Infinite Iterated Function Systems Research Articles

Shrinking target problems in d-decaying Gauss-like iterated function systems

Let Φ={ϕn}n≥1 be an infinite iterated function system (IFS) on [0,1] satisfying the open set condition with the open set (0,1) and let J be its attractor. For any x∈J, there is a unique integer sequence {ωn(x)}n≥1, called the digit sequence of x, such thatx=limn→∞⁡ϕω1(x)∘⋯∘ϕωn(x)(1) except at countably many points. In this paper, we investigate the shrinking target problem in the infinite iterated function system with a polynomial decay of the derivative. More precisely, we consider the size of the setW(T,x0)={x∈J:Tn(x)∈Itn(x0) for infinitely many n∈N}, where T:J⟼J is the expanding map induced by the left shift and Itn(x0) denotes the tn-th cylinder of x0∈J. As a continuation of one result of Li et al. (2014) [11], we obtain the exact Hausdorff dimension of the set W(T,x0) in some general infinite function systems.

Parametrizability of infinitely generated attractors

An infinite iterated function system (IIFS) is a countable collection of contraction maps on a compact metric space. In this paper we study the conditions under which the attractor of such a system admits a parameterization by a continuous or Hölder continuous map of the unit interval.

Brownian motion on the golden ratio Sierpinski gasket

We construct a strongly local regular Dirichlet form on the golden ratio Sierpinski gasket, which is a self-similar set without a finitely ramified cell structure, via a study on the trace of an electrical network on an infinite graph. The Dirichlet form is the unique one that is self-similar in the sense of an infinite iterated function system, and is decimation invariant with respect to a graph-directed construction. The proof is based on a fixed point problem of a renormalization map, inspired by Sabot's celebrated work for finitely ramified fractals. Lastly, the Hunt process associated with the Dirichlet form satisfies a two-sided sub-Gaussian heat kernel estimate.

Intermediate dimensions of infinitely generated attractors

We study the dimension theory of limit sets of iterated function systems consisting of a countably infinite number of contractions. Our primary focus is on the intermediate dimensions: a family of dimensions depending on a parameter θ ∈ [ 0 , 1 ] \theta \in [0,1] which interpolate between the Hausdorff and box dimensions. Our main results are in the case when all the contractions are conformal. Under a natural separation condition we prove that the intermediate dimensions of the limit set are the maximum of the Hausdorff dimension of the limit set and the intermediate dimensions of the set of fixed points of the contractions. This builds on work of Mauldin and Urbański concerning the Hausdorff and upper box dimension. We give several (often counter-intuitive) applications of our work to dimensions of projections, fractional Brownian images, and general Hölder images. These applications apply to well-studied examples such as sets of numbers which have real or complex continued fraction expansions with restricted entries. We also obtain several results without assuming conformality or any separation conditions. We prove general upper bounds for the Hausdorff, box and intermediate dimensions of infinitely generated attractors in terms of a topological pressure function. We also show that the limit set of a ‘generic’ infinite iterated function system has box and intermediate dimensions equal to the ambient spatial dimension, where ‘generic’ can refer to any one of (i) full measure; (ii) prevalent; or (iii) comeagre.

The Hutchinson–Barnsley theory for generalized iterated function systems by means of infinite iterated function systems

The Hutchinson–Barnsley theory for generalized iterated function systems by means of infinite iterated function systems

Big Birkhoff sums in $d$-decaying Gauss like iterated function systems

The increasing rate of the Birkhoff sums in infinite iterated function systems with polynomial decay of the derivative (for example the Gauss map) is studied. For different unbounded potential functions, the Hausdorff dimensions of the sets of points whos

Dimension spectrum of infinite self-affine iterated function systems

Given an infinite iterated function system (IFS) {mathcal {F}}, we define its dimension spectrum D({mathcal {F}}) to be the set of real numbers which can be realised as the dimension of some subsystem of {mathcal {F}}. In the case where {mathcal {F}} is a conformal IFS, the properties of the dimension spectrum have been studied by several authors. In this paper we investigate for the first time the properties of the dimension spectrum when {mathcal {F}} is a non-conformal IFS. In particular, unlike dimension spectra of conformal IFS which are always compact and perfect (by a result of Chousionis, Leykekhman and Urbański, Selecta 2019), we construct examples to show that D({mathcal {F}}) need not be compact and may contain isolated points.

