Small Divisors Theory And Analytic Solutions Of Iterative Functional Equations

The study of dynamical systems is the study of how things change over time. Especially in astronomy, physics, biology, we can find many mathematical models about it. The means for studying change is to find a relationship between what is happening now and what will happen in the future. By analyzing this relationship, we can predict what will happen in the distant future. Usually, by regulation of changes in systems, there are two basic forms of dynamical systems:continuous dynamical systems and discrete dynamical systems. Iteration theory plays an important role in dynamical systems.Many problems of stability in the theory of dynamical systems face the dif-ficulty of small divisors. The most famous example is the KAM theory which introduced by Kolmogorov, Arnold, Moser in the past century. This theory fo-cus on the persistence of quasi-periodic solutions for quasi-integrable Hamiltonian systems. There are some results about one-dimensional analytic small divisor problem. What is worth mentioning, S. Marmi and J.C. Yoccoz [98,99,186], who are the mathematician from Italy and France, made contributions to the intro-duction and development in this field. Where the optimal arithmetical condition to impose is now known:in the local case it is still the same condition which introduced by Brjuno about 40 years ago (hereinafter Brjuno condition); whereas in the global case one has to impose a more restrictive condition by J.C. Yoccoz, and how to reduce these conditions is still a problem. In fact, one-dimensional small divisor problem can be seen as a conjugate of maps in the local field, i.e., f(h(z))= h(g(z)), where g(z)= qz, and can be reduced to linearization of f. Thus, the theory of continued fraction expansion of real number is an impor-tant method to study this problem and we will introduce some concepts about continued fraction.In this thesis, we study the local analytic solutions of several equations, and we discuss the existence of analytic solutions and the explicit structure of such solutions for four kinds of equations. We use the Schroder transformation and power series theory to discuss the analytic solutions. In method requires the eigenvalues of the solutions at their fixed point is off the unit or lies on the unit circle with the Brjuno condition. If an eigenvalue is on the unit circle, we can find 1/1-qn in the majorant series of the formal solutions, it is hard for us to determine the convergence of formal solutions. Then, we can overcome this difficulty by Brjuno condition, and also, we consider the existence of analytic solutions under the resonance condition. In conclusion, we will discuss the analytic solutions of several equations under the following conditions, where q is an eigenvalue of the solution at their fixed point:(C1) 0<|q|< 1;(C2) q= e2πiθ,θ∈R\Q, andθis a Brjuno number [28,99]:B(θ)=∑n=0∞logqn+1/qn<∞, where {pn/qn} denotes the sequence of partial fraction of the continued fraction expansion ofθ.(C3) q= eπiq'/p', for some integer p'∈N with p'≥2, q'∈Z\{0}, q≠e2πiξ/v for all 1≤v≤p'-1, andξ∈Z\{0}.The structure of the thesis as follows:In ChapterⅠ, concepts of small divisor, iterative functional equation, iterative differential equation and q-difference equation will be introduced. Many known results on these fields and the basic theory are also provided.In ChapterⅡ, we investigate the analytic solutions of a polynomial-like itera-tive equation with variable coefficients. There are many results about polynomial-like iterative equations with variable coefficients([34,84,134,156,193,197]), we improve and extend these results. Consider the local invertible analytic solutions of a more general polynomial-like iterative equation with variable coefficients on complex field: We can obtain the equation refer to above articles if we take vi(z):i= 1,...,n are 0, z, f(z). We use power series theory and small divisor theory to discuss the existence of analytic solutions when the eigenvalues of the solutions of equations at the different position. And we obtain the similar result by Abel transformation as Schroder transformation.In chapters III and IV, we consider the local analytic solutions of two kinds functional differential equations: which extend the previous result of Jianguo Si and author, improve the others' results. First equation is a general form which from number theory, it has close relation to Golomb sequence([54,58,100,103], [123]-[126]). We can use the same method to discuss this equation as author's previous work. For the second equa-tion, because "inner" function has second derivative, we transform the second equation into an iterative equation by change in variables, integral transformation and other techniques. Use the same method as we mentioned above, we can study the analytic solutions of this iterative equation. In the end, we illustrate how to find an analytic solution of original equation in this process, and also, we can show the explicit structure of this analytic solution.The research for analytic solutions of q-difference equations has a long his-tory. In chapter V, we recall some results about q-difference equations, in further, consider the analytic solutions of a q-difference equation near the origin: Since functions Ct,j(z), G(z) have poles, we should transform original equation into a q-difference equation without poles, and discuss q with different location. However, the position of q have relation withθandθrelates to continued fraction. It is enough for us to discussθwhen we consider about whether formal solution is convergent. We are not only study the convergence of formal solutions, by choose a particular {pn/qn} (continued fraction ofθ), we can construct a formal solution is divergence. It is improve the results of Weinian Zhang and Bing Xu. In fact, under the following suppose:(Hj) Origin z=0 is the pole of functions Ct,j(z), (t∈N, j= 0,1,…,k) and G(z) respective with h1 order and h2 order, i.e., are convergent for |z|<σ(σ> 0), and for each j= 0,1,…,k, the series∑t=1∞Ct,j(z1)z2 converges for a definite pair of nonzero complex Z1, z2 with |z|<σ, and P(z)=∑j=0kC1,j(0)zj.We study the local analytic solutions of auxiliary equation and get the results of original q-difference equation.