Recent zbMATH articles in MSC 37Bhttps://zbmath.org/atom/cc/37B2021-05-28T16:06:00+00:00WerkzeugOn turbulent relations.https://zbmath.org/1459.030782021-05-28T16:06:00+00:00"López, Jesús A. Álvarez"https://zbmath.org/authors/?q=ai:alvarez-lopez.jesus-a"Candel, Alberto"https://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.Gradient flow line near birth-death critical points.https://zbmath.org/1459.370272021-05-28T16:06:00+00:00"Antony, Charel"https://zbmath.org/authors/?q=ai:antony.charelAuthor's abstract: Near a birth-death critical point in a one-parameter family of gradient flows, there are precisely two Morse critical points of index difference one on the birth side. This paper gives a self-contained proof of the folklore theorem that these two critical points are joined by a unique gradient trajectory up to time-shift. The proof is based on the Whitney normal form, a Conley index construction, and an adiabatic limit analysis for an associated fast-slow differential equation.
Reviewer: Roman Srzednicki (Kraków)Dimension and entropy in compact topological groups.https://zbmath.org/1459.220012021-05-28T16:06:00+00:00"Dikranjan, Dikran"https://zbmath.org/authors/?q=ai:dikranjan.dikran-n"Sanchis, Manuel"https://zbmath.org/authors/?q=ai:sanchis.manuelThe authors consider the e-spectrum \(\mathbf{E}_{\text{top}}(K)\) for a compact-like group \(K\), where the set of all values \(h(f)\), when \(f\) runs over the set \(\text{End}(K)\) of all continuous endomorphism of \(K\). Particular attention is paid to the class \(\mathfrak{C}_{<\infty}\) of topological groups without continuous endomorphism of finite entropy as well as the subclass \(\mathfrak{C}_0\) of \(\mathfrak{C}_{<\infty}\) consisting of those groups \(K\) with \(\mathbf{E}_{\text{top}}(K)=\{0\}\). They show among many others that a compact connected group \(K\) with finite dimensional commutator subgroup belongs to \(\mathfrak{C}_{<\infty}\) if and only if \(\dim K<\infty\). Since the class \(\mathfrak{C}_{<\infty}\) is not stable under closed subgroups or quotients, they study the largest subclass \(S(\mathfrak{C}_{<\infty})\) and \(Q(\mathfrak{C}_{<\infty})\), respectively, of \(\mathfrak{C}_{<\infty}\) having these stability properties. Some other related features are taken into account.
Reviewer: T. M. G. Ahsanullah (Riyadh)Introduction to hierarchical tiling dynamical systems.https://zbmath.org/1459.370162021-05-28T16:06:00+00:00"Frank, Natalie Priebe"https://zbmath.org/authors/?q=ai:priebe-frank.natalieThe paper is devoted to some classical results on dynamical systems arising from substitutions: ordinary (one-dimensional) symbolic systems, one-dimensional tiling systems, multidimensional \(\mathbb{Z}^d\)-systems, and multidimensional tiling systems. The main topics are: various types of substitution systems and relationships between them, dynamical systems arising from these objects, interpretation of standard dynamical properties, super-tiling construction methods, transition matrices, dynamical and diffraction spectra.
For the entire collection see [Zbl 1454.37001].
Reviewer: Anton Shutov (Vladimir)Delone sets and dynamical systems.https://zbmath.org/1459.370172021-05-28T16:06:00+00:00"Solomyak, Boris"https://zbmath.org/authors/?q=ai:solomyak.borisThe paper is a survey of some (mainly classical) results about substitution Delone sets and some of their generalizations, such as substitution Delone \(m\)-sets, self-affine and pseudo-self-affine tilings. The main focus is on the description of the expansion constants and expansion maps, but some other topics such as connections with dynamical systems and diffraction are also briefly discussed. The paper ends with a short list of open problems on the characterization of expansion maps for tilings with inflation symmetry.
For the entire collection see [Zbl 1454.37001].
