$-\frac{\log{\tau}}{\log{|\kappa|}} \approx 1.579354467$.

See Figure \[fig:Dragon\] for the attractor (grey) and points of uniqueness (black). It is interesting to note that since the Hausdorff dimension of the boundary here is approximately $1.093$ (see [@IK]), “most” points of uniqueness of the Rauzy fractal are interior points, whereas our general construction only uses boundary points - see Section \[sec:unique\].

\[ex:twindragon\]

![The twin dragon curve $A_{\frac{1+i}2}$[]{data-label="fig:twindragon"}](twindragon.eps){width="300pt"}

Another famous complex fractal is the twin dragon curve which in our notation is $A_{\kappa}$ with ${\kappa}=\frac{1+i}2$ – see Figure \[fig:twindragon\]. The grey half corresponds to all points in $A_{\kappa}$ whose address begins with $m$ and the black half – with $p$. Their intersection is a part of the boundary of either half, which has the same Hausdorff dimension as the boundary of $A_{\kappa}$, approximately $1.524$ (see, e.g., [@dragon]).

Clearly, if a point in $A_{\kappa}$ has a non-unique address $a_0a_1\dots$, then $\pi(a_na_{n+1}\dots)$ must lie in the aforementioned intersection for some $n$. This means that the complement of the set of uniqueness in this case has dimension $\approx1.524$; on the other hand, it is well known that $A_{\kappa}$ has non-empty interior (see, e.g. [@Gilbert] and references therein). Consequently, a.e. point of $A_{\kappa}$ has a unique address.

Notation {#sec:notation}
