On Poincare series of unicritical polynomials at the critical point

TL;DR

This paper investigates the properties of conformal measures at the critical point of unicritical polynomials, establishing conditions under which atoms occur, and enhancing understanding of their dynamical behavior.

Contribution

It provides new criteria for the presence of atoms in conformal measures at the critical point of unicritical polynomials with a priori bounds.

Findings

Unique conformal measure of minimal exponent has no atom at the critical point.

Necessary and sufficient condition for atoms at the critical point based on derivative growth.

Clarifies the relationship between measure atoms and polynomial dynamics.

Abstract

In this paper, we show that for a unicritical polynomial having a priori bounds, the unique conformal measure of minimal exponent has no atom at the critical point. For a conformal measure of higher exponent, we give a necessary and sufficient condition for the critical point to be an atom, in terms of the growth rate of the derivatives at the critical value.