A `replacement sequence' method for finding the largest real root of an integer monic polynomial

To every integer monic polynomial of degree m can be associated a ‘replacement rule’ that generates a word W* from another word W consisting of symbols belonging to a finite ‘alphabet’ of size 2m. This rule applied iteratively on almost any initial word W0, yields a sequence of words {Wi}. From a count of different symbols in the word Wi, one can obtain a rational approximate to the largest real root of the polynomial. Let p(x) = x a1 x m-1 ..... am , where ai are integers, be the given monic polynomial. Let A = {A1, A + 2, ..., A + m} be a finite set of m symbols. Let A ~ = {A1, A ~ 2, ..., Am} be another set of m symbols associated with the set A so that with each element Ai of A + is associated the element Ai of A . Let A = A ∪ A. Elements of A are called 'letters' belonging to the 'alphabet' A. Let A = A x A x ..... x A (p times). Elements of A are called 'words' of length p that can be formed from 'letters' of A. Let A = ∪p A. Clearly, A consists of all 'words' that can be made from the 'alphabet' A. A 'letter' Ai repeated k times in a word will be denoted by (Ai) . Similarly a 'letter' Ai repeated k times in a word will be denoted by (Ai) . We define (αi) by (αi) = (Ai) if k ≥ 0 (αi) = (Ai) if k < 0 (1) (αi) = (Ai) if k ≥ 0 (αi) = (Ai) if k < 0 Consider the replacement rule R: A → A* given by A1 → (α1) A1 A2 A2 → (α1) A2 A3 . Ai → (α1) Ai Ai+1 . . Am → (α1) Am (2a) A1 → (α1) A1 A2 A2 → (α1) A2 A3 . Ai → (α1) Ai A i+1 . . Am → (α1) Am Let W = As1As2.....Asq, where sk (k=1,2,........,q) ∈ {1,2,........,m}. Then W ∈ A*. The replacement rule R induces a mapping R*:A* → A* defined by