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1.
We show that there is a continuous map of the unit interval into itself of type which has a trajectory disjoint from the set of recurrent points of , but contained in the closure of . In particular, is not closed. A function of type , with nonclosed set of recurrent points, was found by H. Chu and J. Xiong [Proc. Amer. Math. Soc. 97 (1986), 361-366]. However, there is no trajectory contained in , since any point in is eventually mapped into . Moreover, our construction is simpler. We use to show that there is a continuous map of the interval of type for which the set of recurrent points is not an set. This example disproves a conjecture of A. N. Sharkovsky et al., from 1989. We also provide another application of . 相似文献
2.
Let(T, d) be a dendrite with finite branch points and f be a continuous map from T to T. Denote byω(x,f) and P(f) the ω-limit set of x under f and the set of periodic points of,respectively. Write Ω(x,f) = {y| there exist a sequence of points x_k E T and a sequence of positive integers n_1 n_2 … such that lim_(k→∞)x_k=x and lim_(k→∞)f~(n_k)(x_k) =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)~∞f~n(T) = P(f),and ω(x,f)is a periodic orbit for every x ∈ T and map h : x→ω(x,f)(x ET)is continuous.(4) Ω(x,f) is a periodic orbit for any x∈T. 相似文献
3.
Let f be a tree map,P(f) the set of periodic points of f and CR(f) the set of chain recurrent points of f. In this paper,the notion of division for invariant closed subsets of a tree map is introduced. It is proved that: (1) fhas zero topological entropy if and only if for any x∈CR(f)-P(f) and each natural number s the orbit of x under f^5 has a division; (2) If f has zero topological entropy,then for any xECR(f)--P(f) the w-limit set of x is an infinite minimal set. 相似文献
4.
The objective of this work is to present sufficient conditions for having positive topological entropy for continuous self-maps defined on a closed surface by using the action of this map on the homological groups of the closed surface. 相似文献
5.
The dynamics of catalytic networks have been widely studied over the last decades because of their implications in several fields like prebiotic evolution, virology, neural networks, immunology or ecology. One of the most studied mathematical bodies for catalytic networks was initially formulated in the context of prebiotic evolution, by means of the hypercycle theory. The hypercycle is a set of self-replicating species able to catalyze other replicator species within a cyclic architecture. Hypercyclic organization might arise from a quasispecies as a way to increase the informational containt surpassing the so-called error threshold. The catalytic coupling between replicators makes all the species to behave like a single and coherent evolutionary multimolecular unit. The inherent nonlinearities of catalytic interactions are responsible for the emergence of several types of dynamics, among them, chaos. In this article we begin with a brief review of the hypercycle theory focusing on its evolutionary implications as well as on different dynamics associated to different types of small catalytic networks. Then we study the properties of chaotic hypercycles with error-prone replication with symbolic dynamics theory, characterizing, by means of the theory of topological Markov chains, the topological entropy and the periods of the orbits of unimodal-like iterated maps obtained from the strange attractor. We will focus our study on some key parameters responsible for the structure of the catalytic network: mutation rates, autocatalytic and cross-catalytic interactions. 相似文献
7.
This article is concerned with a fourth-order parabolic equation. Based on the regularity estimates for the semigroups and the classical existence theorem of global attractors, we prove that the fourth-order parabolic equation possesses a global attractor in H k (0?≤? k?5) space, which attracts any bounded subset of H k (Ω) in the H k -norm. 相似文献
8.
The notion of topological free entropy dimension of n-tuple of elements in a unital C ∗ algebra was introduced by Voiculescu. In the paper, we compute topological free entropy dimension of one self-adjoint element and topological free orbit dimension of one self-adjoint element in a unital C ∗ algebra. We also calculate the values of topological free entropy dimensions of any families of self-adjoint generators of some unital C ∗ algebras, including irrational rotation C ∗ algebra, UHF algebra, and minimal tensor product of two reduced C ∗ algebras of free groups. 相似文献
9.
This paper deals with the relationship between the periodic orbits of continuous maps on graphs and the topological entropy of the map. We show that the topological entropy of a graph map can be approximated by the entropy of its periodic orbits. 相似文献
10.
