Growth of matrix products and mixing properties of the horocycle flow

TL;DR

This paper proves the stable quasi-mixing property of the horocycle flow on certain Lie groups and applies the result to identify parameter sets with bounded solutions in a discrete Schrödinger equation.

Contribution

It provides a positive solution to the problem of stable quasi-mixing of the horocycle flow and links this to boundedness properties in a related Schrödinger equation.

Findings

The horocycle flow is stably quasi-mixing on SL(2,R)/Γ.

The set of parameters t with only bounded solutions in the Schrödinger equation has finite measure.

The results connect dynamical systems properties with spectral theory of difference equations.

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 $H(t) = \begin{pmatrix} 1 & t 0 & 1 \end{pmatrix}$ 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 Schr\"odinger 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.