Local mean dimension theory for sofic group actions
arxiv(2024)
摘要
Using a local perspective, we introduce mean dimension pairs and give
sufficient conditions of when every non-trivial factor of a continuous group
action of a sofic group G has positive mean dimension. In addition, we prove
that the mean dimension map is Borel, and that the set of subshifts with
completely positive mean dimension of [0,1]^G, the full G-shift on the
interval, is a complete coanalytic set in the set of all subshifts (hence not
Borel). Our results are new even when the acting group is ℤ.
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