Using topological entropy of automorphisms of -algebras, it is shown that some important facts from the theory of AF algebras do not carry over to the class of algebras. It is shown that in general one cannot perturb a basic building block into a larger one which almost contains it. The same entropy obstruction used to prove this fact also provides a new obstruction to the known fact that two injective homomorphisms from a building block into an algebra need not differ by an (inner) automorphism when they agree on K-theory. 相似文献
11.
In this paper we discuss some basic properties of chain reachable sets and chain equivalent sets of continuous maps. It is proved that if f: T→ T is a tree map which has a chain movable fixed point v, and the chain equivalent set CE( v, f) is not contained in the set P( f) of periodic points of f, then there exists a positive integer p not greater than the number of points in the set End([ CE( v, f)])− Pv( f) such that fp is turbulent, and the topological entropy . This result generalizes the corresponding results given in Block and Coven (1986) [2], Guo et al. (2003) [6], Sun and Liu (2003) [10], Ye (2000) [11], Zhang and Zeng (2004) [12]. In addition, in this paper we also consider metric spaces which may not be trees but have open subsets U such that the closures are trees. Maps of such metric spaces which have chain movable fixed points are discussed. 相似文献
12.
令T: X→ X是紧度量空间 (X,d)上的连续映射.该文给出了 T的拓扑压和 T在非游荡集上的限制的拓扑压相等的不依赖于变分原理的一个直接证明.同时,还讨论了半共轭的两个系统的拓扑压之间的关系,证明了拓扑压在一致有限对一条件下是半共轭不变量. 相似文献
13.
本文讨论了树映射的单侧γ-极限点集与吸引中心的关系,得到了树映射具有正拓扑熵的几个等价条件.此外,还得到了树映射是强非混沌以及逐片单调树映射的拓扑熵为零的几个等价条件. 相似文献
14.
In this paper we introduce an algorithm which allows us to compute the topological entropy of a class of piecewise monotone continuous interval maps. The algorithm can be applied to a class of economic models called duopolies, and it can be useful to compute the topological entropy of periodic sequences of continuous maps which have been used in some population growth models. 相似文献
16.
This paper presents rigorous arguments on existence of chaos and an estimate of topological entropy in a simple power system by means of topological horseshoe theory and computer computations. 相似文献
17.
设f是个端点数为n的树T上的连续自映射.本文得到了f的单侧不稳定流形与拓扑熵的关系,并证明了:(1)如果x∈i=0∞fi(Ω(f))-P(f),那么,x的轨道是无限的;(2)如果f有一组可循环的不动点,那么h(f)≥In2(n-1). 相似文献
18.
For a pair of points x, y in a compact, Riemannian manifold M let n
t
( x, y) (resp. s
t
( x, y)) be the number of geodesic segments with length ≤ t joining these points (resp. the minimal number of point obstacles needed to block these geodesic segments). We study relationships
between the growth rates of n
t
( x, y) and s
t
( x, y) as t → ∞. We obtain lower bounds on s
t
( x, y) in terms of the topological entropy h( M) and the fundamental group π
1( M). For instance, we show that if h( M) > 0 then s
t
grows exponentially, with the rate at least h( M)/2. This strengthens earlier results on blocking of geodesics (Burns and Gutkin Discrete Contin Dyn Syst 21:403–413, 2008;
Lafont and Schmidt Geom Topol 11:867–887, 2007), and puts them in a new perspective.
相似文献
19.
Recently, He et al. [On quasi-weakly almost periodic points. Sci. China Math., 56, 597–606(2013)] constructed two binary sub-shifts to solve an open problem posed by Zhou and Feng in[Twelve open problems on the exact value of the Hausdorff measure and on topological entropy: A brief survey of recent results. Nonlinearity, 17, 493–502(2004)]. In this paper, we study more dynamical properties of those two binary sub-shifts. We show that the first one has zero topological entropy and is transitive but not weakly mixing, while the second one has positive topological entropy and is strongly mixing. 相似文献
20.
Given a topological system on a -compact Hausdorff space and its factor we show the existence of a largest topological factor containing such that for each -invariant measure , . When a relative variational principle holds, . 相似文献
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