Analysis of Infinite Dimensional Dynamic Systems with Nonlinear Observation over a Finite Field

Abstract

In this paper we consider a dynamical system defined on the interval [0,1) of the form $$ \begin{array}{*{20}{c}} {{{x}_{{k + 1}}} = 2{{x}_{k}}\left( {\bmod 1} \right)} \\ {{{y}_{k}} = \chi \left( {{{x}_{k}}} \right),} \\ \end{array} $$ (1.1) where X(x) is the characteristic function of a union of dyadic intervals. This system, or a variant of it, has appeared in many different contexts, see for example [2,11,3]. The first author became aware of this formulation of the problem through the work of DiMasi and Gamboni [2]. We show that the system (1.1) is equivalent to $$ \begin{array}{*{20}{c}} {{{{\overrightarrow x }}_{{k + 1}}} = \sigma \left( {\overrightarrow {{{x}_{k}}} } \right)} \\ {{{y}_{k}} = p\left( {\overrightarrow {{{x}_{k}}} } \right),} \\ \end{array} $$ (1.2) where \( {{\overrightarrow x }_{k}} \) is an infinite sequence of 0’s and l’s, σ is the left-shift, and p is a polynomial map into {0,1}. In this form, the system appears to be quite similar to certain “block map” systems studied in topological dynamics [5,6]. However, there are important distinctions between the systems which appear in this paper and the block maps of topological dynamics. These differences will be noted as they arise.