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Discrete & Continuous Dynamical Systems - A

December 2020 , Volume 40 , Issue 12

Special issue on dynamics at DEA 2019

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Preface
Maciej Capiński, Núria Fagella, Michał Misiurewicz, Weixiao Shen, Benjamin Weiss and Krzysztof Ciepliński
2020, 40(12): ⅰ-ⅱ doi: 10.3934/dcds.2020327 +[Abstract](473) +[HTML](127) +[PDF](74.53KB)
Abstract:
Local limit theorems for suspended semiflows
Jon Aaronson and Dalia Terhesiu
2020, 40(12): 6575-6609 doi: 10.3934/dcds.2020294 +[Abstract](397) +[HTML](75) +[PDF](714.45KB)
Abstract:

We prove local limit theorems for a cocycle over a semiflow by establishing topological, mixing properties of the associated skew product semiflow. We also establish conditional rational weak mixing of certain skew product semiflows and various mixing properties including order 2 rational weak mixing of hyperbolic geodesic flows on the tangent spaces of cyclic covers.

The Mandelbrot set is the shadow of a Julia set
François Berteloot and Tien-Cuong Dinh
2020, 40(12): 6611-6633 doi: 10.3934/dcds.2020262 +[Abstract](484) +[HTML](102) +[PDF](423.41KB)
Abstract:

Working within the polynomial quadratic family, we introduce a new point of view on bifurcations which naturally allows to see the set of bifurcations as the projection of a Julia set of a complex dynamical system in dimension three. We expect our approach to be extendable to other holomorphic families of dynamical systems.

On the measure of KAM tori in two degrees of freedom
Luca Biasco and Luigi Chierchia
2020, 40(12): 6635-6648 doi: 10.3934/dcds.2020134 +[Abstract](854) +[HTML](306) +[PDF](404.79KB)
Abstract:

A conjecture of Arnold, Kozlov and Neishtadt on the exponentially small measure of the "non-torus" set in analytic systems with two degrees of freedom is discussed.

Generic Birkhoff spectra
Zoltán Buczolich, Balázs Maga and Ryo Moore
2020, 40(12): 6649-6679 doi: 10.3934/dcds.2020131 +[Abstract](869) +[HTML](373) +[PDF](493.71KB)
Abstract:

Suppose that \begin{document}$ \Omega = \{0, 1\}^\mathbb{N} $\end{document} and \begin{document}$ \sigma $\end{document} is the one-sided shift. The Birkhoff spectrum \begin{document}$ S_{f}(α) = \dim_{H}\Big \{ \omega\in\Omega:\lim\limits_{N \to \infty} \frac{1}{N} \sum\limits_{n = 1}^N f(\sigma^n \omega) = \alpha \Big \}, $\end{document} where \begin{document}$ \dim_{H} $\end{document} is the Hausdorff dimension. It is well-known that the support of \begin{document}$ S_{f}(α) $\end{document} is a bounded and closed interval \begin{document}$ L_f = [\alpha_{f, \min}^*, \alpha_{f, \max}^*] $\end{document} and \begin{document}$ S_{f}(α) $\end{document} on \begin{document}$ L_{f} $\end{document} is concave and upper semicontinuous. We are interested in possible shapes/properties of the spectrum, especially for generic/typical \begin{document}$ f\in C(\Omega) $\end{document} in the sense of Baire category. For a dense set in \begin{document}$ C(\Omega) $\end{document} the spectrum is not continuous on \begin{document}$ \mathbb{R} $\end{document}, though for the generic \begin{document}$ f\in C(\Omega) $\end{document} the spectrum is continuous on \begin{document}$ \mathbb{R} $\end{document}, but has infinite one-sided derivatives at the endpoints of \begin{document}$ L_{f} $\end{document}. We give an example of a function which has continuous \begin{document}$ S_{f} $\end{document} on \begin{document}$ \mathbb{R} $\end{document}, but with finite one-sided derivatives at the endpoints of \begin{document}$ L_{f} $\end{document}. The spectrum of this function can be as close as possible to a "minimal spectrum". We use that if two functions \begin{document}$ f $\end{document} and \begin{document}$ g $\end{document} are close in \begin{document}$ C(\Omega) $\end{document} then \begin{document}$ S_{f} $\end{document} and \begin{document}$ S_{g} $\end{document} are close on \begin{document}$ L_{f} $\end{document} apart from neighborhoods of the endpoints.

