Abstract
Many results are already known, concerning the palindromic factors and the palindomic prefixes of Standard billiard words, i.e., Sturmian words and billiard words in any dimension, starting at the origin. We give new geometrical proofs of these results, especially for the existence in any dimension of Standard billiard words with arbitrary long palindromic prefixes.
Highlights
1.1 Standard Billiard Words in dimension k = 2Let ρ be any positive number
A1a2a1a2a1a2a2a1a2 . . . Looking at the squares crossed by D, i.e., the grey squares in Fig.1.a, the Standard billiard word cα1,α2 encodes the sequence of horizontal and vertical segments joining the two centers of the consecutive grey unit squares
Standard billiard words and Sturmian words have been intensively studied in any dimension
Summary
The half-line D starting at the origin O with slope ρ divides G into two parts It will be more convenient for a multidimensional generalization, to consider a positive vector α = (α1, α2) parallel to D, we have ρ=. Looking at the squares crossed by D, i.e., the grey squares in Fig.1.a, the Standard billiard word cα1,α2 encodes the sequence of horizontal and vertical segments joining the two centers (i.e., white points in Fig. 1.a) of the consecutive grey unit squares. This word is the Standard billiard word associated to D This Standard billiard word is denoted by cα1,α2,...,αk. In many cases in the following, we use the stronger condition (2), and we say that (α1, α2, . . . , αk) is totally irrational, see [2]
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