Recent zbMATH articles in MSC 37Chttps://www.zbmath.org/atom/cc/37C2021-11-25T18:46:10.358925ZWerkzeugPing-pong configurations and circular orders on free groupshttps://www.zbmath.org/1472.200572021-11-25T18:46:10.358925Z"Malicet, Dominique"https://www.zbmath.org/authors/?q=ai:malicet.dominique"Mann, Kathryn"https://www.zbmath.org/authors/?q=ai:mann.kathryn"Rivas, Cristóbal"https://www.zbmath.org/authors/?q=ai:rivas.cristobal"Triestino, Michele"https://www.zbmath.org/authors/?q=ai:triestino.micheleSummary: We discuss actions of free groups on the circle with ``ping-pong'' dynamics; these are dynamics determined by a finite amount of combinatorial data, analogous to Schottky domains or Markov partitions. Using this, we show that the free group \(F_n\) admits an isolated circular order if and only if \(n\) is even, in stark contrast with the case for linear orders. This answers a question from [\textit{K. Mann} and \textit{C. Rivas}, Ann. Inst. Fourier 68, No. 4, 1399--1445 (2018; Zbl 07002300)]. Inspired by work in [\textit{S. Alvarez} et al., ``Generalized ping-pong partitions and locally discrete groups of real-analytic circle diffeomorphisms. II: Applications'', Preprint, \url{arXiv:2104.03348}], we also exhibit examples of ``exotic'' isolated points in the space of all circular orders on \(F_2\). Analogous results are obtained for linear orders on the groups \(F_n \times \mathbb{Z}\).On laminar groups, Tits alternatives and convergence group actions on \(\mathrm{S}^2\)https://www.zbmath.org/1472.200792021-11-25T18:46:10.358925Z"Alonso, Juan"https://www.zbmath.org/authors/?q=ai:alonso.juan-i|alonso.juan-miguel|alonso.juan-antonio|alonso.juan-j|peral-alonso.juan-carlos"Baik, Hyungryul"https://www.zbmath.org/authors/?q=ai:baik.hyungryul"Samperton, Eric"https://www.zbmath.org/authors/?q=ai:samperton.ericSummary: Following previous work of the second author, we establish more properties of groups of circle homeomorphisms which admit invariant laminations. In this paper, we focus on a certain type of such groups, so-called pseudo-fibered groups, and show that many 3-manifold groups are examples of pseudo-fibered groups. We then prove that torsion-free pseudo-fibered groups satisfy a Tits alternative. We conclude by proving that a purely hyperbolic pseudo-fibered group acts on the 2-sphere as a convergence group. This leads to an interesting question if there are examples of pseudo-fibered groups other than 3-manifold groups.Asymptotic expansions with exponential, power, and logarithmic functions for non-autonomous nonlinear differential equationshttps://www.zbmath.org/1472.340212021-11-25T18:46:10.358925Z"Cao, Dat"https://www.zbmath.org/authors/?q=ai:cao.dat-t"Hoang, Luan"https://www.zbmath.org/authors/?q=ai:hoang.luan-thachSummary: This paper develops further and systematically the asymptotic expansion theory that was initiated by \textit{C. Foias} and \textit{J. C. Saut} [Ann. Inst. Henri Poincaré, Anal. Non Linéaire 4, 1--47 (1987; Zbl 0635.35075)]. We study the long-time dynamics of a large class of dissipative systems of nonlinear ordinary differential equations with time-decaying forcing functions. The nonlinear term can be, but not restricted to, any smooth vector field which, together with its first derivative, vanishes at the origin. The forcing function can be approximated, as time tends to infinity, by a series of functions which are coherent combinations of exponential, power and iterated logarithmic functions. We prove that any decaying solution admits an asymptotic expansion, as time tends to infinity, corresponding to the asymptotic structure of the forcing function. Moreover, these expansions can be generated by more than two base functions and go beyond the polynomial formulation imposed in previous work.Existence and stability of limit cycles in the model of a planar passive biped walking down a slopehttps://www.zbmath.org/1472.340622021-11-25T18:46:10.358925Z"Makarenkov, Oleg"https://www.zbmath.org/authors/?q=ai:makarenkov.olegSummary: We consider the simplest model of a passive biped walking down a slope given by the equations of switched coupled pendula
[\textit{T. McGeer}, ``Passive dynamic walking'', Int. J. Robot. Res. 9, 62--82 (1990; \url{doi:10.1177/027836499000900206})].
Following the fundamental work by
\textit{M. Garcia} et al. [``The simplest walking model: stability, complexity, and scaling'', J. Biomech. Eng. 120, No. 2, 281--288 (1998; \url{doi:10.1115/1.2798313})],
we view the slope of the ground as a small parameter \(\gamma \geq 0\). When \(\gamma = 0\), the system can be solved in closed form and the existence of a family of cycles (i.e. potential walking cycles) can be computed in closed form. As observed in
[Garcia et al., loc. cit.],
the family of cycles disappears when \(\gamma\) increases and only isolated asymptotically stable cycles (walking cycles) persist. However, no mathematically complete proofs of the existence and stability of walking cycles have been reported in the literature to date. The present paper proves the existence and stability of a walking cycle (long-period gait cycle, as termed by McGeer) by using the methods of perturbation theory for maps. In particular, we derive a perturbation theorem for the occurrence of stable fixed points from 1-parameter families in two-dimensional maps that can be of independent interest in applied sciences.On the number of limit cycles in generalized Abel equationshttps://www.zbmath.org/1472.340652021-11-25T18:46:10.358925Z"Huang, Jianfeng"https://www.zbmath.org/authors/?q=ai:huang.jianfeng"Torregrosa, Joan"https://www.zbmath.org/authors/?q=ai:torregrosa.joan"Villadelprat, Jordi"https://www.zbmath.org/authors/?q=ai:villadelprat.jordiConsider the generalized Abel differential equation
\[
\frac{{dx}}{{d\theta}} = A(\theta) x^p + B(\theta) x^q,\tag{1}
\]
where \( p,q \) are natural numbers satisfying \( p \neq q, p,q \ge 2\), \(A\) and \(B \) are trigonometric polynomials of degree \(n \ge 1\) and \(m \ge 1\), respectively. Let the number \(H_{p,q}(n,m) \) denote the maximum number of isolated periodic solutions (limit cycles) of (1). By means of the second order Melnikov function the authors prove a lower bound for \(H_{p,q}(n,m) \), which is better than known ones. Especially, they obtain for the classical Abel equation (i.e. \( p=3, q=2\)) the estimate \( H_{3,2}(n,m) \geq 2(n+m)-1 \).Boundedness of solutions of a quasi-periodic sublinear Duffing equationhttps://www.zbmath.org/1472.340662021-11-25T18:46:10.358925Z"Peng, Yaqun"https://www.zbmath.org/authors/?q=ai:peng.yaqun"Zhang, Xinli"https://www.zbmath.org/authors/?q=ai:zhang.xinli"Piao, Daxiong"https://www.zbmath.org/authors/?q=ai:piao.daxiongIn this paper, the authors study the Lagrangian stability for the sublinear Duffing equations \(x''+e(t)|x|^{\alpha-1}x=p(t)\) with \(0<\alpha<1\), where \(e\) and \(p\) are real analytic quasi-periodic functions with frequency \(\omega\). By using the invariant curve theorem for quasi-periodic mappings established by \textit{P. Huang} et al. [Nonlinearity 29, No. 10, 3006--3030 (2016; Zbl 1378.37078)], they prove that every solution of the equation is bounded provided that the mean value of \(e\) is positive and the frequency \(\omega\) satisfies a Diophantine condition.Periodic forcing on degenerate Hopf bifurcationhttps://www.zbmath.org/1472.340742021-11-25T18:46:10.358925Z"Yuan, Qigang"https://www.zbmath.org/authors/?q=ai:yuan.qigang"Ren, Jingli"https://www.zbmath.org/authors/?q=ai:ren.jingliIn this work, the authors deal with the effect of periodic forcing on a system exhibiting a degenerate Hopf bifurcation. Two methods are employed to investigate bifurcations of periodic solution for the periodically forced system. It is obtained by averaging method that the system undergoes fold bifurcation, transcritical bifurcation, and even degenerate Hopf bifurcation of periodic solution. On the other hand, it is also shown by the Poincaré map that the system will undergo fold bifurcation, transcritical bifurcation, Neimark-Sacker bifurcation and flip bifurcation. Finally, we make a comparison between these two methods. The method is novel and enriches the bifurcation theory of delayed differential equation to some degree.Multiple periodic solutions for one-sided sublinear systems: a refinement of the Poincaré-Birkhoff approachhttps://www.zbmath.org/1472.340762021-11-25T18:46:10.358925Z"Dondè, Tobia"https://www.zbmath.org/authors/?q=ai:donde.tobia"Zanolin, Fabio"https://www.zbmath.org/authors/?q=ai:zanolin.fabioThe paper investigates the existence of periodic solutions, both harmonic and subharmonic, for planar Hamiltonian systems of the type \[ x' = h(y), \qquad y' = - a(t)g(x), \] where \(a(t)\) is a sign-changing periodic function and at least one of \(g\) and \(h\) is bounded on \(\mathbb{R}^-\) or \(\mathbb{R}^+\).
