On the computability of rotation sets and their entropies
On the computability of rotation sets and their entropies
复制标题
关于旋转集及其熵的可计算性
DOI：
10.1017/etds.2018.45
复制
发表时间：
2020
影响因子：
0.9
通讯作者：
WOLF, CHRISTIAN
中科院分区：
文献类型：
--
作者：
BURR, MICHAEL A.;SCHMOLL, MARTIN;WOLF, CHRISTIAN
设f：X→ X是紧致度量空间X上的连续动力系统，X→ Rm是m维连续势.（广义）旋转集Rot（）被定义为的所有μ-积分的集合，其中μ在所有不变概率测度上运行。类似于经典的拓扑熵，我们可以将局部熵H（w）与每个w∈ Rot（）联系起来。在本文中，我们研究了旋转集和局部熵函数的可计算性，推导出的条件，暗示他们的可计算性。然后我们应用我们的结果来研究f是有限类型子移位的情况。我们证明了Rot（）是可计算的，H（w）在旋转集的内部是可计算的。最后，我们构造了一个明确的例子，表明，在一般情况下，H是不连续的边界上的旋转集时，被认为是一个函数和w。特别地，H通常在Rot（）的边界处不可计算。
Let f: X→ X be a continuous dynamical system on a compact metric space X and let: X→ Rm be an m-dimensional continuous potential. The (generalized) rotation set Rot () is defined as the set of all µ-integrals of, where µ runs over all invariant probability measures. Analogous to the classical topological entropy, one can associate the localized entropy H (w) to each w∈ Rot (). In this paper, we study the computability of rotation sets and localized entropy functions by deriving conditions that imply their computability. Then we apply our results to study the case where f is a subshift of finite type. We prove that Rot () is computable and that H (w) is computable in the interior of the rotation set. Finally, we construct an explicit example that shows that, in general, H is not continuous on the boundary of the rotation set when considered as a function of and w. In particular, H is, in general, not computable at the boundary of Rot ().