On difference Riccati equation and continued fractions

TL;DR

This paper explores the existence of continuous solutions to a difference Riccati equation with periodic coefficients by linking continued fraction convergence to hyperbolicity of an associated cocycle, providing new criteria for solution existence.

Contribution

It introduces a novel approach connecting continued fraction convergence with hyperbolicity of cocycles to analyze difference Riccati equations with periodic coefficients.

Findings

Convergence of continued fractions is characterized by cocycle hyperbolicity.

Sufficient conditions for solution existence are established using the critical set method.

The approach links continued fractions, hyperbolic dynamics, and Riccati equations.

Abstract

We study a difference Riccati equation $\Phi(x) + \rho(x)/\Phi(x-\omega) = v(x)$ with $1-$periodic continuos coefficients. Using continued fraction theory we investigate a problem of existence of continuos solutions for this equation. It is shown that convergence of a continued fraction representing a solution of the Riccati equation can be expressed in terms of hyperbolicity of a cocycle naturally associated to this continued fraction. We apply the critical set method to establish the uniform hyperbolicity of the cocycle and to obtain sufficient conditions for the convergence of a continued fraction giving a representation for a solution of the Riccati equation.