At first, by further assuming the global continuability for the solutions, a multiplicity result is proved via the Poincaré-Birkhoff theorem; as usual, solutions are distinguished via their nodal properties. Then, a refinement of this result, obtained with the theory of topological horseshoses, is presented; here, the assumption of global continuability is replaced by a largeness condition on the weight function \(a(t)\) in its negativity intervals. In this latter case, the existence of chaotic dynamics is also ensured.
Applications of the results are finally described for a Minkowksi-curvature equation like \[ \left( \frac{u'}{\sqrt{1-(u')^2}} \right)' + a(t) g(u) = 0 \] as well as for the equation, with exponential nonlinearity, \[ u'' + k(t)e^u = p(t). \]Exact multiplicity and stability of periodic solutions for Duffing equation with bifurcation methodhttps://www.zbmath.org/1472.340772021-11-25T18:46:10.358925Z"Liang, Shuqing"https://www.zbmath.org/authors/?q=ai:liang.shuqingSummary: Under some \(L^p\)-norms \((p\in [1,\infty ])\) assumptions for the derivative of the restoring force, the exact multiplicity and the stability of \(2\pi\)-periodic solutions for Duffing equation are considered. The nontrivial \(2\pi\)-periodic solutions of it are positive or negative, and the bifurcation curve of it is a unique reversed \(S\)-shaped curve. The class of the restoring force is extended, comparing with the class of \(L^{\infty }\)-norm condition. The proof is based on the global bifurcation theorem, topological degree and the estimates for periodic eigenvalues of Hill's equation by \(L^p\)-norms\((p\in [1,\infty ])\).Non-resonance and double resonance for a planar system via rotation numbershttps://www.zbmath.org/1472.340792021-11-25T18:46:10.358925Z"Liu, Chunlian"https://www.zbmath.org/authors/?q=ai:liu.chunlian"Qian, Dingbian"https://www.zbmath.org/authors/?q=ai:qian.dingbian"Torres, Pedro J."https://www.zbmath.org/authors/?q=ai:torres.pedro-joseThe authors consider a general planar periodic system and propose two existence results.
In the first one they compare the nonlinearity with two positively homogeneous functions with ``rotation numbers'' larger than some \(n\) and smaller than \(n+1\). They thus prove that the system has a periodic solution, by the use of the Poincaré-Bohl fixed point theorem. This is a generalization of some classical ``nonresonance'' results.
In the second one the above two functions have rotation numbers exactly equal to \(n\) and \(n+1\). Then, in order to avoid possible resonance phenomena, they add two Landesman-Lazer conditions, and they prove again the existence of a periodic solution.
The proofs involve delicate analysis in the phase-plane, in order to precisely estimate the rotational properties of the solutions.Periodic solutions for a singular Liénard equation with indefinite weighthttps://www.zbmath.org/1472.340802021-11-25T18:46:10.358925Z"Lu, Shiping"https://www.zbmath.org/authors/?q=ai:lu.shiping"Xue, Runyu"https://www.zbmath.org/authors/?q=ai:xue.runyuIn this paper, the authors study the following singular Liénard equation \[ x''(t)+f(x(t))x'(t)+\frac{\alpha(t)}{x^\mu(t)}= h(t),\tag{1} \] where \(f\in C((0, +\infty), \mathbb{R})\) may have a singularity at \(x=0,\, \mu\in(0, +\infty)\) is a constant, \(\alpha\) and \(h\) are \(T\)-periodic functions with \(\alpha,\, h \in L^1 ([0, T], \mathbb{R}).\) The weight function \(\alpha\) may change sign on \([0, T].\) A new method for estimating a priori bounds of all possible positive \(T\)-periodic solutions is obtained. By using a continuation theorem of Mawhin's coincidence degree theory, some new results on the existence of positive periodic solutions for the equation (1) are established.Existence of \(T/k\)-periodic solutions of a nonlinear nonautonomous system whose matrix has a multiple eigenvaluehttps://www.zbmath.org/1472.340812021-11-25T18:46:10.358925Z"Yevstafyeva, V. V."https://www.zbmath.org/authors/?q=ai:yevstafyeva.victoria-v|yevstafyeva.vistoria-vThe author considers the \(n\)-dimensional system of differential equations of the following form
\[
\dot{Y} = AY + BF(\sigma) + Kf(t).
\]
It is assumed that the matrix \(A\) has real nonzero eigenvalues, among which there are at least one positive and one multiple eigenvalues, the vectors \(B\) and \(K\) are nonzero, and the function \(F(\sigma),\)\,\(\sigma = (C,Y)\) describes a nonideal two-position relay with two output values. The perturbation function \(f(t)\) is \(T\)-periodic with \(T = 2\pi/\omega,\,\omega >0\) of the form
\[
f(t) = f_0 +f_1\sin(\omega t + \varphi_1) + f_2\sin (2\omega t + \varphi_2).
\]
Necessary conditions for the existence of a \(T/k\)-periodic solution of the system with \(k \in \mathbb{N}\) having two switching points in the phase space are studied. An existence theorem for a \(T\)-periodic solution is proved. A numerical example is presented.Hyers-Ulam stability for a class of perturbed Hill's equationshttps://www.zbmath.org/1472.341032021-11-25T18:46:10.358925Z"Dragičević, Davor"https://www.zbmath.org/authors/?q=ai:dragicevic.davorSummary: In this note we formulate sufficient conditions under which a certain class of nonlinear and nonautonomous differential equations of second order is Hyers-Ulam stable. This class consists of equations obtained by perturbing Hill's equation of the form \(x''=(\lambda^2(t)-\lambda '(t))x\).Transient probability in basins of noise influenced responses of mono and coupled Duffing oscillatorshttps://www.zbmath.org/1472.341132021-11-25T18:46:10.358925Z"Cilenti, Lautaro"https://www.zbmath.org/authors/?q=ai:cilenti.lautaro"Balachandran, Balakumar"https://www.zbmath.org/authors/?q=ai:balachandran.balakumarThis paper concerns Duffing oscillators perturbed by random noise and modeled using Ito stochastic differential equations (SDE). An improvement of the path integration method is introduced that reduces the number of points used and therefore facilitates less costly computation. The improved method is used in numerical experiments to study the relationship between intensity of random noise in the forcing function and destruction of the high-amplitude mode in the multistability region of hardened Duffing oscillators and also to study the level of intensity of noise that destroys an energy localized mode in arrays of two coupled Duffing oscillators. The paper notes that the new way of selecting a reduced number of points can be applied to lessen Monte-Carlo Euler-Maruyama method simulation costs for the SDEs.Orbital stability investigations for travelling waves in a nonlinearly supported beamhttps://www.zbmath.org/1472.350392021-11-25T18:46:10.358925Z"Nagatou, K."https://www.zbmath.org/authors/?q=ai:nagatou.kaori"Plum, M."https://www.zbmath.org/authors/?q=ai:plum.michael"McKenna, P. J."https://www.zbmath.org/authors/?q=ai:mckenna.patrick-josephThis paper investigates the orbital stability of the traveling waves solutions of the following nonlinear beam equation:
\[
\varphi_{tt} + \varphi_{xxxx} + e^{\varphi} - 1 = 0, \qquad (x, t) \in \mathbb{R} \times \mathbb{R}.
