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 |
ASJC Scopus subject areas
- Statistical and Nonlinear Physics
- Mathematical Physics
- Physics and Astronomy(all)
- Applied Mathematics
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Nakada, H., & Natsui, R. (2008). The non-monotonicity of the entropy of α-continued fraction transformations. Nonlinearity, 21(6), 1207-1225. https://doi.org/10.1088/0951-7715/21/6/003