Balls in minimal systems

Question

If $(X,T)$ is a minimal system uniquely ergodic with $\mu$, is there $p\in X$ such that $\mu(\partial B(p,t))=0$ for all $t>0$?

Answer 1

Let $T$ be an irrational rotation of the circle. We modify the metric on the circle as follows, letting $d(\cdot,\cdot)$ be the standard metric on the circle; and for $C$ a non-empty closed subset of the reals, let $D(x,C)$ denote the distance from $x$ to $C$. Let $C$ be a Cantor set of positive measure contained in $[\frac 14,\frac 12]$ and containing $\frac 14$ and define a new metric by $$ \rho(x,y)= \begin{cases} d(x,y)&\text{if $d(x,y)<\frac 14$}\\ \tfrac 14+D(d(x,y),C)&\text{if $d(x,y)\ge \frac 14$.} \end{cases} $$ This generates the same topology as the original metric. But $\partial B(p,\frac 14)=p\pm C$.