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Diffeomorphisms of the sphere conjugate to a rotation
8 votes

An irational rotation $f$ is arbitrarily distorted in $\text{Diff}^{\infty}S^2$, this was proved by Calegari and Freedman and in a more general setting by Militon,...

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what is the number of paths returning to 0 on the hexagonal lattice
7 votes

If you want a rough answer, it is something of the order of $\frac{3^n}{n}$. This random paths are easier than self avoiding walks, you can think of these paths in this way: If you consider the even ...

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