Stronger Forms of Sensitivity for Measure-Preserving Maps and Semiflows on Probability Spaces

Abstract

This paper is concerned with some stronger forms of sensitivity for measure-preserving maps and semiflows on probability spaces. A new form of sensitivity is introduced, called ergodic sensitivity. It is shown that, on a metric probability space with a fully supported measure, if a measure-preserving map is weak mixing, then it is ergodically sensitive and multisensitive; and if it is strong mixing, then it is cofinitely sensitive, where it is not required that the map is continuous and the space is compact. Similar results for measure-preserving semiflows are obtained, where it is required in a result about ergodic sensitivity that the space is compact in some sense and the semiflow is continuous. In addition, relationships between some sensitive properties of a map and its iterations are discussed, including syndetic sensitivity, cofinite sensitivity, ergodic sensitivity as well as usual sensitivity, 𝑛-sensitivity, and multisensitivity. Moreover, it is shown that multisensitivity, cofinite sensitivity, and ergodic sensitivity can be lifted up by a semiopen factor map.

1. Introduction

One of the most interesting characteristics of a dynamical system is when orbits of nearby points deviate after finite steps. This is also one of the important features of chaotic dynamical behaviors. It is termed as sensitive dependence on initial conditions (briefly, sensitivity). Sensitivity is a key notion when studying the complexity of a dynamical system. So, it is very important to study what systems have sensitive dependence. This problem has gained much attention recently (see [1–14]).

In [1], Abraham et al. proved that if a measure-preserving map 𝑇 on a metric probability space (𝑋,𝑑,ℬ(𝑋),𝜇) with supp𝜇=𝑋 is either topologically mixing or weak mixing and satisfies that for any nonempty open set 𝑈⊂𝑋 and there exists a subsequence {𝑛_𝑘} with positive upper density such that 𝑈_𝑘≥0𝑇^−𝑛𝑘𝑈≠∅, (1) then 𝑇 is sensitive. In the same paper, they proved that if 𝑇 is strong mixing and sup𝜇=𝑋, then it is sensitive; and if 𝑇 is an exact endomorphism and sup𝜇=𝑋, then it is cofinitely sensitive. He et al. [8] showed that if a measure-preserving map 𝑇 (resp., a measure-preserving semiflow 𝜑) on (𝑋,𝑑,ℬ(𝑋),𝜇) with supp𝜇=𝑋 is weak mixing, then it is sensitive. In addition, if 𝑋 is a nontrivial metric space (i.e., a metric space is not reduced to a single point) and a map 𝑓 on 𝑋 is topologically mixing, then 𝑓 is sensitive [7, Proposition  7.2.14].

There are several ways to extend this notion. Here, we only list the following three ways:(1)one may define 𝑛-sensitivity as it was done by Nemiskii and Stepanov in [15] and Ye and Zhang in [16];(2)one may require that in any open subset 𝑈 there is a pair (𝑥;𝑦) which is a Li-Yorke pair as Akin and Kolyada in [17] did (see also recent work by Li et al. in [18], where a stronger form of sensitivity is defined);(3)the third way is what we now consider in the present paper, that is, study 𝑁_𝑓(𝑉,𝛿).Previously, the third way was considered by several scholars. More recently, Moothathu [12] studied continuous self-maps on compact metric spaces and initiated a preliminary study of stronger forms of sensitivity, including syndetic sensitivity, cofinite sensitivity, and multisensitivity. In particular, he showed that any syndetically transitive and nonminimal map is syndetically sensitive. This improves the result that if a continuous map is topologically transitive and has a dense set of periodic points in an infinite metric space, then it is sensitive [3]. Xiong [14] introduced the concept of 𝑛-sensitivity for continuous self-maps of a complete metric space. Later, Shao et al. [13] investigated some properties of 𝑛-sensitivity of continuous and surjective maps on a compact metric space. James et al. [10] introduced a notion, called measurable sensitivity and showed that a totally ergodic and measurably sensitive map is weakly mixing. More recently, Huang et al. [9] introduced the concepts of 𝜇-sensitivity, 𝑛-sensitivity for 𝜇, 𝜇-complexity, and 𝜇-equicontinuity for a measure-preserving and continuous map on a metric probability space (𝑋,𝑑,ℬ(𝑋),𝜇) and presented a sufficient condition for 𝑛-sensitivity for 𝜇, where 𝑋 is a compact metric space. They proved that 𝜇-sensitivity is equivalent to pairwise sensitivity defined by Cadre and Jacob in [4].

In this paper, we introduce a new and stronger form of sensitivity, ergodic sensitivity, and present several sufficient conditions for multisensitivity, cofinite sensitivity, and ergodic sensitivity of measure-preserving maps and semiflows, where it is not required that maps and semiflows are continuous and spaces are compact. We show that, for a measure-preserving map on a metric probability space with a fully supported measure, if it is weak mixing, then it is ergodically sensitive and multisensitive; and if it is strong mixing, then it is cofinitely sensitive. Related problems for measure-preserving semiflow are also discussed. In addition, we consider the relationships between five forms of sensitivity (i.e., sensitivity, multisensitivity, cofinite sensitivity, syndetic sensitivity, ergodic sensitivity, and 𝑛-sensitivity) of a map 𝑓 and its iterations 𝑓^𝑚 for 𝑚≥2.

The rest of this paper is organized as follows. In Section 2, we recall some basic concepts and lemmas and introduce a new and stronger form of sensitivity, called ergodic sensitivity. In Section 3, we give several sufficient conditions for multisensitivity, cofinite sensitivity, and ergodic sensitivity. Finally, we discuss the relationships between five forms of sensitivity of a map and its iterations in Section 4.

2. Preliminaries

In this section, we first introduce some notations and basic concepts, including a new and stronger form of sensitivity, called ergodic sensitivity, and then give two useful lemmas.

By 𝐍 denote the set of all positive integers. Denote 𝐙⁺∶={0,1,…}, 𝐑⁺∶=[0,+∞), and 𝐍_𝑛∶={0,1,…,𝑛−1}. We will use |𝐴| to denote the cardinality of a set 𝐴.

