Recent zbMATH articles in MSC 37Ahttps://www.zbmath.org/atom/cc/37A2021-05-28T16:06:00+00:00WerkzeugNonequilibrium molecular dynamics, fractal phase-space distributions, the Cantor set, and puzzles involving information dimensions for two compressible baker maps.https://www.zbmath.org/1459.370802021-05-28T16:06:00+00:00"Hoover, William G."https://www.zbmath.org/authors/?q=ai:hoover.william-graham"Hoover, Carol G."https://www.zbmath.org/authors/?q=ai:hoover.carol-gSummary: Deterministic and time-reversible nonequilibrium molecular dynamics simulations typically generate ``fractal'' (fractional-dimensional) phase-space distributions. Because these distributions and their time-reversed twins have zero phase volume, stable attractors ``forward in time'' and unstable (unobservable) repellors when reversed, these simulations are consistent with the second law of thermodynamics. These same reversibility and stability properties can also be found in compressible baker maps, or in their equivalent random walks, motivating their careful study. We illustrate these ideas with three examples: a Cantor set map and two linear compressible baker maps, \(N2(q,p)\) and \(N3(q,p)\). The two baker maps' information dimensions estimated from sequential mappings agree, while those from pointwise iteration do not, with the estimates dependent upon details of the approach to the maps' nonequilibrium steady states.Nonuniformly hyperbolic attractors. Geometric and probabilistic aspects.https://www.zbmath.org/1459.370012021-05-28T16:06:00+00:00"Alves, José F."https://www.zbmath.org/authors/?q=ai:alves.jose-ferreiraThe Palis conjecture says that typical dynamical systems on finite-dimensional Riemannian manifolds have a finite number of measures -- often called physical measures -- that describe time averages of almost all orbits with respect to Lebesgue measure. The notion of a typical system is a bit vague, but it includes, for example, systems that are members of open and dense subsets of a space of transformations in some appropriate topology.
The main purpose of this book is to explore some classes of dynamical systems for which the Palis conjecture has already been proved, and to consider both the existence of physical measures and some of their statistical properties. The question of the existence of physical measures is related in some respects to the existence of attractors for the related dynamical systems.
A Borel measure \(\mu\) on a compact finite-dimensional Riemannian manifold \(M\) is a physical measure for \(f: M \rightarrow M\) if there exists a set of initial states \(x \in M\) of positive Lebesgue measure such that, for any continuous \(\varphi : M \rightarrow \mathbb{R}\), \(\lim_{n \to \infty} \frac{1}{n} \sum_{j = 0} ^ {n-1} \varphi( f^j (x)) = \int \varphi d\mu\). Such a physical measure also satisfies \(\mu(f^{-1}(A)) = \mu (A)\) for every Borel set \(A \subset M\).
The first part of the book provides introductory material and establishes the background and setting. The Palis conjecture has been established for uniformly hyperbolic structures. From the work of Sinai, Ruelle and Bowen (SRB) it follows that uniformly hyperbolic systems have a finite number of physical measures whose basins cover almost all points of the ambient manifold in the sense of Lebesgue measure. Coding systems with a finite number of symbols using finite Markov partitions and then conjugating to a subshift of finite type was important in establishing the SRB results.
The problem of the existence and finiteness of physical measures is open for nonuniformly hyperbolic systems. The work [\textit{L.-S. Young}, Ann. Math. (2) 147, No. 3, 585--650 (1998; Zbl 0945.37009)] for example, introduced Markov-like partitions with infinitely many symbols and an inducing scheme that defines new dynamical systems with uniformly hyperbolic behavior. This procedure allows retrieval of statistical properties from the original system.
The main part of the book then begins with a discussion of expanding structures. Gibbs-Markov maps are introduced and Young's construction of a tower extension associated with an induced Gibbs-Markov map is described. Here the author derives the existence of ergodic invariant probability measures that are absolutely continuous with respect to a reference measure, and draws conclusions regarding the exponential decay of correlations with respect to these measures. Following this Young structures and their corresponding tower extensions are introduced. For systems with Young structures the author shows the existence of SRB measures and estimates the decay of correlations with respect to these measures.
The author then establishes a general framework that is appropriate for both nonuniformly expanding and partially hyperbolic systems. Inducing schemes and a proof of the integrability of inducing times are important pieces. Results on the estimation of the tails of recurrence times are also described here.
