# Uniform Convergence and Transitive Subsets

Authors: Lei Liu; Shuli Zhao; Hongliang Liang
Journal: Discrete Dynamics in Nature and Society
 (2012)

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## Abstract

Let(X,d) be a metric space and a sequence of continuous maps fn:X→X that converges uniformly to a map f. We investigate the transitive subsets of fn whether they can be inherited by f or not. We give sufficient conditions such that the limit map f has a transitive subset. In particular, we show the transitive subsets of fn that can be inherited by f if fn converges uniformly strongly to f.


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## Body

## 1. Introduction

A topological dynamical system is a pair(X,f), where X is a compact metric space with metric d and f:X→X is a continuous map. When X is finite, it is a discrete space and there is no any nontrivial convergence. Hence, we assume that X contains infinitely many points. Define ℕ by the set of all positive integers.In [1], Blanchard and Huang introduced the concepts of weakly mixing subset and partial weak mixing, derived from a result given by Xiong and Yang [2] and showed “partial weak mixing implies Li-Yorke chaos” and “Li-Yorke chaos does not imply partial weak mixing”. A closed set A with at least two elements is said to be weakly mixing if for any k∈ℕ, any choice of nonempty open subsets V1,V2,…,Vk of A and nonempty open subsets U1,U2,…,Uk of X with A∩Ui≠∅, i=1,2,…,k, there exists a m∈ℕ such that fm(Vi)∩Ui≠∅ for 1≤i≤k. A topological dynamical system (X,f) is called partial weak mixing if X contains a weakly mixing subset. Motivated by the idea of Blanchard and Huang’s notion of “weakly mixing subset”, Oprocha and Zhang [3] extended the notion of weakly mixing subset and gave the concept of “transitive subset” and discussed its basic properties.It is a well-known fact that if a sequence of continuous maps converges uniformly, then the uniform limit map is continuous. Abu-Saris and Al-Hami [4] studied uniform convergence and chaotic behavior. Later Abu-Saris et al. [5] pointed out some wrong claims in [4] and corrected them. Román-Flores [6] gave sufficient conditions for the topological transitivity of uniform limit map f:X→X of a sequence of continuous maps fn:X→X, where X is a compact metric space. Fedeli and Le Donne [7] studied the dynamical behavior of the uniform limit of a sequence of continuous self-maps on a compact metric space satisfying topological transitivity or other related properties and gave some conditions for the transitivity of a limit. Bhaumik and Choudhury [8] investigated the chaotic behavior of the uniform limit map f:I→I of a sequence of continuous topologically transitive maps fn:I→I, where I is a compact interval. Recently, Yan, Zeng, and Zhang et al. [9] studied transitivity and sensitive dependence on initial conditions for uniform limits.In this paper, motivated by the idea of Román-Flores [6], we give sufficient conditions such that the limit map f has a transitive subset. In particular, we prove that A is a transitive subset of (X,f) if A is a transitive subset of (X,fn) for every n∈ℕ when a sequence of continuous maps fn converges strongly uniformly to a map f, where (X,d) is a compact metric space. Moreover, we give an example to show that if A is a transitive subset of (X,f), then A cannot be a transitive subset of (X,fn) for some n∈ℕ.


