Set Of Recurrent Points Research Articles

Uniqueness of Denjoy minimal sets for twist maps with zero entropy *

For monotone twist maps with zero topological entropy, we show that the set of recurrent points with irrational rotation number α can be described by a single orientation preserving circle homeomorphism and hence there is either an invariant circle, or a unique Denjoy minimal set, of rotation number α.

The centre and the depth of the centre for continuous maps on dendrites with unique branch point

Let D be a dendrite with unique branch point and f:D→D be continuous. Denote by R(f) and Ω(f) the set of recurrent points and the set of non-wandering points of f, respectively. Let Ω0(f)=D and Ωk(f)=Ω(f|Ωk−1(f)) for any positive integer k. The minimal k such that Ωk(f)=Ωk+1(f) is called the depth of f, where k is a positive integer or ∞. In this note, we show that Ω2(f)=R(f)‾ and the depth of f is at most 2.

Characteristic of pointwise-recurrent maps on a dendrite

Let f be a continuous map on a dendrite D with f (D) = D. Denote by R(f ) and AP (f ) the set of recurrent points and the set of almost periodic points of f , respectively, and denote by ω(x, f ), Λ(x, f ), Γ(x, f ) and Ω(x, f ) the set of ω-limit points, the set of α-limit points, the set of γ-limit points and the set of weak ω-limit points of x under f , respectively. In this paper, we show that the following statements are equivalent: (1) D = R(f ). (2) D = AP (f ). (3) Ω(x, f ) = ω(x, f ) for any x ∈ D. (4) Ω(x, f ) = Γ(x, f ) for any x ∈ D. (5) f is equicontinuous. (6) [c, d] ⊄ Ω(x, f ) for any c, d,x ∈ D with c ≠ (7) Ω(x, f ) is minimal for any x ∈ D. (8) Card(Λ − 1(x, f ) ∩ (D−End(D))) < ∞ for any x ∈ D, where Λ − 1(x, f ) = {y : x ∈ Λ(y, f )}, End(D) is the set of endpoints of D and Card(A) is the cardinal number of set A. (9) If x ∈ Λ(y, f ) with x, y ∈ D, then y ∈ ω(x, f ). (10) Map h : x → ω(x, f ) (x ∈ D) is continuous and for any x, y ∈ D with x ∉ ω(y, f ), ω(x, f ) ≠ ω(y, f ). Besides, we also study characteristic of pointwise-recurrent maps on a dendrite with the number of branch points being finite.

Recurrence in non-autonomous dynamical systems

ABSTRACT We consider a sequence of continuous maps on a compact metric space X uniformly converging to a function f. This sequence forms a non-autonomous discrete dynamical system. In such case, the set of omega-limit points is invariant with respect to the limit function f. Here we give negative answer to questions whether the sets of recurrent points and non-wandering points are also invariant. We also discuss the relation of the set of recurrent points of and its limit function f.

Topological and Metric Recurrence for General Markov Chains

Using ideas borrowed from topological dynamics and ergodic theory we introduce topological and metric versions of the recurrence property for general Markov chains. The main question of interest here is how large is the set of recurrent points. We show that under some mild technical assumptions the set of non recurrent points is of zero reference measure. Necessary and sufficient conditions for a reference measure $m$ (which needs not to be dynamically invariant) to satisfy this property are obtained. These results are new even in the purely deterministic setting.

Special $$\alpha $$ α -Limit Points and $$\gamma $$ γ -Limit Points of a Dendrite Map

Let (X, d) be a compact metric space and f be a continuous map from X to X. Denote by R(f), \({ SA}(f)\) and \(\Gamma (f)\) the set of recurrent points, the set of special \(\alpha \)-limit points and the set of \(\gamma \)-limit points of f, respectively. It is well known that for an interval map f, the following three statements hold: (1) \(R(f)\subset { SA}(f)\cap \Gamma (f)\); (2) \({ SA}(f)= \Gamma (f)\); (3) \({ SA}(f)\cup \Gamma (f)\subset \overline{R(f)}\). The aim of this paper is to show that the above statement (1) holds for maps of dendrites with the number of endpoints being \(\aleph _{\mathbf{0}}\) (the cardinal number of the set of positive integers) and the above statements (2) and (3) do not hold for maps of dendrites with the number of endpoints being \(\aleph _{\mathbf{0}}\). Besides, we also study unilateral \(\gamma \)-limit points for maps of dendrites with the number of branch points being finite.