Reviewer: Anton Shutov (Vladimir)Equilibrium-like solutions of asymptotically autonomous differential equations.https://zbmath.org/1459.370252021-05-28T16:06:00+00:00"Jänig, Axel"https://zbmath.org/authors/?q=ai:janig.axelThe paper deals with skew-product semiflows, especially those ones generated by nonautonomous
ordinary equations \(\dot x=f(t,x)\) and semilinear parabolic evolution equations \(\dot u=Ax+f(t,x)\). The autonomous
asymptoticity of \(f\) is assumed, i.e., \(f^t\to f^{\pm}\) as \(t\to \pm\infty\), where \(f^t(s,x)=f(s+t,x)\),
both \(f^{\pm}\) are \(t\)-independent, and the convergence is determined by a suitable topology.
Under further hypothesis, the author presents some constructions and results related to
Morse decompositions and to the corresponding nonautonomous homology Conley indices. The notion
of the nonautonomous Conley index was introduced by the author in
[J. Fixed Point Theory Appl. 19, No. 3, 1825--1870 (2017; Zbl 1379.37033)].
The results of this paper are used to find theorems on the existence of solutions of the considered
evolution equations which connect hyperbolic equilibria of the asymptotic autonomous systems.
Reviewer: Roman Srzednicki (Kraków)Entropy of transcendental entire functions.https://zbmath.org/1459.370322021-05-28T16:06:00+00:00"Benini, Anna Miriam"https://zbmath.org/authors/?q=ai:benini.anna-miriam"Fornæss, John Erik"https://zbmath.org/authors/?q=ai:fornass.john-erik"Peters, Han"https://zbmath.org/authors/?q=ai:peters.hanThe paper takes under consideration the topological entropy of transcendental entire maps. The results of [\textit{W. Bergweiler}, Conform. Geom. Dyn. 4, No. 2, 22--34 (2000; Zbl 0954.30012); \textit{J. P. R. Christensen} and \textit{P. Fischer}, Acta Math. Hung. 73, No. 3, 213--218 (1996; Zbl 0928.28005)] lead to the conclusion that the topological entropy of a transcendental function is always strictly positive. As transcendental entire maps have infinite topological degree, it could be expected that the topological entropy is also infinite. The main result of the paper is as follows.
Theorem. Let \(f\) be a transcendental entire function and let \(N\in\mathbb{N}.\) There exists a non-empty bounded open set \(V\subset\mathbb{C}\) so that \(V\subset f(V)\) and such that any point in \(V\) has at least \(N\) preimages in \(V\), counted with multiplicity.
It follows from this result that in the case of transcendental entire maps their topological entropy is, in fact, infinite. It should be mentioned that the same conclusion was reached independently in [\textit{M. Wendt}, Zufällige Juliamengen und invariante Maße mit maximaler Entropie. University of Kiel (PhD Thesis) (2005)].
Reviewer: Ewa Ciechanowicz (Szczecin)Univoque bases of real numbers: local dimension, devil's staircase and isolated points.https://zbmath.org/1459.110222021-05-28T16:06:00+00:00"Kong, Derong"https://zbmath.org/authors/?q=ai:kong.derong"Li, Wenxia"https://zbmath.org/authors/?q=ai:li.wenxia"Lü, Fan"https://zbmath.org/authors/?q=ai:lu.fan"Wang, Zhiqiang"https://zbmath.org/authors/?q=ai:wang.zhi-qiang|wang.zhiqiang"Xu, Jiayi"https://zbmath.org/authors/?q=ai:xu.jiayiSummary: Given a positive integer \(M\) and a real number \(x>0\), let \(\mathcal{U}(x)\) be the set of all bases \(q\in(1,M+1]\) for which there exists a unique sequence \((d_i)=d_1d_2\dots\) with each digit \(d_i\in\{0,1,\dots,M\}\) satisfying
\[
x=\sum_{i=1}^\infty\frac{d_i}{q^i}.
