  The entropy $h(T_\alpha)$ of $\alpha$-continued fraction transformations is
known to be locally monotone outside a closed, totally disconnected set $\EE$.
We will exploit the explicit description of the fractal structure of $\EE$ to
investigate the self-similarities displayed by the graph of the function
$\alpha \mapsto h(T_\alpha)$. Finally, we completely characterize the plateaux
occurring in this graph, and classify the local monotonic behaviour.
