A Note on Palindromicity

Abstract

Two results on palindromicity of bi-infinite words in a finite alphabet are presented. The first is a simple, but efficient criterion to exclude palindromicity of minimal sequences and applies, in particular, to the Rudin-Shapiro sequence. The second provides a constructive method to build palindromic minimal sequences based upon regular, generic model sets with centro-symmetric window. These give rise to diagonal tight-binding models in one dimension with purely singular continuous spectrum.

Total palindrome complexity of finite words

“…A nonexhaustive list of these papers contains [3,9,4,1,8,6] and [7]. In [2] some properties related to the palindrome complexity of finite words are considered.…”

Spectral Classification of One‐Dimensional Binary Aperiodic Crystals: An Algebraic Approach

“…[59][60][61][62][63] We recall that a word is a palindrome if it reads the same forwards and backwards, and that a symbolic sequence is called palindromic if it contains palindromes of arbitrary length. [64] A particularly relevant example of BAC exhibiting palindromic symmetry is based on the Thue-Morse sequence and it displays both Fourier lattice and energy singular continuous spectra (see Fig.1).…”

Strictly ergodic subshifts and associated operators

“…7 We first describe general results that can be obtained using the two general methods we discussed and then turn to some specific examples, where more can be said.…”

“…We mentioned earlier that the proof of Theorem 6.8 cannot give a stronger result since the set of strongly palindromic sequences is always of zero measure 7 There are also papers dealing with Schrödinger operators associated with non-primitive substitutions [45,61,62,110]. The subshifts considered in these papers are, however, linearly recurrent and hence strictly ergodic, so that the theory is quite similar.…”