We refer to [12, 19, 20] for the following basic concepts. Let (𝑋,𝑑) be a metric space, ℬ(𝑋) the sigma-algebra of Borel subsets of 𝑋, and 𝜇 a probability measure on (𝑋,ℬ(𝑋)). Then the space 𝑋 is called to be a metric probability space, denoted by the quadruple (𝑋,𝑑,ℬ(𝑋),𝜇), which is often briefly denoted by the triple (𝑋,ℬ(𝑋),𝜇).

A measurable map 𝑇 is called measure-preserving on (𝑋,ℬ(𝑋),𝜇) if 𝜇(𝐵)=𝜇(𝑇⁻¹(𝐵)) for any 𝐵∈ℬ(𝑋). A measurable semiflow 𝜑 is called measure-preserving on (𝑋,ℬ(𝑋),𝜇) if 𝜇(𝐵)=𝜇(𝜑_𝑡⁻¹(𝐵)) for any 𝐵∈ℬ(𝑋) and for any 𝑡∈𝐑⁺.

The following concepts are about mixing properties of maps and semiflows in the measure-theoretical sense.

Definition 1. (i) A measure-preserving map 𝑇 on (𝑋,ℬ(𝑋),𝜇) is called weak mixing and strong mixing if, for any 𝐴,𝐵∈ℬ(𝑋), the following two equalities hold, respectively: lim_𝑛→∞1𝑛^𝑛−1_𝑖=0𝜇𝐴𝑇^−𝑖(𝐵)−𝜇(𝐴)𝜇(𝐵)=0,lim_𝑛→∞𝜇𝐴𝑇^−𝑛(𝐵)=𝜇(𝐴)𝜇(𝐵). (2)
(ii) A measure-preserving semiflow 𝜑 on (𝑋,ℬ(𝑋),𝜇) is called weak mixing and strong mixing if, for any 𝐴,𝐵∈ℬ(𝑋), the following two equalities hold, respectively: lim_𝑡→∞1𝑡^𝑡₀𝜇𝐴𝜑_𝑠⁻¹(𝐵)−𝜇(𝐴)𝜇(𝐵)𝑑𝑠=0,lim_𝑡→∞𝜇𝐴𝜑_𝑡⁻¹(𝐵)=𝜇(𝐴)𝜇(𝐵). (3)

The following concepts describe three different forms of transitivity of a map 𝑓∶𝑋→𝑋 and a semiflow 𝜑∶𝐑⁺×𝑋→𝑋 in the topological sense. For convenience, denote 𝑁_𝑓(𝑈,𝑉)∶=𝑛∈𝐍∶𝑇^𝑛(𝑈),𝑁∩𝑉≠∅_𝜑(𝑈,𝑉)∶=𝑡>0∶𝜑_𝑡(𝑈)∩𝑉≠∅ (4) for any sets 𝑈,𝑉⊂𝑋.

Definition 2. Let 𝑓∶𝑋→𝑋 be a map and 𝑋 a metric space.(i)The map 𝑓 is said to be topologically transitive and topologically mixing on 𝑋 if, for any pair of nonempty open sets 𝑈,𝑉⊂𝑋, the following conditions hold, respectively: 𝑁_𝑓(𝑈,𝑉)≠∅ and 𝑁_𝑓(𝑈,𝑉)⊃{𝑘,𝑘+1,…} for some integer 𝑘≥1.(ii)The map 𝑓 is said to be topologically weakly mixing on 𝑋 if 𝑓×𝑓 is topologically transitive on the product space 𝑋×𝑋.

Clearly, topological mixing is stronger than topologically weak mixing, and topologically weak mixing is stronger than topological transitivity. There are other two different forms of transitivity: syndetic transitivity [15] and topological ergodicity [21], which are not considered in the present paper.

Their corresponding concepts to semiflows are given as follows.

Definition 3. Let 𝜑∶𝐑⁺×𝑋→𝑋 be a semiflow and 𝑋 a metric space.(i)The semiflow 𝜑 is said to be topologically transitive and topologically mixing on 𝑋 if, for any pair of nonempty open sets 𝑈,𝑉⊂𝑋, the following conditions hold, respectively: 𝑁_𝜑(𝑈,𝑉)≠∅ and 𝑁_𝜑(𝑈,𝑉)⊃[𝐿,+∞) for some constant 𝐿>0.(ii)A semiflow 𝜑 is said to be topologically weakly mixing on 𝑋 if 𝜑×𝜑 is topologically transitive on the product space 𝑋×𝑋.

Let 𝑆 be a subset of 𝐙⁺ (resp., a Lebesgue measurable subset of 𝐑⁺). Its upper and lower densities are defined, respectively, by 𝑑(𝑆)∶=limsup_𝑘→∞1𝑘𝑆𝑁_𝑘,𝑑(𝑆)∶=liminf_𝑘→∞1𝑘𝑆𝑁_𝑘 (5) (resp., limsup_𝑡→∞(1/𝑡)𝑙(𝑆∩[0,𝑡]) and liminf_𝑡→∞(1/𝑡)𝑙(𝑆∩[0,𝑡]), where 𝑙(𝑆) is the Lebesgue measure of 𝑆 [8]), and its density is defined by 𝑑(𝑆)∶=lim_𝑘→∞1𝑘𝑆𝑁_𝑘 (6) (resp., lim_𝑡→∞(1/𝑡)𝑙(𝑆∩[0,𝑡])) and if it exists.

