Nonwandering Research Articles

Sum of the exponential and a polynomial: singular values and Baker wandering domains

This article studies the singular values of entire functions of the form E k ( z ) + P ( z ) where E k denotes the k − times composition of e z with itself and P is any non-constant polynomial. It is proved that the full preimage of each neighbourhood of ∞ is an infinitely connected domain without having any unbounded boundary component. Following the literature, the point at ∞ is called a Baker omitted value for the function in such a situation. More importantly, there are infinitely many critical values and no finite asymptotic value and in fact, the set of all critical values is found to be unbounded for these functions. We also investigate the iteration of three examples of entire functions with Baker omitted value and prove that these do not have any Baker wandering domain. There is a conjecture stating that the number of completely invariant domains of a transcendental entire function is at most one. How some of these maps are right candidates to work upon in view of this conjecture is demonstrated.

No bounded geometry wandering domains for sufficiently regular automorphisms

A question whether sufficiently regular manifold automorphisms may have wandering domains with controlled geometry is answered in the negative for quasiconformal or smooth homeomorphisms of n n -tori, n ≥ 2 n\ge 2 , and hyperbolic surfaces. Besides control on geometry of wandering domains, the assumptions are either analytic, e.g., minimal sets having measure zero or supporting invariant conformal structures, or geometric, such as uniform relative separation of wandering domains.

Stable/unstable holonomies, density of periodic points, and transitivity for continuum-wise hyperbolic homeomorphisms

We discuss different regularities on stable/unstable holonomies of cw-hyperbolic homeomorphisms and prove that if a cw-hyperbolic homeomorphism has continuous joint stable/unstable holonomies, then it has a dense set of periodic points in its non-wandering set. For that, we prove that the hyperbolic cw-metric (introduced in Artigue et al (2024 J. Differ. Equ. 378 512–38)) can be adapted to be self-similar (as in Artigue (2018 Ergodic Theory Dyn. Syst. 38 2422–46)) and, in this case, continuous joint stable/unstable holonomies are pseudo-isometric. We also prove transitivity of cw-hyperbolic homeomorphisms assuming that the stable/unstable holonomies are isometric. In the case the ambient space is a surface, we prove that a cw F -hyperbolic homeomorphism has continuous joint stable/unstable holonomies when every bi-asymptotic sector is regular.

On Morse–Smale diffeomorphisms on simply connected manifolds

We study relations between a structure of non-wandering set of a Morse–Smale diffeomorphism f and its carrying closed manifold Mn. We prove that if f has no any saddle periodic points with one-dimensional unstable manifolds, and for any periodic point σ of Morse index (n−1) its unstable manifolds do not intersect stable invariant manifolds of saddle periodic points different from σ, then Mn is simply connected. This fact does not follow from Morse inequalities that give only restrictions on homology groups of Mn.

On non-contractible periodic orbits and bounded deviations

We present a dichotomy for surface homeomorphisms in the isotopy class of the identity. We show that, in the absence of a degenerate fixed point set, either there exists a uniform bound on the diameter of orbits of non-wandering points for the lifted dynamics in the universal covering space, or the map has non-contractible periodic orbits. We then use this new tool to characterize the dynamics of area preserving homeomorphisms of the torus without non-contractible periodic orbits, showing that if the fixed point set is non-degenerate, then either the lifted dynamics is uniformly bounded, or it has a single strong irrational dynamical direction.

Pattern Analysis of Periodic Sediment Elevation Data: A Decision Support Framework for Deposited Sediment Management in Coastal Environment

This research explores sediment deposition patterns in coastal areas by employing synthetic data representing wandering set and recurrent patterns, utilizing Fourier transformation and power-law analysis to characterize frequency content and scaling behavior. The study employs fundamental models to interpret observed patterns and investigates the relevance of Poincaré's theorem in understanding recurrent patterns. Leveraging Python-based computation systems and open-source satellite data, the findings reveal contrasting behaviors in power-law exponents between scenarios, with wandering set patterns exhibiting scale-free characteristics and recurrent patterns displaying structured frequency distributions. This holistic approach advances scientific knowledge on sediment dynamics, informing evidence-based decision-making for effective sediment management and coastal resilience planning. Through its methodology and insights, the research contributes to understanding sediment deposition processes, elucidating underlying mechanisms driving coastal sedimentation, and providing practical approaches for monitoring and studying sediment dynamics in coastal environments, ultimately enhancing strategies for coastal management and conservation.

