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On mixing diffeomorphisms of the disc. (English) Zbl 1448.37052

A pseudo-rotation \(f\) is a homeomorphism of \(\mathbb{D} = \{(x,y) \in \mathbb{R}^2 \colon x^2+y^2 \leq 1\}\) that preserves the Lebesgue measure, fixes the origin, and has no other periodic points.
B. Bramham showed in [Invent. Math. 199, No. 2, 561–580 (2015; Zbl 1353.37007)] that when the rotation number of a pseudo-rotation is very well approximated by rationals then the pseudo-rotation is \(C^0\)-rigid.
In this paper the authors generalize, on the basis of Franks’ free disc Lemma, the above result:
(i) They extend the result to non-Brjuno rotation numbers;
(ii) They determine rigidity in higher regularity.
Further, the authors give a quantitive gap of \(||D f^m||\) between exponential and sub-exponential growth for an area-preserving disc diffeomorphism of class \(C^2\), thus proving rigidity for non-Brjuno rotation numbers.
It is pointed out by the authors that their approach to rigidity of pseudo-rotations cannot go much beyond non-Brjuno type numbers, irrespective of the regularity of the pseudo-rotation. Thus, to cover the case of less Liouvillean numbers, they use KAM theory.
Last but not least, the introduction of this article ends with a series of questions and comments about the rigidity of pseudo-rotations, which emphasize the importance and vitality of this topic in dynamical systems theory.

MSC:

37E30 Dynamical systems involving homeomorphisms and diffeomorphisms of planes and surfaces
37E45 Rotation numbers and vectors
37C05 Dynamical systems involving smooth mappings and diffeomorphisms
37E10 Dynamical systems involving maps of the circle
37B40 Topological entropy
37A25 Ergodicity, mixing, rates of mixing
37J40 Perturbations of finite-dimensional Hamiltonian systems, normal forms, small divisors, KAM theory, Arnol’d diffusion

Citations:

Zbl 1353.37007
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References:

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