Computer assisted proofs of two-dimensional attracting invariant tori for ODEs
Maciej J. Capiński, Emmanuel Fleurantin and J. D. Mireles James
2020, 40(12): 6681-6707 doi: 10.3934/dcds.2020162 +[Abstract](869) +[HTML](363) +[PDF](3490.25KB)
Abstract:

This work studies existence and regularity questions for attracting invariant tori in three dimensional dissipative systems of ordinary differential equations. Our main result is a constructive method of computer assisted proof which applies to explicit problems in non-perturbative regimes. We obtain verifiable lower bounds on the regularity of the attractor in terms of the ratio of the expansion rate on the torus with the contraction rate near the torus. We consider separately two important cases of rotational and resonant tori. In the rotational case we obtain \begin{document}$ C^k $\end{document} lower bounds on the regularity of the embedding. In the resonant case we verify the existence of tori which are only \begin{document}$ C^0 $\end{document} and neither star-shaped nor Lipschitz.

Time dependent center manifold in PDEs
Hongyu Cheng and Rafael de la Llave
2020, 40(12): 6709-6745 doi: 10.3934/dcds.2020213 +[Abstract](626) +[HTML](250) +[PDF](475.68KB)
Abstract:

We consider externally forced equations in an evolution form. Mathematically, these are skew systems driven by a finite dimensional dynamical system. Two very common cases included in our treatment are quasi-periodic forcing and forcing by a stochastic process. We allow that the evolution is a PDE and even that it is not well-posed and that it does not define a flow (not all initial conditions lead to a solution).

We first establish a general abstract theorem which, under suitable (spectral, non-degeneracy, smoothness, etc) assumptions, establishes the existence of a "time-dependent invariant manifold" (TDIM). These manifolds evolve with the forcing. They are such that the original equation is always tangent to a vector field in the manifold. Hence, for initial data in the TDIM, the original equation is equivalent to an ordinary differential equation. This allows us to define families of solutions of the full equation by studying the solutions of a finite dimensional system. Note that this strategy may apply even if the original equation is ill posed and does not admit solutions for arbitrary initial conditions (the TDIM selects initial conditions for which solutions exist). It also allows that the TDIM is infinite dimensional.

Secondly, we construct the center manifold for skew systems driven by the external forcing.

Thirdly, we present concrete applications of the abstract result to the differential equations whose linear operators are exponential trichotomy subject to quasi-periodic perturbations. The use of TDIM allows us to establish the existence of quasi-periodic solutions and to study the effect of resonances.

On globally hypoelliptic abelian actions and their existence on homogeneous spaces
Danijela Damjanovic, James Tanis and Zhenqi Jenny Wang
2020, 40(12): 6747-6766 doi: 10.3934/dcds.2020164 +[Abstract](645) +[HTML](314) +[PDF](576.23KB)
Abstract:

We define globally hypoelliptic smooth \begin{document}$ \mathbb R^k $\end{document} actions as actions whose leafwise Laplacian along the orbit foliation is a globally hypoelliptic differential operator. When \begin{document}$ k = 1 $\end{document}, strong global rigidity is conjectured by Greenfield-Wallach and Katok: every globally hypoelliptic flow is smoothly conjugate to a Diophantine flow on the torus. The conjecture has been confirmed for all homogeneous flows on homogeneous spaces [9]. In this paper we conjecture that among homogeneous \begin{document}$ \mathbb R^k $\end{document} actions (\begin{document}$ k\ge 2 $\end{document}) on homogeneous spaces globally hypoelliptic actions exist only on nilmanifolds. We obtain a partial result towards this conjecture: we show non-existence of globally hypoelliptic \begin{document}$ \mathbb R^2 $\end{document} actions on homogeneous spaces \begin{document}$ G/\Gamma $\end{document}, with at least one quasi-unipotent generator, where \begin{document}$ G = SL(n, \mathbb R) $\end{document}. We also show that the same type of actions on solvmanifolds are smoothly conjugate to homogeneous actions on nilmanifolds.

On the regularity of the Green current for semi-extremal endomorphisms of $ \mathbb{P}^2 $
Christophe Dupont and Axel Rogue
2020, 40(12): 6767-6781 doi: 10.3934/dcds.2020163 +[Abstract](588) +[HTML](285) +[PDF](376.35KB)
Abstract:

We study the regularity of the Green current for semi-extremal endomorphisms of \begin{document}$ \mathbb{P}^2 $\end{document}. Under suitable assumptions, we show that the pointwise lower Radon-Nikodym derivative of stable slices with respect to the one dimensional Lebesgue measure is bounded at almost every point for the equilibrium measure. This provides a weak amount of metric regularity for the Green current along holomorphic discs.

