Growth of matrix products and mixing properties of the horocycle flow

Fëdor Nazarov; Ekaterina Shulman

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arXiv · 1008.3077

Growth of matrix products and mixing properties of the horocycle flow

Fëdor Nazarov·Ekaterina Shulman

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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 $SL(2,\mathbb{R})/Γ$?} Equivalently it can be reformulated in terms of boundedness of the sequences of products $ P_n(t) = Φ_n H(t)Φ_{n-1} H(t) \, ... \, Φ_1 H(t) $ where $H(t) = \begin{pmatrix} 1 & t 0 & 1 \end{pmatrix}$ and $Φ=\{Φ_n\} \subset SL(2,\mathbb{R})$. 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.

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Fëdor Nazarov, Ekaterina Shulman. 2010-08-18. Growth of matrix products and mixing properties of the horocycle flow. https://arxiv.org/abs/1008.3077

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