On the Riccati Difference Equation and Continued Fractions

Abstract

We consider a Riccati difference equation \(\Phi(x) + \rho(x)/\Phi(x-\omega) = v(x)\) under the assumption that coefficients \(\rho\), \(v\) are \(1\)-periodic continuous functions of a real variable and \(\omega\) is an irrational parameter. By using a connection between continued fraction theory and theory of \(SL(2,\mathbb{R})\)-cocycles over irrational rotation, we investigate the problem of existence of continuous solutions to this equation. It is shown that the convergence of a continued fraction representing a solution to the Riccati equation can be expressed in terms of hyperbolicity of the cocycle naturally associated to this continued fraction. We establish sufficient conditions for the uniform hyperbolicity of a \(SL(2,\mathbb{R})\)-cocycle, which imply the convergence of the corresponding continued fraction. The results thus obtained, along with the critical set method, have been applied to a special class of Riccati equations \(\rho(x)\equiv 1, v(x) = g b(x), g\gg 1,\) to obtain sufficient conditions for the existence of continuous solutions in this case.

DOI 10.1134/S1061920824601538