Growth of matrix products and mixing properties of the horocycle flow

Abstract

Let

$$H(t) = \left( {\begin{array}{*{20}c} 1 & t \\ 0 & 1 \\ \end{array} } \right)$$

and Φ_* = {Φ _n } ^∞ _n=1 ⊂ SL(2, ℝ). We consider the sequence of products P _n (t) = Φ _n H(t)Φ _n−1 H(t) ...Φ₁ H(t), and denote by B(Φ_*) the set of those t ∈ ℝ₊ for which the sequence {P _n (t)} is bounded. The question is how large can the set B(Φ_*) be? We show that

• the measure |B(Φ_*)| of B(Φ_*) is finite

• for every countable set S ⊂ ℝ₊, there is a sequence Φ_* with S ⊂ B(Φ_*)

• there exists a sequence Φ_* for which the set B(Φ_*) is essentially unbounded, that is, |B(Φ_*) ∩ [a, +∞)| > 0 for all a > 0.

The first of these statements implies the positive answer to the question by L. Polterovich and Z. Rudnick on the stability of quasi-mixing for the horocycle flow on SL(2, ℝ)/Γ.