\]
More precisely, this paper rigorously prove the orbital stability for one traveling wave, and orbital instability for 15 traveling waves among the 36 known traveling waves, solutions to this equation. The employed method to achieve these results combines analytical and computed-assisted techniques.Erratum to: ``Triggered fronts in the complex Ginzburg Landau equation''https://www.zbmath.org/1472.353662021-11-25T18:46:10.358925Z"Goh, Ryan"https://www.zbmath.org/authors/?q=ai:goh.ryan-n"Scheel, Arnd"https://www.zbmath.org/authors/?q=ai:scheel.arndErratum to the authors' paper [ibid. 24, No. 1, 117--144 (2014; Zbl 1297.35224)].A nonsingular action of the full symmetric group admits an equivalent invariant measurehttps://www.zbmath.org/1472.370022021-11-25T18:46:10.358925Z"Nessonov, Nilolay"https://www.zbmath.org/authors/?q=ai:nessonov.nilolaySummary: Let \(\overline{\mathfrak{S}}_\infty\) denote the set of all bijections of natural numbers. Consider an action of \(\overline{\mathfrak{S}}_\infty\) on a measure space \((X,\mathfrak{M},\mu)\), where \(\mu\) is an \(\overline{\mathfrak{S}}_\infty\)-quasi-invariant measure. We prove that there exists an \(\overline{\mathfrak{S}}_\infty\)-invariant measure equivalent to \(\mu\).Abundance of wild historic behaviorhttps://www.zbmath.org/1472.370042021-11-25T18:46:10.358925Z"Araujo, V."https://www.zbmath.org/authors/?q=ai:araujo.vitor|araujo.vanessa-o|araujo.vanilse-s"Pinheiro, V."https://www.zbmath.org/authors/?q=ai:pinheiro.vladia|pinheiro.viltonFor a dynamical system \((X,f)\), a point \(x\in X\) is said to have historic behavior or to be irregular if the sequence \(\frac{1}{n_{k}} \sum_{j=1}^{n_{k}} \delta_{f^{j}(x)}\) does not converge in the weak* topology where \(\delta_{y}\) is the Dirac measure at point \(y.\)
From Birkhoff's ergodic theorem, the irregular set is not detectable from the point of view of any invariant measure.
However, the irregular set may have strong dynamical complexity in the sense of Hausdorff dimension, Lebesgue positive measure, topological entropy, topological pressure, distributional chaos and residual property.
Here the authors prove that a kind of stronger historical behavior, called wild historical behavior, is a topologically generic subset (countable intersection of open and dense subsets)
for wide classes of dynamical models. They consider a set function \[\tau_{x}(A)=\lim \sup _{n \rightarrow \infty} \frac{1}{n} \sum_{j=1}^{n} \delta_{f^{j}(x)}(A)\] that will serve as a pre-measure to obtain a Borel measure \(\eta_{x}\) through a classic well-known construction of Carathéodory. They prove that a point \(x\in X\) has historic behavior if and only if \(\eta_{x}\) is not a probability measure. A point \(x\in X\) is said to have wild historic behavior in \(\Lambda\) or to be a wild historic point if \(\eta_{x}\) gives infinite mass to every open subset of a compact invariant subset \(\Lambda\) for the dynamics. It is clear that a wild historic point has historic behavior.
The paper contains several interesting results. For example, they prove that the set of points with wild historic behavior is a topologically generic subset in:
(1) Every mixing topological Markov chain with a denumerable set of symbols (either one-sided or two-sided);
(2) Every open continuous transitive and positively expansive map of a compact metric space;
(3) Each local homeomorphism defined on an open dense subset of a compact space admitting an induced full branch Markov map;
(4) Suspension semiflows, with bounded roof functions, over the local homeomorphisms of the previous item;
(5) The stable set of any basic set \(\Lambda\) of either an Axiom A diffeomorphism, or an Axiom A vector field;
(6) The support of an expanding measure for a \(C^{1+}\) local diffeomorphism away from a non-flat critical/singular set on a compact manifold;
(7) The support of a non-atomic hyperbolic measure for a \(C^{1+}\) diffeomorphism, or a \(C^{1+}\) vector field of a compact manifold.
They also use properties of the measure \(\eta_{x}\) to deduce some features of the involved dynamical system, like existence of heteroclinic connections from the existence of open sets of historic points.
A well-known example of heteroclinic connections is given by ``Bowen eyes''.
Theorem B gives a sufficient condition for a diffeomorphism to exhibit a heteroclinic attractor as in the example of \textit{R. Bowen} [Am. J. Math. 95, 429--460 (1973; Zbl 0282.58009)].Weak rigidity of entropy spectrahttps://www.zbmath.org/1472.370062021-11-25T18:46:10.358925Z"Nakagawa, Katsukuni"https://www.zbmath.org/authors/?q=ai:nakagawa.katsukuniSummary: In this paper, we consider entropy spectra on topological Markov shifts. We prove that if two measure-preserving dynamical systems of Gibbs measures with Hölder continuous potentials are isomorphic, then their entropy spectra are the same. This result raises a new rigidity problem. We call this problem the weak rigidity problem, contrasting it with the strong rigidity problem proposed by \textit{L. Barreira} and \textit{V. Saraiva} [J. Stat. Phys. 130, No. 2, 387--412 (2008; Zbl 1131.37009)]. We give a complete answer to the weak rigidity problem for Markov measures on a topological Markov shift with a \(2\times 2\) aperiodic transition matrix. Moreover, we show that a `non-rigidity' result holds for a certain topological Markov shift with a \(3\times 3\) aperiodic transition matrix.Topological mixing of Weyl chamber flowshttps://www.zbmath.org/1472.370122021-11-25T18:46:10.358925Z"Dang, Nguyen-Thi"https://www.zbmath.org/authors/?q=ai:dang.nguyen-thi"Glorieux, Olivier"https://www.zbmath.org/authors/?q=ai:glorieux.olivierSummary: In this paper we study topological properties of the right action by translation of the Weyl chamber flow on the space of Weyl chambers. We obtain a necessary and sufficient condition for topological mixing.Characterizing orbit structures of homeomorphisms on Cantor setshttps://www.zbmath.org/1472.370142021-11-25T18:46:10.358925Z"Sherman, Casey"https://www.zbmath.org/authors/?q=ai:sherman.caseyThe author considers the following problem: if \(X\) is a Cantor set and \(T\colon X\to X\) is a homeomorphism, what possible orbit structures can \(T\) have? The {orbit spectrum} of \(T\) is the sequence
\[
\sigma(T)=(\zeta,\sigma_1,\sigma_2,\sigma_3,\dots)
\]
of cardinals, where \(\zeta\) is the number of \(\mathbb{Z}\)-orbits and \(\sigma_n\) is the number of \(n\)-cycles. The {periodic support} of \(\sigma\) is \(\textrm{supp}(\sigma) =\{n\colon \sigma_n\neq 0\}\) and the {\(\mathfrak{c}\)-support} of \(\sigma\) is \(\mathfrak{c}\,\textrm{supp}(\sigma)=\{n\colon \sigma_n=\mathfrak{c}\}\).
We say that \(T\colon X\to X\) is {finitely based} if there exist \(k>0\) and \(n_1,\dots,n_k\in\mathrm{supp}(\sigma)\) such that for every \(j\in \mathrm{supp}(\sigma)\) there is some \(i\leq k\) with \(n_i\mid j\). A number \(j\in \mathrm{supp}(\sigma)\) is called a {stray period for \(T\)} if \(jk\not\in\mathfrak{c}\,\mathrm{supp}(\sigma)\) for all \(k\in\mathbb{N}\).
The main result is the following characterization of the sequence of cardinals which can be realized by the orbit spectrum of homeomorphism on a Cantor set.
Theorem. Let \(\sigma=(\zeta,\sigma_1,\sigma_2,\dots)\) be a sequence of cardinals which are countable or with size \(\mathfrak{c}\). There exists a homeomorphism \(T\colon X\to X\) on a Cantor set \(X\) with \(\sigma(T)=\sigma\) if and only if one of the following properties holds:
(1) \(\zeta=0\), \(\sigma\) is finitely based, and \(\sigma\) has no stray terms;
(2) \(1\leq \zeta<\mathfrak{c}\), \(\mathfrak{c}\,\mathrm{supp}(\sigma)\) is infinite, and \(\sum_{\text{stray } n}\sigma_n\leq \zeta\);
(3) \(\zeta=\mathfrak{c}\).