A multifractal analysis for cuspidal windings on hyperbolic surfaces

In this paper, we investigate the multifractal decomposition of the limit set of a finitely generated, free Fuchsian group with respect to the mean cusp-winding number. We completely determine its multifractal spectrum by means of a certain free energy function and show that the Hausdorff dimension of sets consisting of limit points with the same scaling exponent coincides with the Legendre transform of this free energy function. As a by-product we generalize previously obtained results on the multifractal formalism for infinite iterated function systems to the setting of infinite graph directed Markov systems.

Contractive affine generalized iterated function systems which are topologically contracting

Abstract In this paper we provide alternative descriptions of those generalized possibly infinite iterated function systems whose constitutive functions are affine contractions which are topologically contracting generalized iterated function systems. As a by-product, we establish conditions under which the attractor of a contractive affine generalized possibly infinite iterated function system is compact.

On Hyperbolic Affine Generalized Infinite Iterated Function Systems

The aim of this paper is to provide alternative characterizations of hyperbolic affine generalized infinite iterated function systems. More precisely, we prove that, for such a system $${\mathcal {F}}=((X,\left\| .\right\| ),(f_{i})_{i\in I})$$ , among others, the following statements are equivalent: (a) $${\mathcal {F}}$$ is hyperbolic. (b) $$ {\mathcal {F}}$$ has attractor. (c) $${\mathcal {F}}$$ is strictly topologically contractive. (d) $${\mathcal {F}}$$ is uniformly point-fibred. In this way we generalize the result from the paper by Miculescu and Mihail (J Math Anal Appl 407:56–68, 2013). More equivalent statements are given for the particular case when I is finite and X is finite dimensional.

A remark on big Birkhoff sums in d-decaying Gauss like iterated function systems

Recently, Liao and Rams [10] considered certain infinite iterated function systems that are natural generalizations of Gauss map. They studied the increasing rate of Birkhoff sums in such systems with polynomial decay of the derivative and calculated the Hausdorff dimensions of the sets of points whose Birkhoff sums share the same increasing rate for different unbounded potential functions. In this remark, we investigate the critical cases for the growth rate function which are not discussed in [10].

Extremality and dynamically defined measures, part II: Measures from conformal dynamical systems

AbstractWe present a new method of proving the Diophantine extremality of various dynamically defined measures, vastly expanding the class of measures known to be extremal. This generalizes and improves the celebrated theorem of Kleinbock and Margulis [Logarithm laws for flows on homogeneous spaces. Invent. Math.138(3) (1999), 451–494] resolving Sprindžuk’s conjecture, as well as its extension by Kleinbock, Lindenstrauss, and Weiss [On fractal measures and Diophantine approximation. Selecta Math.10 (2004), 479–523], hereafter abbreviated KLW. As applications we prove the extremality of all hyperbolic measures of smooth dynamical systems with sufficiently large Hausdorff dimension, and of the Patterson–Sullivan measures of all nonplanar geometrically finite groups. The key technical idea, which has led to a plethora of new applications, is a significant weakening of KLW’s sufficient conditions for extremality. In the first of this series of papers [Extremality and dynamically defined measures, part I: Diophantine properties of quasi-decaying measures. Selecta Math.24(3) (2018), 2165–2206], we introduce and develop a systematic account of two classes of measures, which we call quasi-decaying and weakly quasi-decaying. We prove that weak quasi-decay implies strong extremality in the matrix approximation framework, as well as proving the ‘inherited exponent of irrationality’ version of this theorem. In this paper, the second of the series, we establish sufficient conditions on various classes of conformal dynamical systems for their measures to be quasi-decaying. In particular, we prove the above-mentioned result about Patterson–Sullivan measures, and we show that equilibrium states (including conformal measures) of nonplanar infinite iterated function systems (including those which do not satisfy the open set condition) and rational functions are quasi-decaying.

Dimensions in infinite iterated function systems consisting of bi-Lipschitz mappings

We study infinite iterated function systems (IIFSs) consisting of bi-Lipschitz mappings instead of conformal contractions, focussing on IFSs that do not satisfy the open set condition. By assuming the logarithmic distortion property and some cardinality growth condition, we obtain a formula for the Hausdorff, box, and packing dimensions of the limit set in terms of certain topological pressure. By assuming, in addition, the weak separation condition, we show that these dimensions are equal to the growth dimension of the limit set.