## 2. Preliminaries

Topological transitivity (see [10–12]) are global characteristic of topological dynamical systems. Let (X,f) be a topological dynamical system. (X,f) is topologically transitive if for any nonempty open subsets U and V of X there exists a n∈ℕ such that fn(U)∩V≠∅. For a topological dynamical system (X,f), the orbit of x is the set orb(x,f)={fn(x):n∈ℕ} for every x∈X. (X,f) is point transitive if there exists a point x0∈X with dense orbit, that is, orb(x0,f)¯=X. Such a point x0 is called a transitive point of (X,f). By [13], if X is a compact metric space without isolated points, then the topologically transitive and point transitive are equivalent.Definition 2.1 (see[3]).
A closed subsetA is called a transitive subset of (X,f) if for any choice of nonempty open subset VA of A and nonempty open subset U of X with A∩U≠∅, there exists a n∈ℕ such that fn(VA)∩U≠∅.Remark 2.2.
(1) By Definition2.1, (X,f) is transitive if and only if X is a transitive subset of (X,f).
(2) Ifa∈X is a transitive point of (X,f), then {a} is a transitive subset of (X,f).Definition 2.3 (see[14]).
Let(X,τ) be a topological space. A and B are two nonempty subsets of X. B is dense in A if A⊆A∩B¯.In fact, we easily prove thatB is dense in A if and only if VA∩B≠∅ for any nonempty open set VA of A.Proposition 2.4.
Let(X,f) be a topological dynamical system and A be a nonempty closed set of X. Then the following conditions are equivalent.(1)
A is a transitive subset of (X,f).(2)
LetVA be a nonempty open subset of A and U a nonempty open subset of X with A∩U≠∅. Then there exists n∈ℕ such that VA∩f-n(U)≠∅.(3)
LetU be a nonempty open set of X with U∩A≠∅. Then ⋃n∈ℕf-n(U) is dense in A.Proof.
(1)⇒(2) Let A be a transitive subset of (X,f). Then for any choice of nonempty open set VA of A and nonempty open set U of X with A∩U≠∅, there exists n∈ℕ such that fn(VA)∩U≠∅. Since fn(VA∩f-n(U))=fn(VA)∩U, it follows that VA∩f-n(U)≠∅.
(2)⇒(3) Let VA be a nonempty open set of A and U be a nonempty open set of X with A∩U≠∅. By the assumption of (2), there exists n∈ℕ such that VA∩f-n(U)≠∅. Furthermore,
(2.1)VA∩⋃n∈Nf-n(U)=⋃n∈N(VA∩f-n(U))≠∅.
Hence,⋃n∈ℕf-n(U) is dense in A.
(3)⇒(1) Let VA be a nonempty open set of A and U a nonempty open set of X with A∩U≠∅. Since ⋃n∈ℕf-n(U) is dense in A, it follows that VA∩⋃n∈ℕf-n(U)≠∅. Hence, there exists n∈ℕ such that VA∩f-n(U)≠∅. Moreover, fn(VA∩(f-n(U))=fn(VA)∩U, which implies fn(VA)∩U≠∅. Therefore, A is a transitive subset of (X,f).Definition 2.5.
Let(X,d) be a metric space and a sequence of continuous maps fn:X→X, for each n∈ℕ. {fn:n∈ℕ} is said to converge strongly uniformly to f if for any ɛ>0, there exists n0∈ℕ such that for any x∈X, l∈ℕ and n≥n0 satisfying
(2.2)d((fn)l(x),fl(x))<ɛ.
If {fn:n∈ℕ} converges strongly uniformly to f, {fn:n∈ℕ} is called a strong uniform convergent sequence.The following example is from [9, 15]; we show that the example is a strong uniformly convergence example.Example 2.6.
LetI=[0,1]. DenoteIij=[j-1/3i-1,j/3i-1]for anyi∈ℕandj=1,2,…,3i-1. Letfij:Iij→Iijsatisfy(2.3)fij(x)=j-13i-1+fi1(x-j-13i-1)foranyx∈Iij,where(2.4)fi1(x)={0,if0≤x≤13i,3x-13i-1,if13i<x<23i,13i-1,if23i≤x≤13i-1.
For any n∈ℕ, we define fn:I→I satisfying
(2.5)fn(x)=fnj(x)foranyx∈Injandj=1,2,…,3n-1.
Then it is easy to see that fn:I→I is a continuous map for each n∈ℕ and fn converges strongly uniformly to idI, the identity on I.