The weak mixing property on negatively curved manifolds

Given a complete manifold of negative curvature, we show that weak mixing is a generic property in the set of all probability measures invariant by the geodesic flow, as soon as the flow is topologically weakly mixing in restriction to its non-wandering set.

Global Functional Connectivity is Associated with Mind Wandering Domain of Comfort.

The resting-state paradigm is frequently applied to study spontaneous activity of the brain in normal and clinical conditions. To assess the relationship between brain activity and subjective experiences, various questionnaires are used. Previous studies using Amsterdam Resting State Questionnaire were focusing on fMRI functional connectivity or EEG microstates and spectral aspect. Here, we utilized Global Field Synchronization as the parameter to estimate global functional connectivity. By re-analyzing the resting-state data from 226 young healthy participants we showed a strong evidence of relationship between ARSQ domain of Comfort and GFS values in the alpha range (r = 0.210, BF10 = 12.338) and substantial evidence for positive relationship between ARSQ domain of Comfort and GFS in the beta frequency range (r = 196, BF10 = 6.307). Our study indicates the relevance of assessments of spontaneous thought occurring during the resting-state for the understanding of the individual intrinsic electrical brain activity.

On limit sets of simplest skew products defined on multidimensional cells

The purpose of this work is to describe two important types of limit sets of the most simple skew products of interval maps, the phase space of each of which is a compact n-dimensional cell (n ≥ 2): firstly, a non-wandering set and, secondly, ω-limit sets of trajectories. Methods. A method for investigating of a nonwandering set (new even for the two-dimensional case) is proposed, based on the use of the concept of C0-Ω-blow up in continuous closed interval maps, and the concept of C0-Ω-blow up introduced in the work in the family of continuous fibers maps. To describe the ω-limit sets, the technique of special series constructed for the trajectory and containing an information about its asymptotic behavior is used. Results. A complete description is given of the nonwandering set of the continuous simplest skew product of the interval maps, that is, a continuous skew product on a compact n-dimensional cell, the set of (least) periods of periodic points of which is bounded. The results obtained in the description of a nonwandering set are used in the study of ω-limit sets. The paper describes a topological structure of ω-limit sets of the maps under consideration. Sufficient conditions have been found under which the ω-limit set of the trajectory is a periodic orbit, as well as the necessary conditions for the existence of one-dimensional ω-limit sets. Conclusion. Further development of the C0-Ω-blow up technique in the family of maps in fibers will allow us to describe the structure of a nonwandering set of skew products of one-dimensional maps, in particular, with a closed set of periodic points defined on the simplest manifolds of arbitrary finite dimension. Further development of the theory of special divergent series constructed in the work will allow us to proceed to the description of ω-limit sets of arbitrary dimension d, where 2 ≤ d ≤ n - 1, n ≥ 3, in the simplest skew products.

Morse-Smale diffemorphisms with non-wandering points with pairwise different indices on 3-manifolds

В настоящей работе мы рассматриваем класс $G$ сохраняющих ориентацию диффеоморфизмов Морса-Смейла $f$, заданных на замкнутом 3-многообразии $M^3$, неблуждающее множество которых состоит из четырех неподвижных точек с попарно различными индексами Морса. Из результатов работ С. Смейла и К. Мейера следует, что все градиентно-подобные потоки с аналогичными свойствами имеют энергетическую функцию Морса с четырьмя критическими точками попарно различных индексов Морса. Это означает, что несущее многообразие $M^3$ для этих потоков допускает разложение Хегора рода 1, и, следовательно, оно гомеоморфно линзовому пространству $L_{p,q}$. Несмотря на простую структуру неблуждающего множества диффеоморфизмов в классе $G$, существуют диффеоморфизмы с дико вложенными сепаратрисами. Согласно результатам В. З. Гринеса, Ф. Лауденбаха, О. В. Починки такие диффеоморфизмы не обладают энергетической функцией, и вопрос о топологии их несущего многообразия остается открытым. Согласно результатам В. З. Гринеса, Е. В. Жужомы и В. С. Медведева $M^3$ гомеоморфно линзовому пространству $L_{p,q}$ в случае локально плоского вложения замыканий одномерных сепаратрис диффеоморфизма $f\in G$. Более того, блуждающее множество диффеоморфизма $f$ содержит по меньшей мере $p$ некомпактных гетероклинических кривых. В настоящей работе аналогичный результат получен для произвольных диффеоморфизмов класса $G$. На каждом линзовом пространстве $L_{p,q}$ построены диффеоморфизмы из класса $G$ с диким вложением одномерных сепаратрис. Такие примеры ранее были известны только на 3-сфере. Также установлено, что топологическая сопряженность диффеоморфизмов класса $G$ с единственной некомпактной гетероклинической кривой полностью определяется эквивалентностью узлов Хопфа, являющихся проекциями одномерных седловых сепаратрис в пространство орбит бассейна стока. Более того, любой узел Хопфа $L$ реализуется таким диффеоморфизмом. В этом смысле полученный результат подобен классификации диффеоморфизмов Д. Пикстона, полученной Х. Бонатти и В. З. Гринесом. Библиография: 65 названий.

