Uniform Convergence and Transitive Subsets

Abstract

Let (𝑋,𝑑) be a metric space and a sequence of continuous maps 𝑓_𝑛∶𝑋→𝑋 that converges uniformly to a map 𝑓. We investigate the transitive subsets of 𝑓_𝑛 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 𝑓_𝑛 that can be inherited by 𝑓 if 𝑓_𝑛 converges uniformly strongly to 𝑓.

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,…,𝑉_𝑘 of 𝐴 and nonempty open subsets 𝑈₁,𝑈₂,…,𝑈_𝑘 of 𝑋 with 𝐴∩𝑈_𝑖≠∅, 𝑖=1,2,…,𝑘, there exists a 𝑚∈ℕ such that 𝑓^𝑚(𝑉_𝑖)∩𝑈_𝑖≠∅ 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 𝑓_𝑛∶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 𝑓_𝑛∶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,𝑓_𝑛) for every 𝑛∈ℕ when a sequence of continuous maps 𝑓_𝑛 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,𝑓_𝑛) 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 𝑓^𝑛(𝑈)∩𝑉≠∅. For a topological dynamical system (𝑋,𝑓), the orbit of 𝑥 is the set orb(𝑥,𝑓)={𝑓^𝑛(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 subset 𝐴 is called a transitive subset of (𝑋,𝑓) if for any choice of nonempty open subset 𝑉^𝐴 of 𝐴 and nonempty open subset 𝑈 of 𝑋 with 𝐴∩𝑈≠∅, there exists a 𝑛∈ℕ such that 𝑓^𝑛(𝑉^𝐴)∩𝑈≠∅.

Remark 2.2. (1) By Definition 2.1, (𝑋,𝑓) is transitive if and only if 𝑋 is a transitive subset of (𝑋,𝑓).
(2) If 𝑎∈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 that 𝐵 is dense in 𝐴 if and only if 𝑉^𝐴∩𝐵≠∅ for any nonempty open set 𝑉^𝐴 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)Let 𝑉^𝐴 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 𝑉^𝐴∩𝑓^−𝑛(𝑈)≠∅. (3)Let 𝑈 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 𝑉^𝐴 of 𝐴 and nonempty open set 𝑈 of 𝑋 with 𝐴∩𝑈≠∅, there exists 𝑛∈ℕ such that 𝑓^𝑛(𝑉^𝐴)∩𝑈≠∅. Since 𝑓^𝑛(𝑉^𝐴∩𝑓^−𝑛(𝑈))=𝑓^𝑛(𝑉^𝐴)∩𝑈, it follows that 𝑉^𝐴∩𝑓^−𝑛(𝑈)≠∅.
(2)⇒(3) Let 𝑉^𝐴 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 𝑉^𝐴∩𝑓^−𝑛(𝑈)≠∅. Furthermore, 𝑉^𝐴∩_𝑛∈ℕ𝑓^−𝑛(𝑈)=_𝑛∈ℕ𝑉^𝐴∩𝑓^−𝑛(𝑈)≠∅. (2.1)
Hence, ⋃_𝑛∈ℕ𝑓^−𝑛(𝑈) is dense in 𝐴.
(3)⇒(1) Let 𝑉^𝐴 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 𝑉^𝐴∩⋃_𝑛∈ℕ𝑓^−𝑛(𝑈)≠∅. Hence, there exists 𝑛∈ℕ such that 𝑉^𝐴∩𝑓^−𝑛(𝑈)≠∅. Moreover, 𝑓^𝑛(𝑉^𝐴∩(𝑓^−𝑛(𝑈))=𝑓^𝑛(𝑉^𝐴)∩𝑈, which implies 𝑓^𝑛(𝑉^𝐴)∩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 𝑓_𝑛∶𝑋→𝑋, for each 𝑛∈ℕ. {𝑓_𝑛∶𝑛∈ℕ} is said to converge strongly uniformly to 𝑓 if for any 𝜀>0, there exists 𝑛₀∈ℕ such that for any 𝑥∈𝑋, 𝑙∈ℕ and 𝑛≥𝑛₀ satisfying 𝑑𝑓_𝑛^𝑙(𝑥),𝑓^𝑙(𝑥)<𝜀. (2.2) If {𝑓_𝑛∶𝑛∈ℕ} converges strongly uniformly to 𝑓, {𝑓_𝑛∶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. Let 𝐼=[0,1]. Denote 𝐼^𝑗_𝑖=[𝑗−1/3^𝑖−1,𝑗/3^𝑖−1] for any 𝑖∈ℕ and 𝑗=1,2,…,3^𝑖−1. Let 𝑓^𝑗_𝑖∶𝐼^𝑗_𝑖→𝐼^𝑗_𝑖 satisfy𝑓^𝑗_𝑖(𝑥)=𝑗−13^𝑖−1+𝑓¹_𝑖𝑥−𝑗−13^𝑖−1forany𝑥∈𝐼^𝑗_𝑖,where (2.3)𝑓¹_𝑖(𝑥)=10,if0≤𝑥≤3^𝑖,13𝑥−3^𝑖−11,if3^𝑖2<𝑥<3^𝑖,13^𝑖−12,if3^𝑖1≤𝑥≤3^𝑖−1. (2.4) For any 𝑛∈ℕ, we define 𝑓_𝑛∶𝐼→𝐼 satisfying 𝑓_𝑛(𝑥)=𝑓^𝑗_𝑛(𝑥)forany𝑥∈𝐼^𝑗_𝑛and𝑗=1,2,…,3^𝑛−1. (2.5) Then it is easy to see that 𝑓_𝑛∶𝐼→𝐼 is a continuous map for each 𝑛∈ℕ and 𝑓_𝑛 converges strongly uniformly to 𝑖𝑑_𝐼, the identity on 𝐼.