The main result has the following consequence: for a bijection \(T\) on a set \(X\) with the cardinal \(\mathfrak{c}\), there exists a topology making \(X\) a Cantor set and \(T\) a homeomorphism if and only if the orbit spectrum of \(T\) can be realized by the orbit spectrum of homeomorphism on a Cantor set.Extreme partitions of a Lebesgue space and their application in topological dynamicshttps://www.zbmath.org/1472.370152021-11-25T18:46:10.358925Z"Bułatek, Wojciech"https://www.zbmath.org/authors/?q=ai:bulatek.wojciech"Kamiński, Brunon"https://www.zbmath.org/authors/?q=ai:kaminski.brunon"Szymański, Jerzy"https://www.zbmath.org/authors/?q=ai:szymanski.jerzySummary: It is shown that any topological action \(\Phi\) of a countable orderable and amenable group \(G\) on a compact metric space \(X\) and every \(\Phi \)-invariant probability Borel measure \(\mu\) admit an extreme partition \(\zeta\) of \(X\) such that the equivalence relation \(R_{\zeta}\) associated with \(\zeta\) contains the asymptotic relation \(A(\Phi)\) of \(\Phi . As\) an application of this result and the generalized Glasner theorem it is proved that \(A(\Phi)\) is dense for the set \(E_{\mu}(\Phi)\) of entropy pairs.Hausdorff dimension of frequency sets of univoque sequenceshttps://www.zbmath.org/1472.370162021-11-25T18:46:10.358925Z"Li, Yao-Qiang"https://www.zbmath.org/authors/?q=ai:li.yaoqiangSummary: We study the set \(\Gamma\) consisting of univoque sequences, the set \(\Lambda\) consisting of sequences in which the lengths of consecutive zeros and consecutive ones are bounded, and their frequency subsets \(\Gamma_a\), \(\underline{\Gamma}_a\), \(\overline{\Gamma}_a\) and \(\Lambda_a\), \(\underline{\Lambda}_a\), \(\overline{\Lambda}_a\) consisting of sequences respectively in \(\Gamma\) and \(\Lambda\) with frequency, lower frequency and upper frequency of zeros equal to some \(a\in[0,1]\). The Hausdorff dimension of all these sets are obtained by studying the dynamical system \((\Lambda^{(m)},\sigma)\) where \(\sigma\) is the shift map and \(\Lambda^{(m)}=\left\{w\in\{0,1\}^{\mathbb{N}}:w\text{ does not contain }0^m\text{ or }1^m\right\}\) for integer \(m\geq 3\), studying the Bernoulli-type measures on \(\Lambda^{(m)}\) and finding out the unique equivalent \(\sigma\)-invariant ergodic probability measure.New examples of stably ergodic diffeomorphisms in dimension 3https://www.zbmath.org/1472.370212021-11-25T18:46:10.358925Z"Núñez, Gabriel"https://www.zbmath.org/authors/?q=ai:nunez.gabriel"Obata, Davi"https://www.zbmath.org/authors/?q=ai:obata.davi"Rodriguez Hertz, Jana"https://www.zbmath.org/authors/?q=ai:rodriguez-hertz.janaSome remarks on global analytic planar vector fields possessing an invariant analytic sethttps://www.zbmath.org/1472.370222021-11-25T18:46:10.358925Z"García, Isaac A."https://www.zbmath.org/authors/?q=ai:garcia.isaac-aSummary: We study the problem of determining the canonical form that a planar analytic vector field in all the real plane can have to possess a given invariant analytic set. We determine some conditions that guarantee the only solution to this inverse problem is the trivial one.The existence of a 3-cycle implies the existence of all cycles (another proof)https://www.zbmath.org/1472.370232021-11-25T18:46:10.358925Z"Al-Fadhel, Tariq A."https://www.zbmath.org/authors/?q=ai:al-fadhel.tariq-aSummary: It is known that the existence of a 3-cycle of a continuous map implies the existence of all cycles as in [\textit{R. L. Devaney}, A first course in chaotic dynamical systems. Theory and experiment. Reading, MA: Addison-Wesley (1992; Zbl 0768.58001); \textit{T.-Y. Li} and \textit{J. A. Yorke}, Am. Math. Mon. 82, 985--992 (1975; Zbl 0351.92021); \textit{A. N. Sharkovskij}, Int. J. Bifurcation Chaos Appl. Sci. Eng. 5, No. 5, 1263--1273 (1995; Zbl 0890.58012)]. This paper presents another technique using the inverse images of the fixed point of the map to give another proof of this fact and to give details about the phase portraits of each cycle.Asymptotic escape rates and limiting distributions for multimodal mapshttps://www.zbmath.org/1472.370242021-11-25T18:46:10.358925Z"Demers, Mark F."https://www.zbmath.org/authors/?q=ai:demers.mark-f"Todd, Mike"https://www.zbmath.org/authors/?q=ai:todd.mikeSummary: We consider multimodal maps with holes and study the evolution of the open systems with respect to equilibrium states for both geometric and Hölder potentials. For small holes, we show that a large class of initial distributions share the same escape rate and converge to a unique absolutely continuous conditionally invariant measure; we also prove a variational principle connecting the escape rate to the pressure on the survivor set, with no conditions on the placement of the hole. Finally, introducing a weak condition on the centre of the hole, we prove scaling limits for the escape rate for holes centred at both periodic and non-periodic points, as the diameter of the hole goes to zero.Dynamical zeta functions of Reidemeister type and representations spaceshttps://www.zbmath.org/1472.370252021-11-25T18:46:10.358925Z"Fel'shtyn, Alexander"https://www.zbmath.org/authors/?q=ai:felshtyn.alexander"Zietek, Malwina"https://www.zbmath.org/authors/?q=ai:zietek.malwinaLet \(G\) be a countable discrete group and \(\varphi \,:\,G\,\longrightarrow\,G\) be an endomorphism. Two elements \(a,\,b\in G\) are said to be \(\varphi\)-conjugate if there exists \(g\in G\) such that \(b\,=\,ga\varphi (g^{-1})\). The resulting class of equivalence of an element \(x\in G\) is called \(\varphi\)-conjugacy and is denoted by \(\{\,x\,\}_{\varphi }\). The number of \(\varphi\)-conjugacy classes is called ``Reidemeister number'' and is denoted by \(R(\varphi )\). A Reidemeister zeta function can be defined [the first author, Colloq. Math. 62, No. 1, 153--166 (1991; Zbl 0745.58038)] as the power series
\[ R_{\varphi}(z)\,=\,\exp \left(\sum_{n\,=\,1}^{\infty }\frac{R(\varphi {n})}{n}z^{n}\right)\,.\]
For the discrete group \(G\), let \(\hat{G}\) denotes the unitary dual of \(G\), i.e., the space of equivalence classes of unitary irreducible representations of \(G\), equipped with the hull-kernel topology. Let \(\hat{G}_{f}\) denote the subspace of \(\hat{G}\) formed by irreducible finite-dimensional representations and let \(\hat{G}_{ff}\) denote the subspace of \(\hat{G}_{f}\) formed by finite representations.
For an endomorphism of \(G\), although the correspondence \(\hat{\varphi } : \varrho\,\longrightarrow\,\varrho\circ \varphi\) does not define a dynamical system on \(\hat{G}\), one can consider representations \(\varrho\) such that \(\varrho\sim \varrho\circ \varphi\), and in analogy \(R(\varphi)\) can be defined as the number \(RT(\varphi )\) of all \([\varrho]\in \hat{G}\) such that \(\varrho\sim \varrho\circ \varphi\). If \([\varrho]\in \hat{G}_{f}\) (resp., \([\varrho]\in \hat{G}_{ff}\) ), then one defines \(RT^{f}(\varphi )\) (resp., \(RT^{ff}(\varphi )\)).
