On Dynamics Of Asymptotically Minimal Polynomials

Official URL: https://risc01.sabanciuniv.edu/record=b3205725

In this thesis, we study dynamical properities of asymptotically extremal polynomials associated with a non-polar planar compact set E. In particular, we prove that if the zeros of such polynomials are uniformly bounded then their Brolin measures, wn’s, converge weakly to the equilibrium measure of E. For this, we observe that {wn}n is sequentially pre-compact with respect to the weak∗ -topology and if ν is the weak∗ limit, then the support of this measure contained in the support of the equilibrium measure of E. Approximating compact sets by fractals is a fruitful technique and used for different problems in complex analysis such as the universal dimension spectrum for harmonic measures. Another aspect of research in this context is approximating a given planar compact set by polynomial filled Julia sets (respectively Julia sets) with respect to the Hausdorff topology. In the second part of this thesis, we consider the problem of classifying all possible limit sets of a sequence of filled Julia sets of asymptotically minimal polynomials. First, we observe that the sequence of filled Julia sets (respectively Julia sets) of asymptotically minimal polynomials may not converge in the Hausdorff topology. On the other hand, we prove that if E is regular in the sense of Dirichlet problem and the zeros of such polynomials are sufficiently close to E then the filled Julia sets converge to the polynomial convex hull of E in the Klimek topology. Moreover, we prove that for any Hausdorff-limit set of filled Julia sets, the polynomial convex hull of this limit set coincide with the polynomial convex hull of E. Finally, we discuss possible generalizations of these results to multi-dimensional setting.