Not at all. Take $f(x)=x^2$ on the unit interval. For examples of uniquely ergodic homeomorphisms of the Cantor set see The prevalence of uniquely ergodic systems by Jewett.

Not at all. Take $f(x)=x^2$ on the unit interval. For examples of uniquely ergodic homeomorphisms of the Cantor set see The prevalence of uniquely ergodic systems by Jewett. A simple explicit example is provided by the boundary action of any hyperbolic automorphism of a homogeneous tree (attach a binary tree to each integer and look at the action of the integer shift on the space of ends of the whole construction).

Not at all. Take $f(x)=x^2$ on the unit interval.

Not at all. Take $f(x)=x^2$ on the unit interval. For examples of uniquely ergodic homeomorphisms of the Cantor set see The prevalence of uniquely ergodic systems by Jewett.