Some Problems About Geometric Representation Of Ubstitutions

Let T be a set of n letters, T~Zbe the set of doubly infinite word in T, C (？) T~Z. whensome type’s substitution in C meets certain conditions (such as primitive substitution, Pisottype substitution, unimodular substitution), the substitution and its dynamical system havea lot of properties and theorems. But if we only observe and study the word, it may be alittle abstract. So if we present the substitution’s dynamical system in geometry, it may bemore intuitive, have more sense of three-dimensional space and get more conclusions. thispaper studies the representation of Pisot type substitution in Geometry. By studying theproperties and theorems about the Pisot type and the dynamical system of this substitution,the map from a set C to the geometrical space, we get the following conclusions: a Pisot typesubstitution dynamical system have a factor which is isomorphic to a torus’s some minimalrotation. if the Pisot substitution is unimodular satisfies some combination condition, weget the above dynamical system is measurably conjugate to an exchange of domains in anEuclidean space’s self-similar compact subset.The introduction mainly introduced the paper’s study background, current situation,existing problems and structure arrangement, then introduced some basic knowledge, somespecial symbols, substitutions, and some properties and theorems that may be used in thefollowing content, and then introduced the basic vectors generated by the incidence matrix,the map from the word to the geometric space, the semi-conjugate between the Pisot type’sdynamical system and a minimal shift on a torus. in the fourth chapter, from the shallowerto the deeper, by satisfying some conditions, we introduced the measurably one to onemap from a word set to the geometric space, the measure-theoretically conjugate betweenthe dynamical system of Pisot type and an exchange of domains on a self similar compactsubset of Euclid’s space. Because we frequently use the prefix of a word’s prefix-suffixdevelopment, in the end of the paper, we will give a simple introduction about the relationbetween the prefix and suffix.