Invariant Measure Associated with a Generalized Countable Iterated Function System

Generalized countable iterated function systems (GCIFS) are an extension of countable iterated function systems by considering contractions from X × X into X instead of contractions on the compact metric space X into itself. For a GCIFS endowed with a system of probabilities we associate an invariant and normalized Borel measure whose support is just the attractor of the respective GCIFS, extending the classical Hutchinson’s construction.     

Algorithms generating images of attractors of generalized iterated function systems

The paper is devoted to searching algorithms which will allow to generate images of attractors of 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×...×X to X). Two presented algorithms are counterparts of classical deterministic algorithm and so-called chaos game. The third and fourth one is fitted to special kind of GIFSs - to affine GIFS, which are, in turn, also investigated.     

Generalized Iterated Function Systems with Place Dependent Probabilities

In the present paper we introduce the concept of generalized iterated function system with place dependent probabilities (GIFSpdp). In this framework, we provide sufficient conditions under which the Markov operator associated to a GIFSpdp is Lipschitz. We also prove, under certain conditions, the existence of an analogue of Hutchinson measure associated to a GIFSpdp and study its properties.     

Rigidity of Conformal Iterated Function Systems

The paper extends the rigidity of the mixing expanding repellers theorem of D. Sullivan announced at the 1986 IMC. We show that, for a regular conformal, satisfying the Open Set Condition, iterated function system of countably many holomorphic contractions of an open connected subset of a complex plane, the Radon–Nikodym derivative d/dm has a real-analytic extension on an open neighbourhood of the limit set of this system, where m is the conformal measure and is the unique probability invariant measure equivalent with m. Next, we introduce the concept of nonlinearity for iterated function systems of countably many holomorphic contractions. Several necessary and sufficient conditions for nonlinearity are established. We prove the following rigidity result: If h, the topological conjugacy between two nonlinear systems F and G, transports the conformal measure m _F to the equivalence class of the conformal measure m _G , then h has a conformal extension on an open neighbourhood of the limit set of the system F. Finally, we prove that the hyperbolic system associated to a given parabolic system of countably many holomorphic contractions is nonlinear, which allows us to extend our rigidity result to the case of parabolic systems.     

Dual systems of algebraic iterated function systems

We study a class of graph-directed iterated function systems on R$\mathbb{R}$ with algebraic parameters, which we call algebraic GIFS. We construct a dual IFS of an algebraic GIFS, and study the relations between the two systems. We determine when a dual system satisfies the open set condition, which is fundamental. For feasible Pisot systems, we construct the left and right Rauzy–Thurston tilings, and study their multiplicities and decompositions. We also investigate their relation with codings space, domain-exchange transformation, and the Pisot spectrum conjecture. The dual IFS provides a unified and simple framework for Rauzy fractals, β-tilings and related studies, and allows us gain better understanding.     

Attractors of generalized IFSs that are not attractors of IFSs

Mihail and Miculescu introduced the notion of a generalized iterated function system (GIFS in short), and proved that every GIFS generates an attractor. (In our previous paper we gave this notion a more general setting.) In this paper we show that for any m≥2$m\ge 2$, there exists a Cantor subset of the plane which is an attractor of some GIFS of order m  , but is not an attractor of a GIFS of order m−1$m-1$. In particular, this result shows that there is a subset of the plane which is an attractor of some GIFS, but is not an attractor of an IFS. We also give an example of a Cantor set which is not an attractor of a GIFS.     

Separation conditions for conformal iterated function systems

We extend both the weak separation condition and the finite type condition to include finite iterated function systems (IFSs) of injective C ¹ conformal contractions on compact subsets of \mathbbR^d{{\mathbb{R}}^d} . For conformal IFSs satisfying the bounded distortion property, we prove that the finite type condition implies the weak separation condition. By assuming the weak separation condition, we prove that the Hausdorff and box dimensions of the attractor are equal and, if the dimension of the attractor is α, then its α-dimensional Hausdorff measure is positive and finite. We obtain a necessary and sufficient condition for the associated self-conformal measure μ to be singular. By using these we give a first example of a singular invariant measure μ that is associated with a non-linear IFS with overlaps.     