The secant map applied to a real polynomial with multiple roots
Antonio Garijo and Xavier Jarque
2020, 40(12): 6783-6794 doi: 10.3934/dcds.2020133 +[Abstract](709) +[HTML](353) +[PDF](509.6KB)
Abstract:

We investigate the plane dynamical system given by the secant map applied to a polynomial \begin{document}$ p $\end{document} having at least one multiple root of multiplicity \begin{document}$ d>1 $\end{document}. We prove that the local dynamics around the fixed points related to the roots of \begin{document}$ p $\end{document} depend on the parity of \begin{document}$ d $\end{document}.

A general mechanism of instability in Hamiltonian systems: Skipping along a normally hyperbolic invariant manifold
Marian Gidea, Rafael de la Llave and Tere M. Seara
2020, 40(12): 6795-6813 doi: 10.3934/dcds.2020166 +[Abstract](916) +[HTML](313) +[PDF](376.13KB)
Abstract:

We describe a recent method to show instability in Hamiltonian systems. The main hypothesis of the method is that some explicit transversality conditions – which can be verified in concrete systems by finite calculations – are satisfied.

In particular, for several types of perturbations of integrable Hamiltonian systems, the hypothesis can be verified by just checking that some Melnikov-type integrals have non-degenerate zeros. This holds for Baire generic sets of perturbations in the \begin{document}$ C^r $\end{document}-topology, for \begin{document}$ r \in [3, \infty) \cup \{\omega\} $\end{document}. Our method does not require that the unperturbed Hamiltonian system is convex, or that the perturbation is polynomial, which are non-generic properties.

Provided that the transversality conditions are verified, one concludes the existence of orbits which change the action coordinate by a quantity independent of the size of the perturbation. In fact, one can obtain orbits that follow any path in action space, up to an error decreasing with the size of the perturbation.

How to identify a hyperbolic set as a blender
Stefanie Hittmeyer, Bernd Krauskopf, Hinke M. Osinga and Katsutoshi Shinohara
2020, 40(12): 6815-6836 doi: 10.3934/dcds.2020295 +[Abstract](411) +[HTML](146) +[PDF](3623.24KB)
Abstract:

A blender is a hyperbolic set with a stable or unstable invariant manifold that behaves as a geometric object of a dimension larger than that of the respective manifold itself. Blenders have been constructed in diffeomorphisms with a phase space of dimension at least three. We consider here the question of how one can identify, characterize and also visualize the underlying hyperbolic set of a given diffeomorphism to verify whether it actually is a blender or not. More specifically, we employ advanced numerical techniques for the computation of global manifolds to identify the hyperbolic set and its stable and unstable manifolds in an explicit Hénon-like family of three-dimensional diffeomorphisms. This allows to determine and illustrate whether the hyperbolic set is a blender; in particular, we consider as a distinguishing feature the self-similar structure of the intersection set of the respective global invariant manifold with a plane. By checking and illustrating a denseness property, we are able to identify a parameter range over which the hyperbolic set is a blender, and we discuss and illustrate how the blender disappears.

Gaussian iterative algorithm and integrated automorphism equation for random means
Justyna Jarczyk and Witold Jarczyk
2020, 40(12): 6837-6844 doi: 10.3934/dcds.2020135 +[Abstract](732) +[HTML](285) +[PDF](310.32KB)
Abstract:

Gauss-type iterates for random means are considered and their limit behaviour is studied. Among others the invariance of the limit with respect to the given random mean-type mapping \begin{document}$ {\bf{M}} $\end{document} is established under some relatively weak assumptions. The algorithm is applied to prove the existence and uniqueness of solutions \begin{document}$ \varphi $\end{document} of the equation

in the class of (deterministic) means in \begin{document}$ p $\end{document} variables.

Rigidity of random group actions
Woochul Jung, Keonhee Lee, Carlos Morales and Jumi Oh
2020, 40(12): 6845-6854 doi: 10.3934/dcds.2020130 +[Abstract](1016) +[HTML](412) +[PDF](316.07KB)
Abstract:

We prove that if a finitely generated random group action is robustly expansive and has the shadowing property, then it is rigid. We apply this result to analyze the rigidity of certain iterated function systems or actions of the discrete Heisenberg group.

