have a small number of points of uniqueness (see Theorem \[thm:uniq-rational\]). With the exception of this case, all other IFS will have a continuum of points of uniqueness.

Our second result is

\[thm:uniq\] For all non-degenerate IFS not explicitly mentioned in Theorem \[thm:uniq-rational\], the set of points of uniqueness is uncountable, and with positive Hausdorff dimension.

Again, the real case where ${\lambda}>0, \mu > 0$ has been shown in [@HS]. We prove the remaining cases in Section \[sec:unique\].

\[ex:rauzy\]

![Points of uniqueness for the Rauzy fractal[]{data-label="fig:Dragon"}](rauzy-uniq.eps){width="350pt"}

As an example, consider the famous Rauzy fractal introduced in [@Rauzy]. Let ${\kappa}$ be one of the complex roots of $z^3-z^2-z-1$, i.e., ${\kappa}\approx -0.419 + 0.606i$. Consider the attractor $A_{\kappa}$ satisfying $A_{\kappa}= ({\kappa}A_{\kappa}-1) \cup ({\kappa}A_{\kappa}+ 1)$. It follows from the results of [@Mess] that the unique addresses in this case are precisely those which do not contain three consecutive identical symbols.

It is easy to show by induction that the number of $m$-$p$ words of length $n$ with such a property which start with $m$ is the $n$th Fibonacci number. Consequently, the set of unique addresses has topological entropy equal to $\log\tau$, where $\tau=\frac{1+\sqrt5}2$. Hence the Hausdorff dimension of the set of uniqueness is 