由广义Mazur纽结定义的Pixton微分同胚的拟能量函数
A quasi-energy function for Pixton diffeomorphisms defined by generalized Mazur knots
AI总结：
本文为广义Mazur纽结定义的Pixton微分同胚构造了具有最少临界点数的拟能量函数，并给出了Lyapunov函数临界点数量的下界估计。
摘要：
本文给出了Pixton微分同胚（即三维中链复发集由四个点组成的Morse-Smale微分同胚：一个源、一个鞍点和两个汇）的Lyapunov函数临界点数量的下界估计。Ch. Bonatti和V. Grines证明了此类微分同胚$f$的拓扑等价类完全由Hopf纽结$L_{f}$的等价类决定，其中$L_{f}$是流形$\mathbb S^2\times\mathbb S^1$基本群的生成类中的纽结。他们还证明了存在无穷多个这样的类，并且任何Hopf纽结都可以由某个Pixton微分同胚实现。D. Pixton证明了由标准Hopf纽结$L_0=\{s\}\times \mathbb S^1$定义的微分同胚具有能量函数（Lyapunov函数），其临界点集与链复发集重合，而具有非平凡（即不等价于标准的）Hopf纽结的Pixton微分同胚的任何Lyapunov函数的临界点集严格大于该微分同胚的链复发集。具有最少临界点数的Pixton微分同胚的Lyapunov函数被称为拟能量函数。本文为广义Mazur纽结定义的Pixton微分同胚构造了一个拟能量函数。
英文摘要：
In this paper we give a lower estimate for the number of critical points of the Lyapunov function for Pixton diffeomorphisms (i.e. Morse-Smale diffeomorphisms in dimension 3 whose chain recurrent set consists of four points: one source, one saddle and two sinks). Ch. Bonatti and V. Grines proved that the class of topological equivalence of such diffeomorphism $f$ is completely defined by the equivalency class of the Hopf knot $L_{f}$ that is the knot in the generating class of the fundamental group of the manifold $\mathbb S^2\times\mathbb S^1$. They also proved that there are infinitely many such classes and that any Hopf knot can be realized by a Pixton diffeomorphism. D.~Pixton proved that diffeomorphisms defined by the standard Hopf knot $L_0=\{s\}\times \mathbb S^1$ have an energy function (Lyapunov function) whose set of critical points coincide with the chain recurrent set whereas the set of critical points of any Lyapunov function for Pixton diffeomorphism with nontrivial (i.e. non equivalent to the standard) Hopf knot is strictly larger than the chain recurrent set of the diffeomorphism. The Lyapunov function for Pixton diffeomorphism with minimal number of critical points is called the quasi-energy function. In this paper we construct a quasi-energy function for Pixton diffeomorphisms defined by a generalized Mazur knot.
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