符号动力系统的正扩张性与网络维数
Positive expansiveness versus network dimension in symbolic dynamical systems
摘要：
一个“符号动力系统”是A^V的闭完备子集X的连续变换F：X-->X，其中A是有限集，V是可数集。（例子包括子移位、里程计、元胞自动机和自动机网络。函数F在V上导出一个有向图结构，其几何结构揭示了关于动力系统（X，F）的信息。“维数”dim（V）是描述有向图中球的增长率作为其半径的函数的指数。我们展示：若X有正熵且dim（V）>1，且系统（A^V，X，F）满足最小对称和混合条件，则（X，F）不能正扩张，这推广了Shereshevsky关于多维元胞自动机的一个著名结果.我们还构造了一个反例，这个结果的版本没有对称性条件。最后，我们证明了网络维数在拓扑共轭下是不变的，拓扑共轭是霍尔德连续的。
英文摘要：
A `symbolic dynamical system' is a continuous transformation F:X-->X of a closed perfect subset X of A^V, where A is a finite set and V is countable. (Examples include subshifts, odometers, cellular automata, and automaton networks.) The function F induces a directed graph structure on V, whose geometry reveals information about the dynamical system (X,F). The `dimension' dim(V) is an exponent describing the growth rate of balls in the digraph as a function of their radius. We show: if X has positive entropy and dim(V)>1, and the system (A^V,X,F) satisfies minimal symmetry and mixing conditions, then (X,F) cannot be positively expansive; this generalizes a well-known result of Shereshevsky about multidimensional cellular automata. We also construct a counterexample to a version of this result without the symmetry condition. Finally, we show that network dimension is invariant under topological conjugacies which are Holder-continuous.
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