Correlation Measures And Dimensions Of Pattern Sequences

The infinite sequence defined on the finite alphabet{1,-1}is an ancient and interesting object in mathematics.In this dissertation,we focus on a kind of pattern sequences which is called”arithmetic fractal”.Their correlation measure and correlation dimension are mainly discussed.This dissertation is divided into six chapters and organized as follows.In the first chapter,we introduce the background of fractal geometry and dynamical system,and the physics background of these sequences.In the second chapter,we mainly present the definition and basic properties of cor-relation measure and correlation dimension,as well as?₂?{1,-1}?sequences and pattern sequences.Finally,three important lemmas are given.In the third chapter,we give the results of all two-degree pattern sequences.A two-degree pattern sequence{a_n}_n?0is divided into subsegments of length two and satisfies a_n(a_2na_2n+1)=a_m(a_2ma_2m+1),n?m?mod 2?.And we know that by the above formula the sequence is determined by its first four terms a₀,a₁,a₂,a₃.The correlation function is used to study the correlation measure.So by computing the correlation function we find that there are three cases in the result.If a₀a₁a₂a₃=-1,the correlation measure is Lebesgue measure?we call the sequence is noncorrelated?and D₂=1.If a₀a₁a₂a₃=1 and a₂=1,the sequence is periodic and the correlation measure is discrete,D₂=0.If a₀a₁a₂a_?3=1 and a₂=-1,the correlation measure is singular continuous,D₂=3-log₂?1+17?.In the fourth chapter,we characterize that a three-degree pattern sequence is noncorre-lated.It is easy to know that such sequence can be determined by the first eight items and the following relations: a_na_2n+i=a_ma_2m+i?i=0,1?,n?m?mod 2²?.The necessary and sufficient conditions indicate that the sequence is noncorrelated when the first eight items of the sequence satisfy one of the three situations given,as well as D₂=1.In the fifth chapter,we get the conclusion that a special pattern sequence with a special pattern set of degree k and bound l is noncorrelated if and only if l=2.In addition,We study k+1-degree pattern sequences and obtain a sufficient condition for noncorrelated.In the final chapter,we summarize the main results of this dissertation and put forward some questions which can be further studied.