On the entropy of japanese continued fractions

We consider a one-parameter family of expanding interval maps {Tα} 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 Tα for α = 1 n .

We consider a one-parameter family of expanding interval maps {Tα }α∈[0,1] (Japanese continued fractions) which include the Gauss map (α = 1 1) and the nearest integer and by-excess continued fraction maps (α = 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 1 find the natural extension and the invariant densities of the maps Tα for α = n .