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.

Supported in part by NSF Grant #DMS 8905334, NASA Grant #NAG2-89 and NSA Grant #MDA904-90-H-4009.

Supported in part by NSA Grant #MDA904-90-H-4009.