## 3. Main Results

LetC(X,X) denote the set of continuous maps f:X→X. In the sequel, as in usual, d∞(f,g) denotes the uniform metric on C(X,X), that is, d∞(f,g)=supx∈Xd(f(x),g(x)). A topological space X is perfect if X is closed and has no isolated points. Clearly, if X is a perfect space, then any nonempty open set U of X has no isolated points.From the idea of Román-Flores [6], we obtain the following theorem.Theorem 3.1.
Let(X,d) be a compact metric space and a sequence of continuous maps fn:X→X that converges uniformly to a map f. Assume that A is a perfect set of X and A is a transitive subset of (X,fn) for all n∈ℕ. Additionally, suppose that
(1)d∞((fn)n,fn)→0 as n→∞,
(2){(fn)n(x):n∈ℕ} is dense in A, for some x∈X.
ThenA is a transitive subset of (X,f).Proof.
LetVA be a nonempty open set of A and U a nonempty open set of X with A∩U≠∅. Since condition (2), there exists x0∈X such that {(fn)n(x0):n∈ℕ} is dense in A. Furthermore, by condition (1) and A is perfect, we obtain that the sequence {fn(x0):n∈ℕ} is also dense in A. Moreover, VA is a nonempty open set of A; there exists k∈ℕ such that z=fk(x0)∈VA. Let G=(U∩A)∖{f(x0),f2(x0),…,fk(x0)}. Then G is a nonempty open set of A. Since A is a perfect metric space and {fn(x0):n∈ℕ} is dense in A, there exists l>k such that fl(x0)∈G⊆(U∩A). Hence, we have
(3.1)fl(x0)=fl-k(fk(x0))=fl-k(z)∈fl-k(VA)∩(U∩A).
Consequently, fl-k(VA)∩U≠∅. Therefore, A is a transitive subset of (X,f).Theorem 3.2.
Let(X,d) be a compact metric space. Assume a sequence of continuous maps fn:X→X that converges strongly uniformly to a map f and A is a transitive subset of dynamical systems (X,fn) for each n∈ℕ. Then A is a transitive subset of (X,f).Proof.
LetVA be a nonempty open set of A and U a nonempty open set of X with A∩U≠∅. Since X is a compact metric space and A∩U≠∅, there exists a nonempty open set W of X such that W¯⊆U and W∩A≠∅.
LetWn=⋃k=1∞(fn)-k(W) for each n∈ℕ. Since A is a transitive subset of (X,fn) for each n∈ℕ, by Proposition 2.4, then Wn is an open set of X and Wn is dense in A. We denote W∞=⋂n=1∞Wn. By Baire theorem, W∞ is dense in A. Furthermore, we have VA∩W∞≠∅. Take a point y0∈VA∩W∞. There exists kn∈ℕ such that y0∈(fn)-kn(W) for each n∈ℕ. Denote xn=(fn)kn(y0) for each n∈ℕ. Without loss of generality, we may assume limn→∞xn=x∈W¯ because X is a compact metric space. Choose a δ>0 such that B(x,δ)={y∈X:d(x,y)<δ}⊆U. Since maps sequence {fn:n∈ℕ} converges strongly uniformly to f and limn→∞xn=x, there exists n0∈ℕ such that
(3.2)d((fn0)kn0(y0),fkn0(y0))<δ2andd(xn0,x)=d((fn0)kn0(y0),x)<δ2.
It follows thatd(x,fkn0(y0))<δ, which implies fkn0(y0)∈U. Therefore, fkn0(VA)∩U≠∅. This shows that A is a transitive subset of (X,f).The following example is from [4]. We give the example which shows if maps sequence {fn:n∈ℕ} converges uniformly to a map f and A is a transitive subset of (X,fn) for each n∈ℕ, then A cannot be a transitive subset of (X,f).Example 3.3 (see [4]).
LetS1be the unit circle andTλ:S1→S1a translation map such that(3.3)Tλ(θ)=θ+2λπ,λ∈R.
Let λ be an irrational number, λn=λ/n, and Tn=Tλn:S1→S1 such that Tn(θ)=θ+(2λ/n)π. Let maps sequence {Tn:n∈ℕ} converge uniformly to a map T0. Then T0 is not topologically transitive on S1; that is, S1 is not a transitive subset of dynamical system (S1,T0).
It is well known that ifλ=q/p is a rational number, then all points are periodic of period q, and so the set of periodic points is, obviously, dense in S1. Moreover, by Jacobi’s Theorem [16], if λ is an irrational number, then Tλ is topologically transitive on S1. Therefore, S1 is a transitive subset of (S1,Tλ). Since λn=λ/n is an irrational number for each n∈ℕ, then Tn=Tλn:S1→S1 is topologically transitive for each n∈ℕ, which implies S1 is a transitive subset of (S1,Tn) for each n∈ℕ. Moreover, maps sequence {Tλn:n∈ℕ} converges uniformly to a map T0=id, where id is identity map. Therefore, T0 is not topologically transitive on S1, which implies S1 is not a transitive subset of (S1,T0).
Letfn:X→X be a continuous map for each n∈ℕ, and maps sequence {fn:n∈ℕ} converges uniformly to a map f. The following example shows that A is a transitive subset of (X,f), but there exists k∈ℕ such that A is not a transitive subset of (X,fk).Example 3.4.
Let(3.4)fn(x)={2nn-2x,if0≤x≤n-22n,1,ifn-22n≤x≤n+22n,n=3,4,…2nn-2(1-x),ifn+22n≤x≤1.
Observe that the given sequence converges uniformly to tent map
(3.5)f(x)={2x,if0≤x≤12,2(1-x),if12≤x≤1,
Figures 1 and 2, which is known to be topologically transitive on I=[0,1] (see [16]). We will prove that [1/4,3/4] is a transitive subset of (X,f).
LetS(fk) denote the set of extreme value points of fk for every k∈ℕ; then S(fk)={1/2k,2/2k,…,(2k-1)/2k}. Since S(f)={1/2}, f(1/2)=1, f(0)=0, and f(1)=0, we have
(3.6)fk(x)={1,ifx=12k,32k,…,2k-12k,0,ifx=0,22k,42k,…,2k-22k,1.
LetIkj=[j/2k,(j+1)/2k] for 0≤j≤2k-1. Then fk(Ikj)=[0,1]. For any nonempty open set U of [1/4,3/4]. Without loss of generality, we take U=(x0-ɛ,x0+ɛ) for a given ɛ>0 and x0∈int[1/4,3/4], where int[1/4,3/4] denotes the interior of [1/4,3/4]. When l∈ℕ and l>log2(1/ɛ), then there exists j∈ℕ and 0≤j≤2l-1 such that Ilj⊆U. Furthermore, we have fl(U)=[0,1]. Thus, for any nonempty open set U of [1/4,3/4] and nonempty open set V of [0,1] with V∩[1/4,3/4]≠∅, there exists k∈ℕ such that fk(U)∩V≠∅. This shows that [1/4,3/4] is a transitive subset of (I,f). Moreover, f4(x)=1 and (f4)n(x)=0(n≥2) for all x∈[1/4,3/4], which implies that [1/4,3/4] is not a transitive subset of (I,f4).Figure 1Figure 2



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