Orthogonal polynomials for the weakly equilibrium Cantor sets (Q2814400)

Given a sequence \(\gamma=\{\gamma_s\}_{s=1}^\infty\) with \(0<\gamma_s<1/4\) define \(r_0=1\) and \(r_s=\gamma_s r_{s-1}^2\). Let \(P_1(x)=x-1\) and \(P_{2^{s+1}}(x)=P_{2^{s}}(x)\cdot (P_{2^{s}}(x)+r_s)\) for \(s=0,1,\dots\). For \(s=0,1,\dots\) consider a nested sequence of sets \(E_s=\{x\in\mathbb R: P_{2^{s+1}}(x)\leq 0\}\). Then \(K(\gamma):=\cap_{s=0}^\infty E_s\) is a Cantor type set. Let \(\mu_{K(\gamma)}\) denote the equilibrium measure on \(K(\gamma)\). One of the main results is: The monic orthogonal polynomial \(Q_{2^s}\) of degree \(2^s\) with respect to \(\mu_{K(\gamma)}\) coincides with \(P_{2^{s}}(x)+r_s/2\), which is known to be the corresponding Chebyshev polynomial for \({K(\gamma)}\) in [\textit{A. P. Goncharov}, Potential Anal. 40, No. 2, 143--161 (2014; Zbl 1283.31001)]. (A monic polynomial \(P\) of degree \(n\) is a Chebyshev polynomial for a compact set \(K\), if the value \(\| P\|_{L^\infty(K)}\) is minimal among all monic polynomials of degree \(n\).) The authors discuss also \(Q_n\) of all degrees, and related objects.