According to the classical definition, a map 𝑓 (resp., a semiflow 𝜑) is sensitive in 𝑋 if there is a constant 𝛿>0 such that, for any 𝑥∈𝑋 and any open neighborhood 𝑉_𝑥 of 𝑥, there is 𝑛∈𝐙⁺ (resp., 𝑡∈𝐑⁺) such that sup{𝑑(𝑓^𝑛(𝑥),𝑓^𝑛(𝑦))∶𝑦∈𝑉_𝑥}>𝛿 (resp., sup{𝑑(𝜑_𝑡(𝑥),𝜑_𝑡(𝑦))∶𝑦∈𝑉_𝑥}>𝛿), where 𝛿 is called a constant of sensitivity. Now, we write this in a slightly different way. For 𝑉⊂𝑋 and 𝛿>0, denote 𝑁_𝑓(𝑉,𝛿)∶=𝑛∈𝐙⁺𝑓∶thereexist𝑥,𝑦∈𝑉with𝑑^𝑛(𝑥),𝑓^𝑛𝑦,𝑁>𝛿_𝜑(𝑉,𝛿)∶=𝑡∈𝐑⁺𝜑∶thereexist𝑥,𝑦∈𝑉with𝑑_𝑡(𝑥),𝜑_𝑡𝑦>𝛿. (7) In terms of these notations, the above sensitivity properties can be equivalently defined as [12](1)𝑓 (resp., 𝜑) is sensitive in 𝑋 if there is a constant 𝛿>0 such that 𝑁_𝑓(𝑉,𝛿) (resp., 𝑁_𝜑(𝑉,𝛿)) is nonempty for any nonempty open set 𝑉⊂𝑋.In [12], Moothathu gave the following three stronger forms of sensitivity:(2)𝑓 (resp., 𝜑) is cofinitely sensitive in 𝑋 if there is a constant 𝛿>0 such that 𝑁_𝑓(𝑉,𝛿)⊃[𝑛,+∞)⋂𝐍 for some 𝑛≥1 (resp., 𝑁_𝜑(𝑉,𝛿)⊃[𝑡,+∞) for some 𝑡>0) for any nonempty open subset 𝑉⊂𝑋;(3)𝑓 (resp., 𝜑) is multisensitive in 𝑋 if there is a constant 𝛿>0 such that ⋂^𝑘_𝑖=1𝑁_𝑓(𝑉_𝑖,𝛿)≠∅ (resp., ⋂^𝑘_𝑖=1𝑁_𝜑(𝑉_𝑖,𝛿)≠∅) for each 𝑘≥1 and any nonempty open sets 𝑉₁,𝑉₂,…,𝑉_𝑘⊂𝑋;(4)𝑓 is syndetically sensitive in 𝑋 if there is a constant 𝛿>0 such that 𝑁_𝑓(𝑉,𝛿) is a syndetic set for any nonempty open set 𝑉⊂𝑋.Motivated by the idea in the definition of topological ergodicity introduced by Akin [21], we now introduce another stronger form of sensitivity as follows.(5)𝑓 (resp., 𝜑) is called to be ergodically sensitive in 𝑋 if there is a constant 𝛿>0 such that 𝑁_𝑓(𝑉,𝛿) (resp., 𝑁_𝜑(𝑉,𝛿)) has a positive upper density for any nonempty open subset 𝑉⊂𝑋.

For convenience, such a constant 𝛿 in the above definitions is called a sensitive constant of 𝑓 with respect to the corresponding sensitive forms.

Remark 4. In [12], it is required that the map is continuous and the space is compact in the definitions of the concepts in (9)–(25) as well as in the definitions of the three concepts given in Definition 2.

By the above definitions, it can be easily implied that cofinitelysensitive⟹syndeticallysensitive⟹ergodicallysensitive⟹sensitive. (8) So, cofinite sensitivity is the strongest one among the above five different forms of sensitivity.

Definition 5. Let (𝑋,𝑑) be a nontrivial metrics space and 𝑓∶𝑋→𝑋 a map. For a given integer 𝑛≥2, the system (𝑋,𝑓) or the map 𝑓 is said to be 𝑛-sensitive if there exists a constant 𝛿>0 such that, for any nonempty and open set 𝑈, there are distinct points 𝑥₁,𝑥₂,…,𝑥_𝑛∈𝑈 and some 𝑚∈𝑍⁺ satisfying that 𝑑(𝑓^𝑚(𝑥_𝑖),𝑓^𝑚(𝑥_𝑗))≥𝛿 for 1≤𝑖<𝑗≤𝑛. Such a constant 𝛿 is called an 𝑛-sensitive constant of 𝑓.

It is clear that 2-sensitivity is just sensitivity. For any given 𝑛≥2, there exists a minimal system, that is, 𝑛-sensitive, but not (𝑛+1)-sensitive (see [13]).

Remark 6. In the case that 𝑋 is a locally connected and compact nontrivial metric space, Shao et al. [13] showed that if a continuous and surjective map 𝑓∶𝑋→𝑋 is sensitive, then it is 𝑛-sensitive for all 𝑛≥2. Note that if the assumptions that 𝑋 is compact and 𝑓 is surjective are removed, then the result still holds. This can be easily verified by the method used in the proof of Proposition  4.1 in [14]. Consequently, if 𝑓 is multisensitive, then it is 𝑛-sensitive for each 𝑛≥2 in this case.

To end this section, we introduce two useful lemmas. A set 𝑆⊂𝐙⁺ (resp., 𝑆⊂𝐑⁺) is called relatively dense in 𝐙⁺ (resp., 𝐑⁺) if there is 𝑛∈𝐍 (resp., 𝐿>0) such that, for any 𝑘∈𝐙⁺ (resp., 𝑡∈𝐑⁺), we have 𝑆∩{𝑘,𝑘+1,…,𝑘+𝑛−1}≠∅ (resp., 𝑆∩(𝑡,𝑡+𝐿)≠∅). Clearly, the relative density of a set in 𝐙⁺ is equivalent to its syndedicity.

Lemma 7 (see [19]). Let 𝑇 be a measure-preserving map on (𝑋,ℬ(𝑋),𝜇). For any 𝐴∈ℬ(𝑋), if 𝜇(𝐴)>0, then, for any 𝜀>0, the set 𝑆={𝑘∈𝐍∶𝜇(𝐴⋂𝑇^−𝑘(𝐴))≥(𝜇(𝐴))²−𝜀} is relatively dense in 𝐍.

Lemma 8 (see [15]). Let 𝜑 be a measure-preserving semiflow on (𝑋,ℬ(𝑋),𝜇). For any 𝐴∈ℬ(𝑋), if 𝜇(𝐴)>0, then the set 𝑆={𝑡∈𝐑⁺∶𝜇(𝐴⋂𝜑_𝑡⁻¹(𝐴))>𝜆(𝜇(𝐴))²}(𝜆<1) is relatively dense in 𝐑⁺.

