Density Of Periodic Points Research Articles

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.

Topological dynamics of Markov multi-maps of the interval

We study Markov multi-maps of the interval from the point of view of topological dynamics. Specifically, we define the forward trajectory system of a Markov multi-map, and we investigate whether it has properties such as topological transitivity, topological mixing, density of periodic points, and specification. To each Markov multi-map we associate a shift of finite type (SFT), and our main results relate the properties of the SFT with those of the forward trajectory system. Under a general coding condition, we establish necessary and sufficient conditions for topological transitivity and topological mixing of the forward trajectory system. These results complement existing work showing a relationship between the topological entropy of a Markov multi-map and its associated SFT. We also characterize when the inverse limit systems associated to the Markov multi-maps have the properties mentioned above.

Elementary proof for the existence of periodic points with real and simple spectrum for diffeomorphisms in any dimension

Using only elementary techniques we prove that for any $ C^r $ diffeomorphism, $ f $, of a compact manifold of dimension $ d>2 $, $ 1\leq r\leq \infty $, admitting a transverse homoclinic intersection, we can find a $ C^1 $-open neighborhood of $ f $ containing a $ C^1 $-open and $ C^r $-dense set of $ C^r $ diffeomorphisms which have a periodic point with real and simple spectrum. We use this fact to prove that $ C^r $-generically among $ C^r $ diffeomorphisms with horseshoes, we have density of periodic points with real and simple spectrum inside the horseshoe. As a corollary, we obtain that generically in the $ C^1 $-topology the unique obstruction to the existence of periodic points with real and simple spectrum are the Morse-Smale diffeomorphisms with all the periodic points admitting non-real eigenvalues.

On endomorphisms of projective varieties with numerically trivial canonical divisors

Let [Formula: see text] be a klt projective variety with numerically trivial canonical divisor. A surjective endomorphism [Formula: see text] is amplified (respectively, quasi-amplified) if [Formula: see text] is ample (respectively, big) for some Cartier divisor [Formula: see text]. We show that after iteration and equivariant birational contractions, a quasi-amplified endomorphism will descend to an amplified endomorphism. As an application, when [Formula: see text] is Hyperkähler, [Formula: see text] is quasi-amplified if and only if it is of positive entropy. In both cases, [Formula: see text] has Zariski dense periodic points. When [Formula: see text] is an abelian variety, we give and compare several cohomological and geometric criteria of amplified endomorphisms and endomorphisms with countable and Zariski dense periodic points (after an uncountable field extension).

Dynamic analysis and image encryption application of a sinusoidal-polynomial composite chaotic system

Due to complex properties of ergodicity, non-periodic ability and sensitivity to initial states, chaotic systems are widely used in cryptography. In this paper, we propose a sinusoidal-polynomial composite chaotic system (SPCCS), and prove that it satisfies Devaney’s definition of chaos: the sensitivity to initial conditions, topological transitivity and density of periodic points. The experimental results show that SPCCS has better unpredictability and more complex chaotic behavior than the classical chaotic maps. Furthermore, we provide a new image encryption algorithm combining pixel segmentation operation, block chaotic matrix confusing operation and pixel diffusion operation with the proposed SPCCS. The detailed simulation results verify superiority and effectiveness of the proposed image encryption algorithm.

Sensitivity of Nonautonomous Dynamical Systems on Uniform Spaces

We introduce the concepts of sensitivity, multisensitivity, cofinite sensitivity, and syndetic sensitivity for nonautonomous dynamical systems on uniform spaces and obtain some sufficient conditions under which topological transitivity and dense periodic points imply sensitivity for nonautonomous systems on Hausdorff uniform spaces. We also study sensitivity and other stronger versions of sensitivity for the systems induced on hyperspaces and for the product of nonautonomous dynamical systems on uniform spaces.

On periodic points of torus endomorphisms

It is a well-known fact that Anosov endomorphisms of n-torus which are different from automorphisms and expanding endomorphisms are not structurally stable and, in general, are not conjugated to algebraic endomorphisms. Nevertheless, hyperbolic algebraic endomorphisms of torus are conjugated with their C1 perturbations on the set of periodic points. Therefore the study of algebraic toral endomorphisms is very important. This paper is devoted to study of the structure of the sets of periodic and pre-periodic points of algebraic toral endomorphisms. Various group properties of this sets of points are studied. The density of periodic points for algebraic endomorphisms of n-torus is proved; it is clarifief how the number of periodic and pre-periodic points with a fixed denominator depends on the properties of the characteristic polynomial. The Theorem 1.1 is the main result of this paper. It contains an algorithm that allows to split the sets of periodic and pre-periodic points of a given algebraic endomorphism of two-dimensional torus.

