Balanced Word And Its Orbit Majorization

Sturmian sequences play important roles in many field,and from them,the concepts of finite balanced word and most unbalanced word can be derived.In this thesis,we consider the properties of the orbits of these words.The majorization of vectors defines a partial order between the binary orbits of finite words.Under this order,the balanced word and the most unbalanced word reach the extremal values respectively.We study the extremal properties of the r-moments of the words for r?1.When 0<r<1,we make numerical experiment to explore the different nature of r-moment.We also study the transfer problem between the orbits.Define a linear transformation T:RM?RM as x':=T(x),whereThis transformation is induced by the following real invertible symmetric matrix P withUnder this transformation,any word can be converted to a most unbalanced word,or it can be converted to a balanced word.That is,in the case where the slope is given,for any word ??{0,1}m,balance word ??{0,1}m and the most unbalanced word ??{0,1}m satisfying ?<?<?,there is always real invertible symmetric matrices P,Q,such that ?=P?=PQ?.