On the affine random walk on the torus

الأصل · EN

Let μ be a borelian probability measure on G:=SLd(Z) Tᵈ. Define, for x∈ Tᵈ, a random walk starting at x denoting for n∈ N, {arrayrcl X₀ &=&x Xₙ₊₁ &=& aₙ₊₁ Xₙ + bₙ₊₁ array. where ((aₙ,bₙ))∈ Gⁿ is an iid sequence of law μ. Then, we denote by Pₓ the measure on (Tᵈ)ⁿ that is the image of μ⊗ ⁿ by the map ((gₙ) (x,g₁ x, g₂ g₁ x,, gₙ g₁ x,)) and for any φ∈ L¹((Tᵈ)ⁿ, Pₓ), we set Eₓ φ((Xₙ)) = ∫ φ((Xₙ)) dPₓ((Xₙ)). Bourgain, Furmann, Lindenstrauss and Mozes studied this random walk when μ is concentrated on SLd(Z) {0} and this allowed us to study, for any hölder-continuous function f on the torus, the sequence (f(Xₙ)) when x is not too well approximable by rational points. In this article, we are interested in the case where μ is not concentrated on SLd(Z) Qᵈ/Zᵈ and we prove that, under assumptions on the group spanned by the support of μ, the Lebesgue's measure ν on the torus is the only stationary probability measure and that for any hölder-continuous function f on the torus, Eₓ f(Xₙ) converges exponentially fast to ∫ fdν. Then, we use this to prove the law of large numbers, a non-concentration inequality, the functional central limit theorem and it's almost-sure version for the sequence (f(Xₙ)). In the appendix, we state a non-concentration inequality for products of random matrices without any irreducibility assumption.