Title of Seminar: Infosys Chandrasekharan Random Geometry Colloquium
Title of Talk: Brolin’s theorem for finitely generated polynomial semigroups
Speaker: Mayuresh Londhe, IISc
Date: August 23, 2021
Time: 16:00:00 Hours
Venue: AG-77
Abstract: In this talk, we give a description of a natural invariant measure associated with a finitely generated polynomial semigroup (which we shall call the Dinh--Sibony measure) in terms of potential theory. The existence of this measure follows from a very general result of Dinh--Sibony applied to a holomorphic correspondence that one can naturally associate with a semigroup of the above type. We are interested in a precise description of this invariant measure. This requires the theory of logarithmic potentials in the presence of an external field, which, in our case, is explicitly determined by the choice of a set of generators. Our result generalizes the classical result by Brolin. Along the way, we establish the continuity of the logarithmic potential for the Dinh--Sibony measure, which might be of independent interest. If time permits, we shall also present some bounds on the capacity and diameter of the Julia sets of such semigroups, which uses the F-functional of Mhaskar and Saff.
Title of Talk: Brolin’s theorem for finitely generated polynomial semigroups
Speaker: Mayuresh Londhe, IISc
Date: August 23, 2021
Time: 16:00:00 Hours
Venue: AG-77
Abstract: In this talk, we give a description of a natural invariant measure associated with a finitely generated polynomial semigroup (which we shall call the Dinh--Sibony measure) in terms of potential theory. The existence of this measure follows from a very general result of Dinh--Sibony applied to a holomorphic correspondence that one can naturally associate with a semigroup of the above type. We are interested in a precise description of this invariant measure. This requires the theory of logarithmic potentials in the presence of an external field, which, in our case, is explicitly determined by the choice of a set of generators. Our result generalizes the classical result by Brolin. Along the way, we establish the continuity of the logarithmic potential for the Dinh--Sibony measure, which might be of independent interest. If time permits, we shall also present some bounds on the capacity and diameter of the Julia sets of such semigroups, which uses the F-functional of Mhaskar and Saff.