Nonwandering Research Articles

On Quasi-nested wandering domains

In this paper, the nature of the singularity of a meromorphic functions of the form f (z) = 1 h(z) + a for a ? C and h is an entire function having a Baker wandering domain, lying over the Baker omitted value is discussed. Various dynamical issues relating to the singular values of f have been studied. Also following are shown in this paper. If a be the Baker omitted value of f then f has a Quasi-nested wandering domain U if and only if there exists {nk}k>0 such that each Unk surrounds a and Unk ? a as k ? ?. If f is a function having Quasi-nested wandering domain then all the Fatou components of f are bounded. In particular, f has no Baker domain. Also existence of Quasi-nested wandering domain ensures that the Julia component containing ? i.e., J? is a singleton buried component. At the end of the paper a result about the non existence of Quasi-nested wandering domain is given.

Entropy for partial actions of ℤ

In this paper we introduce the definition of topological entropy for a partial Z \mathbb {Z} -action on metric spaces. We show that the definition of partial entropy is an extension of the definition of topological entropy for a Z \mathbb Z -action. We also prove that the partial topological entropy is concentrated on the nonwandering set.

Partially hyperbolic diffeomorphisms and Lagrangian contact structures

AbstractIn this paper, we classify the three-dimensional partially hyperbolic diffeomorphisms whose stable, unstable, and central distributions $E^s$ , $E^u$ , and $E^c$ are smooth, such that $E^s\oplus E^u$ is a contact distribution, and whose non-wandering set equals the whole manifold. We prove that up to a finite quotient or a finite power, they are smoothly conjugated either to a time-map of an algebraic contact-Anosov flow, or to an affine partially hyperbolic automorphism of a nil- ${\mathrm {Heis}}{(3)}$ -manifold. The rigid geometric structure induced by the invariant distributions plays a fundamental part in the proof.

Structure of Julia sets for post-critically finite endomorphisms on $${\mathbb {P}}^2$$

Let $f$ be a post-critically finite endomorphism (PCF map for short) on $\mathbb{P}^2$, let $J_1$ denote the Julia set and let $J_2$ denote the support of the measure of maximal entropy. In this paper we show that: 1. $J_1\setminus J_2$ is contained in the union of the (finitely many) basins of critical component cycles and stable manifolds of sporadic super-saddle cycles. 2. For every $x\in J_2$ which is not contained in the stable manifold of a sporadic super-saddle cycle, there is no Fatou disk containing $x$. Here sporadic means that the super-saddle cycle is not contained in a critical component cycle. Under the additional assumption that all branches of $PC(f)$ are smooth and intersect transversally, we show that there is no sporadic super-saddle cycle. Thus in this case $J_1\setminus J_2$ is contained in the union of the basins of critical component cycles, and for every $x\in J_2$ there is no Fatou disk containing $x$. As consequences of our result: 1.We answer some questions of Fornaess-Sibony about the non-wandering set for PCF maps on $\mathbb{P}^2$ with no sporadic super-saddle cycles. 2. We give a new proof of de Th\'elin's laminarity of the Green current in $J_1\setminus J_2$ for PCF maps on $\mathbb{P}^2$. 3. We show that for PCF maps on $\mathbb{P}^2$ an invariant compact set is expanding if and only if it does not contain critical points, and we obtain characterizations of PCF maps on $\mathbb{P}^2$ which are expanding on $J_2$ or satisfy Axiom A.

Rotation Domains and Stable Baker Omitted Value

A Baker omitted value, in short bov of a transcendental meromorphic function f is an omitted value such that there is a disk D centered at the bov for which each component of the boundary of \(f^{-1}(D)\) is bounded. Assuming all the iterates \(f^n\) to be analytic in a neighborhood of its bov, this article proves a number of results on the rotation domains of the function. The number of p-periodic Herman rings is shown to be finite for each \(p \ge 1\). It is also proved that every Julia component intersects the boundaries of at most finitely many Herman rings. Further, if the bov is the only limit point of the critical values then it is shown that f has infinitely many repelling fixed points. If a repelling periodic point of period p is on the boundary of a p-periodic rotation domain then the periodic point is shown to be on the boundary of infinitely many Fatou components. As a corollary we have shown that if D is a p-periodic rotation domain of f and \(f^p\) is univalent in a neighbourhood of the boundary \( \partial D \) of D then f has no repelling p-periodic point on \( \partial D \). Under additional assumptions on the critical points, a sufficient condition is found for a Julia component to be a singleton. As a consequence, it is proved that if the boundary of a wandering domain W accumulates at some point of the plane under the iteration of f then each limit of \(f^n\) on W is either a parabolic periodic point or in the \(\omega \)-limit set of recurrent critical points. Using the same ideas, the boundaries of rotation domains are shown to be in the \(\omega \)-limit set of recurrent critical points.