In the preprint [the first author et al., ``New zeta functions of Reidemeister type and twisted Burnside-Frobenius theory'', Preprint, \url{arXiv:1804.02874}], by means of the analogy with the Reidemeister zeta function the following dynamical representation functions are defined:
\[ RT_{\varphi}(z)\,=\,\exp \left(\sum_{n\,=\,1}^{\infty }\frac{RT(\varphi {n})}{n}z^{n}\right)\,,\]
\[ RT^{f}_{\varphi}(z)\,=\,\exp \left(\sum_{n\,=\,1}^{\infty }\frac{RT^{f}(\varphi {n})}{n}z^{n}\right)\,,\]
\[ RT^{ff}_{\varphi}(z)\,=\,\exp \left(\sum_{n\,=\,1}^{\infty }\frac{RT^{ff}(\varphi {n})}{n}z^{n}\right)\,.\]
These zeta functions are assumed to be well defined.
In [the first author and \textit{R. Hill}, \(K\)-Theory 8, No. 4, 367--393 (1994; Zbl 0814.58033)] the following problem is posed. For which groups and endomorphisms is the Reidemeister zeta function \(R(\varphi)\) a rational function? Is this zeta function an algebraic function?
This problem has been studied in the literature, see for example [the first author, Colloq. Math. 62, No. 1, 153--166 (1991; Zbl 0745.58038); Topology Appl. 67, No. 2, 119--131 (1995; Zbl 0845.55002); \textit{L. Li}, Adv. Math., Beijing 23, No. 3, 251--256 (1994; Zbl 0883.55003); \textit{K. Dekimpe} and \textit{G.-J. Dugardein}, J. Fixed Point Theory Appl. 17, No. 2, 355--370 (2015; Zbl 1329.55002); the first author and \textit{J. B. Lee}, Topology Appl. 181, 62--103 (2015; Zbl 1346.37021)].
Here we report some of the results obtained in this paper, thus referring for the definitions and the terminology used therein.
For a discrete group \(G\) and an endomorphism \(AM^{f}(\varphi ^{n})\) let us denote the number of isolated \(n\)-periodic points of the dynamical system on \(\hat{G}^{\varphi}_{f}\) by \((\hat{\varphi })^{n}\). The Artin-Mazur representation zeta function is defined as
\[ AM^{\varphi }_{f}(z)\,=\,\exp \left(\sum_{n\,=\,1}^{\infty }\frac{AM^{f}(\varphi ^{n})}{n}z^{n}\right)\,. \]
Let \(Z(\varphi)\) be one of the numbers \(RT(\varphi),\,\,RT^{f}(\varphi),\,\, RT^{ff}(\varphi),\,\, AM^{f}(\varphi)\). Then \(Z_{\varphi }(z)\) represents the corresponding zeta function:
\[ Z_{\varphi}(z)\,=\, \exp \left(\sum_{n\,=\,1}^{\infty }\frac{Z(\varphi ^{n})}{n}z^{n}\right)\,.\] The following theorems are proved.
Theorem 2.9. Let \(\varphi\) be a periodic automorphism of least period \(m\) of a group \(G\). Then the zeta function
\(Z_{\varphi} (z)\) is equal to
\[ Z_{\varphi} (z)\,=\,\prod _{d|m}\sqrt[d]{(1-z^{d})^{-P(d)}}\,,\]
where the product is taken over all the divisors \(d\) of the period \(m\), and \(P(d)\) is the integer
\(P(d)\,=\,\sum_{d_{1}|d}\mu (d_{1})Z(\varphi ^{d/d_{1}})\).
Here \(\mu\) denotes the Möbius function.
Theorem 2.11. Let \(\varphi : G\,\longrightarrow\,G\) be an endomorphism of a group \(G\). Suppose that the subspaces
\(\hat{G}^{\varphi},\,\, \hat{G}^{\varphi}_{f},\,\,\hat{G}^{\varphi}_{ff}\) are finite. Then the zeta function \(Z_{\varphi}(z)\) is a rational function satisfying the functional equation
\[ Z_{\varphi }\left(\frac{1}{z}\right)\,=\,(-1)^{a}z^{b}Z_{\varphi}(z).\]
Here the numbers \(a\) and \(b\) are respectively the number of periodic \(\hat{\varphi}\)-orbits of elements
of \(\hat{G}^{\varphi}\), resp., \(\hat{G}^{\varphi}_{f}\) or \(\hat{G}^{\varphi}_{ff}\), and the number of periodic elements of \(\hat{G}^{\varphi}\), resp., \(\hat{G}^{\varphi}_{f}\) or \(\hat{G}^{\varphi}_{ff}\).
Also for finitely generated torsion-free nilpotent groups and crystallographic groups a theorem is proved.
In Section 3 the Reidemeister torsion is defined and the authors explore a connection between Reidemeister-type zeta functions and the Reidemeister torsion of the mapping torus (Theorem 3.2). If \(G\) is a discrete group and \(\varphi\) an endomorphism, then it is possible (under some conditions) to obtain information for the Reidemeister-type zeta function \(R_{\varphi}(z)\) for the group \(G\) from the (induced) Reidemeister-type zeta function of a subgroup of the group \(G\), or a quotient of the group.
In the last section the authors, following the work of \textit{R. Miles} [Bull. Lond. Math. Soc. 40, No. 4, 696--704 (2008; Zbl 1147.37012)], present results about the Pólya-Carlson dichotomy and study the analytic behavior of the Reidemeister zeta function for a large class of automorphisms of abelian groups.
For the entire collection see [Zbl 1448.37001].Fourier multipliers and transfer operatorshttps://www.zbmath.org/1472.370262021-11-25T18:46:10.358925Z"Pollicott, Mark"https://www.zbmath.org/authors/?q=ai:pollicott.markThe author gives a rigorous proof of a conjectured numerical value proposed by \textit{X. Chen} and \textit{H. Volkmer} [J. Fractal Geom. 5, No. 4, 351--386 (2018; Zbl 1400.37026)] which estimates a quantity related to the spectral radius of a transfer operator. The problem is significantly connected to the theory of Fourier multipliers. More specifically, the author takes the bounded linear operator \(\mathcal{L}: C^{0}([0, 1])\rightarrow C^{0}([0, 1])\) defined by \[(\mathcal{L}u)(t) = \frac{1}{3} \sum_{i=0}^{3}\left|\sin\left(\frac{\pi (t+i)}{3}\right)\right|u\left(\frac{t+i}{3}\right).\] For estimating the conjectured numerical value \(c=\lim_{n\rightarrow +\infty}||\mathcal{L}^{n}||^{1/n}\), the following complex function is used: \[d(z)=\exp\bigg(-\sum_{n=1}^{\infty}\frac{z^{n}}{n}\frac{1}{3^{n}-1}\sum_{j=0}^{3^{n}-1}\prod_{k=0}^{n-1}\sin\bigg(\frac{3^{k}j\pi}{3^{n}-1}\bigg)\bigg), \quad z\in \mathbb{C}.\] Note that \(d(z)\) extends analytically to \(\mathbb{C}\). The smallest positive zero \(\alpha>0\) is the reciprocal of the spectral radius \(c\), i.e., \(c=1/\alpha\). He describes a rigorous computation to determine a better estimate of \(c\), namely \[c= 0.648314752798325682324771447 \dots \pm 10^{-27}.\]
The author also considers a more general form of the above bounded linear operator \(\mathcal{L}\) and estimates its spectral radius. He gives two applications to justify the importance of his results.Pressure inequalities for Gibbs measures of countable Markov shiftshttps://www.zbmath.org/1472.370272021-11-25T18:46:10.358925Z"Rühr, René"https://www.zbmath.org/authors/?q=ai:ruhr.reneSummary: We provide a quantification of the uniqueness of Gibbs measure for topologically mixing countable Markov shifts with locally Hölder continuous potentials. Corollaries for speed of convergence for approximation by finite subsystems are also given.Pressure, Poincaré series and box dimension of the boundaryhttps://www.zbmath.org/1472.370282021-11-25T18:46:10.358925Z"Iommi, Godofredo"https://www.zbmath.org/authors/?q=ai:iommi.godofredo"Velozo, Anibal"https://www.zbmath.org/authors/?q=ai:velozo.anibalOn conditions for rate-induced tipping in multi-dimensional dynamical systemshttps://www.zbmath.org/1472.370292021-11-25T18:46:10.358925Z"Kiers, Claire"https://www.zbmath.org/authors/?q=ai:kiers.claire"Jones, Christopher K. R. T."https://www.zbmath.org/authors/?q=ai:jones.christopher-k-r-tSummary: The possibility of \textit{rate-induced tipping} (R-tipping) away from an attracting fixed point has been thoroughly explored in 1-dimensional systems. In these systems, it is impossible to have R-tipping away from a path of quasi-stable equilibria that is \textit{forward basin stable} (FBS), but R-tipping is guaranteed for paths that are non-FBS of a certain type. We will investigate whether these results carry over to multi-dimensional systems. In particular, we will show that the same conditions guaranteeing R-tipping in 1-dimension also guarantee R-tipping in higher dimensions; however, it is possible to have R-tipping away from a path that is FBS even in 2-dimensional systems. We will propose a different condition, \textit{forward inflowing stability} (FIS), which we show is sufficient to prevent R-tipping in all dimensions. The condition, while natural, is difficult to verify in concrete examples. \textit{Monotone systems} are a class for which FIS is implied by an easily verifiable condition. As a result, we see how the additional structure of these systems makes predicting the possibility of R-tipping straightforward in a fashion similar to 1-dimension. In particular, we will prove that the FBS and FIS conditions in monotone systems reduce to comparing the relative positions of equilibria over time. An example of a monotone system is given that demonstrates how these ideas are applied to determine exactly when R-tipping is possible.On Sinaĭ billiards on flat surfaces with hornshttps://www.zbmath.org/1472.370302021-11-25T18:46:10.358925Z"Bruin, Henk"https://www.zbmath.org/authors/?q=ai:bruin.henkSummary: We show that certain billiard flows on planar billiard tables with horns can be modeled as suspension flows over Young towers [\textit{L.