Sensitivity to initial conditions

[Zafa Pi]

Let M = {x₁, x₂, x₃, ...} U {p} be a perfect metric space.
Let f be continuous, taking M to M with f(x_n) = x_n+1 and f(p) = p.
I would like to know if this dynamical system is necessarily sensitive to initial conditions.

 

[WWGD]

What do you mean by sensitive to initial conditions? How do you define it?

 

[Zafa Pi]

There exists d > 0, such that for each x in M and V a neighborhood of x there is y in V and a positive integer n with the property:
the distance between f⁽ⁿ⁾(x) ( the nth iterate of f) and f⁽ⁿ⁾(y) is > d.