Some Open Problems Concerning Orthogonal Polynomials on Fractals and Related Questions

We discuss several open problems related to analysis on fractals: estimates of the Green functions, the growth rates of the Markov factors with respect to the extension property of compact sets, asymptotics of orthogonal polynomials and the Parreau-Widom condition, Hausdorff measures and the Hausdorff dimension of the equilibrium measure on generalized Julia sets. 1 Background and notation 1.1 Chebyshev and orthogonal polynomials Let K ⊂ C be a compact set containing infinitely many points. We use ‖ · ‖L∞(K) to denote the sup-norm on K, Mn is the set of all monic polynomials of degree n. The polynomial Tn,K that minimizes ‖Qn‖L∞(K) for Qn ∈Mn is called the n-th Chebyshev polynomial on K . Assume that the logarithmic capacity Cap(K) is positive. We define the n-th Widom factor for K by Wn(K) := ‖Tn,K‖L∞(K)/Cap(K). In what follows we consider probability Borel measures μ with non-polar compact support supp(μ) in C. The n-th monic orthogonal polynomial Pn(z;μ) = zn + . . . associated with μ has the property ‖Pn(·;μ)‖2L2(μ) = inf Qn∈Mn ∫ |Qn(z)| dμ(z), where ‖ · ‖L2(μ) is the norm in L2(μ). Then the n-th Widom-Hilbert factor for μ is W 2 n (μ) := ‖Pn(·;μ)‖L2(μ)/(Cap(supp(μ))) . If supp(μ) ⊂ R then a three-term recurrence relation x Pn(x;μ) = Pn+1(x;μ) + bn+1Pn(x;μ) + a 2 n Pn−1(x;μ) is valid for n ∈ N0 := N ∪ {0}. The initial conditions P−1(x;μ) ≡ 0 and P0(x;μ) ≡ 1 generate two bounded sequences (an)n=1, (bn) ∞ n=1 of recurrence coefficients associated with μ. Here, an > 0, bn ∈ R for n ∈ N and ‖Pn(·;μ)‖L2(μ) = a1 · · · an. A bounded two sided C-valued sequence (dn)n=−∞ is called almost periodic if the set {(dn+k) ∞ n=−∞ : k ∈ Z} is precompact in l∞(Z). A one sided sequence (cn)n=1 is called almost periodic if it is the restriction of a two sided almost periodic sequence to N. A sequence (en)n=1 is called asymptotically almost periodic if there is an almost periodic sequence (e ′ n) ∞ n=1 such that |en − e ′ n| → 0 as n→ 0. The class of Parreau-Widom sets plays a special role in the recent theory of orthogonal and Chebyshev polynomials. Let K be a non-polar compact set and gC\K denote the Green function for C \ K with a pole at infinity. Suppose K is regular with respect to the Dirichlet problem, so the set C of critical points of gC\K is at most countable (see e.g. Chapter 2 in [9]). Then K is said to be a Parreau-Widom set if ∑ c∈C gC\K(c)<∞. Parreau-Widom sets on R have positive Lebesgue measure. For different aspects of such sets, see [8, 15, 23]. The class of regular measures in the sense of Stahl-Totik can be defined by the following condition lim n→∞ Wn(μ) 1/n = 1. aBilkent University, 06800, Ankara, Turkey. Alpan · Goncharov 14 For a measure μ supported on R we use the Lebesgue decomposition of μ with respect to the Lebesgue measure: dμ(x) = f (x)d x + dμs(x). Following [9], we define the Szegő class Sz(K) of measures on a given Parreau-Widom set K ⊂ R. Let μK be the equilibrium measure on K . By ess supp(·) we denote the essential support of the measure, that is the set of accumulation points of the support. We have Cap(supp(μ)) = Cap(ess supp(μ)), see Section 1 of [21]. A measure μ is in the Szegő class of K if (i) ess supp(μ) = K . (ii) ∫ K log f (x) dμK(x)> −∞. (Szegő condition) (iii) the isolated points {xn} of supp(μ) satisfy ∑