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

Hitoshi Nakada, Rie Natsui

Research output: Contribution to journalArticle

18 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 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 languageEnglish
Pages (from-to)1207-1225
Number of pages19
JournalNonlinearity
Volume21
Issue number6
DOIs
Publication statusPublished - 2008 Jun 1

Fingerprint

Continued fraction
Entropy
entropy
Tend
Zero
Vary
Monotonic decreasing sequence
One-dimensional Maps
Decay Rate
decay rates
Monotone
Computer Simulation
computerized simulation
intervals
Interval
Computer simulation
estimates
Estimate

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 journalArticle

Nakada, Hitoshi ; Natsui, Rie. / The non-monotonicity of the entropy of α-continued fraction transformations. In: Nonlinearity. 2008 ; Vol. 21, No. 6. pp. 1207-1225.
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