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

Hitoshi Nakada; Rie Natsui

doi:10.1088/0951-7715/21/6/003

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

Hitoshi Nakada, Rie Natsui

Department of Mathematics

Research output: Contribution to journal › Article › peer-review

24 Citations (Scopus)

Abstract

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.

Original language	English
Pages (from-to)	1207-1225
Number of pages	19
Journal	Nonlinearity
Volume	21
Issue number	6
DOIs	https://doi.org/10.1088/0951-7715/21/6/003
Publication status	Published - 2008 Jun 1

ASJC Scopus subject areas

Statistical and Nonlinear Physics
Mathematical Physics
Physics and Astronomy(all)
Applied Mathematics

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abstract = "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 Tα 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), (Jn), (Kn), (Ln) such that (a) 1/n, (b) In+1 < Jn < Kn < L n < In, (c) the entropy of Tα is increasing on In, decreasing on Kn and constant (depends on n) on Jn and Ln.",

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

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