Orthogonal polynomials on several intervals: accumulation points of recurrence coefficients and of zeros

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Let E = ∪ⱼ ₌ ₁ˡ [a₂ⱼ₋₁,a₂ⱼ], a₁ < a₂ <... < a₂ₗ, l ≥ 2 and set ω(∞) =(ω₁(∞),...,ωₗ₋₁(∞)), where ωⱼ(∞) is the harmonic measure of [a₂ ⱼ ₋ ₁, a₂ ⱼ] at infinity. Let μ be a measure which is on E absolutely continuous and satisfies Szegő's-condition and has at most a finite number of point measures outside E, and denote by (Pₙ) and (Qₙ) the orthonormal polynomials and their associated Weyl solutions with respect to dμ, satisfying the recurrence relation √λ₂ ₊ ₙ y₁ ₊ ₙ = (x - α₁ ₊ ₙ) yₙ -√λ₁ ₊ ₙ y₋₁ ₊ ₙ. We show that the recurrence coefficients have topologically the same convergence behavior as the sequence (n ω(∞))ₙ∈ ₙ modulo 1; More precisely, putting (αˡ⁻¹₁ ₊ ₙ, λˡ⁻¹₂ ₊ ₙ) = (α[ₗ ₁/₂]₊₁₊ₙ,..., α₁₊ₙ,..., α₋[ₗ₋₂/₂]₊₁₊ₙ, λ[ₗ₋₂/₂]₊₂₊ₙ,...,λ₂₊ₙ,..., λ₋[ₗ₋₁/₂]₊₂₊ₙ) we prove that (αˡ⁻¹₁ ₊ ₙν, λˡ⁻¹₂ ₊ ₙν)ν∈ ₙ converges if and only if (nνω(∞))ν∈ ₙ converges modulo 1 and we give an explicit homeomorphism between the sets of accumulation points of (αˡ⁻¹₁ ₊ ₙ, λˡ⁻¹₂ ₊ ₙ) and (nω(∞)) modulo 1.