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Substitution can be traced back Thue 's work in 1906 , he introduced a sequence generated by the substitution ( now known as the Thue-Morse sequence ) , followed by more researchers in this area had a strong interest from this sequence was different perspective on research and promotion, after the discovery of quasicrystals was D.Scheehtman , resulting in a periodic structure on related research , but in a non- periodic system of research or less . many researchers dedicated to research in finite alphabet substitution sequence generated automatically generated by non- cyclical nature of the chain due to trace mapping and substitution are closely linked, is such a powerful tool for the study of the last decades it was found that the importance of substitution sequence , linked with number theory , fractal geometry, harmonic analysis , portfolio analysis , and formal languages ??, and widely used in physics and computer science . article is the study of a number of identities in the matrix ring nature and in SL2 (C) on the free group generated maps in physics or geometry related to nature, providing a simple recursive relationship on track and structural relationships in the correlation algorithm , also called trace mapping describes the current power system in a limited substitution alphabet part of the research results , explore the finite alphabet substitution on the properties of dynamical systems , known σ is the two -letter substitution, then there exists a unique medium . Polynomial corresponding to the σ trace mapping , its existence has been proved J.-P.Allouche and J.Peyriere paper studies at Ω = {(X, Y, z) ∈ R3, x2 y2 z2-xyz- 4 = 0 } on Φσn progressive , and a detailed description Φσ iterative and dynamic forms, such as invariant set , fixed point , cycle tracks and disorder , etc. Research Φσ will mainly depend on the nature of the difference divided Ω three branches of the irreducible nature of substitution σ they are determined by the permutation matrix Mσ , and gives three specific substitution verify the surface Ω differential form part of the driving force .
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