Answering two open problems on Banks theorem for non-autonomous dynamical systems

ABSTRACTThis paper shows that (1) there exists a topologically transitive NADS having two disjoint invariant periodic orbits with dense periodic points, which is finitely generated but not periodic; (2) there exists a topologically transitive non-finitely generated NADS having two disjoint invariant periodic orbits with dense periodic points, which is not sensitive. This answers positively the Open Problems 4.1 and 4.2 posed in Yang and Li [A remark on Banks theorem, J. Difference Equ. Appl. 24 (2018), pp. 1777–1782].

A remark on banks theorem

ABSTRACTFor discrete autonomous dynamical systems, it was found that in the three conditions defining Devaney chaos, topological transitivity and dense periodic points together imply sensitive dependence on initial condition (Banks, Brooks, Cairns, Davis and Stacey, On Devaney's definition of chaos, The Amer. Math. Monthly 99 (1992), pp. 332–334). In this paper, we give the definition of finitely generated non-autonomous dynamical systems (NADS) and generalize the Banks et al. theorem to the finitely generated NADS.

Monotone dynamical systems with dense periodic points

In this paper we prove a recent conjecture by M. Hirsch, which says that if (f,Ω) is a discrete time monotone dynamical system, with f:Ω→Ω a homeomorphism on an open connected subset of a finite dimensional vector space, and the periodic points of f are dense in Ω, then f is periodic.

On chaotic properties of new semidynamical systems of Lasota type in two spaces

The present study is concerned with construct two new semidynamical systems which are generated by two partial differential equations of Lasota type. In addition, this study discusses the asymptotic properties: Strong stability, exponential stability, periodic points, the density of periodic points, transitivity and chaos in two spaces: [Formula: see text] space and [Formula: see text] space.

A novel chaotic Jaya algorithm for unconstrained numerical optimization

Jaya algorithm is one of the recent algorithms developed to solve optimization problems. The basic concept of this algorithm consists in moving the obtained solution, for a given problem, toward the best solution and avoiding the worst one. However, it severely suffers from premature convergence problem and therefore can be easily trapped in local optimums. This study aimed to alleviate these drawbacks and improve the performance of the original Jaya algorithm. Here, three new mutation strategies were implemented in the original Jaya to improve both its global and local search abilities. Chaotic maps were proved to be able to boost the search capabilities of meta-heuristic algorithms. Therefore, after demonstrating its chaotic behavior through the sensitivity to initial conditions, topological transitivity and the density of periodic points, we proposed a new 2D cross chaotic map. The chaotic sequences provided by the proposed chaotic map were embedded into the original Jaya algorithm to generate the initial population and control the search equations. It is worth mentioning that the modifications incorporated in the original algorithm did not affect its two essential characteristics, i.e., simplicity and nonrequirement of additional control parameters. As case studies, sixteen benchmark functions were used to evaluate the performance of the proposed chaotic Jaya algorithm (C-Jaya) regarding solution accuracy and convergence speed. Comparisons with some other meta-heuristic algorithms for low-, middle- and high-dimensional benchmark functions show that the proposed C-Jaya algorithm enhances the performance of original Jaya significantly. Moreover, it offers the fastest global convergence, the highest solution quality and it is the most robust on almost all the test functions among all the algorithms. Nonparametric statistical procedures, i.e., Friedman test, Friedman aligned ranks test and Quade test, conducted to analyze the obtained results, show the superiority of the proposed algorithm.

Sensitive dependence for nonautonomous discrete dynamical systems

Given a nonautonomous discrete dynamical system (NDS) (X,f1,∞) we show that transitivity and density of periodic points do not imply sensitivity in general, i.e., in the definition of Devaney chaos there are no redundant conditions for NDS. In addition, we show that if we also assume uniform convergence of the sequence (fn) that induces the NDS, then sensitivity follows. Furthermore, in contrast to the autonomous case, we show that there exist minimal NDS which are neither equicontinuous nor sensitive.

