The Topological Dimension On The Left-cancellative Amenable Semigroup Action

In this paper,we firstly review some important concepts about the topological dimension.After introducing the metric,we mainly study the relation between the mean topological dimension and such mean topological dimension under some metric in the dynamic system.By using the Ornstein-Weiss theorem,we prove the existence of some limitation and consider the properties of the mean topological dimension when the continuous map is a subshift as well.Finally,we expand the corresponding conclusions to the left-cancellative amenable semigroup action by the properties of the amenable semigroup.