On the entropy of Japanese continued fractions

2006/01/24 by Laura Luzzi, Luzzi, Laura, Stefano Marmi +1
Computer Science · Mathematics · #11K50 (Primary) 37A10 #37A35 #37E05 (Secondary) #Chaos-based Image/Signal Encryption #Computability, Logic, AI Algorithms #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals #Number Theory (math.NT) #math.DS #math.NT #msc:11K50 #msc:37A10 #msc:37A35 #msc:37E05

paper · pdf · doi:10.48550/arxiv.math/0601576

42 pages, 12 figures; v2: minor changes

openalex publication_date 2006/01/24 · arxiv created 2006/06/01 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We consider a one-parameter family of expanding interval maps \T_α\_{α∈ [0,1]} (japanese continued fractions) which include the Gauss map (α=1) and the nearest integer and by-excess continued fraction maps (α=1/2,α=0). We prove that the Kolmogorov-Sinai entropy h(α) of these maps depends continuously on the parameter and that h(α) → 0 as α→ 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_α for α=(1)/(n).

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