Graph-directed Iterated Function Systems Research Articles

Purely periodic continued fractions and graph-directed iterated function systems

We describe Gauss-type maps as geometric realizations of certain codes in the monoid of nonnegative matrices in the extended modular group. Each such code, together with an appropriate choice of unimodular intervals in P1R\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\,\ extrm{P}\\,}}^1\\mathbb {R}$$\\end{document}, determines a dual pair of graph-directed iterated function systems, whose attractors contain intervals and constitute the domains of a dual pair of Gauss-type maps. Our framework covers many continued fraction algorithms (such as Farey fractions, Ceiling, Even and Odd, Nearest Integer, …\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ldots $$\\end{document}) and provides explicit dual algorithms and characterizations of those quadratic irrationals having a purely periodic expansion.

Tiling iterated function systems

This paper presents a detailed symbolic approach to the study of self-similar tilings. It uses properties of addresses associated with graph-directed iterated function systems to establish conjugacy properties of tiling spaces. Tiles may be fractals and the tiled set maybe a complicated unbounded subset of RM.

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.

Separation Conditions for Iterated Function Systems with Overlaps on Riemannian Manifolds

We formulate the weak separation condition and the finite type condition for conformal iterated function systems on Riemannian manifolds with nonnegative Ricci curvature, and generalize the main theorems by Lau et al. (Monatsch Math 156:325–355, 2009). We also obtain a formula for the Hausdorff dimension of a self-similar set defined by an iterated function system satisfying the finite type condition, generalizing a corresponding result by Jin and Yau (Commun Anal Geom 13:821–843, 2005) and Lau and Ngai (Adv Math 208:647–671, 2007) on Euclidean spaces. Moreover, we obtain a formula for the Hausdorff dimension of a graph self-similar set generated by a graph-directed iterated function system satisfying the graph finite type condition, extending a result by Ngai et al. (Nonlinearity 23:2333–2350, 2010).

Tilings from graph directed iterated function systems

A new method for constructing self-referential tilings of Euclidean space from a graph directed iterated function system (GIFS), based on a combinatorial structure we call a pre-tree, is introduced. For each GIFS, a family of tilings is constructed indexed by a parameter. For what we call a commensurate GIFS, our method is used to define what we refer to as balanced tilings. Under mild conditions on the commensurate GIFS and the parameter, the resulting balanced tilings have a finite set of prototiles, are self-similar, and are quasiperiodic. A notion of rigidity is defined for a commensurate GIFS, and a necessary and sufficient condition for two rigid balanced tilings to be congruent is provided. For a given rigid GIFS, there are uncountably many balanced tilings, corresponding to uncountably many parameters. All rigid balanced tilings are non-periodic.

On the smallest disks enclosing graph-directed fractals

We consider the graph-directed iterated function systems and give upper bounds for the diameters of the smallest disks enclosing their attractors. We also give an algorithm to obtain these smallest enclosing disks with any proximity.

A contraction principle on gauge spaces with graphs and application to infinite graph-directed iterated function systems

A contraction principle on gauge spaces with graphs and application to infinite graph-directed iterated function systems

Differentiability of Lq-spectrum and multifractal decomposition by using infinite graph-directed IFSs

By constructing an infinite graph-directed iterated function system associated with a finite iterated function system, we develop a new approach for proving the differentiability of the Lq-spectrum and establishing the multifractal formalism of certain self-similar measures with overlaps, especially those defined by similitudes with different contraction ratios. We apply our technique to a well-known class of self-similar measures of generalized finite type.

Graph-directed fractal interpolation functions

It is known that there exists a function interpolating a given data set such that the graph of the function is the attractor of an iterated function system, which is called a fractal interpolation function. We generalize the notion of the fractal interpolation function to the graph-directed case and prove that for a finite number of data sets there exist interpolation functions each of which interpolates a corresponding data set in $\mathbb{R}^2$ such that the graphs of the interpolation functions are attractors of a graph-directed iterated function system.

Connectedness of invariant sets of graph-directed IFS

In this paper, we study the connectedness of the invariant sets of a graph-directed iterated function system (IFS). For a graph-directed IFS with N states, we construct N graphs. We prove that all the invariant sets are connected, if and only if all the N graphs are connected; in this case, the invariant sets are all locally connected and path connected. Our result extends the results on the connectedness of the self-similar sets.

