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.
| Original language | English |
|---|---|
| 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
The non-monotonicity of the entropy of α-continued fraction transformations. / Nakada, Hitoshi; Natsui, Rie.
In: Nonlinearity, Vol. 21, No. 6, 01.06.2008, p. 1207-1225.Research output: Contribution to journal › Article
}
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 -