2025/09/25 by Huang, J., Xiao, Z.
#Dynamical Systems (math.DS) #FOS: Mathematics
paper · doi:10.48550/arxiv.2509.20738
Let (X,G), (Y,G) be two G-systems, where G is an infinite countable discrete amenable group and X, Y are compact metric spaces. Suppose that U is a cover of X. We first introduce the conditional local topological intricacy Inttop (G,U|Y) and average sample complexity Asctop (G,U|Y). Given an invariant measure μ of X, we study the conditional local measure-theoretical intricacy Intμ^±(G,U|Y) and average sample complexity Ascμ^±(G,U|Y). For any Følner sequence \Fn\n∈ℕ, we take \cFnS\S⊆ Fn to be the uniform system of coefficients. We establish the equivalence of Ascμ-(G,U|Y) and Ascμ+(G,U|Y) when G=ℤ. Furthermore, we verified that Ascμ-(G,U) is equal to Ascμ+(G,U) in general case. Finally, we give a local variational principle of average sample complexity.