Growth of matrix products and mixing properties of the horocycle flow

Abstract

\noindent In [1] L. Polterovich and Z. Rudnick considered the behavior of a one-parameter subgroup of a Lie group under the influence of a sequence of kicks. Among others they raise the following problem: {\it is the horocycle flow stably quasi-mixing on $SL(2,\mathbb{R})/\Gamma$?} Equivalently it can be reformulated in terms of boundedness of the sequences of products $P_n(t)={\Phi }_{n}H(t){\Phi }_{n-1}H(t)...{\Phi }_{1}H(t)$ where $H(t)=\left(\begin{array}{ccc}1& t0& 1\end{array}\right)$ and $\Phi =\{{\Phi }_n\}\subset SL(2,\mathbb{R})$. We solve this problem positively and as a consequence obtain the following application to the discrete Schrödinger equation \begin{equation*} q_{k+1} - (2+tc_k)q_k + q_{k-1}=0, \qquad k\geq 1: \end{equation*} the set of values of the parameter $t$ for which the equation has only bounded solutions, has finite measure.