3. Sufficient Conditions for Multisensitivity, Cofinite Sensitivity, and Ergodic Sensitivity

In this section, we will give several sufficient conditions for multisensitivity, cofinite sensitivity, and ergodic sensitivity of measure-preserving maps and semiflows. This section is divided into three subsections.

3.1. Multisensitivity

In this subsection, we first show that multisensitivity can be lifted up by a semiopen factor map and then give a sufficient condition for multisensitivity of measure-preserving maps (resp., semiflows).

In [13], the authors proved that 𝑛-sensitivity can be lifted up by a semiopen factor map, where a map is called semiopen if the image of any nonempty open set contains a nonempty open subset. Now, we show that multisensitivity has the same property.

Let 𝑓∶𝑋→𝑋 and 𝑔∶𝑌→𝑌 be maps, where 𝑋 and 𝑌 are metric spaces. If there exists a continuous and surjective map 𝜋∶𝑋→𝑌 such that 𝜋∘𝑓=𝑔∘𝜋, then (𝑌,𝑔) is said to be a factor of the system (𝑋,𝑓), and (𝑋,𝑓) is said to be an extension of (𝑌,𝑔), while 𝜋 is said to be a factor map between (𝑋,𝑓) and (𝑌,𝑔).

Proposition 9. Let (𝑋,𝑑) and (𝑌,𝑒) be nontrivial metric spaces, let 𝑓∶𝑋→𝑋 and 𝑔∶𝑌→𝑌 be maps, and let 𝜋∶𝑋→𝑌 be a semiopen factor map between (𝑋,𝑓) and (𝑌,g). If 𝑔 is multisensitive, then so is 𝑓.

Proof. Suppose that 𝑔 is multisensitive with sensitive constant 𝛿>0. By the continuity of 𝜋, there exists a constant 𝛿^′>0 such that if 𝑒(𝑦₁,𝑦₂)>𝛿 for 𝑦₁,𝑦₂∈𝑌, then 𝑑(𝑥₁,𝑥₂)>𝛿^′, where 𝜋(𝑥_𝑖)=𝑦_𝑖, 𝑖=1,2. Let 𝑛≥1 be any given integer and 𝑈_𝑖 any nonempty open set in 𝑋 for each 1≤𝑖≤𝑛. Since 𝜋 is semiopen, 𝜋(𝑈_𝑖) contains a nonempty open subset 𝑉_𝑖 for each 1≤𝑖≤𝑛. Therefore, ⋂^𝑛_𝑖=1𝑁_𝑔(𝑉_𝑖,𝛿)≠∅ implies ⋂^𝑛_𝑖=1𝑁_𝑓(𝑈_𝑖,𝛿^′)≠∅. Thus, the proof is complete.

Next, we study sufficient conditions for multisensitivity.

Lemma 10. Let (𝑋,𝑑) be a nontrivial metric space. If a map 𝑓∶𝑋→𝑋 satisfies that 𝑓×𝑓×⋯×𝑓_𝑘 is topologically transitive for each integer 𝑘≥1, then 𝑓 is multisensitive in 𝑋.

Proof. As 𝑋 is not reduced to a single point, there exists a constant 𝛿>0 such that, for every 𝑥∈𝑋, there is 𝑦∈𝑋 satisfying 𝑑(𝑥,𝑦)>3𝛿. We will remark that this claim will be repeatedly used in this section.
Fix any integer 𝑘≥1 and let 𝑉_𝑖⊂𝑋, 1≤𝑖≤𝑘, be any bounded and nonempty open sets with diam(𝑉_𝑖)<𝛿, where diam(𝑉_𝑖)∶=sup_{𝑥,𝑦∈𝑉𝑖}{𝑑(𝑥,𝑦)} is the diameter of 𝑉_𝑖. Then there exists a nonempty open subset 𝑈_𝑖 such that 𝑑𝑈_𝑖,𝑉_𝑖>𝛿,1≤𝑖≤𝑘. (9) Since 𝑓×𝑓×⋯×𝑓_2𝑘 is topologically transitive, one has 𝑓×𝑓×⋯×𝑓^𝑛_2𝑘𝑉₁×𝑉₁×𝑉₂×𝑉₂×⋯×𝑉_𝑘×𝑉_𝑘𝑉₁×𝑈₁×𝑉₂×𝑈₂×⋯×𝑉_𝑘×𝑈_𝑘≠∅ (10) for some integer 𝑛≥1. This implies that 𝑓^𝑛(𝑉_𝑖)∩𝑉_𝑖≠∅ and 𝑓^𝑛(𝑉_𝑖)∩𝑈_𝑖≠∅, and consequently there exist 𝑥_𝑖,𝑥^′_𝑖∈𝑉_𝑖 such that 𝑓^𝑛(𝑥_𝑖)∈𝑉_𝑖 and 𝑓^𝑛(𝑥^′_𝑖)∈𝑈_𝑖 for 1≤𝑖≤𝑘. So, it follows from (9) that 𝑑(𝑓^𝑛(𝑥_𝑖),𝑓^𝑛(𝑥^′_𝑖))>𝛿 for 1≤𝑖≤𝑘. This yields ^𝑘_𝑖=1𝑁_𝑓𝑉_𝑖,𝛿≠∅. (11) Therefore, 𝑓 is multisensitive in 𝑋. The proof is complete.

Remark 11. It is known that a continuous map on a compact space is topologically weak mixing if and only if 𝑓×𝑓×⋯×𝑓_𝑘 is topologically transitive for each 𝑘≥2.(1)In [22, Theorem  3.1], it was shown that if 𝑓 is continuous and topologically mixing on a compact metric space 𝑋, then it is sensitive. Since the topological mixing is stronger than the topological weak mixing, Lemma 10 relaxes the conditions of [22, Theorem  3.1] and improves it by noting that it is not required that the space is compact and the map is continuous.(2)In [12], Moothathu claimed that if a continuous map 𝑓 is topologically weak mixing on a compact metric space, then it is multisensitive in 𝑋. So, Lemma 10 relaxes the conditions of this result.

