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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: carminat@dm.unipi.it)
GIULIO TIOZZO
Affiliation:
Department of Mathematics, Harvard University, 1 Oxford Street, 02138 Cambridge, MA, USA (email: tiozzo@math.harvard.edu)
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Abstract

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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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