Invariant set of two-dimensional dynamics of golden ratio encoders

Abstract

A discrete-time two-dimensional dynamical system appears in a Golden Ratio Encoder (GRE), a type of analog-to-digital converter. One of the essential elements in analyzing a given dynamical system is identifying the invariant set of that system. The invariant set of dynamics of GREs is not known, except in special cases. We herein determine the invariant set of the dynamics of GREs with an amplification factor α and a threshold θ for a wide range of parameters (α, θ). The invariant set is separated into six sub-regions and the transition probabilities between the sub-regions are defined. We show that the uniform distribution on the invariant set is an invariant density for this dynamical system.