114 research outputs found
Combinatorial substitutions and sofic tilings
A combinatorial substitution is a map over tilings which allows to define
sets of tilings with a strong hierarchical structure. In this paper, we show
that such sets of tilings are sofic, that is, can be enforced by finitely many
local constraints. This extends some similar previous results (Mozes'90,
Goodman-Strauss'98) in a much shorter presentation.Comment: 17 pages, 16 figures. In proceedings of JAC 201
The geometry of non-unit Pisot substitutions
Let be a non-unit Pisot substitution and let be the
associated Pisot number. It is known that one can associate certain fractal
tiles, so-called \emph{Rauzy fractals}, with . In our setting, these
fractals are subsets of a certain open subring of the ad\`ele ring
. We present several approaches on how to
define Rauzy fractals and discuss the relations between them. In particular, we
consider Rauzy fractals as the natural geometric objects of certain numeration
systems, define them in terms of the one-dimensional realization of
and its dual (in the spirit of Arnoux and Ito), and view them as the dual of
multi-component model sets for particular cut and project schemes. We also
define stepped surfaces suited for non-unit Pisot substitutions. We provide
basic topological and geometric properties of Rauzy fractals associated with
non-unit Pisot substitutions, prove some tiling results for them, and provide
relations to subshifts defined in terms of the periodic points of , to
adic transformations, and a domain exchange. We illustrate our results by
examples on two and three letter substitutions.Comment: 29 page
Critical connectedness of thin arithmetical discrete planes
An arithmetical discrete plane is said to have critical connecting thickness
if its thickness is equal to the infimum of the set of values that preserve its
-connectedness. This infimum thickness can be computed thanks to the fully
subtractive algorithm. This multidimensional continued fraction algorithm
consists, in its linear form, in subtracting the smallest entry to the other
ones. We provide a characterization of the discrete planes with critical
thickness that have zero intercept and that are -connected. Our tools rely
on the notion of dual substitution which is a geometric version of the usual
notion of substitution acting on words. We associate with the fully subtractive
algorithm a set of substitutions whose incidence matrix is provided by the
matrices of the algorithm, and prove that their geometric counterparts generate
arithmetic discrete planes.Comment: 18 pages, v2 includes several corrections and is a long version of
the DGCI extended abstrac
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