Extremal polynomials and polynomial preimages

Abstract

This article examines the asymptotic behavior of the Widom factors, denoted ${\mathcal{W}}_n$, for Chebyshev polynomials of finite unions of Jordan arcs. We prove that, in contrast to Widom's proposal, when dealing with a single smooth Jordan arc, ${\mathcal{W}}_n$ converges to 2 exclusively when the arc is a straight line segment. Our main focus is on analysing polynomial preimages of the interval $[-2,2]$, and we provide a complete description of the asymptotic behavior of ${\mathcal{W}}_n$ for symmetric star graphs and quadratic preimages of $[-2,2]$. We observe that in the case of star graphs, the Chebyshev polynomials and the polynomials orthogonal with respect to equilibrium measure share the same norm asymptotics, suggesting a potential extension of a conjecture posed by Christiansen, Simon and Zinchenko. Lastly, we propose a possible connection between the $S$-property and Widom factors converging to $2$.