The Renormalization Operator And Kneading Sequence

In 1978, Feigenbaum first discovered an astonishing universal metric property of period-doubling bifurcations in transition to chaotic behavior( i.e. the so-called Feigenbaum phenomenon). Furthermore, he proposed to use the method of renormalization group to explain this phenomenon. So far, the whole theory is built on a certain number of geometric assumptions which Feigenbaum gave for some kinds of function spaces, a key one of which is the supposition that the renormalization operatorT2g = g2(-x) has fixed points, i.e. ,the corresponding functional equationhas solutions. In the following over twenty years, the research on Feigenbaum phenomenon has attracted many people's attention. Up to now, the scientific workers in those fields including physics , chemistry, biology and every branch of mathematics have obtained fairly rich fruits in this respect. This causes that we know not only that renormalization operator has fixed points and has what kind of fixed points, but also be clear about how to construct its continuous,differentiable, even smooth fixed points, and know about some dynamicalbehaviors of fixed points, and so on.Among of them,in 1985,Yang Lu and Zhang Jingzhong changed the original renormalization operator into thisi.e. put forward the second type of Feigenbaum functional equationf(x) = f2(-A:c), (A (0,1) is to determined),It is more convenient to study than the original one, it has the same effect as the original one and the solutions between two equations have very direct contact. They also constructed single-valley continuous and differentiable solutions to this equation.In 1988, Liao Gongfu posed the concept of the continuous single-valley expansion solution to the second type of Feigenbaum functional equation,consequently ,he provided its constructing method and some dynamical properties.But early in 1984 ,Eckmann, Epstein and Wittwer consideredthe renormalization operator Tpf = -fp(Xx) under a broader sense, i.e.,thecorresponding Feigenbaum's equationFor p large enough, they had shown that the equation has a solution similar to the quadratic function /(x) = 1 - 2x2. Further, in 1994,Liao Gongfu posed the following equationand pointed out that the equation (2) has the same effect as the equation (1) and the solutions between two equations have very direct contact. He also constructed single-valley continuous and differentiable solutions to the equation(2). In this dissertation we attest the existence of a continuous nonsingle-valley solution and a continuous single-valley expansion solution to the equation(2), and give a feasible method to construct a continuous single-valley expansion solution further.The unimodal map and kneading sequence are two important concepts in dynamical system.For the former,it is a mathematical model about the number of insects difference .equation,also Feigenbaum discoverd the astonishing universal phenomenon based in the research to it.For the latter,it was introduced by Milnor-Thurston ,and it is a important implement in the research to one: dimension dynamical system.By the research to its internal properties we can deeply learned the dynamical properties of one-dimension map(including topology properties and ergodic properties ),so it is very significative to research the kneading sequence of a unimodal map.In order to explain Feigenbaum phenomenon better and explore the universal principle, we consider the relations of the kneading sequence between /and Tpf. According to the thoughts in fractal theory,we research the sets of admissible kneading sequences of a unimodal map, investigate its Hausdorff dimension and measure.In this dissertation,first, we review the history and some recent results of the research on renormalization operator, give some properties of its fixed points and prove the existence of a continuous single-valley expansion solution with conformation. Second, we mainly investigate the sets of admissible kneading sequences of a unimodal map . Last, we consider the relations of the kneading sequence between /and Tpf,furthermore,the kneading seq...