\]
The sequence \((d_i)\) is called a \(q\)-expansion of \(x\). In this paper we investigate the local dimension of \(\mathcal{U}(x)\) and prove a `variation principle' for unique non-integer base expansions. We also determine the critical values of \(\mathcal{U}(x)\) such that when \(x\) passes the first critical value the set \(\mathcal{U}(x)\) changes from a set with positive Hausdorff dimension to a countable set, and when \(x\) passes the second critical value the set \(\mathcal{U}(x)\) changes from an infinite set to a singleton. Denote by \(\mathbf{U}(x)\) the set of all unique \(q\)-expansions of \(x\) for \(q\in\mathcal{U}(x)\). We give the Hausdorff dimension of \(\mathbf{U}(x)\) and show that the dimensional function \(x \mapsto\dim_H \mathbf{U}(x)\) is a non-increasing Devil's staircase. Finally, we investigate the topological structure of \(\mathcal{U}(x)\). Although the set \(\mathcal{U}(1)\) has no isolated points, we prove that for typical \(x > 0\) the set \(\mathcal{U}(x)\) contains isolated points.Characterization for entropy of shifts of finite type on Cayley trees.https://zbmath.org/1459.370152021-05-28T16:06:00+00:00"Ban, Jung-Chao"https://zbmath.org/authors/?q=ai:ban.jungchao"Chang, Chih-Hung"https://zbmath.org/authors/?q=ai:chang.chih-hungApplications of Conley index theory on difference equations with non-resonance.https://zbmath.org/1459.390292021-05-28T16:06:00+00:00"Zhou, Ben-Xing"https://zbmath.org/authors/?q=ai:zhou.benxing"Liu, Chungen"https://zbmath.org/authors/?q=ai:liu.chungenThe main result of the paper provides the existence of nontrivial \(N\)-periodic
solutions of the difference equation \(\Delta^2x_{n-1}+\nabla F_n(x_n)=0\), where
\(F_n=F_{n+N}\), \(F_n\) is of class \(C^2\), the gradient \(\nabla F_n\) is asymptotically linear both at \(0\)
and at \(\infty\), and the matrices corresponding to the asymptotic linearity satisfy additional conditions related
to non-resonance and to the values of Morse indices and nullity. The proofs are based on
Conley index theory.
Reviewer: Roman Srzednicki (Kraków)On existence and continuation of solutions of the state-dependent impulsive dynamical system with boundary constraints.https://zbmath.org/1459.340632021-05-28T16:06:00+00:00"Chen, Ling"https://zbmath.org/authors/?q=ai:chen.ling"He, Zhi Long"https://zbmath.org/authors/?q=ai:he.zhilong"Li, Chuan Dong"https://zbmath.org/authors/?q=ai:li.chuandong"Umar, Hafiz Gulfam Ahmad"https://zbmath.org/authors/?q=ai:umar.hafiz-gulfam-ahmadSummary: The state-dependent impulsive dynamical system with boundary constraints is a kind of special but common system in nature. But because of the complexity of the geometry or topological structures of the impulsive surface, it is hard to determine when an event or an impulsive surface is reached. Therefore, a general state-dependent impulsive nonlinear dynamical system is rarely studied. This paper presents a class of state-dependent impulsive dynamical systems with boundary constraints. We obtain the existence and continuation of their viable solutions and provide sufficient conditions for the existence and uniqueness of the viable solutions to the system. Finally, two examples are given to illustrate the effectiveness of the results.The Ruelle operator for symmetric \(\beta\)-shifts.https://zbmath.org/1459.370292021-05-28T16:06:00+00:00"Lopes, Artur O."https://zbmath.org/authors/?q=ai:lopes.artur-oscar"Vargas, Victor"https://zbmath.org/authors/?q=ai:vargas.victorSummary: Consider \(m \in \mathbb{N}\) and \(\beta \in (1, m + 1]\). Assume that \(a \in \mathbb{R}\) can be represented in base \(\beta\) using a development in series \(a = \sum^{\infty}_{n = 1}x(n)\beta^{-n}\), where the sequence \(x = (x(n))_{n \in \mathbb{N}}\) takes values in the alphabet \(\mathcal{A}_m := \{0, \dotsc, m\}\). The above expression is called the \(\beta\)-expansion of \(a\) and it is not necessarily unique. We are interested in sequences \(x = (x(n))_{n \in \mathbb{N}} \in \mathcal{A}_m^\mathbb{N}\) which are associated to all possible values \(a\) which have a unique expansion. We denote the set of such \(x\) (with some more technical restrictions) by \(X_{m,\beta} \subset\mathcal{A}_m^\mathbb{N}\). The space \(X_{m, \beta}\) is called the symmetric \(\beta\)-shift associated to the pair \((m, \beta)\). It is invariant by the shift map but in general it is not a subshift of finite type. Given a Hölder continuous potential \(A \colon X_{m, \beta} \to\mathbb{R}\), we consider the Ruelle operator \(\mathcal{L}_A\) and we show the existence of a positive eigenfunction \(\psi_A\) and an eigenmeasure \(\rho_A\) for some values of \(m\) and \(\beta\). We also consider a variational principle of pressure. Moreover, we prove that the family of entropies \((h(\mu_{tA}))_{t > 0}\) converges, when \(t \to \infty\), to the maximal value among the set of all possible values of entropy of all \(A\)-maximizing probabilities.On the information-theoretic structure of distributed measurements.https://zbmath.org/1459.940702021-05-28T16:06:00+00:00"Balduzzi, David"https://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].Resolution of deadlocks in a fine discrete floor field cellular automata model -- modeling of turning and lateral movement at bottlenecks.https://zbmath.org/1459.370122021-05-28T16:06:00+00:00"Fu, Zhijian"https://zbmath.org/authors/?q=ai:fu.zhijian"Deng, Qiangqiang"https://zbmath.org/authors/?q=ai:deng.qiangqiang"Schadschneider, Andreas"https://zbmath.org/authors/?q=ai:schadschneider.andreas"Li, Yanlai"https://zbmath.org/authors/?q=ai:li.yanlai"Luo, Lin"https://zbmath.org/authors/?q=ai:luo.linThin annuli property and exponential distribution of return times for weakly Markov systems.https://zbmath.org/1459.370082021-05-28T16:06:00+00:00"Pawelec, Łukasz"https://zbmath.org/authors/?q=ai:pawelec.lukasz"Urbański, Mariusz"https://zbmath.org/authors/?q=ai:urbanski.mariusz"Zdunik, Anna"https://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)Investigating the use of wasps \textit{Anagyrus lopezi} as a biological control agent for cassava mealybugs: modeling and simulation.https://zbmath.org/1459.370832021-05-28T16:06:00+00:00"Aekthong, Supassorn"https://zbmath.org/authors/?q=ai:aekthong.supassorn"Rattanakul, Chontita"https://zbmath.org/authors/?q=ai:rattanakul.chontitaSummary: In this paper, a cellular automata model is developed in order to investigate the control of cassava mealybugs in a cassava field when the wasp \textit{Anagyrus lopezi} is used as the biological control agent. The model is constructed based upon farmers' usual practices of cassava planting in Thailand. However, the instructions on how to release the \textit{Anagyrus lopezi} wasps in a cassava field are various. In this study, we developed a cellular automata model based upon farmers' usual practices recommended by many organizations in Thailand such as the Department of Agriculture, Ministry of Agriculture and Cooperatives, Thailand, and the Thai Tapioca Development Institute. The effect of the life cycles of cassava mealybugs and the \textit{Anagyrus lopezi} wasps is also taken into account. The available reported data from many sources are utilized so that parameter values in the model are obtained. Computer simulations of different tactics of biological control are carried out so that a guideline for controlling the spread of cassava mealybugs by the \textit{Anagyrus lopezi} wasps is obtained.Pseudorandom number generator by \(p\)-adic chaos and Ramanujan expander graphs.https://zbmath.org/1459.053072021-05-28T16:06:00+00:00"Naito, Koichiro"https://zbmath.org/authors/?q=ai:naito.koichiroSummary: In our previous paper [the author, ``Randomness of \(p\)-adic discrete dynamical systems and its applications to cryptosystems'', in: Proceedings of the 10th international conference on nonlinear analysis and convex analysis, NACA 2017, Chitose, Japan, 2017. Yokohama: Yokohama Publishers (to appear)], applying chaotic properties of the \(p\)-adic dynamical system given by the \(p\)-adic logistic map, we constructed a new pseudorandom number generator. In this paper, using the pseudorandom sequences given by this generator, we construct random adjacency matrices and their random graphs. Then we numerically show that the eigenvalue distributions of these random matrices have the