On the entropy of Japanese continued fractions

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July 2008, 20(3): 673-711. doi: 10.3934/dcds.2008.20.673

On the entropy of Japanese continued fractions

1.	Scuola Normale Superiore, Piazza dei Cavalieri, 7, 56126, Pisa (PI), Italy, Italy

Received February 2007 Revised October 2007 Published December 2007

Abstract
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We consider a one-parameter family of expanding interval maps $\{T_{\alpha}\}_{\alpha \in [0,1]}$ (Japanese continued fractions) which include the Gauss map ($\alpha=1$) and the nearest integer and by-excess continued fraction maps ($\alpha=\frac{1}{2},\,\alpha=0$). We prove that the Kolmogorov-Sinai entropy $h(\alpha)$ of these maps depends continuously on the parameter and that $h(\alpha) \to 0$ as $\alpha \to 0$. Numerical results suggest that this convergence is not monotone and that the entropy function has infinitely many phase transitions and a self-similar structure. Finally, we find the natural extension and the invariant densities of the maps $T_{\alpha}$ for $\alpha=\frac{1}{n}$.

Keywords: continuity of entropy., Japanese continued fractions, natural extension.

Mathematics Subject Classification: Primary: 11K50; Secondary: 37A10, 37A35, 37E0.

Citation: Laura Luzzi, Stefano Marmi. On the entropy of Japanese continued fractions. Discrete & Continuous Dynamical Systems - A, 2008, 20 (3) : 673-711. doi: 10.3934/dcds.2008.20.673

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