Ergodic Theory and Dynamical Systems

Table of Contents - 11 July 2012 - Volume 32, Issue 04  

Research Article

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

CARLO CARMINATIa1 and GIULIO TIOZZOa2

a1 Dipartimento di Matematica, Università di Pisa, Largo Bruno Pontecorvo 5, 56127 Pisa, Italy (email: carminat@dm.unipi.it)

a2 Department of Mathematics, Harvard University, 1 Oxford Street, 02138 Cambridge, MA, USA (email: tiozzo@math.harvard.edu)

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.

(Received May 20 2010)

(Revised May 31 2011)

(Online publication September 14 2011)