Recent zbMATH articles in MSC 37Ahttps://zbmath.org/atom/cc/37A2021-02-12T15:23:00+00:00WerkzeugErgodic theorems for the shift action and pointwise versions of the Abért-Weiss theorem.https://zbmath.org/1452.370042021-02-12T15:23:00+00:00"Bernshteyn, Anton"https://zbmath.org/authors/?q=ai:bernshteyn.antonLet \(\Gamma\) be a countably infinite group with identity element \(1\) and \(\alpha: \Gamma \curvearrowright(X, \mu)\) be a probability measure-preserving action.
Given \(f \in L^{1}(X, \mu),\) the global average is
\[
\mathbb{E}_{\mu} f:=\int_{X} f \mathrm{d} \mu,
\]
and the pointwise averages are
\[
\mathbb{E}_{D} f(x):=\frac{1}{|D|} \sum_{\delta \in D} f(\delta \cdot x),
\]
where \(x \in X\) and \(D\) is a nonempty finite subset of \(\Gamma\). Theorem 2.1 gives a pointwise ergodic theorem for continuous maps \(f\) on the Bernoulli shift action \(\Gamma \curvearrowright\left([0 ; 1]^{\Gamma}, \lambda^{\Gamma}\right)\). Instead of considering the problem of when the limits of those two averages will meet as \(|D| \rightarrow \infty\), the author considers a sequence \(\left(D_{n}\right)_{n \in \mathbb{N}}\) of finite subsets of \(\Gamma\) such that \(\left|D_{n}\right| / \log n \rightarrow \infty .\) This result leads to the mean ergodic theorem for the shift, as a corollary.
Theorem 2.1 follows by a concentration of measure inequality together with the Borel-Cantelli lemma. Note that the Lovász Local Lemma, a useful tool in probabilistic combinatorics, plays an important role in this paper. The author also reviews it and its measurable analogues in Section 6 and goes through one of its proofs in order to prove some results.
Theorem 2.6 is a push-forward ergodic theorem for the shift. Based on the definition of residually finite and on this theorem, the author comes up with a new definition called approximately residually finite.
As a strengthening of the theorem of \textit{M. Abért} and \textit{B. Weiss} [Ergodic Theory Dyn. Syst. 33, No. 2, 323--333 (2013; Zbl 1268.37006)], Theorem 2.11 gives a pointwise (Abért-Weiss) theorem.
The last result, Theorem 2.14, is valid for uniformly subexponential actions. As the author mentions, it is a purely Borel version of the Abért-Weiss theorem for finitely generated groups of subexponential growth.
Reviewer: Meirong Zhang (Beijing)Equilibrium states for a class of skew products.https://zbmath.org/1452.370402021-02-12T15:23:00+00:00"Carvalho, Maria"https://zbmath.org/authors/?q=ai:carvalho.maria-lucilia|carvalho.maria-conceicao|de-carvalho.maria-pires|carvalho.maria-leonor-da-silva"Pérez, Sebastián A."https://zbmath.org/authors/?q=ai:perez.sebastian-aThe authors consider skew products on $M\times T^2$, where $M$ is the two-sphere or the two-torus, which are partially hyperbolic and semi-conjugate to an Axiom A diffeomorphism. The authors present sufficient conditions, both on the skew products and on the potentials, for the existence and uniqueness of equilibrium states, and discuss their statistical stability.
The main properties proved in this paper remain valid if instead of $M\times T^2$ one considers similar dynamical systems on $M\times T^n$ with $n>2$. This is due to the fact that the stable and unstable directions of a partially hyperbolic diffeomorphism are integrable [\textit{M. W. Hirsch} et al., Invariant manifolds. Berlin-Heidelberg-New York: Springer-Verlag (1977; Zbl 0355.58009)], and so is the center bundle without restrictions on its dimension for torus diffeomorphisms isotopic to a linear Anosov automorphism along a path of partially hyperbolic diffeomorphisms [\textit{T. Fisher} et al., Math. Z. 278, No. 1--2, 149--168 (2014; Zbl 1350.37033)].
