Set Of Nonwandering Points Research Articles

Zero entropy and stable rotation sets for monotone recurrence relations

AbstractIn this paper, we show that each element in the convex hull of the rotation set of a compact invariant chain transitive set is realized by a Birkhoff solution, which is an improvement of the fundamental lemma of T. Zhou and W.-X. Qin [Pseudo solutions, rotation sets, and shadowing rotations for monotone recurrence relations. Math. Z.297 (2021), 1673–1692] in the study of rotation sets for monotone recurrence relations. We then investigate the properties of rotation sets assuming the system has zero topological entropy. The rotation set for a Birkhoff recurrence class is a singleton and the forward and backward rotation numbers are identical for each solution in the same Birkhoff recurrence class. We also show the continuity of rotation numbers on the set of non-wandering points. If the rotation set is upper-stable, then we show that each boundary point is a rational number, and we also obtain a result of bounded deviation.

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

Existence of an Invariant Torus for a Degenerate Linear Extension of Dynamical Systems

Under the assumptions that a degenerate system defined on the direct product of a torus and a Euclidean space can be reduced to a central canonical form and that the corresponding homogeneous nondegenerate system is exponentially dichotomous on the semiaxes, we establish a necessary and sufficient condition for the existence of a unique invariant torus of the degenerate linear system. We also establish conditions for the preservation of an asymptotically stable invariant toroidal manifold for a degenerate linear extension of the dynamical system on a torus under small perturbations in the set of nonwandering points.

Lyapunov spectrum for Hénon-like maps at the first bifurcation

For a strongly dissipative Hénon-like map at the first bifurcation parameter at which the uniform hyperbolicity is destroyed by the formation of tangencies inside the limit set, we effect a multifractal analysis, i.e. decompose the set of non-wandering points on the unstable manifold into level sets of an unstable Lyapunov exponent, and give a partial description of the Lyapunov spectrum which encodes this decomposition. We derive a formula for the Hausdorff dimension of the level sets in terms of the entropy and unstable Lyapunov exponent of invariant probability measures, and show the continuity of the Lyapunov spectrum. We also show that the set of points for which the unstable Lyapunov exponents do not exist carries the full Hausdorff dimension.

On Diffeomorphisms of Compact 2-Manifolds with All Nonwandering Points Being Periodic

The aim of the present paper is to study conditions under which all the nonwandering points are periodic points, for a discrete dynamical system of two variables defined on a compact manifold. We include a survey of known results in all dimensions, and study the remaining open question in dimension two. We present two results, one negative and one positive. The negative result: we construct a Kupka–Smale diffeomorphism in [Formula: see text] (which can be extended to a diffeomorphism of the sphere) with a closed set of periodic points that differs from the set of nonwandering points. The positive result: we present a condition on the widely studied Hénon family which guarantees that all nonwandering points are periodic. Finally, we close by describing what future work may be needed to resolve our broad goals.

Birkhoff spectrum for Hénon-like maps at the first bifurcation

We effect a multifractal analysis for a strongly dissipative Hénon-like map at the first bifurcation parameter at which the uniform hyperbolicity is destroyed by the formation of tangencies inside the limit set. We decompose the set of non-wandering points on the unstable manifold into level sets of Birkhoff averages of continuous functions, and derive a formula for the Hausdorff dimension of the level sets in terms of the entropy and unstable Lyapunov exponent of invariant probability measures.

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.

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.

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).

Some properties of positive entropy maps

AbstractWe prove that the stable classes for continuous maps on compact metric spaces have measure zero with respect to any ergodic invariant measure with positive entropy. Then, every continuous map with positive topological entropy on a compact metric space has uncountably many stable classes. We also prove that every continuous map with positive topological entropy of a compact metric space cannot be Lyapunov stable on its recurrent set. For homeomorphisms on compact metric spaces we prove that the sets of heteroclinic points, and sinks in the canonical coordinates case, have zero measure with respect to any ergodic invariant measure with positive entropy. These results generalize those of Fedorenko and Smital [Maps of the interval Ljapunov stable on the set of nonwandering points. Acta Math. Univ. Comenian. (N.S.)60 (1) (1991), 11–14], Huang and Ye [Devaney’s chaos or 2-scattering implies Li–Yorke’s chaos. Topology Appl.117 (3) (2002), 259–272], Reddy [The existence of expansive homeomorphisms on manifolds. Duke Math. J.32 (1965), 627–632], Reddy and Robertson [Sources, sinks and saddles for expansive homeomorphisms with canonical coordinates. Rocky Mountain J. Math.17 (4) (1987), 673–681], Sindelarova [A counterexample to a statement concerning Lyapunov stability. Acta Math. Univ. Comenian. 70 (2001), 265–268], and Zhou [Some equivalent conditions for self-mappings of a circle. Chinese Ann. Math. Ser. A12(suppl.) (1991), 22–27].

