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February  2006, 15(1): 87-119. doi: 10.3934/dcds.2006.15.87

Ergodic properties of signed binary expansions

Karma Dajani 1, , Cor Kraaikamp 2, and  Pierre Liardet 3,

1. 

Universiteit Utrecht, Fac. Wiskunde en Informatica and MRI, Budapestlaan 6, P.O. Box 80.000, 3508 TA Utrecht, Netherlands
2. 

Technische Universiteit Delft, EWI, Thomas Stieltjes Institute for Mathematics, Mekelweg 4, 2628 CD Delft, Netherlands
3. 

Université de Provence, Centre de Mathématiques et Informatique, 39 rue Joliot-Curie, F-13453 Marseille cedex 13, France

Received  September 2005 Revised  December 2005 Published  February 2006

In this paper it is shown that the classical signed binary expansion involves mainly two dynamical systems: the binary odometer and a three state Markov chain. Introducing the notions of additive and multiplicative block functions (e.g., sum-of-digits and Hamming weight functions), we derive dynamical systems which are skew products over the odometer. Their spectral properties are investigated, and applications are given to certain Maharam extensions. The proofs are related to the spectral measure of unitary operators, obtained from cocycles associated to block functions.
Keywords: transducer., shift transformation, ergodic properties, Separated signed binary expansions, Maharam transformation, odometer, spectral measure.
Mathematics Subject Classification: 37Axx, 11K5.
Citation: Karma Dajani, Cor Kraaikamp, Pierre Liardet. Ergodic properties of signed binary expansions. Discrete & Continuous Dynamical Systems, 2006, 15 (1) : 87-119. doi: 10.3934/dcds.2006.15.87
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