Sinai's principle in complex dynamics

We will discuss Sinai's principle for recurrence in the unicritical dynamics, from the point of view of harmonic measure: for almost all 𝑐 with respect to harmonic measure on the boundary of the Mandelbrot set, | 𝑓_𝑐^𝑛(𝑐) − 𝑐 | ≤ 𝑛^-𝑎 for infinitely many 𝑛 if and only if 𝑎 ≤ 1. This talk is based on joint work with Graczyk and Świątek.