3. Main Results

Let 𝐶(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)=sup_𝑥∈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 𝑓_𝑛∶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,𝑓_𝑛) for all 𝑛∈ℕ. Additionally, suppose that
(1) 𝑑_∞((𝑓_𝑛)^𝑛,𝑓^𝑛)→0 as 𝑛→∞,
(2) {(𝑓_𝑛)^𝑛(𝑥)∶𝑛∈ℕ} is dense in 𝐴, for some 𝑥∈𝑋.
Then 𝐴 is a transitive subset of (𝑋,𝑓).

Proof. Let 𝑉^𝐴 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 {(𝑓_𝑛)^𝑛(𝑥₀)∶𝑛∈ℕ} is dense in 𝐴. Furthermore, by condition (1) and 𝐴 is perfect, we obtain that the sequence {𝑓^𝑛(𝑥₀)∶𝑛∈ℕ} is also dense in 𝐴. Moreover, 𝑉^𝐴 is a nonempty open set of 𝐴; there exists 𝑘∈ℕ such that 𝑧=𝑓^𝑘(𝑥₀)∈𝑉^𝐴. Let 𝐺=(𝑈∩𝐴)⧵{𝑓(𝑥₀),𝑓²(𝑥₀),…,𝑓^𝑘(𝑥₀)}. Then 𝐺 is a nonempty open set of 𝐴. Since 𝐴 is a perfect metric space and {𝑓^𝑛(𝑥₀)∶𝑛∈ℕ} is dense in 𝐴, there exists 𝑙>𝑘 such that 𝑓^𝑙(𝑥₀)∈𝐺⊆(𝑈∩𝐴). Hence, we have 𝑓^𝑙𝑥₀=𝑓^𝑙−𝑘𝑓^𝑘𝑥₀=𝑓^𝑙−𝑘(𝑧)∈𝑓^𝑙−𝑘𝑉^𝐴∩(𝑈∩𝐴). (3.1) Consequently, 𝑓^𝑙−𝑘(𝑉^𝐴)∩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 𝑓_𝑛∶𝑋→𝑋 that converges strongly uniformly to a map 𝑓 and 𝐴 is a transitive subset of dynamical systems (𝑋,𝑓_𝑛) for each 𝑛∈ℕ. Then 𝐴 is a transitive subset of (𝑋,𝑓).

Proof. Let 𝑉^𝐴 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≠∅.
Let 𝑊_𝑛=⋃^∞_𝑘=1(𝑓_𝑛)^−𝑘(𝑊) for each 𝑛∈ℕ. Since 𝐴 is a transitive subset of (𝑋,𝑓_𝑛) for each 𝑛∈ℕ, by Proposition 2.4, then 𝑊_𝑛 is an open set of 𝑋 and 𝑊_𝑛 is dense in 𝐴. We denote 𝑊^∞=⋂^∞_𝑛=1W_𝑛. By Baire theorem, 𝑊^∞ is dense in 𝐴. Furthermore, we have 𝑉^𝐴∩𝑊^∞≠∅. Take a point 𝑦₀∈𝑉^𝐴∩𝑊^∞. There exists 𝑘_𝑛∈ℕ such that 𝑦₀∈(𝑓_𝑛)^−𝑘𝑛(𝑊) for each 𝑛∈ℕ. Denote 𝑥_𝑛=(𝑓_𝑛)^𝑘𝑛(𝑦₀) for each 𝑛∈ℕ. Without loss of generality, we may assume lim_𝑛→∞𝑥_𝑛=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 {𝑓_𝑛∶𝑛∈ℕ} converges strongly uniformly to 𝑓 and lim_𝑛→∞𝑥_𝑛=𝑥, there exists 𝑛₀∈ℕ such that 𝑑𝑓_𝑛0^𝑘𝑛0𝑦₀,𝑓^𝑘𝑛0𝑦₀<𝛿2and𝑑𝑥_𝑛0,𝑥=𝑑𝑓_𝑛0^𝑘𝑛0𝑦₀<𝛿,𝑥2. (3.2)
It follows that 𝑑(𝑥,𝑓^𝑘𝑛0(𝑦₀))<𝛿, which implies 𝑓^𝑘𝑛0(𝑦₀)∈𝑈. Therefore, 𝑓^𝑘𝑛0(𝑉^𝐴)∩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 {𝑓_𝑛∶𝑛∈ℕ} converges uniformly to a map 𝑓 and 𝐴 is a transitive subset of (𝑋,𝑓_𝑛) for each 𝑛∈ℕ, then 𝐴 cannot be a transitive subset of (𝑋,𝑓).

