Infinite Iterated Function Systems Research Articles

Finer geometric rigidity of limit sets of conformal IFS

We consider infinite conformal iterated function systems in the phase space R d \mathbb {R}^d with d ≥ 3 d\ge 3 . Let J J be the limit set of such a system. Under a mild technical assumption, which is always satisfied if the system is finite, we prove that either the Hausdorff dimension of J J exceeds the topological dimension k k of the closure of J J or else the closure of J J is a proper compact subset of either a geometric sphere or an affine subspace of dimension k k . A similar dichotomy holds for conformal expanding repellers.

The doubling property of conformal measures of infinite iterated function systems

We provide sufficient conditions for the conformal measures induced by regular conformal infinite iterated function systems to satisfy the doubling property. We apply these conditions to iterated function systems derived from the continued fraction algorithm—continued fractions with restricted entries. For these systems our conditions are expressed in terms of the asymptotic density properties of the allowed entries. As examples, we give some relatively large classes of sets of continued fractions with restricted entries for which the corresponding conformal measures have the doubling property. Similarly, we give some other classes for which the conformal measure does not have the doubling property.

Diophantine Analysis of Conformal Iterated Function Systems

A formula for the Hausdorff dimension of a subset of the limit set of a conformal infinite iterated function system is derived and expressed as a unique zero of the topological pressure of corresponding strongly Holder family of functions.

Hausdorff Dimension of Harmonic Measure for Self-Conformal Sets

Under some technical assumptions it is shown that the Hausdorff dimension of the harmonic measure on the limit set of a conformal infinite iterated function system is strictly less than the Hausdorff dimension of the limit set itself if the limit set is contained in a real–analytic curve, if the iterated function system consists of similarities only, or if this system is irregular. As a consequence of this general result the same statement is proven for hyperbolic and parabolic Julia sets, finite parabolic iterated function systems and generalized polynomial-like mappings. Also sufficient conditions are provided for a limit set to be uniformly perfect and for the harmonic measure to have the Hausdorff dimension less than 1. Some results in the spirit of Przytycki et al. ( Ann. of Math. 130 (1989), 1–40; Stud. Math. 97 (1991), 189–225) are obtained.

Hausdorff dimension estimates for infinite conformal IFSs

We show that in order to estimate the Hausdorff dimension of conformal infinite iterated function systems (IFSs), it is completely sufficient to consider finite subsystems with N<∞ generators. We give estimates on the accuracy of this approximation and present a simple straightforward algorithm which allows us to reduce this setting further to finite iteration time n. Our method allows us to calculate the Hausdorff dimension of an IFS with the same speed of convergence as the algorithm proposed by McMullen (McMullen C 1998 Am. J. Math. 120 691-721) (i.e. the distance of the approximation to the actual value is ~1/n, where n is the iteration depth, hence corresponding to ~Nn calculation steps). This is much slower than Jenkinson/Pollicott's method for bounded continued fractions (which gives superexponential accuracy ~exp (-n3/2) for the same depth) but does not require extensive knowledge about the periodic points of the involved transformations and is applicable in a more general setting. The method is applied in order to perform numerical calculations for certain classes of examples.

Thermodynamic Formalism and Multifractal Analysis of Conformal Infinite Iterated Function Systems

We develop the thermodynamic formalism for equilibrium states of strongly Holder families of functions. These equilibrium states are supported on the limit set generated by iterating a system of infinitely many contractions. The theory of these systems was laid out in an earlier paper of the last two authors. The first five sections of this paper except Section 3 are devoted to developing the thermodynamic formalism for equilibrium states of Holder families of functions. The first three sections provide us with the tools needed to carry out the multifractal analysis for the equilibrium states mentioned above assuming that the limit set is generated by conformal contractions. The theory of infinite systems of conformal contractions is laid out in [13]. The multifractal analysis is then given in Section 7. In Section 8 we apply this theory to some examples from continued fraction systems and Apollonian packing.

