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A canonical thickening of ℚ and the entropy of α-continued fraction transformations

Published online by Cambridge University Press:  14 September 2011

CARLO CARMINATI  and
GIULIO TIOZZO
CARLO CARMINATI
Affiliation:
Dipartimento di Matematica, Università di Pisa, Largo Bruno Pontecorvo 5, 56127 Pisa, Italy (email: [email protected])
GIULIO TIOZZO
Affiliation:
Department of Mathematics, Harvard University, 1 Oxford Street, 02138 Cambridge, MA, USA (email: [email protected])
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Abstract

We construct a countable family of open intervals contained in (0,1] whose endpoints are quadratic surds and such that their union is a full measure set. We then show that these intervals are precisely the monotonicity intervals of the entropy of α-continued fractions, thus proving a conjecture of Nakada and Natsui.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 32 , Issue 4 , August 2012 , pp. 1249 - 1269
Copyright
Copyright © Cambridge University Press 2011

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References

[BDV02]Bourdon, J., Daireaux, B. and Vallée, B.. Dynamical analysis of α-Euclidean algorithms. J. Algorithms 44(1) (2002), 246–285.CrossRefGoogle Scholar
[CMPT10]Carminati, C., Marmi, S., Profeti, A. and Tiozzo, G.. The entropy of α-continued fractions: numerical results. Nonlinearity 23 (2010), 2429–2456.CrossRefGoogle Scholar
[Cas95]Cassa, A.. Dinamiche caotiche e misure invarianti, Tesi di Laurea, Facoltà di Scienze Matematiche, Fisiche e Naturali, University of Florence, 1995.Google Scholar
[CMM99]Cassa, A., Moussa, P. and Marmi, S.. Continued fractions and Brjuno functions. J. Comput. Appl. Math. 105(1–2) (1999), 403–415.Google Scholar
[GS96]Graczyk, J. and Świa̧tek, G.. Critical circle maps near bifurcation. Comm. Math. Phys. 176 (1996), 227–260.CrossRefGoogle Scholar
[Jar28]Jarník, V.. Zur metrischen Theorie der diophantische Approximationen. Prace Mat.-Fiz. 36(1928–9), 91–106.Google Scholar
[KU10]Katok, S. and Ugarcovici, I.. Structure of attractors for (a,b)-continued fraction transformations. J. Mod. Dyn. 4(4) (2010), 637–691.CrossRefGoogle Scholar
[LM08]Luzzi, L. and Marmi, S.. On the entropy of Japanese continued fractions. Discrete Contin. Dyn. Syst. 20 (2008), 673–711.CrossRefGoogle Scholar
[Nak81]Nakada, H.. Metrical theory for a class of continued fraction transformations and their natural extensions. Tokyo J. Math. 4 (1981), 399–426.CrossRefGoogle Scholar
[NN08]Nakada, H. and Natsui, R.. The non-monotonicity of the entropy of α-continued fraction transformations. Nonlinearity 21 (2008), 1207–1225.CrossRefGoogle Scholar
[Sch05]Schuster, H. G. and Jost, W.. Deterministic Chaos: An Introduction, 4th edn. Wiley-VCH, Weinheim, Germany, 2005.CrossRefGoogle Scholar