Theorem 12. Let (𝑋,𝑑) be a nontrivial metric space and let 𝑇 be a measure-preserving map on (𝑋,𝑑,ℬ(𝑋),𝜇). If 𝑇 is weak mixing and supp𝜇=𝑋, then 𝑇^𝑛 is multisensitive in 𝑋 for every integer 𝑛≥1.

Proof. By the definition, one can easily prove that 𝑇 is weak mixing if and only if 𝑇^𝑛 is too for each 𝑛≥1. So it suffices to show that 𝑇 is multisensitive in 𝑋. Since 𝑇 is weak mixing, 𝑇×𝑇×⋯𝑇_2𝑘 is weak mixing for each 𝑘≥1 by Theorem  1.24 in [15]. Further, every nonempty open set in 𝑋 has a positive measure because of supp𝜇=𝑋. It follows that 𝑇×𝑇×⋯𝑇_2𝑘 is topologically transitive for each 𝑘≥1. Therefore, 𝑇 is multisensitive in 𝑋 by Lemma 10. This completes the proof.

In order to extend the above result for measure-preserving maps to measure-preserving semiflows, we first show the following five lemmas.

Lemma 13. Let (𝑋,𝑑) be a nontrivial metric space. If a semiflow 𝜑∶𝐑⁺×𝑋→𝑋 satisfies that 𝜑×𝜑×⋯×𝜑_𝑘 is topologically transitive for each integer 𝑘≥1, then it is multisensitive in 𝑋.

Proof. With a similar argument to that used in the proof of Lemma 10, one can easily prove Lemma 13. So its details are omitted.

Lemma 14. Let 𝜑 be a measure-preserving semiflow on (𝑋,𝑑,ℬ(𝑋),𝜇) and Ω a semialgebra that generates ℬ(𝑋). Then 𝜑 is weak mixing if and only if, for any 𝐴,𝐵∈Ω, we have lim_𝑡→∞1𝑡^𝑡₀𝜇𝐴𝜑_𝑠⁻¹(𝐵)−𝜇(𝐴)𝜇(𝐵)𝑑𝑠=0. (12)

Proof. The proof is similar to that of Theorem  1.17 in [20] and then is omitted.

Lemma 15. The following are equivalent:(i)lim_𝑡→∞(1/𝑡)∫^𝑡₀|𝑓(𝑠)|𝑑𝑠=0;(ii)there exists a subset 𝑆⊂𝐑⁺ of density zero such that lim_𝑡→∞𝑓(𝑡)=0 provided 𝑡∉𝑆;(iii)lim_𝑡→∞(1/𝑡)∫^𝑡₀|𝑓(𝑠)|²𝑑𝑠=0.

Proof. The proof is similar to that of Theorem  1.20 in [20] and then is omitted here.

The following lemma can be directly derived from Lemma 15 with 𝑓(𝑠)=𝜇(𝜑_𝑠⁻¹(𝐴)⋂𝐵)−𝜇(𝐴)𝜇(𝐵).

Lemma 16. Let 𝜑 be a measure-preserving semiflow on (𝑋,𝑑,ℬ(𝑋),𝜇). Then the following are equivalent:(i)𝜑 is weak mixing;(ii)for every pair of sets 𝐴,𝐵∈ℬ(𝑋), there is a subset 𝑆(𝐴,𝐵)⊂𝐑⁺ of density zero such that lim_{𝐑+()⧵𝑆𝐴,𝐵∋𝑡→∞}𝜇𝜑_𝑡⁻¹(𝐴)𝐵=𝜇(𝐴)𝜇(𝐵); (13)(iii)for every pair of sets 𝐴,𝐵∈ℬ(𝑋), one has lim_𝑡→∞1𝑡^𝑡₀𝜇𝜑_𝑠⁻¹(𝐴)𝐵−𝜇(𝐴)𝜇(𝐵)²𝑑𝑠=0. (14)

Lemma 17. Let 𝜑 be a measure-preserving semiflow on (𝑋,𝑑,ℬ(𝑋),𝜇). Then 𝜑 is weak mixing if and only if so is 𝜑×𝜑.

Proof. First consider the necessity. Suppose that 𝜑 is weak mixing. Fix any sets 𝐴,𝐵,𝐶,𝐷∈ℬ(𝑋). Then by Lemma 16 there exist subsets 𝑆₁,𝑆₂⊂𝐑⁺ of density zero such that lim_{𝐑+⧵𝑆1∋𝑡→∞}𝜇𝜑_𝑡⁻¹(𝐴)𝐵=𝜇(𝐴)𝜇(𝐵),lim_{𝐑+⧵𝑆2∋𝑡→∞}𝜇𝜑_𝑡⁻¹(𝐶)𝐷=𝜇(𝐶)𝜇(𝐷), (15) which yield that lim_{𝐑+⧵𝑆1∪𝑆2∋𝑡→∞}𝜇×𝜇𝜑×𝜑_𝑡⁻¹(𝐴×𝐶)(𝐵×𝐷)=lim_{𝐑+⧵(𝑆1∪𝑆2)∋𝑡→∞}𝜇𝜑_𝑡⁻¹(𝐴)𝐵𝜇𝜑_𝑡⁻¹(𝐶)𝐷(𝐴)𝜇(𝐵)𝜇(𝐶)𝜇(𝐷)=)).=𝜇𝜇×𝜇(𝐴×𝐶𝜇×𝜇(𝐵×𝐷 (16) So, by Lemma 15 one has lim_𝑡→∞1𝑡^𝑡₀𝜇×𝜇𝜑×𝜑_𝑠⁻¹()()−()(𝐴×𝐶𝐵×𝐷𝜇×𝜇𝐴×𝐶𝜇×𝜇𝐵×𝐷)𝑑𝑠=0. (17) Since the measurable rectangles form a semialgebra that generates ℬ(𝑋)×ℬ(𝑋). Therefore, 𝜑×𝜑 is weak mixing by Lemma 14.
Next, we consider the sufficiency. Suppose that 𝜑×𝜑 is weak mixing. Fix any sets 𝐴,𝐵∈ℬ(𝑋). It is evident that 𝜇𝜑_𝑠⁻¹(𝐴)𝐵=𝜇×𝜇𝜑×𝜑_𝑠⁻¹()(),))(𝐴)𝜇(𝐵),𝐴×𝑋𝐵×𝑋𝜇×𝜇(𝐴×𝑋𝜇×𝜇(𝐵×𝑋=𝜇 (18) which, together with the assumption that 𝜑×𝜑 is weak mixing, imply that lim_𝑡→∞1𝑡^𝑡₀𝜇𝜑_𝑠⁻¹(𝐴)𝐵−𝜇(𝐴)𝜇(𝐵)𝑑𝑠=lim_𝑡→∞1𝑡^𝑡₀𝜇×𝜇𝜑×𝜑_𝑠⁻¹()()−()(𝐴×𝑋𝐵×𝑋𝜇×𝜇𝐴×𝑋𝜇×𝜇𝐵×𝑋)𝑑𝑠=0. (19) Hence, 𝜑 is weak mixing. The entire proof is complete.