characteristical properties of the adjacency matrices of Ramanujan graphs.Dynamics and topological entropy of 1D Greenberg-Hastings cellular automata.https://zbmath.org/1459.370142021-05-28T16:06:00+00:00"Kesseböhmer, M."https://zbmath.org/authors/?q=ai:kessebohmer.marc"Rademacher, J. D. M."https://zbmath.org/authors/?q=ai:rademacher.jens-d-m"Ulbrich, D."https://zbmath.org/authors/?q=ai:ulbrich.dThe authors deal with the pure pulse-annihilation subsystem and its skew-product representation. They present a detailed analysis of the non-wandering set and determine the topological entropy, including its asymptotics. Let \(\mathcal{A}\) be an alphabet and \(X:=\mathcal{A}^{\mathbb{Z}}.\) Given the cellular automaton \(T : X \rightarrow X\), let \(Z\) consist of configurations which are either purely left- or right-moving under the dynamics of \(T\). The authors show that the topological entropy of \(T\) restricted to \(Z\) is given by \(h(Z, T|_Z)=2 \ln \rho_a\), where \(\rho_a\) denotes the largest eigenvalue of the transition matrix \(A\). Then it is proved that the subsystem \((Z, T|_Z)\) is chaotic in the sense of Devaney. Finally, the authors prove the following theorem:
Theorem. The topological entropy, \(h(X, T)\) of the 1D Greenberg-Hastings cellular automaton with \(e,r\in \mathbb{N}\) is given by \(h(X, T)=h(Z, T|_Z)=2 \ln \rho_a\), where \(\rho_a\) is the positive root of \(x^{a+1}-x^a-1\). Moreover, \(h(\Omega_{const},T|_{const})=0\) and \(0 < h(\Omega_{var},T|_{var}) < 2 \rho_a\) for \(r > e+1\).
Reviewer: Hasan Akin (Gaziantep)Strictly temporally periodic points in cellular automata.https://zbmath.org/1459.370112021-05-28T16:06:00+00:00"Dennunzio, Alberto"https://zbmath.org/authors/?q=ai:dennunzio.alberto"di Lena, Pietro"https://zbmath.org/authors/?q=ai:di-lena.pietro"Margara, Luciano"https://zbmath.org/authors/?q=ai:margara.lucianoSummary: We study the set of strictly periodic points in surjective cellular automata, i.e., the set of those configurations which are temporally periodic for a given automaton but they not spatially periodic. This set turns out to be dense for almost equicontinuous surjective cellular automata while it is empty for the positively expansive ones. In the class of additive cellular automata, the set of strictly periodic points can be either dense or empty. The latter happens if and only if the cellular automaton is topologically transitive.
For the entire collection see [Zbl 1392.68018].Intrinsic simulations between stochastic cellular automata.https://zbmath.org/1459.681262021-05-28T16:06:00+00:00"Arrighi, Pablo"https://zbmath.org/authors/?q=ai:arrighi.pablo"Schabanel, Nicolas"https://zbmath.org/authors/?q=ai:schabanel.nicolas"Theyssier, Guillaume"https://zbmath.org/authors/?q=ai:theyssier.guillaumeSummary: The paper proposes a simple formalism for dealing with deterministic, non-deterministic and stochastic cellular automata in a unifying and composable manner. Armed with this formalism, we extend the notion of intrinsic simulation between deterministic cellular automata, to the non-deterministic and stochastic settings. We then provide explicit tools to prove or disprove the existence of such a simulation between two stochastic cellular automata, even though the intrinsic simulation relation is shown to be undecidable in dimension two and higher. The key result behind this is the caracterization of equality of stochastic global maps by the existence of a coupling between the random sources. We then prove that there is a universal non-deterministic cellular automaton, but no universal stochastic cellular automaton. Yet we provide stochastic cellular automata achieving optimal partial universality.
For the entire collection see [Zbl 1392.68018].Computing by temporal order: asynchronous cellular automata.https://zbmath.org/1459.681322021-05-28T16:06:00+00:00"Vielhaber, Michael"https://zbmath.org/authors/?q=ai:vielhaber.michaelSummary: Our concern is the behaviour of the elementary cellular automata with state set \(\{0,1\}\) over the cell set \(\mathbb{Z}/n\mathbb{Z}\) (one-dimensional finite wrap-around case), under all possible update rules (asynchronicity).