Reviewer: Anatoly Martynyuk (Kyïv)Some applications of algebraic entropy to the proof of Milnor-Wolf theorem.https://zbmath.org/1452.200392021-02-12T15:23:00+00:00"Xi, W."https://zbmath.org/authors/?q=ai:xi.wenfei"Dikranjan, D."https://zbmath.org/authors/?q=ai:dikranjan.dikran-n"Freni, D."https://zbmath.org/authors/?q=ai:freni.domenico"Toller, D."https://zbmath.org/authors/?q=ai:toller.danieleSeveral nice results are given on the algebraic entropy \(h_{alg}(\phi)\) and the growth of a group endomorphism \(\phi\colon G\to G\). These results are applied to give a new proof of the classical Milnor-Wolf theorem, that is, of the fact that there exist no finitely generated solvable group of intermediate growth. More precisely, the authors show that a finitely generated solvable group \(G\) of subexponential growth is virtually nilpotent, and then the Gromov theorem gives that equivalently \(G\) has polynomial growth.
To achieve this goal, the authors study the growth of a finitely generated cascade, namely, a pair \((G,\phi)\) where \(G\) is a group, \(\phi\colon G\to G\) is an automorphism and there exists a finite subset \(F\) of \(G\) such that \(G=\langle\phi^n(F)\colon n\in\mathbb N\rangle\).
The growth of a finitely generated cascade \((G,\phi)\) is related to the classical growth of the semidirect product \(G\rtimes \langle\phi\rangle\). For example, the group \(G\rtimes \langle\phi\rangle\) is finitely generated if and only if the cascade \((G,\phi)\) is finitely generated, and if \(G\rtimes \langle\phi\rangle\) has subexponential growth then \(G\) is finitely generated as well; in order to obtain the latter property the authors apply their theorem stating that in case \(h_{\mathrm{alg}}(\phi)<\log 2\), then \(G\) is finitely generated. Moreover, if \(G\) is nilpotent, then the group \(G\rtimes \langle\phi\rangle\) has the same growth type of the cascade \((G,\phi)\), and this growth type cannot be intermediate.
Reviewer: Anna Giordano Bruno (Udine)Diffusion limit for a slow-fast standard map.https://zbmath.org/1452.370172021-02-12T15:23:00+00:00"Blumenthal, Alex"https://zbmath.org/authors/?q=ai:blumenthal.alex"de Simoi, Jacopo"https://zbmath.org/authors/?q=ai:de-simoi.jacopo"Zhang, Ke"https://zbmath.org/authors/?q=ai:zhang.ke|zhang.ke.1The authors consider the map \(\mathcal{G}_\epsilon:(x, z) \mapsto\left(x+\epsilon^{-\alpha} \sin (2 \pi x)+\epsilon^{-(1+\alpha)} z, z+\epsilon \sin (2 \pi x)\right)\). They show that this map is conjugated to the Chirikov standard map. At first, they prove a theorem (Theorem C) on decay of correlations of the standard map, and introduce the notion of standard pairs. Then they give a central limit theorem for the standard map (Theorem B), which is used to prove their main theorem. In the last part of the article, they get their main result (Theorem A), a central limit theorem for \(\mathcal{G}_\epsilon\) when \(\alpha\) is properly chosen. The author conjecture that the error term in Theorem C can be improved, which may lead to an improved Weak Law of Large Numbers.
Reviewer: Meirong Zhang (Beijing)On the uniform convergence of ergodic averages for \(C^*\)-dynamical systems.https://zbmath.org/1452.370082021-02-12T15:23:00+00:00"Fidaleo, Francesco"https://zbmath.org/authors/?q=ai:fidaleo.francescoThe author investigates ergodic and spectral properties of general (discrete) \(C^*\)-dynamical systems made of a unital \(C^*\)-algebra and a multiplicative identity-preserving map. A notable particular case is when the dynamical system admits the property of unique ergodicity with respect to the fixed-point subalgebra. Applications to quantum probability justify the study.