Banach upper density recurrent points of C 0-flows

Let X denote a compact metric space with distance d and F : X × ℝ → X or Ft : X → X denote a C0-flow. From the point of view of ergodic theory, all important dynamical behaviors take place on a full measure set. The aim of this paper is to introduce the notion of Banach upper density recurrent points and to show that the closure of the set of all Banach upper density recurrent points equals the measure center or the minimal center of attraction for a C0-flow. Moreover, we give an example to show that the set of quasi-weakly almost periodic points can be included properly in the set of Banach upper density recurrent points, and point out that the set of Banach upper density recurrent points can be included properly in the set of recurrent points.

The Centre and the Depth of the Centre for Continuous Maps on Dendrites with Finite Branch Points

Let \(\mathbf{D}\) be a dendrite with finite branch points and \(f:\mathbf{D}\rightarrow \mathbf{D}\) be continuous. Denote by R(f) and \(\Omega (f)\) the set of recurrent points and the set of non-wandering points of f respectively. Let \(\Omega _0 (f)=\mathbf{D}\) and \(\Omega _n (f)=\Omega (f|_{\Omega _{n-1} (f)})\) for all \(n\in \mathbf{N}\). The minimal \(m\in \mathbf{N}\cup \{\infty \}\) such that \(\Omega _{m} (f)=\Omega _{m+1} (f)\) is called the depth of f. In this note, we show that \(\Omega _3(f)=\overline{R(f)}\) and the depth of f is at most 3. Furthermore, we show that there exist a dendrite \(\mathbf{D}\) with finite branch points and \(f\in C^0(\mathbf{D})\) such that \( \Omega _3(f)=\overline{R(f)}\ne \Omega _2(f)\).

On the existence of maximal ω-limit sets for dendrite maps

For interval maps and also for graph maps, every ω-limit set is a subset of a maximal one. In this note we construct a continuous map on a dendrite with no maximal ω-limit set. Moreover, the set of branch points is nowhere dense, every ω-limit set of the map is nowhere dense, the set of periodic points and the set of recurrent points are equal and the set of ω-limit points is not closed (an example with the last property was constructed by the authors already in [Kočan Z, Kornecká-Kurková V, Málek M. On the centre and the set of omega-limit points of continuous maps on dendrites. Topol Appl 2009;156:2923–2931]).

Complexity of a kind of interval continuous self-map of finite type

An interval map is called finitely typal, if the restriction of the map to non-wandering set is topologically conjugate with a subshift of finite type. In this paper, we prove that there exists an interval continuous self-map of finite type such that the Hausdorff dimension is an arbitrary number in the interval (0, 1), discuss various chaotic properties of the map and the relations between chaotic set and the set of recurrent points.

Topological Entropy and Special α‐Limit Points of Graph Maps

Let G a graph and f : G → G be a continuous map. Denote by h(f), R(f), and SA(f) the topological entropy, the set of recurrent points, and the set of special α‐limit points of f, respectively. In this paper, we show that h(f) &gt; 0 if and only if SA(f) − R(f) ≠ ∅.

A classification of explosions in dimension one

AbstractA discontinuous change in the size of an attractor is the most easily observed type of global bifurcation. More generally, anexplosionis a discontinuous change in the set of recurrent points. An explosion often results from heteroclinic and homoclinic tangency bifurcations. We prove that, for one-dimensional maps, explosions are generically the result of either tangency or saddle-node bifurcations. Furthermore, we give necessary and sufficient conditions for generic tangency bifurcations to lead to explosions.

Non-wandering points and the depth for graph maps

Let G be a graph and f: G → G be continuous. Denote by R(f) and Ω(f) the set of recurrent points and the set of non-wandering points of f respectively. Let Ω0(f) = G and Ωn(f) = \(\Omega (f|_{\Omega _{n - 1} (f)} )\) for all n ∈ ℕ. The minimal m ∈ ℕ ∪ {∞} such that Ωm(f) = Ωm+1(f) is called the depth of f. In this paper, we show that \(\Omega _2 (f) = \overline {R(f)} \) and the depth of f is at most 2. Furthermore, we obtain some properties of non-wandering points of f.

Topological Structure of Non–wandering Set of a Graph Map

Let G be a graph (i.e., a finite one–dimensional polyhedron) and f : G → G be a continuous map. In this paper, we show that every isolated recurrent point of f is an isolated non–wandering point; every accumulation point of the set of non–wandering points of f with infinite orbit is a two–order accumulation point of the set of recurrent points of f; the derived set of an ω–limit set of f is equal to the derived set of an the set of recurrent points of f; and the two–order derived set of non–wandering set of f is equal to the two–order derived set of the set of recurrent points of f.

