The Research On F-Maps

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 operator 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 Feigenbaum functional equation has solutions and has what kind of solutions, but also be clear about how to construct its continuous, differentiable. even smooth solutions, and know about some dynamical behaviors of solutions, and so on.In order to explain Feigenbaum phenomenon better and explore the universal principle, in 1984, Eckmann, Epstein and Wittwer considered the Feigen-baum's equation under a broader senseFor p large enough, they had shown that the equation has a solution similar to the quadratic function f(x) =1 - 2x2.Further, Liao Gongfu posed the following equationand pointed out that this equation has the same effect as the equation (1) and the solutions between two equations have very direct contact. He also constructed even unimodal Cl solutions to the equation (1). In this dissertation we attest the existence of a nonsingle-valley continuous solution to this equation and give a feasible method to construct such a solution further.One of the interesting topics in dynamical system is the likely limit set. Milnor introduced the concept of likely limit set in 1985 :Let M be a smooth compact manifold, possibly with boundary, and let / be a continuous map from M into itself. The likely limit set A of / is denned as the smallest invariant closed subset of M with the property that u(x) C A for every point x 6 M except a set of measure zero.The likely limit set is the unique maximal attractor of a system in the sense of Milnor's attractor definition and it also accumulates the asymptotic behavior of almost all points, so it is very significant to investigate this kind of subsets.In 1997, Liao Gongfu ,Huang Guifeng and He Bohe studied the likely limit sets of a class of infinite polymodal Feigenbaum maps, estimated their Hausdorff dimensions and proved that for any s between 0 and 1 there always exist the continuous map on interval which has a likely limit set with Hausdorff dimension s. When p = 2, the continuously extending single-valley solutions of the Feigenbaum's maps only have the periodic points of period a power of 2, so thrir topological entropy must be zero and they aren't Li-Yorke's chaotic, withal since Li Tianyan and J.Yorke pointed out period three implies chaos, in addition, Louis Block presented that map / has a periodic point whose period is not a power of 2 if and only if / has a periodic orbit of period a power of 2 which is not simple. A natural question to ask : " For p > 3, what situation is the likely limit set of a p-order Feigenbaum's map ? " We shall answer this concretely in this paper mainly by considering p = 3 and p = 4. In the following we call a p-order Feigenbaum's map on [0,1] a F-map in short.First, we review the history and some recent results of the research on F-maps, give some properties of F-maps and prove the existence of nonsingle-valley F-maps with conformation. Second, we mainly investigate the likely limit sets with fractal structure of 3-order & 4-order F-maps. Last, we classify 4-order F-maps with topological conjugacy -I-type and II-type.The following are our main results :1 . Let f0 be a continuous function on [A, 1], where 0 < A < 1. If(1) there exists some...