Quasiperiodic spectra and orthogonality for iterated function system measures

We extend classical basis constructions from Fourier analysis to attractors for affine iterated function systems (IFSs). This is of interest since these attractors have fractal features, e.g., measures with fractal scaling dimension. Moreover, the spectrum is then typically quasi-periodic, but non-periodic, i.e., the spectrum is a “small perturbation” of a lattice. Due to earlier research on IFSs, there are known results on certain classes of spectral duality-pairs, also called spectral pairs or spectral measures. It is known that some duality pairs are associated with complex Hadamard matrices. However, not all IFSs X admit spectral duality. When X is given, we identify geometric conditions on X for the existence of a Fourier spectrum, serving as the second part in a spectral pair. We show how these spectral pairs compose, and we characterize the decompositions in terms of atoms. The decompositions refer to tensor product factorizations for associated complex Hadamard matrices. Research supported in part by a grant from the National Science Foundation DMS-0704191.     

Separation conditions for conformal iterated function systems

We extend both the weak separation condition and the finite type condition to include finite iterated function systems (IFSs) of injective C ¹ conformal contractions on compact subsets of . For conformal IFSs satisfying the bounded distortion property, we prove that the finite type condition implies the weak separation condition. By assuming the weak separation condition, we prove that the Hausdorff and box dimensions of the attractor are equal and, if the dimension of the attractor is α, then its α-dimensional Hausdorff measure is positive and finite. We obtain a necessary and sufficient condition for the associated self-conformal measure μ to be singular. By using these we give a first example of a singular invariant measure μ that is associated with a non-linear IFS with overlaps. The authors are supported in part by an HKRGC grant.     

On Exit and Ergodicity of the Spectrally One-Sided Lévy Process Reflected at Its Infimum

Consider a spectrally one-sided Lévy process X and reflect it at its past infimum I. Call this process Y. For spectrally positive X, Avram et al.(2) found an explicit expression for the law of the first time that Y=X–I crosses a finite positive level a. Here we determine the Laplace transform of this crossing time for Y, if X is spectrally negative. Subsequently, we find an expression for the resolvent measure for Y killed upon leaving [0,a]. We determine the exponential decay parameter for the transition probabilities of Y killed upon leaving [0,a], prove that this killed process is -positive and specify the -invariant function and measure. Restricting ourselves to the case where X has absolutely continuous transition probabilities, we also find the quasi-stationary distribution of this killed process. We construct then the process Y confined in [0,a] and prove some properties of this process.     

Invariant measures of Markov operators associated to iterated function systems consisting of φ-max-contractions with probabilities

We prove that the Markov operator associated to an iterated function system consisting of $\phi$-max-contractions with probabilities has a unique invariant measure whose support is the attractor of the system.     

Analysis of orthogonality and of orbits in affine iterated function systems

We introduce a duality for affine iterated function systems (AIFS) which is naturally motivated by group duality in the context of traditional harmonic analysis. Our affine systems yield fractals defined by iteration of contractive affine mappings. We build a duality for such systems by scaling in two directions: fractals in the small by contractive iterations, and fractals in the large by recursion involving iteration of an expansive matrix. By a fractal in the small we mean a compact attractor X supporting Hutchinson’s canonical measure μ, and we ask when μ is a spectral measure, i.e., when the Hilbert space has an orthonormal basis (ONB) of exponentials . We further introduce a Fourier duality using a matched pair of such affine systems. Using next certain extreme cycles, and positive powers of the expansive matrix we build fractals in the large which are modeled on lacunary Fourier series and which serve as spectra for X. Our two main results offer simple geometric conditions allowing us to decide when the fractal in the large is a spectrum for X. Our results in turn are illustrated with concrete Sierpinski like fractals in dimensions 2 and 3. Research supported in part by the National Science Foundation DMS 0457491.     