Non-wandering points for autonomous/periodic parabolic equations on the circle

We study the properties of non-wandering points of the following scalar reaction-diffusion equation on the circle S 1 , u t = u x x + f ( t , u , u x ) , t > 0 , x ∈ S 1 = R / 2 π Z , where f is independent of t or T -periodic in t . Assume that the equation admits a compact global attractor. It is proved that, any non-wandering point is a limit point of the system (that is, it is a point in some ω -limit set). More precisely, in the autonomous case, it is proved that any non-wandering point is either a fixed point or generates a rotating wave on the circle. In the periodic case, it is proved that any non-wandering point is a periodic point or generates a rotating wave on a torus. In particular, if f ( t , u , − u x ) = f ( t , u , u x ) , then any non-wandering point is a fixed point in the autonomous case, and is a periodic point in the periodic case.

Periodic points and measures for a class of skew-products

We consider the C1-open set \(\cal{V}\) of partially hyperbolic diffeomorphisms on the space \(\mathbb{T}^{2}\times\mathbb{T}^{2}\) whose non-wandering set is not stable, introduced by M. Shub in [57]. Firstly, we show that the non-wandering set of each diffeormorphism in \(\cal{V}\) is a limit of horseshoes in the sense of entropy. Afterwards, we establish the existence of a C2-open set \(\cal{U}\) of C2-diffeomorphisms in \(\cal{V}\) and of a C2-residual subset ℜ of \(\cal{U}\) such that any diffeomorphism in ℜ has equal topological and periodic entropies, is asymptotic per-expansive, has a sub-exponential growth rate of the periodic orbits and admits a principal strongly faithful symbolic extension with embedding. Besides, such a diffeomorphism has a unique probability measure with maximal entropy describing the distribution of periodic orbits. Under an additional assumption, we prove that the skew-products in \(\cal{U}\) preserve a unique ergodic SRB measure, which is physical, whose basin has full Lebesgue measure and which coincides with the measure with maximal entropy.

Classifying simply connected wandering domains

While the dynamics of transcendental entire functions in periodic Fatou components and in multiply connected wandering domains are well understood, the dynamics in simply connected wandering domains have so far eluded classification. We give a detailed classification of the dynamics in such wandering domains in terms of the hyperbolic distances between iterates and also in terms of the behaviour of orbits in relation to the boundaries of the wandering domains. In establishing these classifications, we obtain new results of wider interest concerning non-autonomous forward dynamical systems of holomorphic self maps of the unit disk. We also develop a new general technique for constructing examples of bounded, simply connected wandering domains with prescribed internal dynamics, and a criterion to ensure that the resulting boundaries are Jordan curves. Using this technique, based on approximation theory, we show that all of the nine possible types of simply connected wandering domain resulting from our classifications are indeed realizable.

Homeomorphism of Hereditarily Locally Connected Continua

Let $$f:X\rightarrow X$$ be a hereditarily locally connected continuum homeomorphism and denote respectively by P(f), AP(f) and $$\Omega (f)$$ , the sets of periodic points, almost periodic points and the non-wandering points of f. We show that any $$\omega $$ -limit set (resp. $$\alpha $$ -limit set) is minimal. Moreover, we show that $$\Omega (f)=AP(f)$$ . We also prove that if $$P(f)=\emptyset $$ , then there exists a unique minimal set. On the other hand, if $$P(f)\ne \emptyset $$ then we prove that any infinite minimal set has the adding machine structure and the absence of Li-Yorke pairs. Consequently, we partially solve the positive entropy conjecture which remains open even in the case of hereditarily locally connected continuum.

Dissecting a Resonance Wedge on Heteroclinic Bifurcations

This article studies routes to chaos occurring within a resonance wedge for a 3-parametric family of differential equations acting on a 3-sphere. Our starting point is an autonomous vector field whose flow exhibits a weakly attracting heteroclinic network made by two 1-dimensional connections and a 2-dimensional separatrix between two equilibria with different Morse indices. After changing the parameters, while keeping the 1-dimensional connections unaltered, we concentrate our study in the case where the 2-dimensional invariant manifolds of the equilibria do not intersect. We derive the first return map near the ghost of the attractor and we reduce the analysis of the system to a 2-dimensional map on the cylinder. Complex dynamical features arise from a discrete-time Bogdanov-Takens singularity, which may be seen as the organizing center by which one can obtain infinitely many attracting tori, strange attractors, infinitely many sinks and non-trivial contracting wandering domains. These dynamical phenomena occur within a structure that we call resonance wedge. As an application, we may see the "classical" Arnold tongue as a projection of a resonance wedge. The results are general, extend to other contexts and lead to a fine-tuning of the theory.

