无定形复杂性
Amorphic complexity
摘要：
我们引入无定形复杂性作为一个新的拓扑不变量，它可以度量零熵状态下动力系统的复杂性。它的主要目的是检测在渐近行为的混乱的开始。例如，它给Denjoy的例子在圆和Sturmian子移位正值，而为零的所有等距和Morse-Smale系统。在讨论了基本性质和例子之后，我们证明了无定形复杂性和基本的渐近分离数可以用来区分几乎自守极小系统和等度连续系统。对于符号系统，无定形复杂度等于相关贝西科维奇空间的盒维数。在这种情况下，我们专注于定期Toeplitz流，并给出了详细的描述的关系的缩放行为的密度的p-骨架。最后，我们来看看奇怪的非混沌吸引子出现在所谓的捏斜积系统。连续时间系统，更一般的群作用和应用到切割和投影准晶体将在随后的工作。
英文摘要：
We introduce amorphic complexity as a new topological invariant that measures the complexity of dynamical systems in the regime of zero entropy. Its main purpose is to detect the very onset of disorder in the asymptotic behaviour. For instance, it gives positive value to Denjoy examples on the circle and Sturmian subshifts, while being zero for all isometries and Morse-Smale systems. After discussing basic properties and examples, we show that amorphic complexity and the underlying asymptotic separation numbers can be used to distinguish almost automorphic minimal systems from equicontinuous ones. For symbolic systems, amorphic complexity equals the box dimension of the associated Besicovitch space. In this context, we concentrate on regular Toeplitz flows and give a detailed description of the relation to the scaling behaviour of the densities of the p-skeletons. Finally, we take a look at strange non-chaotic attractors appearing in so-called pinched skew product systems. Continuous-time systems, more general group actions and the application to cut and project quasicrystals will be treated in subsequent work.
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