On a structure of non-wandering set of an <inline-formula><tex-math id="M1">$ \Omega $</tex-math></inline-formula>-stable 3-diffeomorphism possessing a hyperbolic attractor

On a structure of non-wandering set of an <inline-formula><tex-math id="M1">$ \Omega $</tex-math></inline-formula>-stable 3-diffeomorphism possessing a hyperbolic attractor

Scenario of stable transition from diffeomorphism of torus isotopic to identity one to skew product of rough transformations of circle

In this paper, we consider gradient-like diffeomorphisms of a two-dimensional torus isotopic to the identical one. The isotopicity of diffeomorphisms $f_0$, $f_1$ on an $n$-manifold $M^n$ means the existence of some arc $\{f_t:M^n\to M^n,t\in[0,1]\}$ connecting them in the space of diffeomorphisms. If isotopic diffeomorphisms are structurally stable (qualitatively not changing their properties with small perturbations), then it is natural to expect the existence of a stable arc (qualitatively not changing its properties under small perturbations) connecting them. In this case, one says that the isotopic diffeomorphisms $f_0$, $f_1$ are stably isotopic or belong to the same class of stable isotopic connectivity. The simplest structurally stable diffeomorphisms on surfaces are gradient-like transformations having a finite hyperbolic non-wandering set, stable and unstable manifolds of various saddle points of which do not intersect. However, even on a two-dimensional sphere, where all orientation-preserving diffeomorphisms are isotopic, gradient-like diffeomorphisms are generally not stably isotopic. The countable number of pairwise different classes of stable isotopic connectivity is constructed on the base of a rough transformation of the circle $\phi_{\frac{k}{m}}$ with exactly two periodic orbits of the period $m$ and the rotation number $\frac{k}{m}$, which can be continued to a diffeomorphism $F_{\frac k m}:\mathbb S^2\to\mathbb S^2$ with two fixed sources at the North and South poles. On the torus $\mathbb T^2$, the model representative in the considered class is the skew products of rough transformations of a circle. We show that any isotopic gradient-like diffeomorphism of a torus is connected by a stable arc with some model transformation.

Morse – Smale 3-Diffeomorphisms with Saddles of the Same Unstable Manifold Dimension

In this paper, we consider a class of Morse – Smale diffeomorphisms defined on a closed 3-manifold (not necessarily orientable) under the assumption that all their saddle points have the same dimension of the unstable manifolds. The simplest example of such diffeomorphisms is the well-known “source-sink” or “north pole – south pole” diffeomorphism, whose non-wandering set consists of exactly one source and one sink. As Reeb showed back in 1946, such systems can only be realized on the sphere. We generalize his result, namely, we show that diffeomorphisms from the considered class also can be defined only on the 3-sphere.

Stability of the Nonwandering Set in the Region of Attraction Boundary under Perturbations with Application to Vulnerability Assessment

Stability of the Nonwandering Set in the Region of Attraction Boundary under Perturbations with Application to Vulnerability Assessment

Uniform hyperbolicity of a class of scattering maps

In recent years fractal Weyl laws and related quantum eigenfunction hypothesis have been studied in a plethora of numerical model systems, called quantum maps. In some models studied there one can easily prove uniform hyperbolicity. Yet, a numerically sound method for computing quantum resonance states, did not exist. To address this challenge, we recently introduced a new class of quantum maps (Mertig and Shudo 2018 Phys. Rev. E 97 042216). For these quantum maps, we showed that, quantum resonance states can numerically be computed using theoretically grounded methods such as complex scaling or weak absorbing potentials (Mertig and Shudo 2018 Phys. Rev. E 97 042216). However, proving uniform hyperbolicty for this class of quantum maps was not straight forward. Going beyond that work this article generalises the class of scattering maps and provides mathematical proofs for their uniform hyperbolicity. In particular, we show that the suggested class of two-dimensional symplectic scattering maps satisfies the topological horseshoe condition and uniform hyperbolicity. In order to prove these properties, we follow the procedure developed in the paper by Devaney and Nitecki (1979 Commun. Math. Phys. 67 137–46). Specifically, uniform hyperbolicity is shown by identifying a proper region in which the non-wandering set satisfies a sufficient condition to have the so-called sector bundle or cone field. Since no quantum map is known where both a proof of uniform hyperbolicity and a methodologically sound method for numerically computing quantum resonance states exist simultaneously, the present result should be valuable to further test fractal Weyl laws and related topics such as chaotic eigenfunction hypothesis.

Energy Function for Direct Products of Discrete Dynamical Systems

This paper is devoted to the construction of an energy function, i.e. a smooth Lyapunov function, whose set of critical points coincides with the chain-recurrent set of a dynamical system — for a cascade that is a direct product of two systems. One of the multipliers is a structurally stable diffeomorphism given on a two-dimensional torus, whose non-wandering set consists of a zero-dimensional non-trivial basic set without pairs of conjugated points and without fixed source and sink, and the second one is an identical mapping on a real axis. It was previously proved that if a non-wandering set of a dynamical system contains a zero-dimensional basic set, as the diffeomorphism under consideration has, then such a system does not have an energy function, namely, any Lyapunov function will have critical points outside the chain-recurrent set. For an identical mapping, the energy function is a constant on the entire real line. In this paper, it is shown that the absence of an energy function for one of the multipliers is not a sufficient condition for the absence of such a function for the direct product of dynamical systems, that is, in some cases it is possible to select the second cascade in such a way that the direct product will have an energy function.

Historic and physical wandering domains for wild blender-horseshoes

We present diffeomorphisms of wild blender-horseshoes which belong to closures of two types of diffeomorphisms, one of which has a historic contracting wandering domain, and the other has a non-trivial Dirac physical measure supported by saddle periodic orbit. It is a non-trivial extension of Colli–Vargas’ model (Colli and Vargas 2001 Ergod. Theory Dyn. Syst. 21 1657–81) to the higher dimensional dynamics with the use of wild blender-horseshoes.

Oriented and standard shadowing properties on closed surfaces

We prove that oriented and standard shadowing properties are equivalent for topological flows on closed surfaces with the nonwandering sets consisting of a finite number of critical elements (i.e., singularities or closed orbits). Moreover, we prove that each isolated singularity of a topological flow on a closed surface with the oriented shadowing property is either asymptotically stable, backward asymptotically stable, or admits a neighborhood which splits into two or four hyperbolic sectors.

Polynomial Entropy of Induced Maps of Circle and Interval Homeomorphisms

We compute the polynomial entropy of the induced maps on hyperspace for a homeomorphism $f$ of an interval or a circle with finitely many non-wandering points.

Wandering domains arising from Lavaurs maps with siegel disks

The classification of Fatou components for rational functions was concluded with Sullivan's proof of the No Wandering Domains Theorem in 1985. In 2016 it was shown, in joint work of the first and last author with Buff, Dujardin and Raissy, that wandering domains do exist in higher dimensions. In fact, wandering domains arise even for a seemingly simple class of maps: polynomial skew products. While the construction gives an infinite dimensional class of examples, and has been extended to polynomial automorphisms of $\mathbb C^4$ by Hahn and the last author, the currently known wandering domains are essentially unique. Our goal in this paper is to construct a second example, arising from similar techniques, but with distinctly different dynamical behavior. Instead of wandering domains arising from a Lavaurs map with an attracting fixed point, we construct a domain arising from a Lavaurs map with a fixed point of Siegel type. Siegel disks are not robust under perturbations, as opposed to attracting fixed points. We prove a necessary and sufficient condition for the existence of a so-called trapping domain for non-autonomous dynamical systems given by sequences of maps converging parabolically towards a Siegel type limit map. Guaranteeing that this condition is satisfied in our current construction requires a reconsideration of the proof of the original wandering domain, as more precise estimates on the rate of convergence towards the Lavaurs map are required. By adapting ideas introduced recently by Bedford, Smillie and Ueda, and by proving the existence of parabolic curves, with control on their domains of definition, we prove that the convergence rate is parabolic.