Rotational Entropy of an Annular Iterated Functions System

In this article, we consider an iterated functions system (IFS) whose functions are homeomorphisms on an annulus. We define rotational spanning and separating sets for the IFS and then provide two definitions for the rotational entropy of the IFS. We show that in the IFSs, the rotational entropy is a topological invariant. We prove that the rotational entropy of an annular IFS is equal to the rotational entropy on its non-wandering set for sequences of functions with a specific condition.

Dynamics of generalized hyperbolic linear operators

It is introduced an open class of linear operators on Banach spaces such that their non-wandering set is an infinite dimensional topologically mixing subspace, where the periodic points are dense (in particular, they are frequently hypercyclic on the non-wandering set when the Banach space is separable). It is also characterized when the non-wandering set coincides with the whole space.

Existence of an energy function for three-dimensional chaotic "sink-source" cascades.

This paper is a continuation of research in the direction of energy function (a smooth Lyapunov function whose set of critical points coincides with the chain recurrent set of a system) construction for discrete dynamical systems. The authors established the existence of an energy function for any A-diffeomorphism of a three-dimensional closed orientable manifold whose non-wandering set consists of a chaotic one-dimensional canonically embedded surface attractor and repeller.

Topological aspects of incompressible flows

In this article we approach some of the basic questions in topological dynamics, concerning periodic points, transitivity, the shadowing and the gluing orbit properties, in the context of C 0 incompressible flows generated by Lipschitz vector fields. We prove that a C 0 -generic incompressible and fixed-point free flow satisfies the periodic shadowing property, it is transitive and has a dense set of periodic points in the non-wandering set. In particular, a C 0 -generic fixed-point free incompressible flow satisfies the reparametrized gluing orbit property. We also prove that C 0 -generic incompressible flows satisfy the general density theorem and the weak shadowing property, moreover these are transitive.

On the geometry of simply connected wandering domains

We study the geometry of simply connected wandering domains for entire functions and we prove that every bounded connected regular open set, whose closure has a connected complement, is a wandering domain of some entire function. In particular such a domain can be realized as an escaping or an oscillating wandering domain. As a consequence, we obtain that every Jordan curve is the boundary of a wandering Fatou component of some entire function.

On -endomorphisms of the two-dimensional torus

It is proved that in each homotopy class of continuous mappings of the two-dimensional torus to itself that induce a hyperbolic action on the fundamental group, as long as it is free of expanding mappings, there exists an -endomorphism whose nonwandering set consists of an attracting hyperbolic sink and a nontrivial one-dimensional collapsing repeller, which is a one-dimensional orientable lamination, locally homeomorphic to the direct product of a Cantor set and a line segment. Moreover, the unstable -invariant subbundle of the tangent space to the repeller has the property of uniqueness. Bibliography: 23 titles.

Periodic Points of Regular Curve Homeomorphisms

We prove that, if $$f:X\rightarrow X$$ is a regular curve homeomorphism then the set of periodic points is either empty or dense in the set of non-wandering points. Moreover, we prove that each infinite minimal set is approximated by periodic orbits and is conjugate to an adding machine when the set of periodic points is not empty. Furthermore, we give a characterization of equicontinuous regular curve homeomorphisms

Escaping points of commuting meromorphic functions with finitely many poles

Let $f$ and $g$ be commuting meromorphic functions with finitely many poles. By studying the behaviour of Fatou components under this commuting relation, we prove that $f$ and $g$ have the same Julia set whenever $f$ and $g$ have no simply connected fast-escaping wandering domains. By combining this with a recent result of Tsantaris', we obtain the strongest statement (to date) regarding the Julia sets of commuting meromorphic functions. In order to highlight the difference to the entire case, we show that transcendental meromorphic functions with finitely many poles have orbits that alternate between approaching a pole and escaping to infinity at strikingly fast rates.

Supremum topological sequence entropy of circle maps

In this paper some formulas of supremum topological sequence entropy htop∞(f) are investigated for circle maps. If f is a circle map then htop∞(f)=htop∞(f|E(f)‾), where E(f)‾ is the closure of the set of eventually periodic points of f; If f is a circle map with zero topological entropy and Fix(f)≠∅ then htop∞(f|Ω(f))=0, where Ω(f) denotes the set of non-wandering points of f.

Recurrence in generalized semigroup

In [arXiv:1904.12333], we introduced the concept of escaping set in general setting for a topological space and extended the notion of limit set and escaping set for the general semigroup generated by continuous self maps. In this paper we continue with extending the other notions of recurrence for the generalized semigroup analogs to their counterpart in the classical theory of dynamics. We discuss the concept of periodic point, nonwandering point and chain recurrent point in the more general setting of a semigroup and establish the correlation between them.