Geometric description of some Loewner chains with infinitely many slits

The chordal Loewner PDE describes the dynamics of a continuous and decreasing family of simply connected domains of the upper half-plane ℍ. In this talk, we present and describe geometrically explicit solutions to the chordal Loewner equation associated with certain driving functions that produce infinitely many slits. Specifically, for a choice of a sequence of positive numbers ⟮𝑏_𝑛⟯_𝑛⩾𝟷 and points of the real line ⟮𝑘_𝑛⟯_𝑛⩾𝟷, we explicitly solve the Loewner PDE

        ∂_𝑡𝑓⟮𝑧,𝑡⟯ = −𝑓⟮𝑧,𝑡⟯ 𝛴_𝑛⩾𝟷 𝟸𝑏_𝑛 ∕  ⟮𝑧 − 𝑘_𝑛 √⟮𝟷 − 𝑡⟯⟯

in ℍ⨯[𝟶,𝟷), with initial value 𝑓⟮𝑧,𝟶⟯ = 𝑧. We also see that there is a close connection to the theory of semigroups of holomorphic self-maps of ℍ, and using techniques involving the harmonic measure, we analyze the geometric
behaviour of its solutions, as 𝑡→𝟷⁻.