Linear Recurrent Double Sequences with Constant Border in M2(F2) are Classified According to Their Geometric Content

Mihai Prunescu

doi:10.3390/sym3030402

Abstract

The author used the automatic proof procedure introduced in [1] and verified that the 4096 homomorphic recurrent double sequences with constant borders defined over Klein’s Vierergruppe K and the 4096 linear recurrent double sequences with constant border defined over the matrix ring M2(F2) can be also produced by systems of substitutions with finitely many rules. This permits the definition of a sound notion of geometric content for most of these sequences, more exactly for those which are not primitive. We group the 4096 many linear recurrent double sequences with constant border I over the ring M2(F2) in 90 geometric types. The classification over Klein’s Vierergruppe Kis not explicitly displayed and consists of the same geometric types like for M2(F2), but contains more exceptions. There are a lot of cases of unsymmetric double sequences converging to symmetric geometric contents. We display also geometric types occurring both in a monochromatic and in a dichromatic version.

Highlights

Recurrent double sequences are double sequences a : N2 → A which are defined by initial conditions a(i, 0) = g(i), a(0, j) = h(j) and by a recurrence a(i, j) = f (a(i, j − 1), a(i − 1, j − 1), a(i−1, j))
Continuing on this direction, the author has found an automatic proof method able to prove that some concrete recurrent double sequences can be obtained by systems of substitutions [1], and applied it for some recurrent double sequences with periodic initial conditions [5]
After a big number of experiments the author conjectured that for all finite Abelian groups G, all homomorphisms f : G3 → G and all periodic initial conditions g, h : N → G, the resulting recurrent double sequence can be generated by an system of substitutions

Summary

Introduction

Recurrent double sequences produced by homomorphisms lying in the same class of conjugation (tend to) have the same geometric content This fact is at the first sight a surprise, and was not foreseen by Lemma 1.6. B A sporadic exception is the situation in which some recurrent double sequences with different geometric content are produced by conjugated homomorphisms. We will prove that homomorphisms lying in the same class of conjugation produce isomorphic recurrent double sequences over M2 (F2 ). The fact that the geometric types are unions of classes of conjugation is no more surprising Both the normal and the sporadic exceptions noticed in homomorphic double sequences over K completely disappear in M2 (F2 ). M2 (F2 ) to be better than the group K in order to study and classify the homomorphic recurrent double sequences over K

Linear Recurrent Double Sequences Over Rings

Convergence and Geometric Content

Primitive Double Sequences

Affine Recurrence Rules and Two Secret Types