-S. Young}, Ann. Math. (2) 147, No. 3, 585--650 (1998; Zbl 0945.37009)] with exponential tails. This implies exponential decay of correlations for the billiard map. Because the height function of the suspension flow itself is polynomial when the horns are Torricelli-like trumpets, one can derive Limit Laws for the billiard flow, including Stable Limits if the parameter of the Torricelli trumpet is chosen in (1, 2).On commuting billiards in higher-dimensional spaces of constant curvaturehttps://www.zbmath.org/1472.370312021-11-25T18:46:10.358925Z"Glutsyuk, Alexey"https://www.zbmath.org/authors/?q=ai:glutsyuk.alexey-aSummary: We consider two nested billiards in \(\mathbb{R}^d\), \(d\geq3\), with \(C^2\)-smooth strictly convex boundaries. We prove that if the corresponding actions by reflections on the space of oriented lines commute, then the billiards are confocal ellipsoids. This together with the previous analogous result of the author in two dimensions solves completely the commuting billiard conjecture due to \textit{S. Tabachnikov} [Geom. Dedicata 53, No. 1, 57--68 (1994; Zbl 0813.52003)]. The main result is deduced from the classical theorem due to \textit{M. Berger} [Geometry. I, II. Transl. from the French by M. Cole and S. Levy. Berlin: Springer (2009; Zbl 1153.51001)]
which says that in higher dimensions only quadrics may have caustics. We also prove versions of Berger's theorem and the main result for billiards in spaces of constant curvature (space forms).Complexity growth of a typical triangular billiard is weakly exponentialhttps://www.zbmath.org/1472.370322021-11-25T18:46:10.358925Z"Scheglov, Dmitri"https://www.zbmath.org/authors/?q=ai:scheglov.dmitriA generalized diagonal of a polygonal billiard is an orbit which connects two vertices, whose length is calculated in terms of the number of billiard reflections. The main result of the paper is that for a typical (in the Lebesgue measure sense) triangular billiards a number of generalized diagonals of length no greater than \(n\) can be estimated from above by \(C\exp(n^\epsilon)\) for any \(\epsilon>0\). This gives a partial answer to a conjecture by \textit{A. Katok} [Commun. Math. Phys. 111, 151--160 (1987; Zbl 0631.58020)].Stable accessibility with \(2\) dimensional centerhttps://www.zbmath.org/1472.370332021-11-25T18:46:10.358925Z"Avila, Artur"https://www.zbmath.org/authors/?q=ai:avila.artur"Viana, Marcelo"https://www.zbmath.org/authors/?q=ai:viana.marceloConsider a partially hyperbolic diffeomorphism \(f:M\rightarrow M\) of a closed manifold \(M\). Given two points \(x, y\in M\) we say that \(x\) is accessible from \(y\) if there is a \(C^1\) path that connects \(x\) to \(y\) and is tangent at every point to the stable or the unstable space. The equivalence classes of this relation are called \(f\)-accessibility classes. The diffeomorphism \(f\) is called accessible if there exists a unique accessibility class. The diffeomorphism \(f\) is called stably accessible if there is a \(C^1\) open neighborhood \(\mathcal U\) of \(f\) such that every \(C^2\) diffeomorphism \(g\in \mathcal U\) is accessible. In this work the authors show that accessible partially hyperbolic diffeomorphisms with a two-dimensional center bundle are stably accessible.
For the entire collection see [Zbl 1446.37001].Transitive partially hyperbolic diffeomorphisms with one-dimensional neutral centerhttps://www.zbmath.org/1472.370342021-11-25T18:46:10.358925Z"Bonatti, Christian"https://www.zbmath.org/authors/?q=ai:bonatti.christian"Zhang, Jinhua"https://www.zbmath.org/authors/?q=ai:zhang.jinhuaThe aim of this paper is to study partially hyperbolic diffeomorphism with some specific properties. A diffeomorphism of a manifold of class \(C^1\) is said to be partially hyperbolic when the tangent bundle decomposes as the direct sum of a uniformly expanding bundle, a uniformly contracting bundle, and a bundle with intermediate behavior, the center. Furthermore it is called topologically neutral along the center if its iterates map short paths tangent to the center subbundle to short paths, uniformly on \(n\). A diffeomorphism is called transitive if it has dense orbits.
The authors classify topologically (namely up to conjugation by a homeomorphism) diffeomorphisms of closed three-manifolds that are partially hyperbolic, transitive, and have one-dimensional topological neutral center. Namely, up to finite lifts and iterates, these are conjugate to either skew products over a linear Anosov diffeomorphism of the torus with a rotation of the circle, or the time 1-map of a transitive topological Anosov flow. In arbitrary dimension, the authors prove that there exists a continuous metric along the center foliation that is dynamically invariant. Then this result is used for the classification in dimension three.Invariance of entropy for maps isotopic to Anosovhttps://www.zbmath.org/1472.370352021-11-25T18:46:10.358925Z"Carrasco, Pablo D."https://www.zbmath.org/authors/?q=ai:carrasco.pablo-d"Lizana, Cristina"https://www.zbmath.org/authors/?q=ai:lizana.cristina"Pujals, Enrique"https://www.zbmath.org/authors/?q=ai:pujals.enrique-ramiro"Vásquez, Carlos H."https://www.zbmath.org/authors/?q=ai:vasquez.carlos-hEquilibrium states for non-uniformly hyperbolic systems: statistical properties and analyticityhttps://www.zbmath.org/1472.370362021-11-25T18:46:10.358925Z"Afonso, Suzete Maria"https://www.zbmath.org/authors/?q=ai:afonso.suzete-m"Ramos, Vanessa"https://www.zbmath.org/authors/?q=ai:ramos.vanessa"Siqueira, Jaqueline"https://www.zbmath.org/authors/?q=ai:siqueira.jaquelineThis paper explores a large family of nonuniformly expanding maps associated with hyperbolic potentials having small variation. It is known that this class has unique equilibrium states. The authors' goal is to show that these equilibrium states also possess strong statistical properties. They show in particular that correlations decay exponentially and that a version of the central limit holds.
The authors prove that the unique equilibrium state for each member of the family is the eigenfunction of the transfer operator and the eigenmeasure of the dual operator. Both of these have the spectral radius as an eigenvalue. Using an approach with projective metrics they prove that the transfer operator has the spectral gap property in a space of Hölder continuous observables. The core of the paper deals with establishing this spectral gap property.
Once the spectral gap property is proved, the authors also study the behavior of the system under small perturbations of the potential function. They prove that the equilibrium state and other thermodynamic quantities -- such as the topological pressure -- vary analytically with the potential.