Stability of chain recurrent points of continuous maps on interval

Recently, J. Smítal and T.H. Steele [J. Difference Equ. Appl. 15 (2009), 77–86] considered the mappings C and NW, which take a function to its centre , and to its set of non-wandering points , respectively. They characterize a function at which maps C and NW are continuous, and put a problem whether the same characterization is true for the map CR, which takes an to its set of chain recurrent points . We show that the answer is positive.

Fractal structure of hyperbolic Lipschitzian dynamical systems

In the paper, dynamical systems admitting no smooth structure are studied. A theorem on the semiconjugacy of a Lipschitzian dynamical system to the corresponding topological Markov chain is proved. A new approach to evaluating bounds for the Hausdorff and the Kolmogorov dimensions of the set of nonwandering points from a Markov partition is suggested.

On calibrated and separating sub-actions

We consider a one-sided transitive subshift of finite type σ: Σ → Σ and a Holder observable A. In the ergodic optimization model, one is interested in properties of A-minimizing probability measures. If Ā denotes the minimizing ergodic value of A, a sub-action u for A is by definition a continuous function such that A ≥ u ○ σ − u + Ā. We call contact locus of u with respect to A the subset of Σ where A = u ○ σ − u + Ā. A calibrated sub-action u gives the possibility to construct, for any point x e Σ, backward orbits in the contact locus of u. In the opposite direction, a separating sub-action gives the smallest contact locus of A, that we call Ω(A), the set of non-wandering points with respect to A.

A symmetry of sphere map implies its chaos*

A well-known example, given by Shub, shows that for any |d| ≥ 2 there is a self-map of the sphere S n , n ≥ 2, of degree d for which the set of non-wandering points consists of two points. It is natural to ask which additional assumptions guarantee an infinite number of periodic points of such a map. In this paper we show that if a continuous map f : S n → S n commutes with a free homeomorphism g : S n → S n of a finite order, then f has infinitely many minimal periods, and consequently infinitely many periodic points. In other words the assumption of the symmetry of f originates a kind of chaos. We also give an estimate of the number of periodic points.

On λ-Dendroids with the ΩEP-Property

A space X has the ΩEP-property if for each self-mapping the set of nonwandering points is contained in the closure of the set of eventually periodic points. A class of λ-dendroids is presented each member of which has the ΩEP-property for mappings satisfying certain additional conditions. These λ-dendroids are obtained as compactifications of some dendrites without finite sets. Some related problems are posed.

An Interval Counterexample on Topological Sequence Entropy

We provide an example showing that, in general, the topological sequence entropy of a continuous interval map cannot be attained on the set of non-wandering points. This proves that there is no connection between the topological sequence entropy of an interval map and its behavior on sets of special dynamical meaning.

On the structure of the set of nonwandering points of a pair of coupled quadratic maps

In the plane of parameters, we indicate values for which plane endomorphisms constructed by coupling two identical one-dimensional unimodal quadratic maps have an absorbing domain that contains an attractor and a nontrivial invariant subset of the set of nonwandering points.

Critical sets and unimodal mappings of the square

A lower estimate for the number of different invariant subsets of the set of nonwandering points for a class of unimodal mappings is given. Sufficient conditions for such a mapping to have periodic points of arbitrarily large period are described. The machinery of the appearance of such points may be of very different nature. The existence of mappings with trajectory behavior chaotic in the Li-York sense is established. Conditions for the domain of these trajectories to be arbitrary small are given. Therefore, such trajectories cannot be found by numerical methods.

On the theory of one‐sided models in spaces with arbitrary cones

The paper presents a way of constructing quasimonotone nonautonomous systems ensuring x‐stability of the nonautonomous system. There are described extensions quasimonotone with respect to an arbitrary cone, Perron condition and invariant surface stability under perturbations U‐stability on the set of non wandering points is proved to imply u‐stability of quasimonotone nonlinear system and exponential u‐stability on minimal attraction center provides u‐stability of the total systems Examples are available.