Solutions to Difference Riccati Equation via Continued Fractions

We study a difference Riccati equation $\Phi(x)+ \rho(x)/\Phi(x-\omega)=v(x)$ with 1-periodic continuous co-efficients. Using continued fraction theory, we inves-tigate a problem of existence of continuous 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.