Recent zbMATH articles in MSC 37A20https://www.zbmath.org/atom/cc/37A202021-05-28T16:06:00+00:00WerkzeugOn turbulent relations.https://www.zbmath.org/1459.030782021-05-28T16:06:00+00:00"López, Jesús A. Álvarez"https://www.zbmath.org/authors/?q=ai:alvarez-lopez.jesus-a"Candel, Alberto"https://www.zbmath.org/authors/?q=ai:candel.albertoSummary: This paper extends the theory of turbulence of Hjorth to certain classes of equivalence relations that cannot be induced by Polish actions. The results are applied to analyze the quasi-isometry relation and finite Gromov-Hausdorff distance relation in the space of isometry classes of pointed proper metric spaces, called the Gromov space.Cocycle superrigidity for translation actions of product groups.https://www.zbmath.org/1459.220062021-05-28T16:06:00+00:00"Gaboriau, Damien"https://www.zbmath.org/authors/?q=ai:gaboriau.damien"Ioana, Adrian"https://www.zbmath.org/authors/?q=ai:ioana.adrian"Tucker-Drob, Robin"https://www.zbmath.org/authors/?q=ai:tucker-drob.robin-dLet \(G\) be either a profinite or a connected compact group, and \(\Gamma, \Lambda\) be finitely generated dense subgroups. Assuming that the left translation action of \(\Gamma\) on \(G\) is strongly ergodic, it is proven that any cocycle for the left-right translation action of \(\Gamma\times\Lambda\) on \(G\) with values in a countable group is ``virtually'' cohomologous to a group homomorphism.
Moreover, it is shown that the same holds if \(G\) is a (not necessarily compact) connected simple Lie group provided that \(\Lambda\) contains an infinite cyclic subgroup with compact closure. Additionally the first examples of compact actions of \(\mathbb F_2\times\mathbb F_2\) which are W\(^*\)-superrigid are obtained.
Reviewer: Michael L. Blank (Moskva)Triangles in Diophantine approximation.https://www.zbmath.org/1459.110182021-05-28T16:06:00+00:00"Mundici, Daniele"https://www.zbmath.org/authors/?q=ai:mundici.danieleSummary: For any point \(x = (x_1, x_2) \in \mathbb{R}^2\) we let \(G_x = \mathbb{Z} x_1 + \mathbb{Z} x_2 + \mathbb{Z}\) be the subgroup of the additive group \(\mathbb{R}\) generated by \(x_1, x_2, 1\). When \(\operatorname{rank}(G_x) = 3\) we say that \(x\) is a \textit{rank 3 point.} We prove the existence of an infinite set \(\mathcal{I} \subseteq \mathbb{R}^2\) of rank 3 points having the following property: For every two-dimensional continued fraction expansion \({\mu}\) and \(x \in \mathcal{I}\), letting \(\mu(x) = T_0 \supseteq T_1 \supseteq \cdots\), it follows that infinitely many triangles \(T_n\) have some angle \(\leq \arcsin(23^{1 / 2} / 6)\approx \pi /(3.3921424) \approx 53^\circ\). Thus \(\lim \inf_{n \rightarrow \infty} \operatorname{area}(T_n) / \operatorname{diam}(T_n)^2 \leq 23^{1 / 2} / 12\).
At the opposite extreme, we construct a two-dimensional continued fraction expansion \({\mu}\) and a dense set \(\mathcal{D} \subseteq \mathbb{R}^2\) of rank 3 points such that for each \(x \in \mathcal{D}\) the sequence \(T_0 \supseteq T_1 \supseteq \cdots\) of triangles of \(\mu(x)\) has the following property: Letting \(\omega_n\) denote the smallest angle of \(T_n\), it follows that \(\omega_0 < \omega_1 < \cdots\) and \(\lim_{n \rightarrow \infty} \omega_n = \pi / 3\). Further, the other two angles of \(T_n\) are \(> \pi / 3\). Thus \(\lim_{n \rightarrow \infty} \operatorname{area}(T_n) / \operatorname{diam}(T_n)^2 = 3^{1 / 2} / 4\), and the vertices of the triangles \(T_n\) strongly converge to \(x\). Our proofs combine a classical theorem of Davenport and Mahler with binary stellar operations of regular fans.Parabolic invariant tori in quasi-periodically forced skew-product maps.https://www.zbmath.org/1459.370512021-05-28T16:06:00+00:00"Guan, Xinyu"https://www.zbmath.org/authors/?q=ai:guan.xinyu"Si, Jianguo"https://www.zbmath.org/authors/?q=ai:si.jianguo"Si, Wen"https://www.zbmath.org/authors/?q=ai:si.wenSummary: We consider the existence of parabolic invariant tori for a class of quasi-periodically forced analytic skew-product maps \(\varphi : \mathbb{R}^n \times \mathbb{T}^d \to \mathbb{R}^n \times \mathbb{T}^d\):
\[
\varphi \begin{pmatrix} z \\ \theta \end{pmatrix} = \begin{pmatrix} z + \phi (z) + h (z, \theta) + \epsilon f (z, \theta) \\ \theta + \omega \end{pmatrix},