Dynamical Systems Generated by Functions with Connected &lt;em&gt;G&lt;sub&gt;δ&lt;/sub&gt;&lt;/em&gt; Graphs

In 2001, Cs\" ornyei, O'Neil and Preiss proved that the composition of any two Darboux Baire-1 functions $[0,1]\rightarrow [0,1]$ possesses a fixed point, solving a long-standing open problem. In 2004 Szuca proved that this result can be generalized to any $f$ in the class $\cal J$ of functions $[0,1]\rightarrow [0,1]$ with connected $G_\delta$ graph. As a consequence, he proved that for such functions the Sharkovsky theorem is satisfied. As the main result of this paper we prove that as for continuous maps of the interval, any $f$ in $\cal J$ has positive topological entropy if and only if it has a periodic point of period different from $2^n$, for any $n\in\mathbb N$. To do this we show that using Bowen's approach it is possible to define topological entropy for discontinuous maps of a compact metric space with almost all of the standard properties. In particular, the variational principle is true, and consequently, topological entropy is supported by the set of recurrent points. We also develop theory of recurrent, $\omega$-limit, and nonwandering points of functions in $\cal J$ since, in general, standard results from the topological dynamics, are not true. For example, there is a Darboux Baire-1 function $f$ (hence, $f\in\cal J$) such that neither the set of recurrent points nor the set of $\omega$-limit points of $f$ are invariant.

Recurrent Points of Fuzzy Dynamical Systems

The main goal of the present paper is to reconsider classical results from the theory of dynamical systems, making them appropriate for treating systems about which we have imprecise, vague, uncertain or incomplete information. Here we study qualitative properties of dynamical systems related to recurrent points. According to the basic Poincaré recurrence theorem, the set of recurrent points has full measure in any subset of the classical dynamical system space. In general, this property is invalid for fuzzy dynamical systems. In this paper, we consider fuzzy dynamical systems of two types. It is demonstrated that for fuzzy dynamical systems of the first type, the Poincaré property of recurrent points remains true if the fuzzy measure in the system space satisfies some, rather weak, additional conditions. It is also proved that infinitely recurrent points have the same property. At the same time, for fuzzy dynamical systems of the second type, the Poincaré property of recurrent points is invalid, and it is possible to find only sufficiently big subsets of recurrent points. In contrast to fuzzy dynamical systems of the first type, infinitely recurrent points of fuzzy dynamical systems of the second type do not have even this property.

The set of recurrent points of a continuous self-map on an interval and strong chaos

In this paper, we discuss a continuous self-map of an interval and the existence of an uncountable strongly chaotic set. It is proved that if a continuous self-map of an interval has positive topological entropy, then it has an uncountable strongly chaotic set in which each point is recurrent, but is not almost periodic.

Recurrence and unique ergodicity

Let ( Σ, ρ) denote the one-sided symbolic space (with two symbols), σ the shift on Σ, A(·) the set of almost periodic points and R(·) the set of recurrent points. It is shown that there exists an uncountable set T with T⊂R(σ)−A(σ) such that σ : T→ T is uniquely ergodic.

Explosions of chaotic sets

Large-scale invariant sets such as chaotic attractors undergo bifurcations as a parameter is varied. These bifurcations include sudden changes in the size and/or type of the set. An explosion is a bifurcation in which new recurrent points suddenly appear at a non-zero distance from any pre-existing recurrent points. We discuss the following. In a generic one-parameter family of dissipative invertible maps of the plane there are only four known mechanisms through which an explosion can occur: (1) a saddle-node bifurcation isolated from other recurrent points, (2) a saddle-node bifurcation embedded in the set of recurrent points, (3) outer homoclinic tangencies, and (4) outer heteroclinic tangencies. (The term “outer tangency” refers to a particular configuration of the stable and unstable manifolds at tangency.) In particular, we examine different types of tangencies of stable and unstable manifolds from orbits of pre-existing invariant sets. This leads to a general theory that unites phenomena such as crises, basin boundary metamorphoses, explosions of chaotic saddles, etc. We illustrate this theory with numerical examples.

Asymptotic entropy, periodic orbits, and pseudo-Anosov maps

In this paper we derive some properties of a variety of entropy that measures rotational complexity of annulus homeomorphisms, called asymptotic or rotational entropy. In previous work [KS] the authors showed that positive asymptotic entropy implies the existence of infinitely many periodic orbits corresponding to an interval of rotation numbers. In our main result, we show that a Hölder $C^1$ diffeomorphism with nonvanishing asymptotic entropy is isotopic rel a finite set to a pseudo-Anosov map. We also prove that the closure of the set of recurrent points supports positive asymptotic entropy for a ($C^0$) homeomorphism with nonzero asymptotic entropy.