Non-rigid disk pseudo-rotations in $C^1$ and $C^{1,\beta }$

Abstract

We construct area-preserving irrational pseudo-rotations of the closed disk that fail to be $C^0$-rigid. For every Brjuno rotation number we obtain $C^1$ examples, and they may be chosen arbitrarily $C^1$-close to the rigid rotation with the same rotation number. For $0<\beta <1$, a weighted Brjuno condition gives $C^{1,\beta }$ examples that are arbitrarily close in the $C^{1,\beta }$ topology; this weighted range contains every rotation number with denominator-growth exponent strictly smaller than $1/\beta$ and also some numbers at the critical exponent. For a suitable Brjuno--Liouville rotation number, we also obtain an example whose derivative growth is not bounded by any polynomial.