Example 3.3 (see [4]). Let 𝑆¹ be the unit circle and 𝑇_𝜆∶𝑆¹→𝑆¹ a translation map such that𝑇_𝜆(𝜃)=𝜃+2𝜆𝜋,𝜆∈ℝ. (3.3) Let 𝜆 be an irrational number, 𝜆_𝑛=𝜆/𝑛, and 𝑇_𝑛=𝑇_𝜆𝑛∶𝑆¹→𝑆¹ such that 𝑇_𝑛(𝜃)=𝜃+(2𝜆/𝑛)𝜋. Let maps sequence {𝑇_𝑛∶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 if 𝜆=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 𝑇_𝜆 is topologically transitive on 𝑆¹. Therefore, 𝑆¹ is a transitive subset of (𝑆¹,𝑇_𝜆). Since 𝜆_𝑛=𝜆/𝑛 is an irrational number for each 𝑛∈ℕ, then 𝑇_𝑛=𝑇_𝜆𝑛∶𝑆¹→𝑆¹ is topologically transitive for each 𝑛∈ℕ, which implies 𝑆¹ is a transitive subset of (𝑆¹,𝑇_𝑛) for each 𝑛∈ℕ. Moreover, maps sequence {𝑇_𝜆𝑛∶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).
Let 𝑓_𝑛∶𝑋→𝑋 be a continuous map for each 𝑛∈ℕ, and maps sequence {𝑓_𝑛∶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,𝑓_𝑘).

Example 3.4. Let𝑓_𝑛(𝑥)=2𝑛𝑛−2𝑥,if0≤𝑥≤𝑛−2,2𝑛1,if𝑛−22𝑛≤𝑥≤𝑛+22𝑛,𝑛=3,4,…2𝑛()𝑛−21−𝑥,if𝑛+22𝑛≤𝑥≤1. (3.4) Observe that the given sequence converges uniformly to tent map 𝑓(𝑥)=12𝑥,if0≤𝑥≤2,2()11−𝑥,if2≤𝑥≤1, (3.5) 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).
Let 𝑆(𝑓^𝑘) denote the set of extreme value points of 𝑓^𝑘 for every 𝑘∈ℕ; then 𝑆(𝑓^𝑘)={1/2^𝑘,2/2^𝑘,…,(2^𝑘−1)/2^𝑘}. Since 𝑆(𝑓)={1/2}, 𝑓(1/2)=1, 𝑓(0)=0, and 𝑓(1)=0, we have 𝑓^𝑘(𝑥)=11,if𝑥=2^𝑘,32^𝑘2,…,^𝑘−12^𝑘,20,if𝑥=0,2^𝑘,42^𝑘2,…,^𝑘−22^𝑘,1. (3.6)
Let 𝐼^𝑗_𝑘=[𝑗/2^𝑘,(𝑗+1)/2^𝑘] for 0≤𝑗≤2^𝑘−1. Then 𝑓^𝑘(𝐼^𝑗_𝑘)=[0,1]. For any nonempty open set 𝑈 of [1/4,3/4]. Without loss of generality, we take 𝑈=(𝑥₀−𝜀,𝑥₀+𝜀) for a given 𝜀>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/𝜀), then there exists 𝑗∈ℕ and 0≤𝑗≤2^𝑙−1 such that 𝐼^𝑗_𝑙⊆𝑈. Furthermore, we have 𝑓^𝑙(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 𝑓^𝑘(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 1 

Figure 2 

Acknowledgments

The authors would like to thank the referees for many valuable and constructive comments and suggestions for improving this paper. This work was supported by the Natural Science Foundation of Henan Province (092300410148), China.