Symmetries of decimation invariant sequences and digit sets

Authors:
A. Barbé;F. von Haeseler
Affiliations:
Katholieke Universiteit Leuven, Department of Electrical Engineering, Kasteelpark Arenberg, 10, B-3001 Leuven, Belgium;CeVis Universität Bremen, Universitätsallee 29, 29359 Bremen, Germany
Venue:
Theoretical Computer Science
Year:
2002

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Abstract

A bi-infinite sequence is called p-decimation invariant if all p-decimations of it reproduce the sequence albeit with a shift. In this paper we discuss symmetry properties of decimation invariant sequences. A symmetry is a composition of a translation and a reflection. We establish the existence of translation invariant, i.e., periodic, decimation invariant sequences. Moreover, we prove that there exist decimation invariant sequences which are left-periodic and right-periodic, i.e., they are partially translation invariant. We present several criteria for the existence of decimation invariant sequences with additional periodicity properties. Finally, we discuss the existence of decimation invariant sequences that are invariant under reflections. Moreover, in passing we demonstrate that properties of decimation invariant sequences are linked with properties of certain digit sets.