In our context, $$L_s$$ is studied in a space of $$C^m$$ functions and is not compact. Nevertheless, it has a strictly positive $$C^m$$ eigenfunction $$v_s$$ with positive eigenvalue $$\lambda _s$$ equal to the spectral radius of $$L_s$$ . Under appropriate assumptions on the IFS, the Hausdorff dimension of the invariant set of the IFS is the value $$s=s_*$$ for which $$\lambda _s =1$$ . To compute the Hausdorff dimension of an invariant set for an IFS associated to complex continued fractions, (which may arise from an infinite iterated function system), we approximate the eigenvalue problem by a collocation method using continuous piecewise bilinear functions. Using the theory of positive linear operators and explicit a priori bounds on the partial derivatives of the strictly positive eigenfunction $$v_s$$ , we are able to give rigorous upper and lower bounds for the Hausdorff dimension $$s_*$$ , and these bounds converge to $$s_*$$ as the mesh size approaches zero. We also demonstrate by numerical computations that improved estimates can be obtained by the use of higher order piecewise tensor product polynomial approximations, although the present theory does not guarantee that these are strict upper and lower bounds. An important feature of our approach is that it also applies to the much more general problem of computing approximations to the spectral radius of positive transfer operators, which arise in many other applications.

Fractal Convolution: A New Operation between Functions

Abstract: In this paper we define an internal binary operation between functions called in the text \emph{fractal convolution}, that applies a pair of mappings into a fractal function. This is done by means of a suitable Iterated Function System. We study in detail the operation in $\mathcal{L}^p$ spaces and in sets of continuous functions, in a different way to previous works of the authors. We develop some properties of the operation and its associated sets. The lateral convolutions with the null function provide linear operators whose characteristics are explored. The last part of the article deals with the construction of convolved fractals bases and frames in Banach and Hilbert spaces of functions.

Entropy of Iterated Function Systems and Their Relations with Black Holes and Bohr-Like Black Holes Entropies

Abstract: In this paper we consider the metric entropies of the maps of an iterated function system deduced from a black hole which are known the Bekenstein-Hawking entropies and its subleading corrections. More precisely, we consider the recent model of a Bohr-like black hole that has been recently analysed in some papers in the literature, obtaining the intriguing result that the metric entropies of a black hole are created by the metric entropies of the functions, created by the black hole principal quantum numbers, i.e., by the black hole quantum levels. We present a new type of topological entropy for general iterated function systems based on a new kind of the inverse of covers. Then the notion of metric entropy for an Iterated Function System ( I F S ) is considered, and we prove that these definitions for topological entropy of IFS's are equivalent. It is shown that this kind of topological entropy keeps some properties which are hold by the classic definition of topological entropy for a continuous map. We also consider average entropy as another type of topological entropy for an I F S which is based on the topological entropies of its elements and it is also an invariant object under topological conjugacy. The relation between Axiom A and the average entropy is investigated.

Fractal attractors in economic growth models with random pollution externalities.

Abstract: We analyze a discrete time two-sector economic growth model where the production technologies in the final and human capital sectors are affected by random shocks both directly (via productivity and factor shares) and indirectly (via a pollution externality). We determine the optimal dynamics in the decentralized economy and show how these dynamics can be described in terms of a two-dimensional affine iterated function system with probability. This allows us to identify a suitable parameter configuration capable of generating exactly the classical Barnsley's fern as the attractor of the log-linearized optimal dynamical system.

Synchronization in Minimal Iterated Function Systems on Compact Manifolds

Abstract: We treat synchronization for iterated function systems generated by diffeomorphisms on compact manifolds. Synchronization here means the convergence of orbits starting at different initial conditions when iterated by the same sequence of diffeomorphisms. The iterated function systems admit a description as skew product systems of diffeomorphisms on compact manifolds driven by shift operators. Under open conditions including transitivity and negative fiber Lyapunov exponents, we prove the existence of a unique attracting invariant graph for the skew product system. This explains the occurrence of synchronization. The result extends previous results for iterated function systems by diffeomorphisms on the circle, to arbitrary compact manifolds.

Bernstein Fractal Rational Approximants with no Condition on Scaling Vectors

Abstract: Fractal functions defined through iterated function system have been successfully used to approximate any continuous real-valued function defined on a compact interval. The fractal dimension is a q...

Iterated function systems consisting of continuous functions satisfying Banach’s orbital condition

Abstract: Abstract We introduce the concept of iterated function system consisting of continuous functions satisfying Banach’s orbital condition and prove that the fractal operator associated to such a system is weakly Picard. Some examples are provided.

Iterated Function Systems with Place-Dependent Probabilities and the Inverse Problem of Measure Approximation Using Moments

Abstract: We are concerned with the approximation of probability measures on a compact metric space (X,d) by invariant measures of iterated function systems with place-dependent probabilities (IFSPDPs). The ...

On topological entropy and topological pressure of non-autonomous iterated function systems

Abstract: In this paper we introduce the notions of topological entropy and topological pressure for non-autonomous iterated function systems (or NAIFSs for short) on countably infinite alphabets. NAIFSs differ from the usual (autonomous) iterated function systems, they are given \cite{LGMU} by a sequence of collections of continuous maps on a compact topological space, where maps are allowed to vary between iterations. Several basic properties of topological pressure and topological entropy of NAIFSs are provided. Especially, we generalize the classical Bowen's result to NAIFSs ensures that the topological entropy is concentrated on the set of nonwandering points. Then, we define the notion of specification property, under which, the NAIFSs have positive topological entropy and all points are entropy points. In particular, each NAIFS with the specification property is topologically chaotic. Additionally, the $\ast$-expansive property for NAIFSs is introduced. We will prove that the topological pressure of any continuous potential can be computed as a limit at a definite size scale whenever the NAIFS satisfies the $\ast$-expansive property. Finally, we study the NAIFSs induced by expanding maps. We prove that these NAIFSs having the specification and $\ast$-expansive properties.

