Central limit theorem for products of toral automorphisms

الأصل · EN

Let (τₙ) be a sequence of toral automorphisms τₙ: x → Aₙ x modᵈ with Aₙ ∈ A, where A is a finite set of matrices in SL(d, Z). Under some conditions the method of "multiplicative systems" of Komlòs can be used to prove a Central Limit Theorem for the sums ∑ₖ₌₁ⁿ f(τₖ ∘ τₖ₋₁ ∘ τ₁ x) if f is a Hölder function on Tᵈ. These conditions hold for 2× 2 matrices with positive coefficients. In dimension d they can be applied when Aₙ= Aₙ(ω), with independent choices of Aₙ(ω) in a finite set of matrices ∈ SL(d, Z), in order to prove a "quenched" CLT.