Porosity in Conformal Infinite Iterated Function Systems

In this paper we deal with the problem of porosity of limit sets of conformal (infinite) iterated function systems. We provide a necessary and sufficient condition for the limit sets of these systems to be porous. We pay special attention to the systems generated by continued fractions with restricted entries and we give a complete description of the subsets I of positive integers such that the set JI of all numbers whose continued fraction expansion entries are contained in I, is porous. We then study such porous sets in greater detail examining their Hausdorff dimensions, Hausdorff measures, packing measures, and other geometric characteristics. We also show that the limit set generated by the complex continued fraction algorithm is not porous, the limit sets of all plane parabolic iterated function systems are porous, and of all real parabolic iterated function systems are not porous. We provide a very effective necessary and sufficient condition for the limit set of a finite conformal iterated function system to be porous.

Parabolic iterated function systems

In this paper we introduce and explore conformal parabolic iterated function systems. We define and study topological pressure, Perron–Frobenius-type operators, semiconformal and conformal measures and the Hausdorff dimension of the limit set. With every parabolic system we associate an infinite hyperbolic conformal iterated function system and we employ it to study geometric and dynamical features (properly defined invariant measures for example) of the limit set.

Hausdorff Measures versus Equilibrium States of Conformal Infinite Iterated Function Systems

Dealing with infinite iterated function systems we introduce and develop the ergodic theory of Holder systems of functions similarly as in [HU] and [HMU]. In the context of conformal infinite iterated function systems we prove the volume lemma for the Hausdorff dimension of the projection onto the limit set of a shift invariant measure. This can be considered as a Billingsley type result. Our cenral goal is to demonstrate the appearance of the "singularity-absolute continuity" dichotomy for equilibrium states of Holder systems of functions which has been observed in [PUZ,I] and [PUZ,II] (see also [DU1] and [DU2]) in the setting of rational functions of the Riemann sphere.

Dimensions and Measures in Infinite Iterated Function Systems

The Hausdorff and packing measures and dimensions of the limit sets of iterated function systems generated by countable families of conformal contractions are investigated. Conformal measures for such systems, reflecting geometric properties of the limit set, are introduced, proved to exist, and to be unique. The existence of a unique invariant probability equivalent to the conformal measure is derived. Our methods employ the concepts of the Perron-Frobenius operator, symbolic dynamics on a shift space with an infinite alphabet, and the properties of the above-mentioned ergodic invariant measure. A formula for the Hausdorff dimension of the limit set in terms of the pressure function is derived. Fractal phenomena not exhibited by finite systems are shown to appear in the infinite case. In particular, a variety of conditions are provided for Hausdorff and packing measures to be positive or finite, and a number of examples are described showing the appearance of various possible combinations for these quantities. One example given special attention is the limit set associated to the complex continued fraction expansion—in particular lower and upper estimates for its Hausdorff dimension are given. A large natural class of systems whose limit sets are 'dimensionless in the restricted sense' is described.

Lipscomb’s universal space is the attractor of an infinite iterated function system

Lipscomb’s one-dimensional space L ( A ) L(A) on an arbitrary index set A A is injected into the Tychonoff cube I A I^A . The image of L ( A ) L(A) is shown to be the attractor of an iterated function system indexed by A A . This system is conjugate, under an injection, with a set of right-shift operators on Baire’s space N ( A ) N(A) regarded as a code space. This view of L ( A ) L(A) extends the fractal nature of L ( A ) L(A) initiated in a 1992 joint paper by the author and S. Lipscomb. In addition, we give a new proof that as a subspace of Hilbert’s space l 2 ( A ) l^2(A) , the space L ( A ) L(A) is complete and hence is closed in l 2 ( A ) l^2(A) .

Infinite Iterated Function Systems

AbstractWe examine iterated function systems consisting of a countably infinite number of contracting mappings (IIFS). We state results analogous to the well‐known case of finitely many mappings (IFS). Moreover, we show that IIFS can be approximated by appropriately chosen IFS both in terms of Hausdorff distance and of Hausdorff dimension. Comparing the descriptive power of IFS and IIFS as mechanisms defining closed and bounded sets, we show that IIFS are strictly more powerful than IFS. On the other hand, there are closed and bounded non‐empty sets not describable by IIFS.