that a language $\mathcal{L}$ has positive topological entropy if the size of the set of prefixes of length $n$ of $\mathcal{L}$ grows exponentially in $n$. In brief, if we consider closure of all the shifts of sequences from $\{uv, uw\}^*$, then this set will clearly have positive topological entropy, and the injective projection $\pi$ of this set will have positive Hausdorff dimension.

The mixed real case
-------------------

We first assume that ${\lambda}\neq\mu$. The case when they are equal is considered in subsection \[sub:equal\] below.

Let $0 < {\lambda}< \mu < 1$. The vertices of the convex hull of $A_{-{\lambda}, \mu}$ are given by $\pi((pm)^k p^\infty), \pi((mp)^k p^\infty), \pi((pm)^k m^\infty)$, and $\pi((mp)^k m^\infty)$, where $k\ge0$.

It suffices to show that the lines from $\pi((pm)^k p^\infty)$ to $\pi((pm)^{k+1} p^\infty)$, and similarly from $\pi((mp)^k p^\infty)$ to $\pi((mp)^{k+1} p^\infty)$, from $\pi((pm)^k m^\infty)$ to $\pi((pm)^{k+1} m^\infty)$, and from $\pi((mp)^k m^\infty)$ to $\pi((mp)^{k+1} m^\infty)$ are support lines for $A_{-{\lambda}, \mu}$ and that their union is homeomorphic to a circle. We will do the first case only. The other cases are similar.

We will proceed by induction. Consider first the line from $\pi(p^\infty)$ to $\pi(pmp^\infty)$. This will be in the direction $\pi(p^\infty) - \pi(pmp^\infty) = (2 {\lambda}, 