In this paper we prove that Sands' topological condition for Collet-Eckmann maps implies Tsujii's metrical condition; on the other hand, if a Collet-Eckmann map satisfies Tsujii's metrical condition, then ...In this paper we prove that Sands' topological condition for Collet-Eckmann maps implies Tsujii's metrical condition; on the other hand, if a Collet-Eckmann map satisfies Tsujii's metrical condition, then it satisfies Sands' topological condition. Thus we obtain three different versions of Benedicks-Carleson Theorem by using topological conditions.展开更多
For the so-called quadratic family, it is proved that, there exists a positive Lebesgue measure set E in the parameter space such that the corresponding map has a dense critical orbit in the support set of the invaria...For the so-called quadratic family, it is proved that, there exists a positive Lebesgue measure set E in the parameter space such that the corresponding map has a dense critical orbit in the support set of the invariant measure for almost all parameters in E; it is also proved that there is a dense set in E such that the critical orbit of the corresponding map enter the reversed fixed point.展开更多
For unimodal maps on the interval we prove that, if the kneading sequences (KS) are eventually periodic, then their formal languages are regular ones. The finite automata for such languages are constructed. Comparing ...For unimodal maps on the interval we prove that, if the kneading sequences (KS) are eventually periodic, then their formal languages are regular ones. The finite automata for such languages are constructed. Comparing with the languages generated by periodic KS, it is shown that the languages here are not finite complement languages.展开更多
It is known that certain one parameter families of unimodal maps of the interval have a topological universality with regard to their dynamic behavior [ 1, 2]. As a parameter is smoothly increased, a fascinating varie...It is known that certain one parameter families of unimodal maps of the interval have a topological universality with regard to their dynamic behavior [ 1, 2]. As a parameter is smoothly increased, a fascinating variety of dynamic behaviors are produced. For some families the behaviors are monotonic in the parameter, while in others they are not [3]. The question is what sort of conditions on a one parameter family will ensure this monotonicity of the behavior with the parameter？ The answer is unknown and will not be given here. What we do instead is to investigate certain geometric-dynamic-combinatorial consequences of assuming that the family has this monotonicity. Specifically, using tools of symbolic dynamics, state space is ＂course grained＂ with a finite alphabet. We decompose a non-invertible map into nonlinear but invertible pieces. From these invertible pieces, we form inverse maps via composition along words. Equations of motion are developed for both forward and inverse orbits （in both the variables of state space and the parameter）, and an equation relating forward and inverse motions at fix-points is exhibited. Finally, we deduce a list of conditions, each of which is equivalent to monotone behavior. One of these conditions states that simple parity characteristics of words correspond to definite dynamics near fixed-points and vice versa.展开更多
Combining the theory of symbolic dynamics with the formal language theory, we determine the minimal deterministic finite automata (DFA) Aaccepting the formal languages generated by eventually periodic kneading sequenc...Combining the theory of symbolic dynamics with the formal language theory, we determine the minimal deterministic finite automata (DFA) Aaccepting the formal languages generated by eventually periodic kneading sequences of unimodal maps on an interval.展开更多
基金the National Natural Science Foundation of China (No.19901035) andTWAS/CNPq associate fellowship.
文摘In this paper we prove that Sands' topological condition for Collet-Eckmann maps implies Tsujii's metrical condition; on the other hand, if a Collet-Eckmann map satisfies Tsujii's metrical condition, then it satisfies Sands' topological condition. Thus we obtain three different versions of Benedicks-Carleson Theorem by using topological conditions.
文摘For the so-called quadratic family, it is proved that, there exists a positive Lebesgue measure set E in the parameter space such that the corresponding map has a dense critical orbit in the support set of the invariant measure for almost all parameters in E; it is also proved that there is a dense set in E such that the critical orbit of the corresponding map enter the reversed fixed point.
基金National Basic Research Project"Nonlinear Science"
文摘For unimodal maps on the interval we prove that, if the kneading sequences (KS) are eventually periodic, then their formal languages are regular ones. The finite automata for such languages are constructed. Comparing with the languages generated by periodic KS, it is shown that the languages here are not finite complement languages.
文摘It is known that certain one parameter families of unimodal maps of the interval have a topological universality with regard to their dynamic behavior [ 1, 2]. As a parameter is smoothly increased, a fascinating variety of dynamic behaviors are produced. For some families the behaviors are monotonic in the parameter, while in others they are not [3]. The question is what sort of conditions on a one parameter family will ensure this monotonicity of the behavior with the parameter？ The answer is unknown and will not be given here. What we do instead is to investigate certain geometric-dynamic-combinatorial consequences of assuming that the family has this monotonicity. Specifically, using tools of symbolic dynamics, state space is ＂course grained＂ with a finite alphabet. We decompose a non-invertible map into nonlinear but invertible pieces. From these invertible pieces, we form inverse maps via composition along words. Equations of motion are developed for both forward and inverse orbits （in both the variables of state space and the parameter）, and an equation relating forward and inverse motions at fix-points is exhibited. Finally, we deduce a list of conditions, each of which is equivalent to monotone behavior. One of these conditions states that simple parity characteristics of words correspond to definite dynamics near fixed-points and vice versa.
文摘Combining the theory of symbolic dynamics with the formal language theory, we determine the minimal deterministic finite automata (DFA) Aaccepting the formal languages generated by eventually periodic kneading sequences of unimodal maps on an interval.
