Set Of Periodic Points Research Articles

Chain recurrent points of monotone graph maps and CR-Pm property on local dendrites

ABSTRACTIn this paper, the set of chain recurrent points of monotone maps on graphs is investigated. A compact metric space X has the CR-Pm property provided that for each monotone self-mapping f of X, , where and are the sets of periodic points and chain recurrent points of f, respectively. We show that a local dendrite X has the CR-Pm property if and only if X is a dendrite containing neither the infinite star nor the dendrite .

Nonwandering points of monotone local dendrite maps revisited

Let X be a local dendrite and let f:X→X be a monotone map. Denote by P(f) and Ω(f) the sets of periodic points and nonwandering points of f, respectively. We show that Ω(f)=P(f)‾, whenever P(f) is nonempty and Ω(f) is the unique minimal set included in a circle which is either a Cantor set or a circle, whenever P(f) is empty. In the case where the set of endpoints of X is countable, we show that Ω(f)=P(f) whenever P(f) is nonempty.

Multi-sensitivity and other stronger forms of sensitivity in non-autonomous discrete systems

In this paper we obtain different sufficient conditions for a non-autonomous discrete system to be multi-sensitivite. We study properties of a multi-sensitive non-autonomous system in detail. It is proved that on a compact metric space every finitely generated non-autonomous system which is topologically transitive having dense set of periodic points is thickly syndetically sensitive. We introduce and study the notion of totally sensitive non-autonomous systems. We also provide counter examples to support our results.

Dynamical systems arising from random substitutions

Random substitutions are a natural generalisation of their classical ‘deterministic’ counterpart, whereby at every step of iterating the substitution, instead of replacing a letter with a predetermined word, every letter is independently replaced by a word from a finite set of possible words according to a probability distribution. We discuss the subshifts associated with such substitutions and explore the dynamical and ergodic properties of these systems in order to establish the groundwork for their systematic study. Among other results, we show under reasonable conditions that such systems are topologically transitive, have either empty or dense sets of periodic points, have dense sets of linearly repetitive elements, are rarely strictly ergodic, and have positive topological entropy.

Descriptive set theory for expansive systems

Kato [5] and Artigue [3] merged the theory of expansive systems [10] and foliations with the continuum theory [14]. Here we merge the expansive systems but with the descriptive set theory [6] instead. More precisely, we define meagre-expansivity for both homeomorphisms and measures by requiring the interior of the dynamical balls up to some prefixed radio to be either empty or with zero measure respectively. We first prove that every cw-expansive homeomorphism of a locally connected metric space without isolated points is meagre-expansive (but not conversely). Second that a homeomorphism of a metric space is meagre-expansive if and only if every Borel probability measure is meagre-expansive. Next that the space of meagre-expansive measures of a homeomorphism of a compact metric space X is an Fσ subset of the space of Borel probability measures equipped with the weak* topology. In the sequel we prove that every homeomorphism with a meagre-expansive measure of a compact metric space has an invariant meagre-expansive measure. Also that the set of periodic points of every meagre-expansive homeomorphism of a compact metric space has empty interior. In the circle or the interval we prove that there are no meagre-expansive homeomorphisms of the circle or the interval. Moreover, the meagre-expansive measures of an interval homeomorphism or a circle homeomorphism with rational rotation number are precisely the finite convex combinations of Dirac measures supported on isolated periodic points. A circle homeomorphism with irrational rotation number has a meagre-expansive measure if and only if it is a Denjoy map. In such a case the meagre-expansive measures are precisely those measures supported on the unique minimal set of the map. To obtain some of our results we will consider a measurable version of the classical Baire Category.

Topological entropy of a graph map

Let G be a graph and f: G → G be a continuous map. Denote by h(f), P(f),AP(f),R(f) and ω(x, f) the topological entropy of f, the set of periodic points of f, the set of almost periodic points of f, the set of recurrent points of f and the ω-limit set of x under f, respectively. In this paper, we show that the following statements are equivalent: (1) h(f) > 0. (2) There exists an x ∈ G such that ω(x, f) ∩ P(f) ≠ O and ω(x, f) is an infinite set. (3) There exists an x ∈ G such that ω(x, f) contains two minimal sets. (4) There exist x, y ∈ G such that ω(x, f) − ω(y, f) is an uncountable set and ω(y, f) ∩ ω(x, f) ≠ O. (5) There exist an x ∈ G and a closed subset A ⊂ ω(x, f) with f(A) ⊂ A such that ω(x, f) − A is an uncountable set. (6) R(f) − AP(f) ≠ O. (7) f| P ( f ) is not pointwise equicontinuous.

