Symmetries of decimation invariant sequences and digit sets
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. (C) 2002 Elsevier Science B.V. All rights reserved.
symmetry digit sets periodicity decimation of sequences renormalization automaticity mahler equations group operations