论文标题
缓慢熵转换的通用性和刚性
Genericity and Rigidity for Slow Entropy Transformations
论文作者
论文摘要
与经典的kolmogorov-sinai熵相比，katok和thouvenot定义了慢速熵和下部慢熵的概念，这是动态系统的复杂性的更精致的量度。对于任何亚指数速率函数$ a_n（t）$，我们证明存在一类通用的可逆度量，以保留系统，使下慢熵零熵为零，而上慢熵是无限的。另外，考虑到任何次指数速率$ a_n（t）$，我们显示存在一个刚性，弱的混合，可逆系统，因此相对于$ a_n（t）$，较低的慢熵是无限的。这给出了关于具有正多项式上慢熵的刚性变换的问题的一般解决方案，最后，我们将慢熵与Blume提出的熵覆盖率的概念联系起来。特别是，我们表现出的慢熵是一个严格的复杂性概念，并给出了零慢熵零的示例，但也具有任意的sublenear阳性熵收敛速度。
The notion of slow entropy, both upper and lower slow entropy, was defined by Katok and Thouvenot as a more refined measure of complexity for dynamical systems, than the classical Kolmogorov-Sinai entropy. For any subexponential rate function $a_n(t)$, we prove there exists a generic class of invertible measure preserving systems such that the lower slow entropy is zero and the upper slow entropy is infinite. Also, given any subexponential rate $a_n(t)$, we show there exists a rigid, weak mixing, invertible system such that the lower slow entropy is infinite with respect to $a_n(t)$. This gives a general solution to a question on the existence of rigid transformations with positive polynomial upper slow entropy, Finally, we connect slow entropy with the notion of entropy covergence rate presented by Blume. In particular, we show slow entropy is a strictly stronger notion of complexity and give examples which have zero upper slow entropy, but also have an arbitrary sublinear positive entropy convergence rate.
