Geometric Description of Some Loewner Chains with Infinitely Many Slits

We study the chordal Loewner equation associated with certain driving functions that produce infinitely many slits. Specifically, for a choice of a sequence of positive numbers \((b_n)_{n\ge 1}\) ${(b_n)}_{n\ge 1}$ and points of the real line \((k_n)_{n\ge 1}\) ${(k_n)}_{n\ge 1}$ , we explicitily solve the Loewner PDE \(\begin{aligned} \dfrac{\partial f}{\partial t}(z,t)=-f'(z,t)\sum _{n=1}^{+\infty }\dfrac{2b_n}{z-k_n\sqrt{1-t}} \end{aligned}\) $$\begin{array}{c}\hfill {\displaystyle \frac{\partial f}{\partial t}}(z,t)=-f^{\prime }(z,t)\sum _{n=1}^{+\infty }{\displaystyle \frac{2b_n}{z-k_n\sqrt{1-t}}}\end{array}$$ in \(\mathbb {H}\times [0,1)\) $\mathbb{H}\times [0,1)$ . Using techniques involving the harmonic measure, we analyze the geometric behaviour of its solutions, as \(t\rightarrow 1^-\) $t\to 1^{-}$ .