Equicontinuous system

Remarks

For a homeomorphism one often asks the same of the family $\set{f^n : n \in \Z}$. Example: every isometry $f$ is equicontinuous with $\delta = \varepsilon$, since $d(f^n(x), f^n(y)) = d(x, y)$; for instance the rotation $x \mapsto x + \alpha \bmod 1$ of the circle. Non-example: the doubling map $f(x) = 2x \bmod 1$ on the circle $\R / \Z$. Given $\delta > 0$, take $x = 0$ and $y$ with $0 < y < \min(\delta, 1/4)$, and let $n$ be the first integer with $2^n y \ge 1/4$. Then $f^n(y) = 2^n y$ lies in $[1/4, 1/2)$, at distance at least $1/4$ from $f^n(0) = 0$, so the condition fails for $\varepsilon = 1/4$.