Tuning and plateaux for the entropy of $α$-continued fractions

The entropy $h(T_α)$ of $α$-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 $α\mapsto h(T_α)$. Finally, we completely characterize the plateaux occurring in this graph, and classify the local monotonic behaviour.

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