An alternative proof for the minimal dimension of the trace map: The intersection of algebraic varieties

Abstract

Recently, Avishai, Berend, and Glaubman [Phys. Rev. Lett. 72 (1994) 1842] obtained the minimal dimension 3 r − 3 of the trace maps in the theory of quasiperiodic and aperiodic lattices constructed by deterministic substitution schemes of r distinct letters. We first present an alternative proof of the minimal dimension, which is given by 1 2 r(r + 1) for 1 ≤ r ≤ 3 and 3 r − 3 for 3 < r (Theorem 1), using the sequence of Grammians in the three-dimensional vector space, since transfer matrices can be regarded as three-dimensional vectors. Second, we generalize the proof to systems with r vectors in an n-dimensional linear vector space. The minimal dimension is given as 1 2 r(r + 1) for 1 ≤ r ≤ n and nr − 1 2 n(n − 1) for n < r (Theorem 2). Finally, we conclude that the minimal dimension is that of the intersection of algebraic varieties that are defined as a series of Grammians in the system.