We prove that the family of all invariant sets of iterated systems of contractions RN → RN is a nowhere dense Fσ type subset in the space of the nonempty compact subsets of RN equipped with the Hausdorff metric.
Authors:Pawel Hanus, R. Mauldin, and Mariusz Urbański
We develop the thermodynamic formalism for equilibrium states of strongly Hlder 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 Hlder 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 . 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.
We study the set of the representable numbers in base with ρ>1 and n∊ℕ and with digits in an arbitrary finite real alphabet A. We give a geometrical description of the convex hull of the representable numbers in base q and alphabet A and an explicit characterization of its extremal points. A characterizing condition for the convexity of the set of representable numbers is also shown.
Authors:Johan F. Aarnes, Ørjan Johansen, and Alf B. Rustad
This paper introduces a novel idea: the concept of an image transformation. We also introduce the closely related concept of a quasi-homomorphism, and study the properties of these mathematical objects, and give several examples. In particular we investigate iterated systems of image transformations, which we believe give a more realistic approach to the study of so called self-similar structures in nature than what is obtained by iterated function systems.