In a final section the authors broaden their scope to develop comparable results for equilibrium states associated with families of nonuniformly hyperbolic skew products and hyperbolic Hölder continuous potentials.Unique equilibrium states, large deviations and Lyapunov spectra for the Katok maphttps://www.zbmath.org/1472.370382021-11-25T18:46:10.358925Z"Wang, Tianyu"https://www.zbmath.org/authors/?q=ai:wang.tianyuThe Katok map is a nonuniformly hyperbolic diffeomorphism of the torus \(\mathbb{T}^2\) of class \(C^\infty\). This map is generated as a slow-down of the trajectories of a uniformly hyperbolic automorphism of the torus in a small neighborhood near the fixed point. The author studies its thermodynamic formalism using an approach of orbit decomposition following the technique introduced in [\textit{V. Climenhaga} and \textit{D. J. Thompson}, Adv. Math. 303, 745--799 (2016; Zbl 1366.37084)].
The author's approach is to generalize the dynamical properties of the map and the regularity conditions for the potential function described in [\textit{R. Bowen}, Math. Syst. Theory 8, 193--202 (1975; Zbl 0299.54031)] and make them hold at what the author calls an ``essential collection of orbit segments'' that dominates in topological pressure and has ``enough uniformly hyperbolic behavior''.
It is known that because the Katok map is expansive, it has an equilibrium state for any continuous potential. Indeed the Katok map is topologically conjugate to a linear automorphism of the torus by a homeomorphism. From \textit{R. Bowen}'s work it is known that the Katok map has a unique measure of maximal entropy. However, the homeomorphism for the conjugacy is neither differentiable nor Hölder continuous, so the thermodynamic formalism of the Katok map is non-trivial.
The author proves that, for an appropriate choice of a \(t\)-potential, there is an orbit decomposition with the required regularity on a collection of orbit segments that dominates the pressure. Then the author can prove that for any such potentials the equilibrium states are unique.Lorenz attractors and the modular surfacehttps://www.zbmath.org/1472.370402021-11-25T18:46:10.358925Z"Bonatti, Christian"https://www.zbmath.org/authors/?q=ai:bonatti.christian"Pinsky, Tali"https://www.zbmath.org/authors/?q=ai:pinsky.taliStochastic potentials of intermittent mapshttps://www.zbmath.org/1472.370492021-11-25T18:46:10.358925Z"Li, Huaibin"https://www.zbmath.org/authors/?q=ai:li.huaibinThe author considers an intermittent map \(f_{\kappa}:[0,1] \to [0,1]\), \[x \mapsto \begin{cases} x(1+x^{\kappa}), & x(1+x^{\kappa}) <1; \\
x(1+x^{\kappa}) -1, & x(1+x^{\kappa}) \ge 1, \end{cases}\] where \(\kappa>0\) is a positive number. A continuous function \(\varphi: [0,1] \to \mathbb{R}\), also known as potential, is said to be stochastic for \(f_{\kappa}\) if there is a unique equilibrium state \(\nu\) for the topological pressure \(P(f_{\kappa}, \phi)\) and the equilibrium state \(\nu\) is exponentially mixing for the map \(f_{\kappa}\). The main result of the paper is that a Hölder continuous potential \(\varphi: [0,1] \to \mathbb{R}\) is stochastic for \(f_{\kappa}\) if and only \(P(f_{\kappa}, \phi)> \varphi(0)\). As a consequence, the author proves the following:
(1) If \(0<\beta \le \min\{\kappa, 1\}\), then the set of non-stochastic \(\beta\)-Hölder continuous potentials has nonempty interior.
(2) If \(\kappa < 1\) and \(\kappa<\beta \le 1\), then all \(\beta\)-Hölder continuous potentials are stochastic.Winning property of distal set for \(\beta\)-transformationshttps://www.zbmath.org/1472.370502021-11-25T18:46:10.358925Z"Yang, Qianqian"https://www.zbmath.org/authors/?q=ai:yang.qianqian"Wang, Shuailing"https://www.zbmath.org/authors/?q=ai:wang.shuailingSummary: For any \(\beta>1\), let \(([0,1),T_\beta)\) be the \(\beta\)-transformation dynamical system. For any \(y\in[0,1)\), define the distal set of a given point \(y\) as
\[
D_\beta(y)=\bigg\{x\in[0,1):\liminf_{n\to\infty}|T^n_\beta(x)-T^n_\beta(y)|>0\bigg\}
\]
\textit{Q. Yang} et al. [J. Math. Anal. Appl. 464, No. 1, 188--200 (2018; Zbl 1386.37011)]
proved that the Hausdorff dimension of the distal set of any point is one for any \(\beta >1\). In this paper, we study the winning property of the distal set of a given point \(y\). We prove that the distal set of a given point \(y\) is \(\alpha\)-winning for any \(\beta>1\) and \(y\in[0,1)\), where \(\alpha<\frac{1}{64}\) is a constant. By the definition of winning set, it's obvious that the distal set of a given point y is a dense set.Correction to: ``Rel leaves of the Arnoux-Yoccoz surfaces''https://www.zbmath.org/1472.370542021-11-25T18:46:10.358925Z"Hooper, W. Patrick"https://www.zbmath.org/authors/?q=ai:hooper.w-patrick"Weiss, Barak"https://www.zbmath.org/authors/?q=ai:weiss.barakFrom the text: Our arguments in \S6 of our paper [ibid. 24, No. 2, 875--934 (2018; Zbl 1388.37053)] contain an error. In this note we explain the error
and how to fix it. We are grateful to Florent Ygouf for both pointing out the mistake, and for indicating the correct argument included below. All of the results stated in the introduction of the paper remain valid, as a consequence of the amended argument which will be given below. In the recent preprint [``A criterion for density of the isoperiodic leaves in rank 1 affine-invariant orbifolds'', Preprint, \url{arXiv:2002.01186}], \textit{F. Ygouf} proves related results
about existence of dense rel leaves in other loci.Hénon-Devaney like mapshttps://www.zbmath.org/1472.370552021-11-25T18:46:10.358925Z"Leal, Bladismir"https://www.zbmath.org/authors/?q=ai:leal.bladismir"Muñoz, Sergio"https://www.zbmath.org/authors/?q=ai:munoz.sergio-rThe Hénon nonlinear mapping defined by \(H(x,y) = (x +\frac{1}{y}, y -\frac{1}{y} - x)\) maps \(\mathbb{R}^2\) to itself (except that the line \(y = 0\) is omitted from the domain and the line \(y = -x\) is omitted from the range). It can be considered as an asymptotic form of the equations of motion for the restricted three-body problem. \textit{R. L. Devaney} [Commun. Math. Phys. 80, 465--476 (1981; Zbl 0463.28012)]
proved that this map is topologically conjugate to the baker transformation, so it has come to be called the Hénon-Devaney map. It is transitive with dense periodic orbits.
Here the authors establish a general theorem that characterizes the transitivity of homeomorphisms with singularities in the plane. They also provide examples where their theorem applies, and one of these is the Hénon-Devaney map. Their result applies to a general class of systems that they call Hénon-Devaney-like maps, which generalize the domain and range of the original Hénon-Devaney map to replace the excluded lines of that map by two ``long curves''. (A long curve is defined as a proper topological embedding of the real line in \(\mathbb{R}^2\) that divides the plane into two open unbounded regions having the long curve as boundary.) They also add assumptions that the long curves intersect transversely at a single point and four other technical conditions.