Over the torus \(\mathbb{Z}/n\mathbb{Z}\) (\(n\leq 10\)),we will see that the ECA with Wolfram rule 57 maps any \(v\in\mathbb{F}^n_2\) to any \(w\in\mathbb{F}^n_2\), varying the update rule.
We furthermore show that all even (element of the alternating group) bijective functions on the set \(\mathbb{F}^n_2\cong\{0,\dots,2^n-1\}\), can be computed by ECA57, by iterating it a sufficient number of times with varying update rules, at least for \(n\leq 10\). We characterize the non-bijective functions computable by asynchronous rules.
The thread of all this is a novel paradigm:
The algorithm is neither hard-wired (in the ECA), nor in the program or data (initial configuration), but in the temporal order of updating cells, and temporal order is pattern-universal.
For the entire collection see [Zbl 1392.68018].Universality of one-dimensional reversible and number-conserving cellular automata.https://zbmath.org/1459.681312021-05-28T16:06:00+00:00"Morita, Kenichi"https://zbmath.org/authors/?q=ai:morita.kenichiSummary: We study one-dimensional reversible and number-conserving cellular automata (RNCCA) that have both properties of reversibility and number-conservation. In the case of 2-neighbor RNCCA, García-Ramos proved that every RNCCA shows trivial behavior in the sense that all the signals in the RNCCA do not interact each other. However, if we increase the neighborhood size, we can find many complex RNCCAs. Here, we show that for any one-dimensional 2-neighbor reversible partitioned CA (RPCA) with \(s\) states, we can construct a 4-neighbor RNCCA with 4\(s\) states that simulates the former. Since it is known that there is a computationally universal 24-state 2-neighbor RPCA, we obtain a universal 96-state 4-neighbor RNCCA.
For the entire collection see [Zbl 1392.68018].On the conjugacy problem of cellular automata.https://zbmath.org/1459.370132021-05-28T16:06:00+00:00"Jalonen, Joonatan"https://zbmath.org/authors/?q=ai:jalonen.joonatan"Kari, Jarkko"https://zbmath.org/authors/?q=ai:kari.jarkkoIt is well known that a natural notion of isomorphism in topological dynamics is topological conjugacy. Let \(f : A ^{\mathbb{Z}}\rightarrow A^{\mathbb{Z}}\) be a one-dimensional cellular automaton. In [Topological conjugacies between cellular automata. Dresden: Fakultät Mathematik und Naturwissenschaften der Technischen Universität (PhD Thesis, 2017)], \textit{J. Epperlein} proved that if \(T_1 , \dots, T_n\) are tori, there are cellular automata \(g,h : A ^{\mathbb{Z}}\rightarrow A^{\mathbb{Z}}\) such that \(g\) is surjective while \(h\) is not surjective and such that \(f_{T_k} , g_{T_k}\) and \(h_{T_k}\) are pairwise conjugate for all \(k \in \{1, \dots , n\}\).
In this paper, the authors investigate the conjugacy problem of two-dimensional cellular automata, i.e., the problem of deciding whether two cellular automata are conjugate or not. Firstly they prove the following result:
Theorem. The following two sets of pairs of one-dimensional one-sided cellular automata are recursively inseparable:
(i) pairs where the first cellular automaton has entropy strictly higher than the second one, and
(ii) pairs that are strongly conjugate and both have zero topological entropy.
Taking into account the above theorem, the authors present the same inseparability result for reversible two-dimensional two-sided cellular automata. They prove the following result:
Theorem. The following two sets of pairs of reversible two-dimensional cellular automata are recursively inseparable:
(i) pairs where the first cellular automaton has entropy strictly higher than the second one, and
(ii) pairs that are strongly conjugate and both have zero entropy.
