1,924 research outputs found
p-adic path set fractals and arithmetic
This paper considers a class C(Z_p) of closed sets of the p-adic integers
obtained by graph-directed constructions analogous to those of Mauldin and
Williams over the real numbers. These sets are characterized as collections of
those p-adic integers whose p-adic expansions are describeed by paths in the
graph of a finite automaton issuing from a distinguished initial vertex. This
paper shows that this class of sets is closed under the arithmetic operations
of addition and multiplication by p-integral rational numbers. In addition the
Minkowski sum (under p-adic addition) of two set in the class is shown to also
belong to this class. These results represent purely p-adic phenomena in that
analogous closure properties do not hold over the real numbers. We also show
the existence of computable formulas for the Hausdorff dimensions of such sets.Comment: v1 24 pages; v2 added to title, 28 pages; v3, 30 pages, added
concluding section, v.4, incorporate changes requested by reviewe
On certain families of planar patterns and fractals
This survey article is dedicated to some families of fractals that were
introduced and studied during the last decade, more precisely, families of
Sierpi\'nski carpets: limit net sets, generalised Sierpi\'nski carpets and
labyrinth fractals. We give a unifying approach of these fractals and several
of their topological and geometrical properties, by using the framework of
planar patterns.Comment: survey article, 10 pages, 7 figure
Intersections of multiplicative translates of 3-adic Cantor sets
Motivated by a question of Erd\H{o}s, this paper considers questions
concerning the discrete dynamical system on the 3-adic integers given by
multiplication by 2. Let the 3-adic Cantor set consist of all 3-adic integers
whose expansions use only the digits 0 and 1. The exception set is the set of
3-adic integers whose forward orbits under this action intersects the 3-adic
Cantor set infinitely many times. It has been shown that this set has Hausdorff
dimension 0. Approaches to upper bounds on the Hausdorff dimensions of these
sets leads to study of intersections of multiplicative translates of Cantor
sets by powers of 2. More generally, this paper studies the structure of finite
intersections of general multiplicative translates of the 3-adic Cantor set by
integers 1 < M_1 < M_2 < ...< M_n. These sets are describable as sets of 3-adic
integers whose 3-adic expansions have one-sided symbolic dynamics given by a
finite automaton. As a consequence, the Hausdorff dimension of such a set is
always of the form log(\beta) for an algebraic integer \beta. This paper gives
a method to determine the automaton for given data (M_1, ..., M_n).
Experimental results indicate that the Hausdorff dimension of such sets depends
in a very complicated way on the integers M_1,...,M_n.Comment: v1, 31 pages, 6 figure
Path sets in one-sided symbolic dynamics
Path sets are spaces of one-sided infinite symbol sequences associated to
pointed graphs (G_v_0), which are edge-labeled directed graphs G with a
distinguished vertex v_0. Such sets arise naturally as address labels in
geometric fractal constructions and in other contexts. The resulting set of
symbol sequences need not be closed under the one-sided shift. this paper
establishes basic properties of the structure and symbolic dynamics of path
sets, and shows they are a strict generalization of one-sided sofic shifts.Comment: 16 pages, 6 figures; v2, 22pages, 6 figures; title change, adds a new
Theorem 1.5, and a second Appendix, v3, 21 pages, revisions to exposition; v4
revised introduction; v5, 22 pages, changed title, revised introductio
Self-Assembly of Infinite Structures
We review some recent results related to the self-assembly of infinite
structures in the Tile Assembly Model. These results include impossibility
results, as well as novel tile assembly systems in which shapes and patterns
that represent various notions of computation self-assemble. Several open
questions are also presented and motivated
Wavelets and graph -algebras
Here we give an overview on the connection between wavelet theory and
representation theory for graph -algebras, including the higher-rank
graph -algebras of A. Kumjian and D. Pask. Many authors have studied
different aspects of this connection over the last 20 years, and we begin this
paper with a survey of the known results. We then discuss several new ways to
generalize these results and obtain wavelets associated to representations of
higher-rank graphs. In \cite{FGKP}, we introduced the "cubical wavelets"
associated to a higher-rank graph. Here, we generalize this construction to
build wavelets of arbitrary shapes. We also present a different but related
construction of wavelets associated to a higher-rank graph, which we anticipate
will have applications to traffic analysis on networks. Finally, we generalize
the spectral graph wavelets of \cite{hammond} to higher-rank graphs, giving a
third family of wavelets associated to higher-rank graphs
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