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A dynamical system approach to the Kakutani–Fibonacci sequence

Published online by Cambridge University Press:  03 June 2013

INGRID CARBONE ,
MARIA RITA IACÒ  and
ALJOŠA VOLČIČ
INGRID CARBONE
Affiliation:
Department of Mathematics, University of Calabria, Ponte P. Bucci Cubo 30B, 87036 Arcavacata di Rende (Cosenza), Italy email i.carbone@unical.itiaco@mat.unical.itvolcic@unical.it
MARIA RITA IACÒ
Affiliation:
Department of Mathematics, University of Calabria, Ponte P. Bucci Cubo 30B, 87036 Arcavacata di Rende (Cosenza), Italy email i.carbone@unical.itiaco@mat.unical.itvolcic@unical.it
ALJOŠA VOLČIČ
Affiliation:
Department of Mathematics, University of Calabria, Ponte P. Bucci Cubo 30B, 87036 Arcavacata di Rende (Cosenza), Italy email i.carbone@unical.itiaco@mat.unical.itvolcic@unical.it
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Abstract

In this paper we consider the sequence of Kakutani’s $\alpha $-refinements corresponding to the inverse of the golden ratio (which we call the Kakutani–Fibonacci sequence of partitions) and associate to it an ergodic interval exchange (which we call the Kakutani–Fibonacci transformation) using the ‘cutting–stacking’ technique. We prove that the orbit of the origin under this map coincides with a low discrepancy sequence (which we call the Kakutani–Fibonacci sequence of points), which has also been considered by other authors.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 34 , Issue 6 , December 2014 , pp. 1794 - 1806
Copyright
© Cambridge University Press, 2013 

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