弱混沌动力系统的全局和局部复杂性
Global and local Complexity in weakly chaotic dynamical systems
摘要：
在拓扑动力系统中，轨道的复杂性是描述轨道所需的信息量（算法信息量）的度量。这个指标是不变的拓扑共轭。我们认为这个指标的局部复杂性的动态，并提供不同的例子，其行为，显示它如何可以是有用的，以表征各种弱混沌动力学。我们还提供了标准，以找到系统的非平凡轨道复杂性（系统的整个轨道的描述需要一个无限的信息量）。我们还考虑了系统复杂性的全球指标。这个全局指标概括了拓扑熵，考虑到系统的本质上不同的轨道的数量增加小于指数。然后，我们证明，如果系统是建设性的（粗略地说：如果地图可以定义到任何给定的精度使用有限的信息量）的轨道复杂性是处处小于或等于广义拓扑熵。相反，也有一些紧凑的非构造性例子，其中不等式被颠倒，这表明这个概念在这种复杂性问题中很自然地出现。
英文摘要：
In a topological dynamical system the complexity of an orbit is a measure of the amount of information (algorithmic information content) that is necessary to describe the orbit. This indicator is invariant up to topological conjugation. We consider this indicator of local complexity of the dynamics and provide different examples of its behavior, showing how it can be useful to characterize various kind of weakly chaotic dynamics. We also provide criteria to find systems with non trivial orbit complexity (systems where the description of the whole orbit requires an infinite amount of information). We consider also a global indicator of the complexity of the system. This global indicator generalizes the topological entropy, taking into account systems were the number of essentially different orbits increases less than exponentially. Then we prove that if the system is constructive (roughly speaking: if the map can be defined up to any given accuracy using a finite amount of information) the orbit complexity is everywhere less or equal than the generalized topological entropy. Conversely there are compact non constructive examples where the inequality is reversed, suggesting that this notion comes out naturally in this kind of complexity questions.
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