Consider the Loewner equation associated to the upper-half plane. This is an equation originated from an extremal problem in complex analysis. Nowadays, it attracts a lot of attention due to its connection to probability. Normally this equation is driven by a real-valued function. In this talk, we show that when the driving function is complex-valued with small 1/2-Hölder norm, the corresponding hull is a quasi-arc, hence is a simple curve. We also study how the hull changes with respect to complex parameters and make a connection between the Loewner equation and complex dynamics.
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