In this paper we study the topological characteristic factors along cubes of minimal systems. It is shown that up to proximal extensions the pronilfactors are the topological characteristic factors along cubes of minimal systems. In particular, for a distal minimal system, the maximal (d − 1)-step pro-nilfactor is the topological cubic characteristic factor of order d.

Let (X, T) be a topological dynamical system and μ ∈ M(X, T). We show that (X, B, μ, T) is rigid if and only if there exists some subsequence A of N such that (X, T) is μ-A-equicontinuous if and only if there exists some IP-set A such that (X, T) is μ-A-equicontinuous. We show that if there exists some subsequence A of N with positive upper density such that (X, T) is μ-A-mean-equicontinuous, then (X, B, μ, T) is rigid. We also give results with respect to a function.

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