Dynamical systems defined by polynomials with algebraic properties

TL;DR

This paper explores the relationship between polynomial-defined dynamical systems over the torus and the roots of the polynomials, focusing on streams satisfying convolution equations.

Contribution

It introduces a framework linking streams over the torus to polynomial roots, highlighting algebraic properties of these dynamical systems.

Findings

Streams over R/Z satisfying convolution with P(z) relate to roots of P(z)=0.

The study characterizes the algebraic structure of these streams.

Connections between polynomial roots and dynamical behavior are established.

Abstract

Let (x_n; n\in Z) be a bisequence of elements x_n in the 1-dimensional torus R/Z, which is called a stream over R/Z. Let P(z)=a_k z^k+...+a_1 z+a_0 be a polynomial with integer coefficients. Define the set of streams over R/Z such that the convolution product P(z)\times(x_n; n\in Z)=(\sum_{i=0}^k a_i x_{n-i}; n\in Z)=(0; n\in Z), which is called the stream 0 of P. We study similarities between stream 0 of P and the roots of P(z)=0.