Reviewer: Hasan Akin (Gaziantep)Dynamics of the ultra-discrete Toda lattice via Pitman's transformation.https://zbmath.org/1459.390442021-05-28T16:06:00+00:00"Croydon, David A."https://zbmath.org/authors/?q=ai:croydon.david-a"Sasada, Makiko"https://zbmath.org/authors/?q=ai:sasada.makiko"Tsujimoto, Satoshi"https://zbmath.org/authors/?q=ai:tsujimoto.satoshiSummary: By encoding configurations of the ultra-discrete Toda lattice by piecewise linear paths whose gradient alternates between \(-1\) and 1, we show that the dynamics of the system can be described in terms of a shifted version of Pitman's transformation (that is, reflection in the past maximum of the path encoding). This characterisation of the dynamics applies to finite configurations in both the non-periodic and periodic cases, and also admits an extension to infinite configurations. The latter point is important in the study of invariant measures for the ultra-discrete Toda lattice, which is pursued in the parallel work \textit{D. A. Croydon} and \textit{M. Sasada} [J. Math. Phys. 60, No. 8, 083301, 25 p. (2019; Zbl 1426.37013)]. We also describe a generalisation of the result to a continuous version of the box-ball system, whose states are described by continuous functions whose gradient may take values other than \(\pm 1\).Shifts, rotations and distributional chaos.https://zbmath.org/1459.370092021-05-28T16:06:00+00:00"Xu, Dongsheng"https://zbmath.org/authors/?q=ai:xu.dongsheng"Xiang, Kaili"https://zbmath.org/authors/?q=ai:xiang.kaili"Liang, Shudi"https://zbmath.org/authors/?q=ai:liang.shudiSummary: Let \(R_{r_0}, R_{r_1}: \mathbb{S}^1\longrightarrow \mathbb{S}^1\) be rotations on the unit circle \(\mathbb{S}^1\) and define \(f: \varSigma_2\times \mathbb{S}^1\longrightarrow \varSigma_2\times \mathbb{S}^1\) as \(f(x, t)=\bigl(\sigma (x), R_{r_{x_1}}(t)\bigr),\) for \(x=x_1x_2\cdots \in \varSigma_2:=\{0, 1\}^{\mathbb{N}}, t\in \mathbb{S}^1\), where \(\sigma: \varSigma_2\longrightarrow \varSigma_2\) is the shift, and \(r_0\) and \(r_1\) are rotational angles. It is first proved that the system \((\varSigma_2\times \mathbb{S}^1, f)\) exhibits maximal distributional chaos for any \(r_0, r_1\in \mathbb{R} \) (no assumption of \(r_0, r_1\in \mathbb{R}\setminus \mathbb{Q})\), generalizing Theorem 1 in [\textit{X. Wu} and \textit{G. Chen}, Topology Appl. 162, 91--99 (2014; Zbl 1290.37015)]. It is also obtained that \((\varSigma_2\times \mathbb{S}^1, f)\) is cofinitely sensitive and \((\hat{\mathscr{M}}^1, \hat{\mathscr{M}}^1)\)-sensitive and that \((\varSigma_2\times \mathbb{S}^1, f)\) is densely chaotic if and only if \(r_1-r_0 \in \mathbb{R}\setminus \mathbb{Q} \).Bounds for multiple recurrence rate and dimension.https://zbmath.org/1459.370032021-05-28T16:06:00+00:00"Hirayama, Michihiro"https://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)The topological sensitivity with respect to Furstenberg families.https://zbmath.org/1459.370102021-05-28T16:06:00+00:00"Wang, Tengfei"https://zbmath.org/authors/?q=ai:wang.tengfei"Jing, Kai"https://zbmath.org/authors/?q=ai:jing.kai"Yin, Jiandong"https://zbmath.org/authors/?q=ai:yin.jiandongSummary: In this work, a dynamical system \((X, f)\) means that \(X\) is a topological space and \(f:X\longrightarrow X\) is a continuous map. The aim of the article is to introduce the conceptions of topological sensitivity with respect to Furstenberg families, \(n\)-topological sensitivity, and multisensitivity and present some of their basic features and sufficient conditions for a dynamical system to possess some sensitivities. Actually, it is proved that every topologically ergodic but nonminimal system is syndetically sensitive and a weakly mixing system is \(n\)-thickly topologically sensitive and multisensitive under the assumption that \(X\) admits some separability.