Tuning and plateaux for the entropy of α-continued fractions

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