A simple, yet unifying method is provided for the construction of tilings by tiles obtained from the attractor of an iterated function system (IFS). Many examples appearing in the literature in ad hoc ways, as well as new examples, can be constructed by this method. These tilings can be used to extend a fractal transformation defined on the attractor of a contractive IFS to a fractal transforma...

We consider a class of iterated function systems (IFSs) of contracting similarities of Rn, introduced by Hutchinson, for which the invariant set possesses a natural Hölder continuous parameterization by the unit interval. When such an invariant set is homeomorphic to an interval, we give necessary conditions in terms of the similarities alone for it to possess a quasisymmetric (and as a corolla...

We study the decidability of some properties of self-affine sets specified by a graph-directed iterated function system (GIFS) with rational coefficients. We focus on topological properties and we prove that having empty interior is undecidable in dimension two. These results are obtained by studying a particular class of self-affine sets associated with multitape automata. We first establish t...

Using techniques introduced by C. Güntürk, we prove that the attractors of a family of overlapping self-affine iterated function systems contain a neighbourhood of zero for all parameters in a certain range. This corresponds to giving conditions under which a single sequence may serve as a ‘simultaneous β-expansion’ of different numbers in different bases.

We establish integral tests and laws of the iterated logarithm for the lower envelope of positive self-similar Markov processes at 0 and +∞. Our proofs are based on the Lamperti representation and time reversal arguments. These results extend laws of the iterated logarithm for Bessel processes due to Dvoretsky and Erdös [11], Motoo [17] and Rivero [18]. time reversal, integral test, law of the ...

In this paper I continue the study of iterated integrals of modular forms and noncommutative modular symbols for Γ ⊂ SL(2,Z) started in [Ma3]. Main new results involve a description of the iterated Shimura cohomology and the image of the iterated Shimura cocycle class inside it. The concluding section of the paper contains a concise review of the classical modular symbols for SL(2) and a discus...

We introduce a geometric condition of Bloch type which guarantees that a subset of a bounded convex domain in several complex variables is degenerate with respect to every iterated function system. Furthermore we discuss the relations of such a Bloch type condition with the analogous hyperbolic Lipschitz condition.

The main theorem of this paper establishes conditions under which the ‘chaos game’ algorithm almost surely yields the attractor of an iterated function system. The theorem holds in a very general setting, even for non-contractive iterated function systems, and under weaker conditions on the random orbit of the chaos game than obtained previously.

We introduce a Fourier based harmonic analysis for a class of discrete dynamical systems which arise from Iterated Function Systems. Our starting point is the following pair of special features of these systems. (1) We assume that a measurable space X comes with a finite-to-one endomorphism r : X → X which is onto but not one-to-one. (2) In the case of affine Iterated Function Systems (IFSs) in...

It is known that every hyperbolic Iterated Function System containing affine mappings can be represented in barycentric form by which it becomes affine invariant. Some properties were surveyed and some new ones were established. Examples of fractal sets supplement the theory.