Imagine you're exploring the fascinating world of mathematics, specifically something called "rational maps" on something called ${\mathbb P}^1_{\mathbb C}$. Now, some clever folks like Epstein have already figured out some neat principles for these maps. But what if we could extend these principles to something even bigger and more complex? That's where Thurston comes in.

Thurston found a way to take Epstein's ideas and apply them to a special kind of set called the "Fatou set". The trick? Using something called a "topos of $f$ invariant sheaves" and Grothendieck's six operations. It might sound complicated, but think of it like having a magical toolbox that helps you count something called "non-repelling invariant cycles".

Now, here's where it gets really interesting. By applying this magical toolbox, Thurston discovered that the count of these cycles is even better than what Epstein and Shishikura had come up with. It's like finding a shortcut that makes the whole process more efficient.

But wait, there's more! While playing around with these ideas at something called "parabolic fixed points", Thurston found something amazing. He calculated something called the "dualising sheaf" of a real blow up. This is like finding a hidden treasure that not only helps with the main problem but also simplifies a whole other area of mathematics called "resurgent functions and Stokes' phenomenon".

The Hidden Beauty of Algebraic Transversality

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