### 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 |
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Pages (from-to) | 1207-1225 |

Number of pages | 19 |

Journal | Nonlinearity |

Volume | 21 |

Issue number | 6 |

DOIs | |

Publication status | Published - 2008 Jun 1 |

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### ASJC Scopus subject areas

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

### Cite this

*Nonlinearity*,

*21*(6), 1207-1225. https://doi.org/10.1088/0951-7715/21/6/003

**The non-monotonicity of the entropy of α-continued fraction transformations.** / Nakada, Hitoshi; Natsui, Rie.

Research output: Contribution to journal › Article

*Nonlinearity*, vol. 21, no. 6, pp. 1207-1225. https://doi.org/10.1088/0951-7715/21/6/003

}

TY - JOUR

T1 - The non-monotonicity of the entropy of α-continued fraction transformations

AU - Nakada, Hitoshi

AU - Natsui, Rie

PY - 2008/6/1

Y1 - 2008/6/1

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=44949152465&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=44949152465&partnerID=8YFLogxK

U2 - 10.1088/0951-7715/21/6/003

DO - 10.1088/0951-7715/21/6/003

M3 - Article

AN - SCOPUS:44949152465

VL - 21

SP - 1207

EP - 1225

JO - Nonlinearity

JF - Nonlinearity

SN - 0951-7715

IS - 6

ER -