Reviewer: George Stoica (Saint John)Ergodic optimization for hyperbolic flows and Lorenz attractors.https://zbmath.org/1452.370272021-02-12T15:23:00+00:00"Morro, Marcus"https://zbmath.org/authors/?q=ai:morro.marcus"Sant'Anna, Roberto"https://zbmath.org/authors/?q=ai:santanna.roberto"Varandas, Paulo"https://zbmath.org/authors/?q=ai:varandas.pauloSummary: In this article we consider the ergodic optimization for hyperbolic flows and Lorenz attractors with respect to both continuous and Hölder continuous observables. In the context of hyperbolic flows we prove that a Baire generic subset of continuous observables have a unique maximizing measure, with full support and zero entropy, and that a Baire generic subset of Hölder continuous observables admit a unique and periodic maximizing measure. These results rely on a relation between ergodic optimization for suspension semiflows and ergodic optimization for the Poincaré map with respect to induced observables, which allow us to reduce the problem for the context of maps. Using that singular-hyperbolic attractors are approximated by hyperbolic sets, we obtain related results for geometric Lorenz attractors.A note on minimal models for pmp actions.https://zbmath.org/1452.370032021-02-12T15:23:00+00:00"Zucker, Andy"https://zbmath.org/authors/?q=ai:zucker.andyLet \(G\) be a countable group. A metrizable \(G\)-flow \(Y\) is said to be a \textit{model-universal} if, given the invariant measures \(\nu\) on \(Y\), the \(G\)-systems \((Y,\nu)\) recover every free measure-preserving \(G\)-system up to isomorphisms. \textit{B. Weiss} [Contemp. Math. 567, 249--264 (2012; Zbl 1279.37010)] constructed, for every countable group, a minimal model-universal flow.
In this paper, the author presents a new streamlined construction, which shows that a minimal model-universal flow is far from unique. More precisely, let us say that a family \(\{Y_i : i \in I\}\) of minimal \(G\)-flows is \textit{mutually disjoint} if the product \(\prod_{i \in I} Y_i\) is minimal (this implies that the \(Y_i\) are pairwise non-isomorphic \(G\)-flows). The main result in the paper asserts that for any countable group \(G\), there is a mutually disjoint family \(\{Y_i : i < \mathfrak{c}\}\) of free, minimal, model-universal flows.
Reviewer: Nilson C. Bernardes Jr. (Rio de Janeiro)Extensive numerical results for integrable case of standard map.https://zbmath.org/1452.370812021-02-12T15:23:00+00:00"Tirnakli, Ugur"https://zbmath.org/authors/?q=ai:tirnakli.ugur"Tsallis, Constantino"https://zbmath.org/authors/?q=ai:tsallis.constantinoSummary: In recent years, conservative dynamical systems have become a vivid area of research from the statistical mechanical characterization viewpoint. With this respect, several area-preserving maps have been studied. It has been numerically shown that the probability distribution of the sum of the suitable random variable of these systems can be well approximated by a Gaussian (\(q\)-Gaussian) when the initial conditions are randomly selected from the chaotic sea (region of stability islands) in the available phase space. In this study, we will summarize these results and discuss a special case for the standard map, a paradigmatic example of area-preserving maps, for which the map is totally integrable.Nilsystems and ergodic averages along primes.https://zbmath.org/1452.370052021-02-12T15:23:00+00:00"Eisner, Tanja"https://zbmath.org/authors/?q=ai:eisner.tanjaBy using an anti-correlation result for the von Mangoldt function (due to
\textit{B. Green} and \textit{T. Tao} [Ann. Math. (2) 171, No. 3, 1753--1850 (2010; Zbl 1242.11071); Ann. Math. (2) 175, No. 2, 465--540 (2012; Zbl 1251.37012); Ann. Math. (2) 175, No. 2, 541--566 (2012; Zbl 1347.37019)]),
the author proves everywhere convergence of ergodic averages along primes for nilsystems and continuous functions. This result complements a celebrated result by \textit{J. Bourgain} [Lect. Notes Math. 1317, 204--223 (1988; Zbl 0662.47006); Publ. Math., Inst. Hautes Étud. Sci. 69, 5--45 (1989; Zbl 0705.28008)] and \textit{M. Wierdl} [Isr. J. Math. 64, No. 3, 315--336 (1989; Zbl 0695.28007)], which states that ergodic averages along primes converge almost everywhere for \(L^p\)-functions, \(p > 1\), including a polynomial version by \textit{R. Nair} [Ergodic Theory Dyn. Syst. 11, No. 3, 485--499 (1991; Zbl 0751.28008); Stud. Math. 105, No. 3, 207--233 (1993; Zbl 0871.11051)] and \textit{M. Wierdl} [Isr. J. Math. 64, No. 3, 315--336 (1989; Zbl 0695.28007)].
