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

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

2011-11-10

arxiv.org/abs/1111.2554v1

Mathematics

Dynamical Systems

9 pages, 1 figure

Scientific paper

The entropy $h(T_\alpha)$ of $\alpha$-continued fraction transformations is known to be locally monotone outside a closed, totally disconnected set E. We will exploit the explicit description of the fractal structure of E to investigate the self-similarities displayed by the graph of the function $\alpha \mapsto h(T_\alpha)$. Finally, a complete characterization of the plateaux occurring in this graph is provided, using the H\"older-continuity of the entropy.

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