Filled Julia sets of Chebyshev polynomials

TL;DR

This paper investigates the limiting behavior of Julia and filled Julia sets of dual Chebyshev polynomials associated with a compact set in the complex plane, and shows convergence of certain measures to the equilibrium measure.

Contribution

It provides new insights into the limits of Julia sets and the convergence of maximal entropy measures for dual Chebyshev polynomials.

Findings

Hausdorff limits of Julia sets are characterized

Measures of maximal entropy converge weak* to equilibrium measure

Comparison between Julia set limits and the original set K

Abstract

We study the possible Hausdorff limits of the Julia sets and filled Julia sets of subsequences of the sequence of dual Chebyshev polynomials of a non-polar compact set K in C and compare such limits to K. Moreover, we prove that the measures of maximal entropy for the sequence of dual Chebyshev polynomials of K converges weak* to the equilibrium measure on K.