Theorem 18. Let (𝑋,𝑑) be a nontrivial metric space and 𝜑 a measure-preserving semiflow on (𝑋,𝑑,ℬ(𝑋),𝜇). If 𝜑 is weak mixing and supp𝜇=𝑋, then 𝜑 is multisensitive in 𝑋.

Proof. With a similar argument to that used in the proof of Theorem 12, one can easily show this theorem by Lemmas 13 and 17. This completes the proof.

3.2. Cofinite Sensitivity

In this subsection, we first show that cofinite sensitivity can be lifted by a semiopen factor map and then give a sufficient condition for cofinite sensitivity of measure-preserving maps (resp., semiflows).

Proposition 19. Let 𝑋 and 𝑌 be nontrivial metric spaces, let 𝑓∶𝑋→𝑋 and 𝑔∶𝑌→𝑌 be maps, and let 𝜋∶𝑋→𝑌 be a semiopen factor map between (𝑋,𝑓) and (𝑌,𝑔). If 𝑔 is cofinitely sensitive, then so is 𝑓.

Proof. The proof is similar to that of Proposition 9 and so its details are omitted.

Lemma 20. Let (𝑋,𝑑) be a nontrivial metric space. If a map 𝑓∶𝑋→𝑋 is topologically mixing, then it is cofinitely sensitive in 𝑋.

Proof. As is shown in Lemma 10, there exists a constant 𝛿>0 such that, for every nonempty open set 𝑉⊂𝑋 with diam(𝑉)<𝛿, there exists a nonempty open set 𝑈 satisfying 𝑑(𝑈,𝑉)>𝛿. (20) Since 𝑓∶𝑋→𝑋 is topologically mixing, there is an integer 𝑛₀≥1 such that 𝑓^𝑛(𝑉)𝑉≠∅,𝑓^𝑛(𝑉)𝑈≠∅,∀𝑛≥𝑛₀, (21) which, together with (20), yield that 𝑁_𝑓(𝑉,𝛿)⊃[𝑛₀,+∞)⋂𝐍. Therefore, 𝑓 is cofinitely sensitive in 𝑋. This completes the proof.

Remark 21. Proposition  2 in [12] shows that if a map 𝑓 is topologically mixing and continuous in a compact metric space 𝑋, then 𝑓 is cofinitely sensitive in 𝑋. Note that the compactness of the space 𝑋 and the continuity of the map 𝑓 are not required in Lemma 20. So Lemma 20 relaxes the conditions of this proposition.

This result can be extended to semiflows.

Lemma 22. Let (𝑋,𝑑) be a nontrivial metric space. If a semiflow 𝜑∶𝐑⁺×𝑋→𝑋 is topologically mixing, then it is cofinitely sensitive in 𝑋.

Proof. The proof is similar to that of Lemma 20 and so it is omitted.

Theorem 23. Let (𝑋,𝑑) be a nontrivial metric space and let 𝑇 be a measure-preserving map on (𝑋,𝑑,ℬ(𝑋),𝜇). If 𝑇 is strong mixing and supp𝜇=𝑋, then 𝑇^𝑛 is cofinitely sensitive in 𝑋 for each integer 𝑛≥1.

The above theorem follows from Proposition  2.2 in [10]. For completeness, we now give a different proof here.

Proof. By the definition of strong mixing, it can be easily seen that 𝑇 is strong mixing if and only if 𝑇^𝑛 is too for each 𝑛≥1. So it suffices to show that 𝑇 is cofinitely sensitive in 𝑋. Because of supp𝜇=𝑋, every nonempty open set in 𝑋 has a positive measure. So 𝑇 is topologically mixing. Consequently, by Lemma 20, 𝑇 is cofinitely sensitive in 𝑋. Thus, the proof is complete.

Theorem 24. Let (𝑋,𝑑) be a nontrivial metric space and let 𝜑 be a measure-preserving semiflow on (𝑋,𝑑,ℬ(𝑋),𝜇). If 𝜑 is strong mixing and supp𝜇=𝑋, then 𝜑 is cofinitely sensitive in 𝑋.

Proof. With a similar argument to that used in the proof of Theorem 23 and by Lemma 22, one can easily show that this theorem holds. The proof is complete.

3.3. Ergodic Sensitivity

In the final subsection, we will first show that ergodic sensitivity can be lifted by a semiopen factor map and then consider ergodic sensitivity for measure-preserving maps and semiflows on a probability space and give a sufficient condition for each of them.

Proposition 25. Let 𝑋 and 𝑌 be nontrivial metric spaces, let 𝑓∶𝑋→𝑋 and 𝑔∶𝑌→𝑌 be maps, and let 𝜋∶𝑋→𝑌 be a semiopen factor map between (𝑋,𝑓) and (𝑌,𝑔). If 𝑔 is ergodically sensitive, then so is 𝑓.

Proof. The proof is similar to that of Proposition 9 and so its details are omitted.

