Kneading Sequences for Unimodal Expanding Maps of the Interval

Let P and AC be two primary sequences with min{P, AC} RLR∞,ρ（P） and p（AC） be the eigenvalues of P and AC, respectively. Let f ∈ C0（I, I） be a unimodal expanding map with expanding constant λ and m be a nonegative integer. It is proved that f has the kneading sequence K（f） （RC）*m * P if λ （ρ（P））1/2m, and K（f）】（RC）*m * AC * E for any shift maximal sequence E if λ】（ρ（AC））1/2m. The value of （ρ（P））1/2m or （ρ（AC））1/2m is the best possible in the sense that the related conclusion may not be true if it is replaced by any smaller one.

Some Dynamic and Combinatorial Properties of One Parameter Families of Unimodal Maps with Monotonicity

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.

Case of stenosing tenosynovitis of the thumb in children

A female patient,13 months old,suffering from limited extension of the right thumb for 7 days,diagnosed as stenosing tenosynovitis of the thumb in children.Tuina(Chinese massage) therapy was adopted,in which,stimulation was applied with fingers instead of needles,including pressing and kneading,twisting and kneading,plucking,dorsal extending,rotating as well as pushing and rubbing manipulations.The whole procedure of manipulation lasted around 15 min,once every two days,3 times weekly.Meanwhile,the parents were advised to bath the child patient's right hand into warm water every day and fix the affected thumb with splint at night.After 12 treatments with tuina therapy,the flexion and extension function of the affected thumb were returned to normal and the thumb motor function was recovered.In 2 months of follow-up,the thumb motor function was normal and the stenosing tenosynovitis was not recurred.In this case,tuina therapy is adopted in treatment of stenosing tenosynovitis of the thumb in children.As an alternative method of surgery,tuina therapy has a satisfactory clinical effect.

Combinatorics on Words in Symbolic Dynamics:The Quadratic Map

This paper is contributed to the combinatorial properties of the MSS sequences, which are the periodic kneading words of quadratic maps defined on a interval. An explicit expression of adjacency relations on MSS sequences of given lengths is established.

Combinatorics on Words in Symbolic Dynamics:the Antisymmetric Cubic Map

This paper is contributed to the combinatorial properties of the periodic kneading words of antisymmetric cubic maps defined on a interval. The least words of given lengths, the adjacency relations on the words of given lengths and the parity-alternative property in some sets of such words are established.

TOPOLOGICAL AND METRICAL CONDITIONS FOR COLLET-ECKMANN UNIMODAL MAPS

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.