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Michal Rams: Path-dependant shrinking targets

Tid: To 2022-11-24 kl 13.00

Plats: Room 3418, Lindstedtsvägen 25

Föreläsare: Michal Rams (Institute of Mathematics, Polish Academy of Sciences, IMPAN)

Abstract: I will present my work, joint with Henna Koivusalo (Bristol) and Lingmin Liao (Wuhan), on the shrinking target sets, in which the size of the target depends not only on the iteration but also on the trajectory we take to hit it. That is, given a point $y$ and a function $\phi: X\to \mathbb R$, we look at the set X_\phi := \{x\in X; f^n(x) \in B(y, e^{S_n \phi(x)}) {\rm i.o.}\} and try to estimate the size of this set (for a special case of nonconformal locally affine expanding map). In the theory of shrinking target sets their size is usually related to finding zero of a properly chosen pressure function, and it will turn out that the same is true in our case; however, the potential of this pressure function is not going to be additive, so we will have to introduce some technical innovation (which might be of an independent interest).
This result is strongly related to (and only slightly more general than) a recent paper by Barany and Troscheit.