Hostname: page-component-cd9895bd7-8ctnn Total loading time: 0 Render date: 2024-12-26T00:37:29.406Z Has data issue: false hasContentIssue false
  • English
  • Français

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])
Get access Rights & Permissions [Opens in a new window]

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

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