The book concludes with treatments of nonuniformly expanding attractors and partially hyperbolic attractors. The author describes results for a class of nonuniformly expanding maps and for partially hyperbolic attractors with a uniformly contacting direction and nonuniform expansion in the center-unstable direction. Specifically, he proves the existence of SRB measures and the rates of decay correlations with respect to SRB measures.
Reviewer: William J. Satzer Jr. (St. Paul)Bounds for multiple recurrence rate and dimension.https://www.zbmath.org/1459.370032021-05-28T16:06:00+00:00"Hirayama, Michihiro"https://www.zbmath.org/authors/?q=ai:hirayama.michihiroFor a probability measure-preserving system \((X,\mathcal{B},\mu,T)\) and a set \(A\in\mathcal{B}\) with \(\mu(A)>0\), the Poincaré recurrence theorem states that \(\mu\)-almost every point in \(A\) returns to \(A\) infinitely often. If the space \(X\) has a compatible metric \(d\) for which \(\mathcal{B}\) is the Borel \(\sigma\)-algebra then a result of \textit{M. D. Boshernitzan} [Invent. Math. 113, No. 3, 617--631 (1993; Zbl 0839.28008)] gives qualitative information about the closeness of returns, showing that \(\liminf_{n\to\infty}d(x,T^nx)=0\) for \(\mu\)-almost every \(x\in X\) and, under the geometric hypothesis that the \(\alpha\)-Hausdorff measure of \(X\) is finite for some \(\alpha>0\), goes on to show that \(\liminf_{n\to\infty}\bigl(n^{1/\alpha}d(x,T^nx)\bigr)<\infty\) for \(\mu\)-almost every \(x\in X\). \textit{L. Barreira} and \textit{B. Saussol} [Commun. Math. Phys. 219, No. 2, 443--463 (2001; Zbl 1007.37012)] gave estimates for the first return time to a metric ball, and \textit{D. H. Kim} [Nonlinearity 22, No. 1, 1--9 (2009; Zbl 1167.37006)] generalized Boshernitzan's results to actions of countable discrete groups.
Here a multiple and simultaneous analogue of these results are found; the results are too complicated to state here but the flavor is to find quantitative versions of statements of the form \(\liminf_{n\to\infty}\mathrm{diam} \{x,T^nx,T^{2n}x,\dots,T^{Ln}x\}=0\) for \(\mu\)-almost every \(x\in X\). The methods used are diverse and they include results of \textit{W. T. Gowers} [Geom. Funct. Anal. 11, No. 3, 465--588 (2001); Erratum 11, No. 4, 869 (2001; Zbl 1028.11005)] on the quantitative Szemerédi theorem to study long simultaneous return.
Reviewer: Thomas B. Ward (Leeds)On 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.Thin annuli property and exponential distribution of return times for weakly Markov systems.https://www.zbmath.org/1459.370082021-05-28T16:06:00+00:00"Pawelec, Łukasz"https://www.zbmath.org/authors/?q=ai:pawelec.lukasz"Urbański, Mariusz"https://www.zbmath.org/authors/?q=ai:urbanski.mariusz"Zdunik, Anna"https://www.zbmath.org/authors/?q=ai:zdunik.annaA common feature of many results on return times in metric or geometric settings is a need for subtle upper estimates on the measure of shrinking annuli. Specifically, many settings in which a proof of the exponential law is proved involves proving that the measure of thin annuli are small compared to the measure of the inner ball. Indeed, in some settings this is a hypothesis required for the result, and a consequence is that exponential limiting laws are generally only known in settings of measures equivalent to the Lebesgue measure or if the measure of every ball is known to be bounded above by its radius raised to a power larger than \(d-1\) for some other reason. Here a very general class of settings are defined for which this problem can be resolved, giving a solution to the problem of asymptotic distribution of first return times to shrinking balls for a large class of dynamical systems (the weakly Markov systems). Applications include conformal iterated function systems, rational functions on the Riemann sphere, transcendental meromorphic functions on the complex plane, and to the settings of expanding repellers and holomorphic endomorphisms of complex projective spaces. For conformal iterated function systems, the ``full thin annuli property'' is established, giving the same estimate for all radii and proving the exponential law along all radii for essentially all conformal iterated function systems.