Reviewer: George Stoica (Saint John)Uniform entropy scalings of filtrations.https://zbmath.org/1452.600222021-02-12T15:23:00+00:00"Laurent, Stéphane"https://zbmath.org/authors/?q=ai:laurent.stephaneSummary: We study \textit{A. M. Vershik} and \textit{A. D. Gorbulsky}'s notion of entropy scalings [Theory Probab. Appl. 52, No. 3, 493--508 (2008; Zbl 1161.28005); translation from Teor. Veroyatn. Primen. 52, No. 3, 446--467 (2007)] for filtrations in the particular case when the scaling is not \(\epsilon \)-dependent, and is then termed as uniform scaling. Among our main results, we prove that the scaled entropy of the filtration generated by the Vershik progressive predictions of a random variable is equal to the scaled entropy of this random variable. Standardness of a filtration is the case when the scaled entropy with a constant scaling is zero, thus our results generalize some known results about standardness. As a case-study we consider a family of next-jump time filtrations. We also provide some results about the entropy of poly-adic filtrations, rephrasing or generalizing some old results.
For the entire collection see [Zbl 1427.60005].Open sets of exponentially mixing Anosov flows.https://zbmath.org/1452.370352021-02-12T15:23:00+00:00"Butterley, Oliver"https://zbmath.org/authors/?q=ai:butterley.oliver"War, Khadim"https://zbmath.org/authors/?q=ai:mbacke-war.khadimThe authors show that if \(\phi^t\) is a transitive \(C^{1+\alpha}\) Anosov flow such that the stable bundle is \(C^{1+\alpha}\) and the stable and unstable bundles are not jointly integrable, then \(\phi^t\) mixes exponentially with respect to the unique Sinai-Ruelle-Bowen measure. This allows them to show that if a flow is sufficiently close to a volume-preserving Anosov flow and \(\dim E^s=1\), \(\dim E^u \geq 2\) then the flow mixes exponentially if the stable and unstable bundles are not jointly integrable. This implies the existence of non-empty open sets of exponentially mixing Anosov flows.
Reviewer: Miguel Paternain (Montevideo)Examples in the entropy theory of countable group actions.https://zbmath.org/1452.370062021-02-12T15:23:00+00:00"Bowen, Lewis"https://zbmath.org/authors/?q=ai:bowen.lewis-phylipIn this survey article, the author deals with examples, many of which have not appeared before, that highlight differences and similarities between classical ergodic theory and ergodic theory for group actions.
The author gives a brief review of classical entropy theory and motivates the interest in studying entropy theory for actions of general countable groups. Then, he provides an intuitive approach to \(f\)-invariance, sofic groups, sofic entropy and Rokhlin entropy. Automorphisms (and more generally, group actions) are classified up to measure conjugacy. A more general definition based on maps into the symmetric group is presented. The author provides a list of examples in which the entropy has been computed. An example showing that the sofic entropy is not necessarily additive under direct products is given. A finite-to-one factor map from a zero entropy action to a Bernoulli shift is presented. The author deals with generalizations of Ornstein theory for non-amenable groups including the isomorphism theorem, Krieger's generator theorem and Sinai's factor theorem. A proof of the variational principle for the sofic entropy is given. Sofic pressure, equilibrium states for actions of sofic groups and Gibbs measures on random regular graphs are considered. An Abramov-Rokhlin formula for actions of free groups is presented.
Reviewer: Hasan Akin (Gaziantep)Ergodicity and central limit theorem for random interval homeomorphisms.https://zbmath.org/1452.370462021-02-12T15:23:00+00:00"Czudek, Klaudiusz"https://zbmath.org/authors/?q=ai:czudek.klaudiusz"Szarek, Tomasz"https://zbmath.org/authors/?q=ai:szarek.tomasz-jakub|szarek.tomasz-zacharyThe authors are interested in iterated function systems generated by orientation-preserving homeomorphisms on the interval \([0, 1]\). They obtain a simple proof of unique ergodicity on the open interval \((0, 1)\) for a wide class of iterated function systems and establish a quenched central limit theorem for random interval homeomorphisms.
In order to prove the theorem, the authors consider the Maxwell-Woodroofe approach to ergodic stationary Markov chains. This generalises the martingale approximation method given by \textit{M. I. Gordin} and \textit{B. A. Lifshits} [Sov. Math., Dokl. 19, 392--394 (1978; Zbl 0395.60057); translation from Dokl. Akad. Nauk SSSR 239, 766--767 (1978)]. The existence of a unique invariant measure \(\mu \in \mathcal{M}_1((0, 1))\) is proved.