What is topological about topological dynamics?

We consider various notions from the theory of dynamical systems from a topological point of view. Many of these notions can be sensibly defined either in terms of (finite) open covers or uniformities. These Hausdorff or uniform versions coincide in compact Hausdorff spaces and are equivalent to the standard definition stated in terms of a metric in compact metric spaces. We show for example that in a Tychonoff space, transitivity and dense periodic points imply (uniform) sensitivity to initial conditions. We generalise Bryant's result that a compact Hausdorff space admitting a $c$-expansive homeomorphism in the obvious uniform sense is metrizable. We study versions of shadowing, generalising a number of well-known results to the topological setting, and internal chain transitivity, showing for example that $ω$-limit sets are (uniform) internally chain transitive and weak incompressibility is equivalent to (uniform) internal chain transitivity in compact spaces.

Chaos in hyperspaces of nonautonomous discrete systems

We study the interaction of some dynamical properties of a nonautonomous discrete dynamical system (X, f∞) and its induced nonautonomous discrete dynamical system (K(X),f∞¯), where K(X) is the hyperspace of non-empty compact sets in X, endowed with the Vietoris topology. We consider properties like transitivity, weakly mixing, points with dense orbit, density of periodic points, among others. We also present examples of nonautonomous discrete dynamical systems showing that transitivity, density of periodic points and sensitive dependence on initial conditions are independents on the unit interval, i.e., unlike autonomous discrete dynamical systems, in definition of Devaney chaotic there are not redundant conditions for NDS on the interval. Actually, our examples give an even more precise conclusion: the classical result stating that transitivity is a sufficient condition for an autonomous discrete dynamical system on the interval to be Devaney chaotic fails to be true for nonautonomous dynamical systems.

When are all closed subsets recurrent?

In the paper we study relations of rigidity, equicontinuity and pointwise recurrence between an invertible topological dynamical system (t.d.s.)$(X,T)$and the t.d.s. $(K(X),T_{K})$induced on the hyperspace$K(X)$of all compact subsets of$X$, and provide some characterizations. Among other examples, we construct a minimal, non-equicontinuous, distal and uniformly rigid t.d.s. and a weakly mixing t.d.s. which induces dense periodic points on the hyperspace$K(X)$but itself does not have dense distal points, solving in that way a few open questions from earlier articles by Dong, and Li, Yan and Ye.

Monotone Dynamical Systems with Polyhedral Order Cones and Dense Periodic Points

Let $X\subset \mathbb{R}^{n}$ be a set whose interior is connected and dense in $X$, ordered by a closed convex cone $K\subset \mathbb{R}^{n}$ having nonempty interior. Let $T: X\approx X$ be an order-preserving homeomorphism. The following result is proved: Assume the set of periodic points of $T$ is dense in $X$, and $K$ is a polyhedron. Then $T$ is periodic.

On Devaney Chaos and Dense Periodic Points: Period 3 and Higher Implies Chaos

We look at density of periodic points and Devaney Chaos. We prove that if f is Devaney Chaotic on a compact metric space with no isolated points, then the set of points with prime period at least n is dense for each n. Conversely, we show that if f is a continuous function from a closed interval to itself, for which the set of points with prime period at least n is dense for each n, then there is a decomposition of the interval into closed subintervals on which either f or f2 is Devaney Chaotic. (In fact, this result holds if the set of points with prime period at least 3 is dense.)

Transitivity and Chaoticity in 1-D Cellular Automata

Recent progress in symbolic dynamics of cellular automata (CA) shows that many CA exhibit rich and complicated Bernoulli-shift properties, such as positive topological entropy, topological transitivity and even mixing. Noticeably, some CA are only transitive, but not mixing on their subsystems. Yet, for one-dimensional CA, this paper proves that not only the shift transitivity guarantees the CA transitivity but also the CA with transitive non-trivial Bernoulli subshift of finite type have dense periodic points. It is concluded that, for one-dimensional CA, the transitivity implies chaos in the sense of Devaney on the non-trivial Bernoulli subshift of finite types.

Stability and density of periodic points of Lp-space, 0 < p < 1

Stability and density of periodic points of Lp-space, 0 < p < 1