The authors prove that the maximal invariant of a Hénon-Devaney-like map is topologically conjugate to a shift map, and that every Hénon-Devaney-like map has no fixed points and has dense periodic orbits.Integrability and linearizability of a family of three-dimensional quadratic systemshttps://www.zbmath.org/1472.370602021-11-25T18:46:10.358925Z"Aziz, Waleed"https://www.zbmath.org/authors/?q=ai:aziz.waleed"Amen, Azad"https://www.zbmath.org/authors/?q=ai:amen.azad"Pantazi, Chara"https://www.zbmath.org/authors/?q=ai:pantazi.charaSummary: We consider a three-dimensional vector field with quadratic nonlinearities and in general none of the axis plane is invariant. For our investigation, we are interesting in the case of \((1:-2:1)\) -- resonance at the origin. Hence, we deal with a nine parametric family of quadratic systems and our purpose is to understand the mechanisms of local integrability. By computing some obstructions, knowing as resonant focus quantities, first we present necessary conditions that guarantee the existence of two independent local first integrals at the origin. For this reason Gröbner basis and some other algorithms are employed. Then we examine the cases where the origin is linearizable. Some techniques like existence of invariant surfaces and Jacobi multipliers, Darboux method, properties of linearizable nodes of two dimensional systems and power series arguments are used to prove the sufficiency of the obtained conditions. For a particular three-parametric subfamily, we provide conditions on the parameters to guarantee the non-existence of a polynomial first integral.Ground state homoclinic orbits for a class of asymptotically periodic second-order Hamiltonian systemshttps://www.zbmath.org/1472.370652021-11-25T18:46:10.358925Z"Lv, Ying"https://www.zbmath.org/authors/?q=ai:lv.ying"Xue, Yan-Fang"https://www.zbmath.org/authors/?q=ai:xue.yanfang"Tang, Chun-Lei"https://www.zbmath.org/authors/?q=ai:tang.chun-lei|tang.chunleiA class of second-order Hamiltonian systems of the form
\[ \ddot{u}(t)-L(t)u(t) +\nabla W(t,u(t))=0 \]
is considered. Here \(L:\mathbb R\to \mathbb R ^{N^2}\) and \(W\in C^1(\mathbb R \times \mathbb R^N, \mathbb R )\) are asymptotically periodic in \(t\) at infinity. Under a series of conditions, called by the authors ``reformative perturbation conditions'' and some weak super-quadratic conditions on the nonlinearity, the existence of a ground state homoclinic orbit is proved. Recall that a ground state homoclinic orbit is a nonzero solution to the above problem such that \(u(t)\to 0\) when \(|t|\to \infty\). The result extends and unifies some earlier results in [\textit{C. O. Alves} et al., Appl. Math. Lett. 16, No. 5, 639--642 (2003; Zbl 1041.37032); \textit{Z. Chen} and \textit{W. Zou}, Ann. Mat. Pura Appl. (4) 194, No. 1, 183--220 (2015; Zbl 1319.35236); \textit{Z. Liu} et al., Nonlinear Anal., Real World Appl. 36, 116--138 (2017; Zbl 1365.37052); \textit{M.-H. Yang} and \textit{Z.-Q. Han} [Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 74, No. 7, 2635--2646 (2011; Zbl 1218.37082)].
A linear part can be taken as \(L(t)=(l^\infty-\frac{1}{1+t^2})\mathrm{Id}\) and the nonlinear one either \( W(t,x)=|x|^2\ln (1+|x|^2)+\frac{1}{1+t^2}|x|^2\) or \(W(t,x)=e^{2|x|^3} -e^{|x|^3}+\frac{1}{1+t^2}|x|^3\). These do not satisfy some of the earlier conditions.
The method of the proof is based on the concentration-compactness principle by \textit{P.-L. Lions} [Ann. Inst. Henri Poincaré, Anal. Non Linéaire 1, 109--145 (1984; Zbl 0541.49009); Ann. Inst. Henri Poincaré, Anal. Non Linéaire 1, 223--283 (1984; Zbl 0704.49004)] and on the Mountain Pass theorem.Analytic invariant curves for an iterative equation related to Ricker-type second-order equationhttps://www.zbmath.org/1472.390052021-11-25T18:46:10.358925Z"Zhao, Hou Yu"https://www.zbmath.org/authors/?q=ai:zhao.houyu"Fečkan, Michal"https://www.zbmath.org/authors/?q=ai:feckan.michalThe authors show the existence of analytic invariant curves of the difference equation
\[
x_{n+1}=x_{n-1}e^{a-x_{n-1}-x_{n}},
\]
or equivalently of the mapping \(T(x,y)=(y,xe^{a-x-y})\). They seek for analytic invariant curves of \(T\) in the form \(y=f(x)\). The considered system can be written in the form of the iterative equation
\[
f(f(x))=xe^{a-x-f(x)},
\]
where \(x\in\mathbb{C}\) and \(a\in\mathbb{R}\) is a fixed number. This equation is reduced, with \(f(x)=g(\alpha g^{-1}(x) )\), to the auxiliary equation
\[
g(\alpha^{2}x)=g(x)e^{a-g(x)-g(\alpha x)}.
\]
Thus, by proving the existence of analytic solutions for this last equation, the analytic invariant curves of original equation can be determined.
Set \(\alpha = \pm e^{a/2}\). The authors distinguish three different cases for \(\alpha\):
\begin{itemize}
\item[(1)] \(0< |\alpha| < 1\);
\item[(2)] \(\alpha = e^{2\pi i\theta}, \theta\in \mathbb{R}\setminus\mathbb{Q}\) and \(\theta\) defines a Brjuno number, i.e.,
\[
\sum_{n=0}^{\infty}\frac{\log q_{n+1}}{q_{n}}<\infty,
\]
where \(\{p_{n}/q_{n}\}\) denotes the sequence of partial fractions of the continued fraction expansion of \(\theta\);
\item[(3)] \(\alpha = e^{2\pi iq/p}\) for some integer \(p\in \mathbb{N}\) with \(p\geq 2\) and \(q\in \mathbb{Z}\setminus\{ 0 \}\) and \(\alpha \neq e^{2\pi i\xi/\upsilon}\) for all \(1\leq \upsilon\leq p-1\) and \(\xi\in \mathbb{Z}\setminus\{ 0 \}\).
\end{itemize}On compact abelian Lie groups of homeomorphisms of \(\mathbb{R}^m\)https://www.zbmath.org/1472.570342021-11-25T18:46:10.358925Z"Ben Rejeb, Khadija"https://www.zbmath.org/authors/?q=ai:ben-rejeb.khadijaThe continuous actions of compact abelian finite and Lie groups \(G\) on \(\mathbb{R}^m\) are investigated. It is known that in general such groups are not necessarily conjugate to subgroups of \(O(m)\). Here some special case is considered. Let \(S = S(K_1) \times \dots \times S(K_q)\), where \(K_i = \mathbb{R}\) or \(\mathbb{C}\) and \(S(K_i) = \{x \in K_i : |x| = 1 \}\) for \(1 \le i \le q\). These groups act naturally by homeomorphisms on \(\mathbb{R}^m = K_1 \oplus \dots \oplus K_q\). Let \(G\) be a compact Lie group of homeomorphisms of \(\mathbb{R}^m\). Is is shown that such \(G\) is contained in \(S\) if and only if every element of \(G\) centralizes \(S\). Corollary: \(G\) is conjugate to some subgroup of \(S\) if and only if for some homeomorphism \(\alpha\) of \(\mathbb{R}^m\) every element of \(\alpha G \alpha ^{-1}\) centralizes \(S\). Two examples showing the importance of the condition ``centralizes \(S\)'' are given.Absolute and convective instabilities of semi-bounded spatially developing flowshttps://www.zbmath.org/1472.760432021-11-25T18:46:10.358925Z"Brevdo, Leo"https://www.zbmath.org/authors/?q=ai:brevdo.leoSummary: We analyse the absolute and convective instabilities of, and spatially amplifying waves in, semi-bounded spatially developing flows and media by applying the Laplace transform in time to the corresponding initial-value linear stability problem and treating the resulting boundary-value problem on \(\mathbb{R}^+\) for a vector equation as a dynamical system. The analysis is an extension of our recently developed linear stability theory for spatially developing open flows and media with algebraically decaying tails and for fronts to flows in a semi-infinite domain. We derive the global normal-mode dispersion relations for different domains of frequency and treat absolute instability, convectively unstable wave packets and signalling. It is shown that when the limit state at infinity, i.e. the associated uniform state, is stable, the inhomogeneous flow is either stable or absolutely unstable. The inhomogeneous flow is absolutely stable but convectively unstable if and only if the flow is globally stable and the associated uniform state is convectively unstable. In such a case signalling in the inhomogeneous flow is identical with signalling in the associated uniform state.Infinite DLR measures and volume-type phase transitions on countable Markov shiftshttps://www.zbmath.org/1472.820102021-11-25T18:46:10.358925Z"Beltrán, Elmer R."https://www.zbmath.org/authors/?q=ai:beltran.elmer-r"Bissacot, Rodrigo"https://www.zbmath.org/authors/?q=ai:bissacot.rodrigo"Endo, Eric O."https://www.zbmath.org/authors/?q=ai:endo.eric-ossami