arXiv:1111.2554 Abstract | arXiv Analytics

arXiv:1111.2554 [math.DS]Abstract References Reviews Resources

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

Published 2011-11-10, updated 2012-05-31Version 2

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.

Comments: 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

Journal: Nonlinearity 26 (2013), pp. 1049-1070

DOI: 10.1088/0951-7715/26/4/1049

Categories: math.DS

Subjects: 11A55, 11K50, 37A10

Keywords: local monotonic behaviour, locally monotone outside, continued fraction transformations, explicit description, fractal structure

Tags: journal article

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