典型动作的Sofic均维数及比较定理
Sofic mean dimension of typical actions and a comparison theorem
AI总结：
本文通过改进两个结果，确定了全移位和有限群作用的Sofic均维数精确值，并得出了Sofic均维数与经典均维数比较的定理。
摘要：
我们通过Hanfeng Li的论文
英文摘要：
We refine two results in the paper entitled "Sofic mean dimension" by Hanfeng Li, improving two inequalities with two equalities, respectively, for sofic mean dimension of typical actions. On the one hand, we study sofic mean dimension of full shifts, for which, Li provided an upper bound which however is not optimal. We prove a more delicate estimate from above, which is optimal for sofic mean dimension of full shifts over arbitrary alphabets (i.e. compact metrizable spaces). Our refinement, together with the techniques (in relation to an estimate from below) in the paper entitled "Mean dimension of full shifts" by Masaki Tsukamoto, eventually allows us to get the exact value of sofic mean dimension of full shifts over any finite dimensional compact metrizable spaces. On the other hand, we investigate finite group actions. In contrast to the case that the acting group is infinite (and amenable), Li showed that if a finite group acts continuously on a finite dimensional compact metrizable space, then sofic mean dimension may be different from (strictly less than) the classical (i.e. amenable) mean dimension (an explicitly known value in this case). We strengthen this result by proving a sharp lower bound, which, combining with the upper bound, gives the exact value of sofic mean dimension for all the actions of finite groups on finite dimensional compact metrizable spaces. Furthermore, this equality leads to a satisfactory comparison theorem for those actions, deciding when sofic mean dimension would coincide with classical mean dimension. Moreover, our two results, in particular, verify for a typical class of sofic group actions that sofic mean dimension does not depend on sofic approximation sequences.
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