Weighted upper metric mean dimension for amenable group actions

On relative metric mean dimension with potential and variational principles

“…For 0 < ǫ, ρ < 1 and µ ∈ M(X), denote δ ǫ,ρ,µ := sup{δ > 0 : ∃α ∈ P X such that diamα < ǫ, µ(U δ (∂α)) < ρ} and δ ǫ,ρ := inf µ∈M(X) δ ǫ,ρ,µ . The following definition is a variant of the ones introduced in [13, Definition 4.1] and [24]. Moreover, both M T (X) in the above equalities can be replaced by E T (X).…”

Upper metric mean dimensions with potential on subsets

“…As the name suggested, the mean topological dimension is a topological invariant of dynamical systems. There are many important works around this quantity [1,8,14,17,19,20,23,[25][26][27][28]. Applications of the mean dimension to the embedding problem of dynamical systems can be found in [10,15,16,18].…”