Reviewer: Thomas B. Ward (Leeds)A slow triangle map with a segment of indifferent fixed points and a complete tree of rational pairs.https://www.zbmath.org/1459.370072021-05-28T16:06:00+00:00"Bonanno, Claudio"https://www.zbmath.org/authors/?q=ai:bonanno.claudio"Del Vigna, Alessio"https://www.zbmath.org/authors/?q=ai:del-vigna.alessio"Munday, Sara"https://www.zbmath.org/authors/?q=ai:munday.saraA two-dimensional version of a continued fraction introduced by \textit{T. Garrity} [J. Number Theory 88, No. 1, 86--103 (2001; Zbl 1015.11031)] is studied. This defines a map defined on a triangle, and the associated expansions are referred to as triangle sequences. The results here give further parallels between the classical Gauss map and this triangle map. In particular, the role played by the Farey map for the Gauss map finds a parallel in a certain piecewise linear fractional map on the triangle constructed here and shown to preserve an ergodic infinite absolutely continuous measure. Results from infinite ergodic theory are applied to derive weak laws of large numbers and thence an analogue of Khinchin's weak law for the digits of the triangle sequences.
Reviewer: Thomas B. Ward (Leeds)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.Lyapunov exponents of two stochastic Lorenz 63 systems.https://www.zbmath.org/1459.370662021-05-28T16:06:00+00:00"Geurts, Bernard J."https://www.zbmath.org/authors/?q=ai:geurts.bernard-j"Holm, Darryl D."https://www.zbmath.org/authors/?q=ai:holm.darryl-d"Luesink, Erwin"https://www.zbmath.org/authors/?q=ai:luesink.erwinSummary: Two different types of perturbations of the Lorenz 63 dynamical system for Rayleigh-Bénard convection by multiplicative noise -- called stochastic advection by Lie transport (SALT) noise and fluctuation-dissipation (FD) noise -- are found to produce qualitatively different effects, possibly because the total phase-space volume contraction rates are different. In the process of making this comparison between effects of SALT and FD noise on the Lorenz 63 system, a stochastic version of a robust deterministic numerical algorithm for obtaining the individual numerical Lyapunov exponents was developed. With this stochastic version of the algorithm, the value of the sum of the Lyapunov exponents for the FD noise was found to differ significantly from the value of the deterministic Lorenz 63 system, whereas the SALT noise preserves the Lorenz 63 value with high accuracy. The Lagrangian averaged version of the SALT equations (LA SALT) is found to yield a closed deterministic subsystem for the expected solutions which is isomorphic to the original Lorenz 63 dynamical system. The solutions of the closed chaotic subsystem, in turn, drive a linear stochastic system for the fluctuations of the LA SALT solutions around their expected values.On the Legendre and the Lenstra constants for complex continued fractions introduced by J. Hurwitz.https://www.zbmath.org/1459.111632021-05-28T16:06:00+00:00"Nakada, Hitoshi"https://www.zbmath.org/authors/?q=ai:nakada.hitoshiThe present article deals with complex continued fraction expansions (JHCF). A brief description of investigations of these expansions is given.
The author notes that ``the purpose of this paper is to determine the Legendre constant of JHCF following the idea of \textit{H. Nakada} [J. Eur. Math. Soc. (JEMS) 12, No. 1, 55--70 (2010; Zbl 1205.11091)] in the case of Rosen's continued fractions associated with the Hecke group:
\begin{itemize}
\item showing the existence of the Legendre constant and
\item calculating the Lenstra constant of JHCF.
\end{itemize}
Then it turns out that the Legendre constant is equal to the Lenstra constant.''
Some basic properties and notions which are useful for proving the main results, are considered. The natural extension of the JHCF map is constructed. Several auxiliary statements are formulated. The main results from the main goal of this paper are proven with explanations. Connections between obtained results, used notions, and known researches, are considered.
