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Isomorphisms between positive and negative $\beta $-transformations

Published online by Cambridge University Press:  09 November 2012

CHARLENE KALLE
CHARLENE KALLE*
Affiliation:
Mathematical Institute, Leiden University, Postbus 9512, 2300 RA Leiden, The Netherlands (email: [email protected])
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Abstract

We compare a piecewise linear map with constant slope $\beta \gt 1$ and a piecewise linear map with constant slope $-\beta $. These maps are called the positive and negative $\beta $-transformations. We show that for a certain set of $\beta $s, the multinacci numbers, there exists a measurable isomorphism between these two maps. We further show that for all other values of $\beta $between 1 and 2 the two maps cannot be isomorphic.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 34 , Issue 1 , February 2014 , pp. 153 - 170
Copyright
Copyright © 2012 Cambridge University Press 

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