Weakly divergent partial quotients

We study the real numbers with partial quotients diverging to infinity in a subsequence. We show that if the subsequence has positive density then such sets have Hausdorff dimension equal to 1/2. This generalizes one of the results obtained in [C. Y. Cao, B. W. Wang and J. Wu, The growth speed of digits in infinite iterated function systems, Studia. Math. 217(2) (2013) 139–158; I. J. Good, The fractional dimensional theory of continued fractions, Proc. Cambridge Philos. Soc. 37 (1941) 199–228].

On a class of Rauzy fractals without the finiteness property

We present some topological and arithmetical aspects of a class of Rauzy fractals $\mathcal{R}_{a,b}$ related to the polynomials of the form $P_{a,b}(x)=x^{3}-ax^{2}-bx-1$, where $a$ and $b$ are integers satisfying $-a+1 \leq b \leq -2$. This class has the property that $0$ lies on the boundary of $\mathcal{R}_{a,b}$. We construct explicit finite automata that recognize the boundaries of these fractals. This allows to establish the number of neighbors of $\mathcal{R}_{a,b}$ in the tiling it generates. Furthermore, we prove that if $2a+3b+4 \leq 0$ then $\mathcal{R}_{a,b}$ is not homeomorphic to a topological disk. We also show that the boundary of the set $\mathcal{R}_{3,-2}$ is generated by two infinite iterated function systems.

The dimension spectrum of conformal graph directed Markov systems

In this paper we study the dimension spectrum of general conformal graph directed Markov systems modeled by countable state symbolic subshifts of finite type. We perform a comprehensive study of the dimension spectrum addressing questions regarding its size and topological structure. As a corollary we prove that the dimension spectrum of infinite conformal iterated function systems is compact and perfect. On the way we revisit the role of the parameter \(\theta \) in graph directed Markov systems and we show that new phenomena arise. We also establish topological pressure estimates for subsystems in the abstract setting of symbolic dynamics with countable alphabets. These estimates play a crucial role in our proofs regarding the dimension spectrum, and they allow us to study Hausdorff dimension asymptotics for subsystems. Finally, we narrow our focus to the dimension spectrum of conformal iterated function systems and we prove, among other things, that the iterated function system resulting from the complex continued fractions algorithm has full dimension spectrum. As a result, we provide a positive answer to the Texan conjecture for complex continued fractions.

Attractors of Compactly Generated Semigroups of Regular Polynomial Mappings

We investigate the metric space of pluriregular sets as well as the contractions on that space induced by infinite compact families of proper polynomial mappings of several complex variables. The topological semigroups generated by such families, with composition as the semigroup operation, lead to the constructions of a variety of Julia-type pluriregular sets. The generating families can also be viewed as infinite iterated function systems with compact attractors. We show that such attractors can be approximated both deterministically and probabilistically in a manner of the classic chaos game.

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

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.

Spectral asymptotics of one-dimensional fractal Laplacians in the absence of second-order identities

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 ].

Extremality and dynamically defined measures, part I: Diophantine properties of quasi-decaying measures

We present a new method of proving the Diophantine extremality of various dynamically defined measures, vastly expanding the class of measures known to be extremal. This generalizes and improves the celebrated theorem of Kleinbock and Margulis (’98) resolving Sprindžuk’s conjecture, as well as its extension by Kleinbock, Lindenstrauss, and Weiss (’04), hereafter abbreviated KLW. As applications we prove the extremality of all hyperbolic measures of smooth dynamical systems with sufficiently large Hausdorff dimension, of the Patterson–Sullivan measures of all nonplanar geometrically finite groups, and of the Gibbs measures (including conformal measures) of infinite iterated function systems. The key technical idea, which has led to a plethora of new applications, is a significant weakening of KLW’s sufficient conditions for extremality. In Part I, we introduce and develop a systematic account of two classes of measures, which we call quasi-decaying and weakly quasi-decaying. We prove that weak quasi-decay implies strong extremality in the matrix approximation framework, thus proving a conjecture of KLW. We also prove the “inherited exponent of irrationality” version of this theorem, describing the relationship between the Diophantine properties of certain subspaces of the space of matrices and measures supported on these subspaces. In subsequent papers, we exhibit numerous examples of quasi-decaying measures, in support of the thesis that “almost any measure from dynamics and/or fractal geometry is quasi-decaying”. We also discuss examples of non-extremal measures coming from dynamics, illustrating where the theory must halt.