Reviewer: Symon Serbenyuk (Kyïv)Subadditive ergodic theorem for double sequences.https://www.zbmath.org/1459.370052021-05-28T16:06:00+00:00"Kazakevičius, Vytautas"https://www.zbmath.org/authors/?q=ai:kazakevicius.vytautasSummary: A functional ergodic theorem is proved for subadditive families of measurable functions \((h_{k,n}\mid (k,n)\in D)\), where \(D\subset{\mathbb{N}}^2\) is an additive semigroup and subadditivity means that \(h_{k_1+k_2,n_1+n_2}\le h_{k_1,n_1}+h_{k_2,n_2}\circ f^{n_1}\) for some measure-preserving transformation \(f\).Recurrence for measurable semigroup actions.https://www.zbmath.org/1459.280142021-05-28T16:06:00+00:00"Blank, Michael"https://www.zbmath.org/authors/?q=ai:blank.michaelIn the paper under review, the author studies qualitative properties of the set of recurrent points of finitely generated free semigroups of measurable maps. In the case of a semigroup generated by a single element, the classical Poincaré recurrence theorem shows that these properties are closely related to the presence of an invariant measure. The author considers that for a general semigroup the assumption about the existence of a common invariant measure for all generators is somewhat unnatural. Instead, he proposes abstract conditions of conservativity type for this question, under which a (non-necessarily invariant) measure plays the same role as in the Poincaré recurrence case, together with a weaker version of the recurrent property. He reduces the study to the analysis of the recurrence of a specially constructed Markov process. In fact, the same approach was used for Markov chains in the author's previous work [Mosc. Math. J. 19, No. 1, 37--50 (2019; Zbl 1418.37010)]. The author deals in detail with questions of inheritance of a recurrence property from the semigroup generators to the entire semigroup and vice versa.
Reviewer: Athanase Papadopoulos (Strasbourg)A recurrence-weighted prediction algorithm for musical analysis.https://www.zbmath.org/1459.000112021-05-28T16:06:00+00:00"Colucci, Renato"https://www.zbmath.org/authors/?q=ai:colucci.renato"Leguizamon Cucunuba, Juan Sebastián"https://www.zbmath.org/authors/?q=ai:leguizamon-cucunuba.juan-sebastian"Lloyd, Simon"https://www.zbmath.org/authors/?q=ai:lloyd.simonSummary: Forecasting the future behaviour of a system using past data is an important topic. In this article we apply nonlinear time series analysis in the context of music, and present new algorithms for extending a sample of music, while maintaining characteristics similar to the original piece. By using ideas from ergodic theory, we adapt the classical prediction method of Lorenz analogues so as to take into account recurrence times, and demonstrate with examples, how the new algorithm can produce predictions with a high degree of similarity to the original sample.On the information-theoretic structure of distributed measurements.https://www.zbmath.org/1459.940702021-05-28T16:06:00+00:00"Balduzzi, David"https://www.zbmath.org/authors/?q=ai:balduzzi.davidSummary: The internal structure of a measuring device, which depends on what its components are and how they are organized, determines how it categorizes its inputs. This paper presents a geometric approach to studying the internal structure of measurements performed by distributed systems such as probabilistic cellular automata. It constructs the quale, a family of sections of a suitably defined presheaf, whose elements correspond to the measurements performed by all subsystems of a distributed system. Using the quale we quantify (i) the information generated by a measurement; (ii) the extent to which a measurement is context-dependent; and (iii) whether a measurement is decomposable into independent submeasurements, which turns out to be equivalent to context-dependence. Finally, we show that only indecomposable measurements are more informative than the sum of their submeasurements.
For the entire collection see [Zbl 1445.68018].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.The work of Lewis Bowen on the entropy theory of non-amenable group actions.https://www.zbmath.org/1459.370042021-05-28T16:06:00+00:00"Thouvenot, Jean-Paul"https://www.zbmath.org/authors/?q=ai:thouvenot.jean-paulThe aim of this paper is to present the achievements of Lewis Bowen. The main focus is on the problem of isomorphism of Bernoulli actions of countable non-amenable groups which he solved brilliantly in the remarkable papers [\textit{L. P. Bowen}, Ann. Math. (2) 171, No. 2, 1387--1400 (2010; Zbl 1201.37007); \textit{L. Bowen}, J. Am. Math. Soc. 23, No. 1, 217--245 (2010; Zbl 1201.37005)]. Here two invariants were introduced, which led to many developments. One of them is the extension of the definition of sofic measure theoretic entropy. There are also nice implications related to spectral theory.
Reviewer: Michael L. Blank (Moskva)Fourier approximation of the statistical properties of Anosov maps on tori.https://www.zbmath.org/1459.370812021-05-28T16:06:00+00:00"Crimmins, Harry"https://www.zbmath.org/authors/?q=ai:crimmins.harry"Froyland, Gary"https://www.zbmath.org/authors/?q=ai:froyland.garyThe authors investigate the stability of statistical properties of Anosov maps \(T : \mathbb{T}^{d} \circlearrowleft\) on tori by examining the stability of the spectrum of an analytically twisted Perron-Frobenius operator on the anisotropic Banach spaces of \textit{S. Gouëzel} and \textit{C. Liverani} [Ergodic Theory Dyn. Syst. 26, No. 1, 189--217 (2006; Zbl 1088.37010)].
