On the entropy of japanese continued fractions
| Autor: | Laura Luzzi, Stefano Marmi |
|---|---|
| Přispěvatelé: | Luzzi, L, Marmi, Stefano |
| Předmět: |
Gauss map
Mathematics - Number Theory Applied Mathematics Mathematical analysis Dynamical Systems (math.DS) Japanese continued fractions Combinatorics Binary entropy function 11K50 (Primary) 37A10 37A35 37E05 (Secondary) Monotone polygon natural extension FOS: Mathematics Discrete Mathematics and Combinatorics Entropy (information theory) Nearest integer function Number Theory (math.NT) Mathematics - Dynamical Systems continuity of entropy Analysis Mathematics |
| Zdroj: | Scuola Normale Superiore di Pisa-IRIS |
| Popis: | 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 ($\alpha={1/2},\alpha=0$). 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 $\alpha=\frac{1}{n}$. Comment: 42 pages, 12 figures; v2: minor changes |
| Databáze: | OpenAIRE |
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