The dynamical Manin–Mumford problem for plane polynomial automorphisms

Let f be a polynomial automorphism of the affine plane. In this paper we consider the possibility for it to possess infinitely many periodic points on an algebraic curve C . We conjecture that this happens if and only if f admits a time-reversal symmetry; in particular the Jacobian Jac (f) must be a root of unity. As a step towards this conjecture, we prove that the Jacobian and all its Galois conjugates lie on the unit circle in the complex plane. Under mild additional assumptions we are able to conclude that indeed Jac (f) is a root of unity. We use these results to show in various cases that any two automorphisms sharing an infinite set of periodic points must have a common iterate, in the spirit of recent results by Baker–DeMarco and Yuan–Zhang.

Periods of continuous mapson closed surfaces

The objective of the present work is to present information on the set of periodic points of a continuous self-map on a closed surface which can be obtained using the action of this map on homological groups of the closed surface.

Chaotic extensions of continuous maps on compact manifolds

Let X be a compact connected Lipschitz manifold, with or without boundary. We show that for every continuous map and every nowhere dense closed subset K of X, there is a topologically transitive continuous map having a dense set of periodic points in X such that . Combined with a deep theorem of Sullivan, our result extends to all compact connected topological manifolds of dimension .

Indecomposability and Devaney’s chaoticity of semiflows with an arbitrary acting abelian topological semigroup

We prove that a semiflow with an arbitrary acting abelian topological semigroup is Devaney chaotic if and only if it is nonminimal, indecomposable and has a dense set of periodic points. This generalizes a result of X. Wang and Y. Huang.

Periodic points of the planar area preserving Poincaré map inside isolating segment

The aim of this note is to study the structure of the set of periodic points of the area preserving Poincaré map P associated to T-periodic local planar process. We will use the approach based on the method of periodic isolating segments (isolating chains) and the fixed point index theory. The number of the applications to the processes generated by periodic in time non-autonomous planar ODE's are given.

Wieler solenoids, Cuntz–Pimsner algebras and -theory

We study irreducible Smale spaces with totally disconnected stable sets and their associated$K$-theoretic invariants. Such Smale spaces arise as Wieler solenoids, and we restrict to those arising from open surjections. The paper follows three converging tracks: one dynamical, one operator algebraic and one$K$-theoretic. Using Wieler’s theorem, we characterize the unstable set of a finite set of periodic points as a locally trivial fibre bundle with discrete fibres over a compact space. This characterization gives us the tools to analyse an explicit groupoid Morita equivalence between the groupoids of Deaconu–Renault and Putnam–Spielberg, extending results of Thomsen. The Deaconu–Renault groupoid and the explicit Morita equivalence lead to a Cuntz–Pimsner model for the stable Ruelle algebra. The$K$-theoretic invariants of Cuntz–Pimsner algebras are then studied using the Cuntz–Pimsner extension, for which we construct an unbounded representative. To elucidate the power of these constructions, we characterize the Kubo–Martin–Schwinger (KMS) weights on the stable Ruelle algebra of a Wieler solenoid. We conclude with several examples of Wieler solenoids, their associated algebras and spectral triples.

The beta-transformation's companion map for Pisot or Salem numbers and their periodic orbits

ABSTRACTThe β-transformation of the unit interval is defined by Tβ(x) ≔ βx (mod 1). Its eventually periodic points are a subset of [0, 1] intersected with the field extension . If β > 1 is an algebraic integer of degree d > 1, then is a -vector space isomorphic to ; therefore, the intersection of [0, 1] with is isomorphic to a domain in . The transformation from this domain which is conjugate to the β-transformation is called the companion map, given its connection to the companion matrix of β's minimal polynomial. The companion map and the proposed notation provide a natural setting to reformulate a classic result concerning the set of periodic points of the β-transformation for numbers. It also allows to visualize orbits in a d-dimensional space. Finally, we refer connections with arithmetic codings and symbolic representations of hyperbolic toral automorphisms.