Maximal equicontinuous generic factors and weak model sets
Gerhard Keller
2020, 40(12): 6855-6875 doi: 10.3934/dcds.2020132 +[Abstract](728) +[HTML](348) +[PDF](466.14KB)
Abstract:

The orbit closures of regular model sets generated from a cut-and-project scheme given by a co-compact lattice \begin{document}$ {\mathcal L}\subset G\times H $\end{document} and compact and aperiodic window \begin{document}$ W\subseteq H $\end{document}, have the maximal equicontinuous factor (MEF) \begin{document}$ (G\times H)/ {\mathcal L} $\end{document}, if the window is toplogically regular. This picture breaks down, when the window has empty interior, in which case the MEF is trivial, although \begin{document}$ (G\times H)/ {\mathcal L} $\end{document} continues to be the Kronecker factor for the Mirsky measure. As this happens for many interesting examples like the square-free numbers or the visible lattice points, a weaker concept of topological factors is needed, like that of generic factors [24]. For topological dynamical systems that possess a finite invariant measure with full support (\begin{document}$ E $\end{document}-systems) we prove the existence of a maximal equicontinuous generic factor (MEGF) and characterize it in terms of the regional proximal relation. This part of the paper profits strongly from previous work by McMahon [33] and Auslander [2]. In Sections 3 and 4 we determine the MEGF of orbit closures of weak model sets and use this result for an alternative proof (of a generalization) of the fact [34] that the centralizer of any \begin{document}$ {\mathcal B} $\end{document}-free dynamical system of Erdős type is trivial.

Automatic sequences are orthogonal to aperiodic multiplicative functions
Mariusz Lemańczyk and Clemens Müllner
2020, 40(12): 6877-6918 doi: 10.3934/dcds.2020260 +[Abstract](455) +[HTML](164) +[PDF](593.34KB)
Abstract:

Given a finite alphabet \begin{document}$ \mathbb{A} $\end{document} and a primitive substitution \begin{document}$ \theta: \mathbb{A}\to \mathbb{A}^\lambda $\end{document} (of constant length \begin{document}$ \lambda $\end{document}), let \begin{document}$ (X_\theta,S) $\end{document} denote the corresponding dynamical system, where \begin{document}$ X_\theta $\end{document} is the closure of the orbit via the left shift \begin{document}$ S $\end{document} of a fixed point of the natural extension of \begin{document}$ \theta $\end{document} to a self-map of \begin{document}$ \mathbb{A}^{ {\mathbb{Z}}} $\end{document}. The main result of the paper is that all continuous observables in \begin{document}$ X_\theta $\end{document} are orthogonal to any bounded, aperiodic, multiplicative function \begin{document}$ \boldsymbol{u}: {\mathbb{N}}\to {\mathbb{C}} $\end{document}, i.e.

for all \begin{document}$ f\in C(X_\theta) $\end{document} and \begin{document}$ x\in X_\theta $\end{document}. In particular, each primitive automatic sequence, that is, a sequence read by a primitive finite automaton, is orthogonal to any bounded, aperiodic, multiplicative function.

Euler integral and perihelion librations
Gabriella Pinzari
2020, 40(12): 6919-6943 doi: 10.3934/dcds.2020165 +[Abstract](715) +[HTML](304) +[PDF](760.42KB)
Abstract:

We discuss dynamical aspects of an analysis of the two–centre problem started in [15]. The perturbative nature of our approach allows us to foresee applications to the three–body problem.

Finding polynomial roots by dynamical systems – A case study
Sergey Shemyakov, Roman Chernov, Dzmitry Rumiantsau, Dierk Schleicher, Simon Schmitt and Anton Shemyakov
2020, 40(12): 6945-6965 doi: 10.3934/dcds.2020261 +[Abstract](493) +[HTML](171) +[PDF](1150.32KB)
Abstract:

We investigate two well known dynamical systems that are designed to find roots of univariate polynomials by iteration: the methods known by Newton and by Ehrlich–Aberth. Both are known to have found all roots of high degree polynomials with good complexity. Our goal is to determine in which cases which of the two algorithms is more efficient. We come to the conclusion that Newton is faster when the polynomials are given by recursion so they can be evaluated in logarithmic time with respect to the degree, or when all the roots are all near the boundary of their convex hull. Conversely, Ehrlich–Aberth has the advantage when no fast evaluation of the polynomials is available, and when roots are in the interior of the convex hull of other roots.

Interval homeomorphic solutions of a functional equation of nonautonomous iterations
Xiao Tang, Yingying Zeng and Weinian Zhang
2020, 40(12): 6967-6984 doi: 10.3934/dcds.2020214 +[Abstract](560) +[HTML](236) +[PDF](422.1KB)
Abstract:

In this paper we study a functional equation of linear combination of nonautonomous iterations, which can be reduced from invariance of an interval homeomorphism under nonautonomous iteration. We discuss existence, uniqueness and continuous dependence for increasing Lipschitzian solutions on a compact interval, and also discuss for bounded increasing Lipschitzian solutions and unbounded ones on the whole \begin{document}$ \mathbb{R} $\end{document}.

2019  Impact Factor: 1.338

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