Compact Sets in Petals and their Backward Orbits under Semigroups of Holomorphic Functions

Abstract

Let (φt) t≥0 be a semigroup of holomorphic functions in the unit disk D and K a compact subset of D. We investigate the conditions under which the backward orbit of K under the semigroup exists. Subsequently, the geometric characteristics, as well as, potential theoretic quantities for the backward orbit of K are examined. More specifically, results are obtained concerning the asymptotic behavior of its hyperbolic area and diameter, the harmonic measure and the capacity of the condenser that K forms with the unit disk.