Fourier frequencies in affine iterated function systems

We examine two questions regarding Fourier frequencies for a class of iterated function systems (IFS). These are iteration limits arising from a fixed finite families of affine and contractive mappings in Rd, and the “IFS” refers to such a finite system of transformations, or functions. The iteration limits are pairs (X,μ) where X is a compact subset of Rd (the support of μ), and the measure μ is a probability measure determined uniquely by the initial IFS mappings, and a certain strong invariance axiom. The two questions we study are: (1) existence of an orthogonal Fourier basis in the Hilbert space L²(X,μ); and (2) explicit constructions of Fourier bases from the given data defining the IFS.     

SOME NEW ITERATED FUNCTION SYSTEMS CONSISTING OF GENERALIZED CONTRACTIVE MAPPINGS

Iterated function systems (IFS) were introduced by Hutchinson in 1981 as a natural generalization of the well-known Banach contraction principle.In 2010,D.R.Sahu and A.Chakraborty introduced K-Iterated Function System using Kannan mapping which would cover a larger range of mappings.In this paper,following Hutchinson,D.R.Sahu and A.Chakraborty,we present some new iterated function systems by using the so-called generalized contractive mappings,which will also cover a large range of mappings.Our purpose is to prove the existence and uniqueness of attractors for such class of iterated function systems by virtue of a Banach-like fixed point theorem concerning generalized contractive mappings.     

A fixed point theorem for moment matrices of self-similar measures

We consider the self-similar measure on the complex plane C$\mathbb{C}$ associated to an iterated function system (IFS) with probabilities. From this IFS we define an operator in a complete metric space of infinite matrices. Using the expression obtained in a previous work of the authors, we prove that this operator has as fixed point the moment matrix of the self-similar measure. As a consequence, we obtain a very efficient algorithm to compute the moment matrix of the self-similar measure. Finally, in order to estimate the rate of convergence of the algorithm, we find an upper bound of the norm of this contractive operator.     

Equidistribution over function fields

We prove equidistribution of a generic net of small points in a projective variety X over a function field K. For an algebraic dynamical system over K, we generalize this equidistribution theorem to a small generic net of subvarieties. For number fields, these results were proved by Yuan and we transfer here his methods to function fields. If X is a closed subvariety of an abelian variety, then we can describe the equidistribution measure explicitly in terms of convex geometry.     

A functional LIL for integrated α stable process

Let {X（t）,t ∈ R＋} be an integrated α stable process. In this paper, a functional law of the iterated logarithm （LIL） is derived via estimating the small ball probability of X. As a corollary,, the classical Chung LIL of X is obtained. Furthermore, some results about the weighted occupation measure of X（t） are established.     

Nilpotence and finite H-spaces

Zabrodsky asked when is the iterated commutator mapX ⁿ →X for a connected associative H-spaceX a null map. In this paper we reduce this question to a cohomological question and answer it in several cases. Supported by an NSF Presidential Young Investigator award and the Sloan Foundation.     

Analytic Multifunctions, Holomorphic Motions and Hausdorff Dimension in IFSs

We study iterated function systems of contractions which depend holomorphically on a complex parameter λ. We first restrict our attention to systems which consist of similarities that satisfy the OSC. In this setting, we prove that the Hausdorff dimension of the limit set J(λ) is a continuous, subharmonic function of λ. In the remainder of the paper, systems consisting of conformal contractions are considered. We give conditions under which J(λ) and A(λ) = describe a holomorphic motion, and construct an example that shows that this is not the case in general. We finally show that A(λ) is best described as an analytic multifunction of λ, a notion that generalizes that of holomorphic motion. This research was supported by the Natural Sciences and Engineering Research Council of Canada (NSERC) and by the Fonds Québécois de Recherche sur la Nature et les Technologies (FQRNT). This research was supported by the FQRNT.     

Geometry of the common dynamics of flipped Pisot substitutions

In this article we study the common dynamics of two different Pisot substitutions σ ₁ and σ ₂ having the same incidence matrix. This common dynamics arises in the study of the adic systems associated with the substitutions σ ₁ and σ ₂. Since the adic systems considered here have geometric realizations given by solutions to graph-directed iterated function systems, we actually study topological and measure-theoretic properties of the solution of those iterated function systems which describe the common dynamics. We also consider generalizations of these results to the nonunimodular case, the case of more than two substitutions and the case of two substitutions with different incidence matrices.