Lemma 26. Let (𝑋,𝑑) be a nontrivial metric space and 𝑇 a measure-preserving map on (𝑋,𝑑,ℬ(𝑋),𝜇). If 𝑇 is not ergodically sensitive in 𝑋 and supp𝜇=𝑋, then there exist a constant 𝛿>0 and two disjoint and nonempty open sets 𝑈 and 𝑉 in 𝑋 such that 𝑑𝑆𝐙⁺⧵𝑁_𝑇(𝑊,𝛿)>0 (22) for some nonempty open set 𝑊⊂𝑋, where 𝑆=𝑖∈𝐙⁺∶𝑇^𝑖(𝑉)𝑈=∅. (23)

Proof. As is shown in the proof of Lemma 10, there exists a constant 𝛿>0 such that, for every 𝑥∈𝑋, there is 𝑦∈𝑋 with 𝑑(𝑥,𝑦)>3𝛿. Since 𝑇 is not ergodically sensitive in 𝑋, there exists a nonempty open set 𝑊⊂𝑋 such that 𝑑(𝑁_𝑇(𝑊,𝛿))=0 and so 𝑑𝑁_𝑇(𝑊,𝛿)=0. (24) It is clear that diam𝑇^𝑛(𝑊)≤𝛿,∀𝑛∈𝐙⁺⧵𝑁_𝑇(𝑊,𝛿). (25) Fix a point 𝑥∈𝑊 and take a constant 0<𝜀<𝛿 with the open ball 𝐵(𝑥,𝜀)⊂𝑊. Then 𝜇(𝐵(𝑥,𝜀))>0 because of supp𝜇=𝑋. By Lemma 7, the set 𝑆₁=𝑖∈𝐙⁺∶𝐵(𝑥,𝜀)𝑇^−𝑖(𝐵(𝑥,𝜀))≠∅ (26) is relatively dense in 𝐙⁺. Now, for any 𝑖∈𝑆₁, take 𝑧_𝑖∈𝐵(𝑥,𝜀)𝑇^−𝑖(𝐵(𝑥,𝜀)), (27) which implies that 𝑇^𝑖𝑧_𝑖∈𝐵(𝑥,𝜀)𝑇^𝑖(𝐵(𝑥,𝜀)). (28) So it follows from (25) that, for any 𝑦∈𝐵(𝑥,𝜀) and any 𝑘∈𝑆₁⋂(𝐙⁺⧵𝑁_𝑇(𝑊,𝛿)), 𝑑𝑇^𝑘𝑦,𝑥≤𝑑𝑇^𝑘𝑦,𝑇^𝑘𝑧_𝑘+𝑑𝑇^𝑘𝑧_𝑘,𝑥≤𝛿+𝜀<2𝛿. (29) This means that 𝑇^𝑘(𝐵(𝑥,𝜀))⊂𝐵(𝑥,2𝛿) for any 𝑘∈𝑆₁⋂(𝐙⁺⧵𝑁_𝑇(𝑊,𝛿)).
Set 𝑉=𝐵(𝑥,𝜀) and 𝑈=𝑋⧵𝐵(𝑥,2𝛿). Then 𝑈 and 𝑉 are disjoint and nonempty open sets, and 𝑇^𝑘(𝑉)⋂𝑈 = ∅ for any 𝑘 ∈ 𝑆₁⋂(𝐙⁺∖𝑁_𝑇(𝑊,𝛿)), which implies that 𝑆₁𝐙⁺⧵𝑁_𝑇(𝑊,𝛿)⊂𝑆, (30) and consequently 𝑆𝐙⁺⧵𝑁_𝑇(𝑊,𝛿)⊃𝑆₁𝐙⁺⧵𝑁_𝑇(𝑊,𝛿). (31) As the lower density of 𝑆₁ is positive which implies that the upper density of 𝑆∩𝑍⁺⧵𝑁_𝑇(𝑊,𝛿) (32) is positive, since the upper density of 𝑍⁺⧵𝑁_𝑇(𝑊,𝛿) (33) is  1, the proof is complete.

By the Birkhoff ergodic theorem and Lemma 26 one can prove the following theorem. For completeness, we give another proof of Theorem 27.

Theorem 27. Let (𝑋,𝑑) be a nontrivial metric space and let 𝑇 be a measure-preserving map on (𝑋,𝑑,ℬ(𝑋),𝜇). If 𝑇 is weak mixing and supp𝜇=𝑋, then 𝑇^𝑛 is ergodically sensitive in 𝑋 for each integer 𝑛≥1.

Proof. As is shown in the proof of Theorem 12, 𝑇 is weak mixing if and only if 𝑇^𝑛 is too for each 𝑛≥1. So it suffices to show that 𝑇 is ergodically sensitive in 𝑋.
Suppose on the contrary that 𝑇 is not ergodically sensitive in 𝑋. Then, by Lemma 26 there exist a constant 𝛿>0 and two disjoint and nonempty open sets 𝑈 and 𝑉 in 𝑋 such that the set 𝑆⋂(𝐙⁺⧵𝑁_𝑇(𝑊,𝛿)) has a positive upper density for some nonempty open set 𝑊⊂𝑋, where 𝑆={𝑖∈𝐙⁺∶𝑇^𝑖(𝑉)⋂𝑈=∅}. Then 𝑉𝑇^−𝑘(𝑈)=∅,∀𝑘∈𝑆, (34) which implies that 𝜇(𝑉⋂𝑇^−𝑘(𝑈))=0 for each 𝑘∈𝑆. Thus, one has that ^𝑛−1_𝑖=0𝜇𝑉𝑇^−𝑖(𝑈)(𝑉)𝜇(𝑈≥𝑆𝑁−𝜇)_𝑛𝜇(𝑉)𝜇(𝑈)≥𝑆𝐙⁺⧵𝑁_𝑇(𝑊,𝛿)𝑁_𝑛𝜇(𝑉)𝜇(𝑈). (35) Further, 𝜇(𝑉)𝜇(𝑈)>0 since supp𝜇=𝑋, and 𝑈 and 𝑉 are nonempty open sets. Consequently, limsup_𝑛→∞1𝑛^𝑛−1_𝑘=0𝜇𝑉𝑇^−𝑘(𝑈)(𝑉)𝜇(𝑈(𝑉)𝜇(𝑈)−𝜇)≥𝜇𝑑𝑆𝐙⁺⧵𝑁_𝑇(𝑊,𝛿)>0. (36) This is a contradiction since 𝑇 is weak mixing. Therefore, 𝑇 is ergodically sensitive in 𝑋. This completes the proof.