Let \((f_1, \ldots , f_N; p_1, \ldots, p_N)\) be an admissible iterated function system, the authors prove that the corresponding Markov operator \(P\) has a unique invariant measure \(\mu_{*} \in \mathcal{M}_1((0, 1))\). Moreover \(\mu_{*}\) is atomless. Let \(\mu_{*} \in\mathcal{M}_1((0,1))\) be its unique invariant measure. In this case, the authors prove that for any measure \(\mu \in \mathcal{M}_1((0, 1))\) there holds
\[
\lim_{n\rightarrow \infty}\langle P^n\mu,\varphi\rangle=\langle\mu_{*},\varphi\rangle
\]
for \(\varphi\in C([0,1])\). Let \((X_n)\) be the Markov chain corresponding to the Markov operator \(P\). They give the proof of the quenched central limit theorem for the random process \((\varphi(X_n))\), where \(\varphi\) is a Lipschitz function with \(\int_{[0,1]}\varphi d\mu_{*}=0\).
Reviewer: Hasan Akin (Gaziantep)Factoriality, Connes' type III invariants and fullness of amalgamated free product von Neumann algebras.https://zbmath.org/1452.460462021-02-12T15:23:00+00:00"Houdayer, Cyril"https://zbmath.org/authors/?q=ai:houdayer.cyril"Isono, Yusuke"https://zbmath.org/authors/?q=ai:isono.yusukeSummary: We investigate factoriality, Connes' type III invariants and fullness of arbitrary amalgamated free product von Neumann algebras using Popa's deformation/rigidity theory. Among other things, we generalize many previous structural results on amalgamated free product von Neumann algebras and we obtain new examples of full amalgamated free product factors for which we can explicitely compute Connes' type III invariants.Investigation of stickiness influence in the anomalous transport and diffusion for a non-dissipative Fermi-Ulam model.https://zbmath.org/1452.370772021-02-12T15:23:00+00:00"Livorati, André L. P."https://zbmath.org/authors/?q=ai:livorati.andre-luis-prando"Palmero, Matheus S."https://zbmath.org/authors/?q=ai:palmero.matheus-s"Díaz-I, Gabriel"https://zbmath.org/authors/?q=ai:diaz-i.gabriel"Dettmann, Carl P."https://zbmath.org/authors/?q=ai:dettmann.carl-p"Caldas, Iberê L."https://zbmath.org/authors/?q=ai:caldas.ibere-l"Leonel, Edson D."https://zbmath.org/authors/?q=ai:leonel.edson-dennerSummary: We study the dynamics of an ensemble of non interacting particles constrained by two infinitely heavy walls, where one of them is moving periodically in time, while the other is fixed. The system presents mixed dynamics, where the accessible region for the particle to diffuse chaotically is bordered by an invariant spanning curve. Statistical analysis for the root mean square velocity, considering high and low velocity ensembles, leads the dynamics to the same steady state plateau for long times. A transport investigation of the dynamics via escape basins reveals that depending of the initial velocity ensemble, the decay rates of the survival probability present different shapes and bumps, in a mix of exponential, power law and stretched exponential decays. After an analysis of step-size averages, we found that the stable manifolds play the role of a preferential path for faster escape, being responsible for the bumps and different shapes of the survival probability.Limiting entry and return times distribution for arbitrary null sets.https://zbmath.org/1452.370072021-02-12T15:23:00+00:00"Haydn, Nicolai"https://zbmath.org/authors/?q=ai:haydn.nicolai-t-a"Vaienti, Sandro"https://zbmath.org/authors/?q=ai:vaienti.sandroThe authors describe how the limiting return times distribution for arbitrary sets reduce to the compound Poisson distribution, by using clusters, i.e., portions of points that have finite return times in the limit where random return times go to infinity. For periodic points, this is essentially the Pólya-Aeppli distribution, associated with geometrically distributed cluster sizes. Their method applies to synchronisation of coupled map lattices.
