Palindromic language of thin discrete planes

We work on the Réveillès hyperplane P(v,0,ω)P(v,0,ω) with normal vector  v∈Rdv∈Rd, shift  μ=0μ=0 and thickness  ω∈Rω∈R. Such a hyperplane is connected as soon as ω   is greater than some value Ω(v,0)Ω(v,0), called the connecting thickness of v with null shift. In the case where v satisfies the so called Kraaikamp and Meester criterion, at the connecting thickness the hyperplane has very specific properties. First of all the adjacency graph of the voxels forms a tree. This tree appeared in many works both in discrete geometry and in discrete dynamical systems. In addition, it is well known that for a finite coding of length n   of discrete lines, the number of palindromes in the language is exactly n+1n+1. We extend this notion of language to labeled trees and we compute the number of distinct palindromes. In fact for our voxel adjacency trees with n   letters we show that the number of palindromes in the language is also n+1n+1. This result establishes a first link between combinatorics on words, palindromic languages, voxel adjacency trees and connecting thickness of Réveillès hyperplanes. It also provides a better understanding of the combinatorial structure of discrete planes.