Tuning and plateaux for the entropy of $\alpha$-continued fractions

Abstract

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.

20 pages, 2 figures. This version has been considerably expanded, and contains a new result (Theorem 3) classifying local behaviour of the entropy. We have also added several statements and proofs in order to avoid external references. Last but not least, we added an appendix which explains how the techniques we use are derived from Douady-Hubbard tuning for quadratic polynomials

References in corpus (2)

• Natural extensions and entropy of $\alpha$-continued fractions
• The entropy of alpha-continued fractions: numerical results

Cited by in corpus (4)

• The bifurcation locus for numbers of bounded type
• $\alpha$-Expansions with odd partial quotients
• Tanaka-Ito $\alpha$-continued fractions and matching
• Purely periodic continued fractions and graph-directed iterated function systems