Reviewer: George Stoica (Saint John)Approximation orders of real numbers by \(\beta\)-expansions.https://zbmath.org/1452.110972021-02-12T15:23:00+00:00"Fang, Lulu"https://zbmath.org/authors/?q=ai:fang.lulu"Wu, Min"https://zbmath.org/authors/?q=ai:wu.min.1|wu.min.2|wu.min"Li, Bing"https://zbmath.org/authors/?q=ai:li.bing.1The authors note the following investigations:
``We prove that almost all real numbers (with respect to Lebesgue measure) are approximated by the convergents of their \(\beta\)-expansions with the exponential order \(\beta^{-n}\). Moreover, the Hausdorff dimensions of sets of the real numbers which are approximated by all other orders, are determined. These results are also applied to investigate the orbits of real numbers under \(\beta\)-transformation, the shrinking target type problem, the Diophantine approximation and the run-length function of \(\beta\)-expansions.''
In this paper, a survey is devoted to known results related with \(\beta\)-expansions. Basic definitions and properties for \(\beta\)-expansions are given. The separate attention is given to \(n\)-th cylinders defined in terms of \(\beta\)-expansions.
One can note the following main result of this research.
Let \(\beta>1\) be a fixed number and \(\lambda\) be the Lebesgue measure on \([0, 1]\),
\[
[0,1]\ni x=\sum^{\infty} _{k=1}{\frac{\varepsilon_k(x)}{\beta^k}}
\]
and
\[
\omega_n(x)=\sum^{n} _{k=1}{\frac{\varepsilon_k(x)}{\beta^k}}.
\]
Theorem. Let \(\beta>1\) be a real number. Then for \(\lambda\)-almost all \(x\in [0, 1)\),
\[
\lim_{n\to\infty}{\frac{1}{n}\log_{\beta}{(x-\omega_n(x))}}=-1.
\]
Several main results are related to the following set
\[
\left\{x\in[0,1): \liminf_{n\to\infty}{\frac{1}{\phi(n)}\log_{\beta}{(x-\omega_n(x))}}=-1\right\},
\]
where \(\phi\) is a positive function defined on the set of all positive integers.
All proofs are given with explanations.
Reviewer: Symon Serbenyuk (Kyïv)Matching for a family of infinite measure continued fraction transformations.https://zbmath.org/1452.110962021-02-12T15:23:00+00:00"Kalle, Charlene"https://zbmath.org/authors/?q=ai:kalle.charlene"Langeveld, Niels"https://zbmath.org/authors/?q=ai:langeveld.niels-daniel-simon"Maggioni, Marta"https://zbmath.org/authors/?q=ai:maggioni.marta"Munday, Sara"https://zbmath.org/authors/?q=ai:munday.saraOne can begin with authors' abstract:
``As a natural counterpart to Nakada's \(\alpha\)-continued fraction maps, we study a one-parameter family of continued fraction transformations with an indifferent fixed point. We prove that matching holds for Lebesgue almost every parameter in this family and that the exceptional set has Hausdorff dimension 1. Due to this matching property, we can construct a planar version of the natural extension for a large part of the parameter space. We use this to obtain an explicit expression for the density of the unique infinite \(\sigma\)-finite absolutely continuous invariant measure and to compute the Krengel entropy, return sequence and wandering rate of the corresponding maps.''
In the present paper, the main attention is given to a certain family of maps \(\{T_\alpha\}_{\alpha\in (0,1)}\) calling
flipped \(\alpha\)-continued fraction maps. In this research, among other results can be noted that the set of parameters \(\alpha\in (0,1)\) for which the transformation \(T_\alpha\) does not have matching is a Lebesgue null set of full Hausdorff dimension and a fact that the map \(T_\alpha\) is an AFN-map for each \(\alpha\in (0,1)\).
The notion of the matching property of a map is recalled and some known results in this topic are described. In addition, the notions of semi-regular continued fraction expansions and AFN-maps are considered.
The attention is also given to the natural extensions for non-invertible dynamical systems, especially for continued fraction transformations. Several dynamical quantities associated to the systems \(T_\alpha\) are calculated. In other words, the Krengel entropy, return sequence and wandering rate of \(T_\alpha\) for a large part of the parameter space \((0, 1)\) are computed. It is noted that that the Krengel entropy, return sequence, and wandering rate obtained in this paper, do not display any dependence on \(\alpha\). ``These quantities give isomorphism invariants for dynamical systems with infinite invariant measures''.