Equicontinuity of maps on a dendrite with finite branch points

Let (T, d) be a dendrite with finite branch points and f be a continuous map from T to T. Denote by ω(x, f) and P(f) the ω-limit set of x under f and the set of periodic points of f, respectively. Write Ω(x, f) = {y| there exist a sequence of points xk ∈ T and a sequence of positive integers n1 < n2 < ··· such that limk→∞xk = x and \({\lim _{k \to \infty }}{f^{{n_k}}}\left( {{x_k}} \right) = y\)}. In this paper, we show that the following statements are equivalent: (1) f is equicontinuous. (2) ω(x, f) = Ω(x, f) for any x ∈ T. (3) ∩n=1∞fn(T) = P(f), and ω(x, f) is a periodic orbit for every x ∈ T and map h: x → ω(x, f) (x ∈ T) is continuous. (4) Ω(x, f) is a periodic orbit for any x ∈ T.

Periods of continuous maps on some compact spaces

The objective of this paper is to provide information on the set of periodic points of a continuous self-map defined in the following compact spaces: (the n-dimensional sphere), (the product space of the n-dimensional with the m-dimensional spheres), (the n-dimensional complex projective space) and (the n-dimensional quaternion projective space). We use as main tool the action of the map on the homology groups of these compact spaces.

Transitive and series transitive maps on [formula omitted

Motivated by the behavior of topologically transitive homomorphisms of Polish abelian groups, we say a continuous map f:Rd→Rd is ‘series transitive’ if for any two nonempty open sets U,V⊂Rd, there exist x∈U and n∈N such that ∑j=0n−1fj(x)∈V. We show that any map on a discrete and closed subset of Rd can be extended to a mixing map of Rd, and use this result to produce a mixing map f:Rd→Rd (for each d∈N) which is also series transitive. We have examples to say that transitivity and series transitivity are independent properties for continuous self-maps of Rd. We also construct a chaotic map (i.e., a transitive map with a dense set of periodic points) f:Rd→Rd such that f is arbitrarily close to and asymptotic to the identity map. Finally, we make a few observations about topological transitivity of continuous homomorphisms of Polish abelian groups.

Li-Yorke chaos for dendrite maps with zero topological entropy and &lt;i&gt;ω&lt;/i&gt;-limit sets

Let $X$ be a dendrite with set of endpoints $E(X)$ closed and let $f:~X \to X$ be a continuous map with zero topological entropy. Let $P(f)$ be the set of periodic points of $f$. We prove that if $L$ is an infinite $\omega$-limit set of $f$ then $L\cap P(f)\subset E(X)^{\prime}$, where $E(X)^{\prime}$ is the set of all accumulations points of $E(X)$. Furthermore, if $E(X)$ is countable and $L$ is uncountable then $L\cap P(f)=\emptyset$. We also show that if $E(X)^{\prime}$ is finite then any uncountable $\omega$-limit set of $f$ has a decomposition and as a consequence if $f$ has a Li-Yorke pair $(x,y)$ with $\omega_f(x)$ or $\omega_f(y)$ is uncountable then $f$ is Li-Yorke chaotic.

A note on periodic points of dendrite maps and their induced maps

Let X be a compact metric space and 2X be the hyperspace of all nonempty closed subsets of X endowed with the Hausdorff metric. It is well known that for each continuous map f:X→X, the density of periodic points of f implies the density of periodic points of the induced map 2f:2X→2X. Méndez (2010) conjectured in [6] that the converse is true when the phase space X is a dendrite. In [8], Ŝpitalský (2015) constructed a continuous transitive map F:X→X on a dendrite X with only two periodic points. We prove in this note that the set of periodic points of its induced map 2F is dense in 2X. This answers in the negative Méndez's conjecture.

Induced dynamics on the hyperspaces

&lt;p&gt; &lt;/p&gt;&lt;p&gt;In this paper, we study the dynamics induced by finite commutative relation on the hyperspaces. We prove that the dynamics induced on the hyperspace by a non-trivial commutative family of continuous self maps cannot be transitive and hence cannot exhibit higher degrees of mixing. We also prove that the dynamics induced on the hyperspace by such a collection cannot have dense set of periodic points. We also give example to show that the induced dynamics in this case may or may not be sensitive.&lt;/p&gt;

Homeomorphisms on compact metric spaces with finite derived length

&lt;p&gt;The sets of periodic points of self homeomorphisms on an ordinal of finite derived length are characterised, thus characterising the same for homeomorphisms on compact metric spaces with finite derived length. A partition of ordinal is introduced to study this problem which is also used to solve two more problems: one about an equivalence relation and the other about a group action, both on an ordinal of finite derived length.&lt;/p&gt;