Self-similar measures for iterated function systems driven by weak contractions

Abstract: We show the existence and uniqueness for self-similar measures for iterated function systems driven by weak contractions. Our main idea is using the duality theorem of Kantorovich-Rubinstein and equivalent conditions for weak contractions established by Jachymski. We also show collage theorems for such iterated function systems.

Applications of Variable Discounting Dynamic Programming to Iterated Function Systems and Related Problems

Abstract: We study existence and uniqueness of the fixed points solutions of a large class of non-linear variable discounted transfer operators associated to a sequential decision-making process. We establish regularity properties of these solutions, with respect to the immediate return and the variable discount. In addition, we apply our methods to reformulating and solving, in the setting of dynamic programming, some central variational problems on the theory of iterated function systems, Markov decision processes, discrete Aubry-Mather theory, Sinai-Ruelle-Bowen measures, fat solenoidal attractors, and ergodic optimization.

Extremal exponents of random products of conservative diffeomorphisms

Abstract: We show that for a $C^1$-open and $C^{r}$-dense subset of the set of ergodic iterated function systems of conservative diffeomorphisms of a finite-volume manifold of dimension $d\geq 2$, the extremal Lyapunov exponents do not vanish. In particular, the set of non-uniform hyperbolic systems contains a $C^1$-open and $C^r$-dense subset of ergodic random products of i.i.d. conservative surface diffeomorphisms.

Convex hulls of sierpiński relatives

Abstract: This paper presents an investigation of the convex hulls of the Sierpinski relatives These fractals all have the same fractal dimension but different topologies We prove that the relatives have c

Periodic representations in Salem bases

Abstract: We prove that all algebraic bases $\beta$ allow an eventually periodic representations of the elements of $\mathbb Q(\beta)$ with a finite alphabet of digits $\mathcal A$. Moreover, the classification of bases allowing that those representations have the so-called weak greedy property is given. The decision problem whether a given pair $(\beta,\mathcal A)$ allows eventually periodic representations proves to be rather hard, for it is equivalent to a topological property of the attractor of an iterated function system.

A Positive Operator-Valued Measure for an Iterated Function System

Abstract: Given an iterated function system (IFS) on a complete and separable metric space $Y$ , there exists a unique compact subset $X \subseteq Y$ satisfying a fixed point relation with respect to the IFS. This subset is called the attractor set, or fractal set, associated to the IFS. The attractor set supports a specific Borel probability measure, called the Hutchinson measure, which itself satisfies a fixed point relation. P. Jorgensen generalized the Hutchinson measure to a projection-valued measure, under the assumption that the IFS does not have essential overlap (Jorgensen in Adv. Appl. Math. 34(3):561–590, 2005; Operator Theory, Operator Algebras, and Applications, pp. 13–26, 2006). In previous work, we developed an alternative approach to proving the existence of this projection-valued measure (Davison in Acta Appl. Math. 140(1):11–22, 2015; Acta Appl. Math. 140(1):23–25, 2015; Generalizing the Kantorovich metric to projection-valued measures: with an application to iterated function systems, 2015). The situation when the IFS exhibits essential overlap has been studied by Jorgensen and colleagues in Jorgenson et al. (J. Math. Phys. 48(8):083511, 35, 2007). We build off their work to generalize the Hutchinson measure to a positive-operator valued measure for an arbitrary IFS, that may exhibit essential overlap. This work hinges on using a generalized Kantorovich metric to define a distance between positive operator-valued measures. It is noteworthy to mention that this generalized metric, which we use in our previous work as well, was also introduced by R.F. Werner to study the position and momentum observables, which are central objects of study in the area of quantum theory (Werner in J. Quantum Inf. Comput. 4(6):546–562, 2004). We conclude with a discussion of Naimark’s dilation theorem with respect to this positive operator-valued measure, and at the beginning of the paper, we prove a metric space completion result regarding the classical Kantorovich metric.

Non-hyperbolic Iterated Function Systems: semifractals and the chaos game

Abstract: 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 semifractals. Finally, we show that random orbits generated by IFSs draws target sets that are "stable".

Fast approximation of the affinity dimension for dominated affine iterated function systems

Abstract: In 1988 K. Falconer introduced a formula which predicts the value of the Hausdorff dimension of the attractor of an affine iterated function system. The value given by this formula -- sometimes referred to as the affinity dimension -- is known to agree with the Hausdorff dimension both generically and in an increasing range of explicit cases. It is however a nontrivial problem to estimate the numerical value of the affinity dimension for specific iterated function systems. In this article we substantially extend an earlier result of M. Pollicott and P. Vytnova on the computation of the affinity dimension. Pollicott and Vytnova's work applies to planar invertible affine contractions with positive linear parts under several additional conditions which among other things constrain the affinity dimension to be between 0 and 1. We extend this result by passing from planar self-affine sets to self-affine sets in arbitrary dimensions, relaxing the positivity hypothesis to a domination condition, and removing all other constraints including that on the range of values of the affinity dimension. We provide some explicit examples of two- and three-dimensional affine iterated function systems for which the affinity dimension can be calculated to more than 30 decimal places.