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

On the entropy of Japanese continued fractions

Laura Luzzi 1, and  Stefano Marmi 1,

1. 

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

Received  February 2007 Revised  October 2007 Published  December 2007

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