Reviewer: Symon Serbenyuk (Kyïv)Curie-Weiss type models for general spin spaces and quadratic pressure in ergodic theory.https://zbmath.org/1452.370092021-02-12T15:23:00+00:00"Leplaideur, Renaud"https://zbmath.org/authors/?q=ai:leplaideur.renaud"Watbled, Frédérique"https://zbmath.org/authors/?q=ai:watbled.frederiqueThe present paper is the continuation of the recent work [Bull. Soc. Math. Fr. 147, No. 2, 197--219 (2019; Zbl 1430.37007)] of the authors on the generalised Curie-Weiss model and the related quadratic pressure notion in ergodic theory. Therein, close resemblances (affinities) have been established between the theory of phase transitions in model systems of statistical mechanics and that of the thermodynamical formalism of dynamical systems. These have been codified in a ``dictionary'' translating the standard statistical physics lore to this appropriate for the theory of dynamical systems and viceversa, provided one keeps in memory a distinction between probabilistic Gibbs measures (PGM, usually defined on finite lattices) and dynamical Gibbs measures (DGM, invariant measures in the ergodic theory of dynamical systems, which are infinite). With this distinction the notion of quadratic pressure is introduced for dynamical systems and convergence properties of the PGM of a slight generalisation of the Curie-Weiss model are investigated.
The present paper is devoted to the study of PGM convergence properties with respect to the quadratic pressure, while admitting more general interactions between sites for mean field interactions. As an example, the equilibrium state for the mean-field XY model as the number of particles goes to $+\infty$ is constructed.
Reviewer: Piotr Garbaczewski (Opole)Skew product Smale endomorphisms over countable shifts of finite type.https://zbmath.org/1452.370412021-02-12T15:23:00+00:00"Mihailescu, Eugen"https://zbmath.org/authors/?q=ai:mihailescu.eugen-gh"Urbański, Mariusz"https://zbmath.org/authors/?q=ai:urbanski.mariuszThe authors investigate skew product Smale endomorphisms modeled on subshifts of finite type with countable alphabets. The authors study the thermodynamic formalism for skew product Smale endomorphisms over countable-to-1 maps. They provide a thermodynamic formalism of Hölder continuous summable potentials with respect to two sided subshifts of finite type. Some fundamental ergodic results concerning one-sided symbolic dynamics, thermodynamic formalism of Hölder continuous summable potentials with respect to two sided subshifts of finite type, skew product Smale endomorphisms modeled on countable alphabet subshifts of finite type are presented.
By developing new methods suited for the countable alphabet case, the authors prove that projections of almost everywhere conditional measures of equilibrium states of summable Hölder continuous potentials are exact dimensional. They prove a version of Bowen's formula giving the Hausdorff dimension of each fiber as the zero of a pressure function. They also show that the pressure function has no zero. New results about skew products over countable-to-1 endomorphisms are obtained. Under a condition on \(\mu\)-injectivity for the coding of the base map, the authors prove the exact dimensionality of conditional measures of equilibrium measures in stable manifold fibers. Expanding Markov-Rényi maps \(f: I\rightarrow I\) and conformal Smale skew product endomorphisms \(F:I\times Y\rightarrow I \times Y\) over \(f\) are investigated. The authors apply their results to Diophantine approximation of irrational numbers \(x\).
Reviewer: Hasan Akin (Gaziantep)Domination, almost additivity, and thermodynamic formalism for planar matrix cocycles.https://zbmath.org/1452.370392021-02-12T15:23:00+00:00"Bárány, Balázs"https://zbmath.org/authors/?q=ai:barany.balazs"Käenmäki, Antti"https://zbmath.org/authors/?q=ai:kaenmaki.antti"Morris, Ian D."https://zbmath.org/authors/?q=ai:morris.ian-dSummary: In topics such as the thermodynamic formalism of linear cocycles, the dimension theory of self-affine sets, and the theory of random matrix products, it has often been found useful to assume positivity of the matrix entries in order to simplify or make feasible certain types of calculation. It is natural to ask how positivity may be relaxed or generalised in a way which enables similar calculations to be made in more general contexts. On the one hand one may generalise by considering almost additive or asymptotically additive potentials which mimic the properties enjoyed by the logarithm of the norm of a positive matrix cocycle; on the other hand one may consider matrix cocycles which are dominated, a condition which includes positive matrix cocycles but is more general. In this article we explore the relationship between almost additivity and domination for planar cocycles. We show in particular that a locally constant linear cocycle in the plane is almost additive if and only if it is either conjugate to a cocycle of isometries, or satisfies a property slightly weaker than domination which is introduced in this paper. Applications to matrix thermodynamic formalism are presented.