The non-monotonicity of the entropy of α-continued fraction transformations

We consider the α-continued fraction transformations T _α, 0 < α ≤ 1, the one parameter family of one-dimensional maps. Recently, Luzzi and Marmi showed that the entropy of T_α varies continuously as α varies and tends to zero as α tends to zero. They also observed by computer simulation that the entropy is not monotone as a function of α. In this paper, we first give an estimate of the decay rate of the entropy as α tends to zero. Then we show that there exist decreasing sequences of intervals of α, (I _n), (J_n), (K_n), (L_n) such that (a) 1/n, (b) I_n+1 < J_n < K_n < L _n < I_n, (c) the entropy of T_α is increasing on I_n, decreasing on K_n and constant (depends on n) on J_n and L_n.