Geometric Palindromic Closure

http://www.boku.ac.at/MATH/udt/vol07/no2/06DomVuillon13-12.pdfInternational audienceWe define, through a set of symmetries, an incremental construction of geometric objects in Z^d. This construction is directed by a word over the alphabet {1,...,d}. These objects are composed of d disjoint components linked by the origin and enjoy the nice property that each component has a central symmetry as well as the global object. This construction may be seen as a geometric palindromic closure. Among other objects, we get a 3 dimensional version of the Rauzy fractal. For the dimension 2, we show that our construction codes the standard discrete lines and is equivalent to the well known palindromic closure in combinatorics on words

Periodic graphs and connectivity of the rational digital hyperplanes

AbstractGiven a digital hyperplane of Zn defined by a double-inequality h⩽∑i=1naixi<h+δ, we want to determine whether it is connected. The problem consists of computing the connectivity of a graph whose set of vertices is not finite. The classical algorithms of labelling are not deterministic in this framework but we can think of using the properties of the digital hyperplanes and in particular their periodicity to provide a deterministic method. It leads to introduce a special kind of graphs that we call periodic and whose properties allow to compute connective components of infinite size. It provides a deterministic algorithm determining whether a given rational digital hyperplane is connected