On the properties of the mean orbital pseudo-metric

On Feldman–Katok metric and entropy formulas

“…Remark 3.4. (1) In [5], the authors proved that F-equicontinuity is equivalent to a stronger statement, called {F n }-equicontinuity: for every ϵ> 0, there exists δ > 0 such that for every x, y ∈ X, d(x, y) < δ ⇒ F n (x, y) < ϵ, ∀ n ∈ N. (2) In [24], the authors proved that a TDS is uniquely ergodic if and only if F(x, y) = 0 for all x, y ∈ X. In particular, it is {F n }-equicontinuous.…”