American Institute of Mathematical Sciences

Advanced
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 - A, 2006, 15 (1) : 87-119. doi: 10.3934/dcds.2006.15.87
[1]

Lyndsey Clark. The $\beta$-transformation with a hole. Discrete & Continuous Dynamical Systems - A, 2016, 36 (3) : 1249-1269. doi: 10.3934/dcds.2016.36.1249
[2]

Marc Chamberland, Victor H. Moll. Dynamics of the degree six Landen transformation. Discrete & Continuous Dynamical Systems - A, 2006, 15 (3) : 905-919. doi: 10.3934/dcds.2006.15.905
[3]

Oğul Esen, Partha Guha. On the geometry of the Schmidt-Legendre transformation. Journal of Geometric Mechanics, 2018, 10 (3) : 251-291. doi: 10.3934/jgm.2018010
[4]

Neil S. Trudinger, Xu-Jia Wang. Quasilinear elliptic equations with signed measure. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 477-494. doi: 10.3934/dcds.2009.23.477
[5]

Sze-Bi Hsu, Bernold Fiedler, Hsiu-Hau Lin. Classification of potential flows under renormalization group transformation. Discrete & Continuous Dynamical Systems - B, 2016, 21 (2) : 437-446. doi: 10.3934/dcdsb.2016.21.437
[6]

N. Kamran, K. Tenenblat. Periodic systems for the higher-dimensional Laplace transformation. Discrete & Continuous Dynamical Systems - A, 1998, 4 (2) : 359-378. doi: 10.3934/dcds.1998.4.359
[7]

Hyukjin Kwean. Kwak transformation and Navier-Stokes equations. Communications on Pure & Applied Analysis, 2004, 3 (3) : 433-446. doi: 10.3934/cpaa.2004.3.433
[8]

E. García-Toraño Andrés, Bavo Langerock, Frans Cantrijn. Aspects of reduction and transformation of Lagrangian systems with symmetry. Journal of Geometric Mechanics, 2014, 6 (1) : 1-23. doi: 10.3934/jgm.2014.6.1
[9]

Andrey Kochergin. A Besicovitch cylindrical transformation with Hölder function. Electronic Research Announcements, 2015, 22: 87-91. doi: 10.3934/era.2015.22.87
[10]

Xian Chen, Zhi-Ming Ma. A transformation of Markov jump processes and applications in genetic study. Discrete & Continuous Dynamical Systems - A, 2014, 34 (12) : 5061-5084. doi: 10.3934/dcds.2014.34.5061
[11]

Hong-Gunn Chew, Cheng-Chew Lim. On regularisation parameter transformation of support vector machines. Journal of Industrial & Management Optimization, 2009, 5 (2) : 403-415. doi: 10.3934/jimo.2009.5.403
[12]

Hongyu Liu, Ting Zhou. Two dimensional invisibility cloaking via transformation optics. Discrete & Continuous Dynamical Systems - A, 2011, 31 (2) : 525-543. doi: 10.3934/dcds.2011.31.525
[13]

Fioralba Cakoni, Shari Moskow, Scott Rome. Asymptotic expansions of transmission eigenvalues for small perturbations of media with generally signed contrast. Inverse Problems & Imaging, 2018, 12 (4) : 971-992. doi: 10.3934/ipi.2018041
[14]

Alexandre I. Danilenko, Mariusz Lemańczyk. Spectral multiplicities for ergodic flows. Discrete & Continuous Dynamical Systems - A, 2013, 33 (9) : 4271-4289. doi: 10.3934/dcds.2013.33.4271
[15]

Tomáš Roubíček. Modelling of thermodynamics of martensitic transformation in shape-memory alloys. Conference Publications, 2007, 2007 (Special) : 892-902. doi: 10.3934/proc.2007.2007.892
[16]

Hongming Yang, Dexin Yi, Junhua Zhao, Fengji Luo, Zhaoyang Dong. Distributed optimal dispatch of virtual power plant based on ELM transformation. Journal of Industrial & Management Optimization, 2014, 10 (4) : 1297-1318. doi: 10.3934/jimo.2014.10.1297
[17]

Oliver Jenkinson. Every ergodic measure is uniquely maximizing. Discrete & Continuous Dynamical Systems - A, 2006, 16 (2) : 383-392. doi: 10.3934/dcds.2006.16.383
[18]

Miklós Horváth. Spectral shift functions in the fixed energy inverse scattering. Inverse Problems & Imaging, 2011, 5 (4) : 843-858. doi: 10.3934/ipi.2011.5.843
[19]

Almut Burchard, Gregory R. Chambers, Anne Dranovski. Ergodic properties of folding maps on spheres. Discrete & Continuous Dynamical Systems - A, 2017, 37 (3) : 1183-1200. doi: 10.3934/dcds.2017049
[20]

Gerhard Knieper, Norbert Peyerimhoff. Ergodic properties of isoperimetric domains in spheres. Journal of Modern Dynamics, 2008, 2 (2) : 339-358. doi: 10.3934/jmd.2008.2.339

2018 Impact Factor: 1.143

Tools

Metrics

  • PDF downloads (31)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences