Nonwandering Research Articles

Wandering domains for composition of entire functions

C. Bishop in [10, Theorem 17.1] constructs an example of an entire function f in class B with at least two grand orbits of oscillating wandering domains. In this paper we show that his example has exactly two such orbits, that is, f has no unexpected wandering domains. We apply this result to the classical problem of relating the Julia sets of composite functions with the Julia set of its members. More precisely, we show the existence of two entire maps f and g in class B such that the Fatou set of f∘g has a wandering domain, while all Fatou components of f or g are preperiodic. This complements a result of A. Singh in [22, Theorem 4] and results of W. Bergweiler and A. Hinkkanen in [6] related to this problem.

Note on wandering domains in the dynamics of solutions of certain difference equations

In this note we study the value distribution of solutions of certain difference equations analogous to differential equations, the finite order solutions of which do not have wandering domains. Meanwhile, the nonexistence of wandering domains of solutions with finite order of these difference equations is proved. Thus the nonexistence of wandering domains of solutions of these difference and differential equations is similar in some extent.

Dynamical convergence of a certain polynomial family to f_a(z) = z + e^z + a

A transcendental entire function fa(z) = z +e z +a may have a Baker domain or a wandering domain, which never appear in the dynamics of polynomials. We consider a sequence of polynomials Pa,d(z) = (1+a/d)z +(1+z/d) d+1 +a, which converges uniformly on compact sets to fa as d → ∞. We show its dynamical convergence under a certain assumption, even though fa has a Baker domain or a wandering domain. We also investigate the parameter spaces of fa and Pa,d.

Constructing entire functions by quasiconformal folding

We give a method for constructing transcendental entire functions with good control of both the singular values of f and the geometry of f. Among other applications, we construct a function f with bounded singular set, whose Fatou set contains a wandering domain.

Non-wandering Sets for Dendrite Maps

Let $$(X,d)$$ be a metric space, and $$f$$ be a continuous map from $$X$$ to $$X$$ , and denote by $$P(f)$$ , $$\omega (f)$$ and $$\Omega (f)$$ the sets of periodic points, $$\omega $$ -limit points and non-wandering points of $$f$$ , respectively. It is well known that for an interval map $$f$$ , the following statements hold: (1) $$\omega (f)=\bigcap _{n=1}^{+\infty }f^n(\Omega (f))$$ . (2) Any isolated point of $$P( f )$$ is also an isolated point of $$\Omega ( f )$$ . (3) $$x\in \Omega ( f )$$ if and only if there exist points $$ x_k\longrightarrow x$$ and positive integers $$n_k\longrightarrow \infty $$ such that $$ f^{ n_k} (x_k) = x$$ . In [Mai and Sun (Topol Appl, 154, 2306–2311, 2007); Mai et al. (J Math Anal Appl, 383, 553–559, 2011)], we generalized those results to graph maps. It is natural to ask whether those results can be generalized to dendrite maps. The aim of this paper is to show that the answer is negative.

Invariant Probability Measures and Non-wandering Sets for Impulsive Semiflows

We consider impulsive dynamical systems defined on compact metric spaces and their respective impulsive semiflows. We establish sufficient conditions for the existence of probability measures which are invariant by such impulsive semiflows. We also deduce the forward invariance of their non-wandering sets except the discontinuity points.

New types of chaos and non-wandering points in topological spaces

In this paper, we will define a new class of chaotic maps on locally compact Hausdorff spaces called α-type chaotic maps defined by α-type transitive maps. This new definition coincides with Devaney's definition for chaos when the topological space happens to be a metric space. Furthermore, we will study new types of non-wandering points called α-type nonwandering points. We have shown that the α-type nonwandering points imply nonwandering points but not conversely. Finally, we have defined new concepts of chaotic on topological space. Relationships with some other type of chaotic maps are given.

On the dynamical coherence of structurally stable 3-diffeomorphisms

We prove that each structurally stable diffeomorphism f on a closed 3-manifold M3 with a two-dimensional surface nonwandering set is topologically conjugated to some model dynamically coherent diffeomorphism.

Nonwandering sets of interval skew products

In this paper we consider a class of skew products over transitive subshifts of finite type with interval fibres. For a natural class of 1-parameter families we prove that for all but countably many parameter values the nonwandering set (in particular, the union of all attractors and repellers) has zero measure. As a consequence, the same holds for a residual subset of the space of skew products.

Some Aspects of Stability for Semigroup Actions and Control Systems

The present paper introduces both the notions of Lagrange and Poisson stabilities for semigroup actions. Let \(S\) be a semigroup acting on a topological space \(X\) with mapping \(\sigma :S\times X\rightarrow X\), and let \(\mathcal {F}\) be a family of subsets of \(S\). For \(x\in X\) the motion \(\sigma _{x}:S\rightarrow X\) is said to be forward Lagrange stable if the orbit \(Sx\) has compact closure in \(X\). The point \(x\) is forward \(\mathcal {F}\)-Poisson stable if and only if it belongs to the limit set \(\omega \left( x,\mathcal {F}\right) \). The concept of prolongational limit set is also introduced and used to describe nonwandering points. It is shown that a point \(x\) is \( \mathcal {F}\)-nonwandering if and only if \(x\) lies in its forward \(\mathcal {F} \)-prolongational limit set \(J\left( x,\mathcal {F}\right) \). The paper contains applications to control systems.

On essential entropy-carrying sets

In this paper we study properties of essential entropy-carrying sets of a continuous map on a compact metric space. If f:X→X is continuous on a compact metric space X, then the intersection of all essential entropy-carrying sets of f may or may not be an essential entropy-carrying set of f. When this intersection is an essential entropy-carrying set we denote it by E(f), the least essential entropy-carrying set, otherwise we say that E(f) does not exist. We present an example where E(f) does not exist but also find a sufficient condition for E(f) to exist. If f is a piecewise monotone map, we show that E(f) exists and is the finite union of the entropy-carrying sets in the Nitecki Decomposition of the nonwandering set of f intersected with the closure of the periodic points of f. When E(f) exists we study how it relates to other entropy-carrying sets of f including subsets of itself.

Classification of the Regular Components of Two-Dimensional Inner Maps

We propose a topological classification of dynamical systems generated by two-dimensional inner maps on the fully invariant regular components of a wandering set with special attracting boundary (to within the topological conjugacy).

On Preservation of the Invariant torus for Multifrequency Systems

We establish new conditions for the preservation of an asymptotically stable invariant toroidal manifold of the linear extension of a dynamical system on a torus under small perturbations in a set of nonwandering points. The proposed approach is applied to the investigation of the existence and stability of the invariant tori of linear extensions of the dynamical systems with simple structures of limit sets and recurrent trajectories.

SPECTRAL DECOMPOSITION OF k-TYPE NONWANDERING SETS FOR ℤ2-ACTIONS

We prove that the set of k-type nonwandering points of a Z2-action T can be decomposed into a disjoint union of closed and T-invariant sets <TEX>$B_1,{\ldots},B_l$</TEX> such that <TEX>$T|B_i$</TEX> is topologically k-type transitive for each <TEX>$i=1,2,{\ldots},l$</TEX>, if T is expansive and has the shadowing property.

Different forms of entropy dimension for zero entropy systems

The aim of this paper is to introduce the lower s-topological entropy to distinguish zero entropy systems. That this quantity is an invariant factor under topological conjugacy and a power rule is shown. Some examples are given to show that the lower entropy dimension can attain any value in (0, 1), and are different with the upper one and the entropy dimension in the sense of Bowen. A counterexample is used to indicate that the product rule does not hold, and the lower s-topological entropy of the subsystem for the non-wandering set can be strictly less than that of the system when 0 < s < 1. Finally, this study also constructs a dynamical system to show that the transitive system with zero entropy dimension may not be minimal.

Shadowing Property on Finitely Generated Group Actions

In this paper, we study shadowing property for actions of finitely generated groups and show that conjugacy preserves shadowing property and in the actions with shadowing property, topologically transitive is equivalent to chain transitive and the set of non-wandering points is equal to the set of chain recurrent points. For actions of finitely generated free groups, we prove that on a compact metric space of non-zero dimension, every action with chain transitive and shadowing property is sensitive.

Estimates of Hausdorff Dimension for Non-wandering Sets of Higher Dimensional Open Billiards

AbstractThis article concerns a class of open billiards consisting of a finite number of strictly convex, non-eclipsing obstacles K. The non-wandering set M0 of the billiard ball map is a topological Cantor set, and its Hausdorff dimension has been previously estimated for billiards in ℝ2 using well-known techniques. We extend these estimates to billiards in ℝn and make various refinements to the estimates. These refinements also allow improvements to other results. We also show that in many cases, the nonwandering set is confined to a particular subset of ℝn formed by the convex hull of points determined by period 2 orbits. This allows more accurate bounds on the constants used in estimating Hausdorff dimension.

Recurrence properties of a class of nonautonomous discrete systems

We study recurrence properties for the nonautonomous discrete system given by a sequence $(f_n)_{n=1}^\infty$ of continuous selfmaps on a compact metric space. In particular, our attention is paid to the case when the sequence $(f_n)_{n=1}^\infty$ converges uniformly to a map $f$ or forms an equicontinuous family. In the first case we investigate the structure and behavior of an $\omega$-limit set of $(f_n)$ by a dynamical property of the limit map $f$. We also present some examples of $(f_n)$ and $f$ on the closed interval: (a) $\omega(x, (f_n)) \smallsetminus \Omega(f)\not=\emptyset$ for some point $x$; or (b) the set of periodic points of $f$ is closed and for some point $x$, $\omega(x, (f_n))$ is infinite. In the second case we create a perturbation of $(f_n)$ whose nonwandering set has small measure.

Absence of wandering domains for some real entire functions with bounded singular sets

Let f be a real entire function whose set S(f) of singular values is real and bounded. We show that, if f satisfies a certain function-theoretic condition (the "sector condition"), then $f$ has no wandering domains. Our result includes all maps of the form f(z)=\lambda sinh(z)/z + a, where a is a real constant and {\lambda} is positive. We also show the absence of wandering domains for certain non-real entire functions for which S(f) is bounded and the iterates of f tend to infinity uniformly on S(f). As a special case of our theorem, we give a short, elementary and non-technical proof that the Julia set of the complex exponential map f(z)=e^z is the entire complex plane. Furthermore, we apply similar methods to extend a result of Bergweiler, concerning Baker domains of entire functions and their relation to the postsingular set, to the case of meromorphic functions.

Generic fractal structure of finite parts of trajectories of piecewise smooth Hamiltonian systems

Piecewise smooth Hamiltonian systems with tangent discontinuity are studied. A new phenomenon is discovered, namely, the generic chaotic behavior of finite parts of trajectories. The approach is to consider the evolution of Poisson brackets for smooth parts of the initial Hamiltonian system. It turns out that, near second-order singular points lying on a discontinuity stratum of codimension two, the system of Poisson brackets is reduced to the Hamiltonian system of the Pontryagin Maximum Principle. The corresponding optimization problem is studied and the topological structure of its optimal trajectories is constructed (optimal synthesis). The synthesis contains countably many periodic solutions on the quotient space by the scale group and a Cantor-like set of nonwandering points (NW) having fractal Hausdorff dimension. The dynamics of the system is described by a topological Markov chain. The entropy is evaluated, together with bounds for the Hausdorff and box dimension of (NW).