Ergodic properties of signed binary expansions
Karma Dajani, Cor Kraaikamp, Pierre Liardet · Discrete and Continuous Dynamical Systems · 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.