The authors provide briefly review results concerning the spectral stability of twisted quasi-compact operators. They give the abstract requirements for the Nagaev-Guivarc'h method to guarantee a central limit theorem and a large deviation principle, and mention an abstract hypothesis that yields stability of the variance and rate function using a modification of the abstract stability. To extend their results in [Ann. Henri Poincaré 20, No. 9, 3113--3161 (2019; Zbl 1431.37023)] to general perturbations, they find the stability of various statistical properties of Anosov maps, including new classes of numerical approximations. In particular, the authors get new results on the stability of the rate function for Anosov diffeomorphisms under deterministic perturbations. The authors consider the Sinai-Ruelle-Bowen (SRB) measure \(\mu\) on \(\mathbb{T}^{d}\), the limiting variance of scaled Birkhoff sums of a smooth observation function \(g : \mathbb{T}^{d} \rightarrow \mathbb{R}\) guaranteed by the central limit theorem (CLT), and the rate function associated with large deviations of Birkhoff sums of \(g\) from \(\mathbb{E}_{\mu}(g)\) as guaranteed by a large deviation principle (LDP).
The authors introduce two new rigorous and computationally practical approaches. Each is based on Fourier approximation, which is a natural basis for the periodic domain \(\mathbb{T}^{d}\), and can exploit the smoothness in the map \(T\). Their first Fourier-analytic scheme builds finite-rank approximations of the Perron-Frobenius-operator in a two-step process involving convolution, but the convolution is cheaply implementable via Fourier methods. This scheme is widely applicable due to additional mollification, allowing the scheme to smooth away the complications of hyperbolic dynamics. Their second scheme builds a single sequence of finite-rank approximations of the Perron-Frobenius operator that avoids convolution altogether. This scheme removes the mollification step for Anosov maps with approximately constant stable and unstable directions; they attain a pure Fourier projection method based on the Fej'er kernel. In both cases, the authors prove that the approximate SRB measure produced by the scheme converges to the actual measure as the accuracy of the scheme improves.
The authors implement both approaches and find convergent numerical approximations of the SRB measure for a nonlinear perturbation of Arnold's cat map. They provide the Fourier approximation of mollified transfer operators and a direct Fourier approximation via Fej'er kernels. For numerical results, they consider a small perturbation of a linear toral automorphism \(T(x_{1}, x_{2})=(2x_{1}+x_{2}+2\delta \cos(2\pi x_{1}), x_{1}+x_{2}+\delta \sin(4\pi x_{2} +1))\) with \(\delta=0.01\). Further, they also generate images to visualize their results. Moreover, they provide a comparison on Fourier-based estimates of the SRB measure and the variance with estimates from a (non-rigorous) pure Ulam method. Furthermore, they compute Fourier-based estimates of the rate function. The authors also mention that, in their knowledge, it is the first rigorous (in the sense of convergence) computation of the variance and rate function of an Anosov map.
Reviewer: Mohammad Sajid (Buraidah)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)On the anisotropic stable JCIR process.https://www.zbmath.org/1459.601232021-05-28T16:06:00+00:00"Friesen, Martin"https://www.zbmath.org/authors/?q=ai:friesen.martin"Jin, Peng"https://www.zbmath.org/authors/?q=ai:jin.pengSummary: We investigate the anisotropic stable JCIR process which is a multidimensional extension of the stable JCIR process but also a multi-dimensional analogue of the classical JCIR process. We prove that the heat kernel of the anisotropic stable JCIR process exists and it satisfies an a-priori bound in a weighted anisotropic Besov norm. Based on this regularity result we deduce the strong Feller property and prove, for the subcritical case, exponential ergodicity in total variation. Also, we show that in the one-dimensional case the corresponding heat kernel is smooth.Limiting curves for the dyadic odometer.https://www.zbmath.org/1459.370062021-05-28T16:06:00+00:00"Minabutdinov, A. R."https://www.zbmath.org/authors/?q=ai:minabutdinov.a-rThe author proves that limiting curves (i.e., a relatively new approach to the study of fluctuations of ergodic sums), can arise for transformations of finite rank, as well as for the dyadic odometer (i.e., the simplest transformation of rank one with discrete spectrum). This seems to be the first example of limiting curves in the class of uniquely ergodic transformations.
Reviewer: George Stoica (Saint John)