Sensitivity to initial conditions

1. May 19, 2015

Zafa Pi

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

2. May 21, 2015

WWGD

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

3. May 21, 2015

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.