Sturm 3球全局吸引子1：Sturm-Smale复形和曲折
Sturm 3-ball global attractors 1: Thom-Smale complexes and meanders
摘要：
这是第一个三个文件的几何和组合特征的全球斯特姆吸引力，其中包括一个单一的封闭3球。基本的标量偏微分方程是抛物型的，$$ u_t = u_{xx} + f(x,u,u_x)\,, $$在单位区间$0 < x<1$上，具有Neumann边界条件。平衡被假定为双曲线的。在几何上，我们研究了由此产生的Escherich-Smale动态复杂的细胞所定义的非稳定流形的平衡。该复合体是一种规则的胞腔复形。我们的几何描述涉及一个双极方向的1-骨架，半球分解的边界2-球体的两个极子午线，和子午线重叠的某些2-细胞面对相反的半球。用Sturm置换的方法对平衡常微分方程边值问题的射击曲线进行了组合描述，即射击曲线的曲折性质。它涉及到极端的二维不稳定平衡在诺依曼边界的相对定位$x=0$和$x=1$，分别，和重叠达到极蛇射击曲流。在本文中，我们显示的影响$$ \text{Sturm attractor}\quad \Longrightarrow \quad \text{Thom-Smale complex} \quad \Longrightarrow \quad \text{meander}\,.$$的续集，第2部分，关闭循环的等价的影响$$ \text{meander} \quad \Longrightarrow \quad \text{Sturm attractor}\,.$$许多明确的例子将在第3部分进行讨论。目前的3球三部曲扩展了我们以前的三部曲对平面Sturm全球吸引子仍然难以捉摸的目标的几何和组合特征的任意维度的所有Sturm全球吸引子。
英文摘要：
This is the first of three papers on the geometric and combinatorial characterization of global Sturm attractors which consist of a single closed 3-ball. The underlying scalar PDE is parabolic, $$ u_t = u_{xx} + f(x,u,u_x)\,, $$ on the unit interval $0 < x<1$ with Neumann boundary conditions. Equilibria are assumed to be hyperbolic. Geometrically, we study the resulting Thom-Smale dynamic complex with cells defined by the unstable manifolds of the equilibria. The Thom-Smale complex turns out to be a regular cell complex. Our geometric description involves a bipolar orientation of the 1-skeleton, a hemisphere decomposition of the boundary 2-sphere by two polar meridians, and a meridian overlap of certain 2-cell faces in opposite hemispheres. The combinatorial description is in terms of the Sturm permutation, alias the meander properties of the shooting curve for the equilibrium ODE boundary value problem. It involves the relative positioning of extreme 2-dimensionally unstable equilibria at the Neumann boundaries $x=0$ and $x=1$, respectively, and the overlapping reach of polar serpents in the shooting meander. In the present paper we show the implications $$ \text{Sturm attractor}\quad \Longrightarrow \quad \text{Thom-Smale complex} \quad \Longrightarrow \quad \text{meander}\,.$$ The sequel, part 2, closes the cycle of equivalences by the implication $$ \text{meander} \quad \Longrightarrow \quad \text{Sturm attractor}\,.$$ Many explicit examples will be discussed in part 3. The present 3-ball trilogy extends our previous trilogy on planar Sturm global attractors towards the still elusive goal of geometric and combinational characterizations of all Sturm global attractors of arbitrary dimension.
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