\]
where \(\phi : \mathbb{R}^n \to \mathbb{R}^n\) is a homogeneous function of degree \(l\) with \(l \geq 2\) and \(h = \mathcal{O}(|z|^{l + 1})\). We obtain the following results: (a) For \(n = 1, l\) being odd and \(\varepsilon\) sufficiently small, parabolic invariant tori exist if \(\omega\) satisfies the Brjuno-Rüssmann's non-resonant condition. (b) For \(n = 1\), and \(\varepsilon\) sufficiently small, parabolic invariant tori also exist if one of the following conditions holds: (i) First order average is non-zero, first order non-average part is small enough and the forcing frequency \(\omega\) does not need any arithmetic condition. (ii) First order average is non-zero and \(\omega\) satisfies the Brjuno-type weak non-resonant condition; (iii) \(l = 2\), first order average is zero, both first and second order non-average parts are small enough and \(\omega\) satisfying Brjuno-type weak non-resonant condition; (iv) \(l > 2\), first order average is zero, the second order average is non-zero, both first and second order non-average parts are small enough and \(\omega\) satisfies the Brjuno-type weak non-resonant condition. (c) In the case \(n > 1\), if first order average belongs to the interior of the range of \(\varphi, Spec(D \phi) \cap \text{i} \mathbb{R} = \emptyset\), first order non-average part is small enough and the forcing frequency \(\omega\) does not need any arithmetic condition, then the quasi-periodically forced skew-product maps above admit parabolic invariant tori for \(\varepsilon\) sufficiently small. The main methods of this paper are KAM theory and fixed point theorem, which are finally shown that it can be directly applied to the existence problem of quasi-periodic response solutions of degenerate harmonic oscillators.Classification of regular subalgebras of the hyperfinite II\(_1\) factor.https://www.zbmath.org/1459.460542021-05-28T16:06:00+00:00"Popa, Sorin"https://www.zbmath.org/authors/?q=ai:popa.sorin-teodor"Shlyakhtenko, Dimitri"https://www.zbmath.org/authors/?q=ai:shlyakhtenko.dimitri-l"Vaes, Stefaan"https://www.zbmath.org/authors/?q=ai:vaes.stefaanThe present articles achieves three aims. First it classifies regular subalgebras of the amenable II\(_1\) factor \(B \subset R\) that satisfy the freeness condition \(B' \cap R = \mathcal{Z}(B)\) in terms of an associated discrete measurable groupoid. Second, in order to obtain this result, it proves a cocycle vanishing theorem for free actions of amenable discrete measured groupoids on II\(_1\) von Neumann algebras. Third, to illustrate the complexity of the resulting classification, the authors obtain results on the model-theoretic complexity of classifying amenable discrete measured groupoids.
It is known by work of \textit{A. Connes} et al. [Ergodic Theory Dyn. Syst. 1, 431--450 (1981; Zbl 0491.28018)] that the amenable II\(_1\) factor \(R\) has a unique Cartan subalgebra up to isomorphism, that is, a regular von Neumann subalgebra \(A \subset R\) such that \(A' \cap R = A\). This is essentially a uniqueness theorem for the amenable ergodic II\(_1\) equivalence relation combined with a cocyle vanishing theorem. Indeed, \textit{J. Feldman} and \textit{C. C. Moore} [Trans. Am. Math. Soc. 234, 289--324 (1977; Zbl 0369.22009); ibid. 234, 325--359 (1977; Zbl 0369.22010)] proved that \(A \subset R\) is isomorphic to \(\mathrm{L}^\infty(X) \subset \mathrm{L}(\mathcal{R}, \sigma)\) for a II\(_1\) equivalence relation \(\mathcal{R}\) twisted by a 2-cocycle \(\sigma\). The starting point of the present classification result is an analogue description of regular inclusions \(B \subset R\) satisfying \(B' \cap R = \mathcal{Z}(B)\) as a twisted crossed product of an amenable discrete measured groupoid acting freely on \(B\). The cocycle vanishing result used to conclude the authors' description of such inclusions subsumes many previously known results in the framework of groups and equivalence relations, and at the same time uses those in its proof.
The article is clearly structured and well written. In particular, it addresses the problem context in a concise way.
Reviewer: Sven Raum (Stockholm)