Can someone provide some examples to illustrate the difference between equicontinuity and mean equicontinuity? Can someone provide a concrete example that is mean equicontinuous but not equicontinuous? Thank you
Definitions:
A dynamical system (X,T) is called equicontinuous if for every $ε>0$ there is a $δ>0$ such that whenever x, y ∈ X with d(x, y) < δ, $d(T^n(x),T^n(y))<ε$ for n = 0,1,2,...
A dynamical system (X,T) is called mean equicontinuous if for every ε > 0, there exists a δ > 0 such that whenever x, y ∈ X with d(x, y) < δ,
$\limsup_{n\to\infty}\frac{1}{n}\sum_{i=0}^{n-1}d(T^n(x),T^n(y))<ε$.