The valuative capacity of subshifts of finite type

The characteristic sequence of a subset E of the integers enables to generalize the factorial. The asymptotic limit of this sequence, called the valuative capacity, is actually related to the transfinite diameter of this set embedded into the set of p-adic integers. We use tools of potential theory to obtain formulas to compute this capacity. As an example, we provide a family of sets, invariant by a family of graph-directed iterated function systems, whose valuative capacities are algebraic numbers.

Applications of Multivalued Contractions on Graphs to Graph-Directed Iterated Function Systems

We apply a fixed point result for multivalued contractions on complete metric spaces endowed with a graph to graph-directed iterated function systems. More precisely, we construct a suitable metric space endowed with a graphGand a suitableG-contraction such that its fixed points permit us to obtain more information on the attractor of a graph-directed iterated function system.

FRACTAL GEOMETRY OF EQUILIBRIUM PAYOFFS IN DISCOUNTED SUPERGAMES

This paper examines the pure-strategy subgame-perfect equilibrium payoffs in discounted supergames with perfect monitoring. It is shown that the equilibrium payoffs can be identified as sub-self-affine sets or graph-directed iterated function systems. We propose a method to estimate the Hausdorff dimension of the equilibrium payoffs and relate it to the equilibrium paths and their graph presentation.

Periodic codings of algebraic graph-directed IFS

We study the points with periodic codings of a class of graph-directed iterated function systems on ℝ with algebraic parameters (which we call algebraic GIFS). It is shown that the set of points with periodic codings is closely related to the Rauzy box. Especially for the feasible Pisot GIFS, this set is completely characterized by the Rauzy box; this result unifies and extends the previous results on the study of periodic points of classical and generalized β-transformations.

Dual systems of algebraic iterated function systems

We study a class of graph-directed iterated function systems on 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.

Finite type, open set conditions and weak separation conditions

We study graph-directed iterated function systems of finite type. We show that such an IFS of finite type induces another graph-directed IFS of finite type where every strongly connected component satisfies the open set condition. We introduce the notions of topological and geometric weak separation properties, and summarize the relationship between the different separation conditions. For the induced IFS, the similarity, growth, box and Hausdorff dimensions coincide. Finally we prove that the generalized finite-type condition for graphs implies the geometric weak separation property.

Graph-directed iterated function systems satisfying the generalized finite type condition

We extend the generalized finite type condition to graph-directed iterated function systems with overlaps. Under this condition, we can compute the Hausdorff dimension of the attractor F in terms of the spectral radius of certain weighted incidence matrix. Moreover, if the Haudorff dimension of F is α, then the α-dimensional Hausdorff and packing measures of F are shown to be strictly positive. By assuming in addition that the graph is strongly connected, we show that the Hausdorff, packing, and box dimensions are equal and the α-dimensional Hausdorff and packing measures are finite.

On single-matrix graph-directed iterated function systems

We study graph-directed function systems where each contraction in the system has the form f e ( x ) = A − 1 ( x + d e ) , where A is an expanding matrix. We show that a certain discreteness implies the open set condition, and the latter implies the strong open set condition. Hausdorff measures and dimensions (w.r.t. a weak norm) of the invariant sets are investigated. The stationary Markov measures of the system are proved to be translation invariant.

Constructing quasisymmetric functions via graph-directed iterated function systems

Using Tukia’s method for representing a quasisymmetric function as a quasisymmetric sieve, we generalize his modification to the Salem scheme and find a sufficient condition for the collection of functions that realize a structure parametrization of a graph-directed function system of a particular form (a one-dimensional multizipper) to consist of quasisymmetric functions. We give an asymptotically sharp estimate for the quasisymmetry coefficient of these functions in terms of the dilation coefficients of the mappings constituting a given multizipper, which yields a substantially more general method for constructing quasisymmetric functions than Tukia’s construction.

Geometry of the common dynamics of flipped Pisot substitutions

In this article we study the common dynamics of two different Pisot substitutions σ1 and σ2 having the same incidence matrix. This common dynamics arises in the study of the adic systems associated with the substitutions σ1 and σ2. 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.