Remark 28. Syndetic sensitivity implies ergodic sensitivity. However, Moothathu gave an example of a sensitive map that is not ergodically sensitive in Theorem 7 in [12].

Lemma 29. Let (𝑋,𝑑) be a nontrivial metric space, whose bounded and closed subsets are compact, and let 𝜑 be a continuous measure-preserving semiflow on (𝑋,𝑑,ℬ(𝑋),𝜇). If 𝜑 is not ergodically sensitive in 𝑋 and supp𝜇=𝑋, then there exist two disjoint and nonempty open sets 𝑈, 𝑉 in 𝑋 such that 𝑑(𝑆)>0, where 𝑆=𝑡∈𝐑⁺∶𝑉𝜑_𝑡⁻¹(𝑈)=∅. (37)

Proof. As is shown in the proof of Lemma 10, there exists a constant 𝛿>0 such that, for every 𝑥∈𝑋, there is 𝑦∈𝑋 with 𝑑(𝑥,𝑦)>3𝛿. By the assumption that 𝜑 is not ergodically sensitive in 𝑋, there exists a nonempty open set 𝑊⊂𝑋 such that 𝑑(𝑁_𝜑(𝑊,𝛿))=0, and so 𝑑(𝑁_𝜑(𝑊,𝛿))=0. Consequently, 𝑑(𝐑⁺⧵𝑁_𝜑(𝑊,𝛿))=1.
Fix any 𝑥∈𝑊. There exists a positive constant 𝜀<𝛿 such that 𝐵(𝑥,𝜀)⊂𝑊. As supp𝜇=𝑋, we have 𝜇(𝐵(𝑥,𝜀))>0. Take 𝑉=𝐵(𝑥,𝜀). By Lemma 8, the set 𝑆₁=𝑡∈𝐑⁺∶𝑉𝜑_𝑡(𝑉)=≠∅𝑡∈𝐑⁺∶𝑉𝜑_𝑡⁻¹(𝑉)≠∅ (38) is relatively dense in 𝐑⁺; that is, there exists 𝐿>0 such that 𝑆₁(𝑡,𝑡+𝐿)≠∅,∀𝑡∈𝐑⁺. (39)
For any 𝑡∈𝑆₁⋂(𝐑⁺⧵𝑁_𝜑(𝑊,𝛿)), 𝑉⋂𝜑_𝑡(𝑉)≠∅ and diam(𝜑_𝑡(𝑉))≤𝛿, and so 𝑉𝜑_𝑡(𝑉)⊂𝐵(𝑥,2𝛿). (40) Since 𝜑 is continuous and 𝐵(𝑥,2𝛿) is compact, 𝜑|_{[0,1]×𝐵(𝑥,2𝛿)} is uniformly continuous. Hence, there exists a constant 0<𝜀^′<min{𝜀,1} such that 𝜑_𝑡𝐵(𝑥,2𝛿)⊂𝐵(𝑥,3𝛿),∀𝑡∈0,𝜀^′, (41) which, together with (40), implies that 𝜑_𝜏(𝑉)⊂𝐵(𝑥,3𝛿) (42) for any 𝑡∈𝑆₁⋂(𝐑⁺⧵𝑁_𝜑(𝑊,𝛿)) and any 𝜏∈[𝑡,𝑡+𝜀^′]. Let 𝑈=𝑋⧵𝐵(𝑥,3𝛿) and 𝑆₂=_{𝑡∈𝑆1⋂(𝐑+⧵𝑁𝜑(𝑊,𝛿))}𝑡,𝑡+𝜀^′. (43) It follows from (42) that, for any 𝜏∈𝑆₂, 𝑈⋂𝜑_𝜏(𝑉)=∅; that is, 𝑉⋂𝜑_𝜏⁻¹(𝑈)=∅. Thus, 𝑆₁𝐑⁺⧵𝑁_𝜑(𝑊,𝛿)⊂𝑆₂⊂𝑆. (44) As the lower density of 𝑆₁ is positive which implies that the upper density of 𝑆∩𝐑⁺⧵𝑁_𝜑(𝑊,𝛿) (45) is positive, since the upper density of 𝐑⁺⧵𝑁_𝜑(𝑊,𝛿) (46) is  1, therefore, the proof is complete.

Theorem 30. Let (𝑋,𝑑) be a nontrivial metric space, whose bounded and closed subsets are compact, and let 𝜑 be a continuous measure-preserving semiflow on (𝑋,𝑑,ℬ(𝑋),𝜇). If 𝜑 is weak mixing and supp𝜇=𝑋, then 𝜑 is ergodically sensitive in 𝑋.

Proof. On the contrary, 𝜑 is not ergodically sensitive in 𝑋. Then, by Lemma 29, there exist two disjoint and nonempty open sets 𝑈, 𝑉 in 𝑋 such that the set 𝑆=𝑡∈𝐑⁺∶𝑉𝜑_𝑡⁻¹(𝑈)=∅ (47) has a positive upper density. Since supp𝜇=𝑋, we have 𝜇(𝑉)𝜇(𝑈)>0, and so lim_𝑡→∞1𝑡^𝑡₀𝜇𝑉𝜑_𝑠⁻¹(𝑈)(𝑉)𝜇(𝑈(𝑉)𝜇(𝑈)−𝜇)𝑑𝑠≥𝜇𝑑(𝑆)>0. (48) This contradicts the assumption that 𝜑 is weak mixing. Therefore, 𝜑 is ergodically sensitive in 𝑋. This completes the proof.

4. Relationships between Sensitive Properties of a Map and Its Iterations

In the final section, we discuss relationships between sensitive properties of a map 𝑓 and its iterations 𝑓^𝑚, including sensitivity, syndetic sensitivity, ergodic sensitivity, cofinite sensitivity, multisensitivity, and 𝑛-sensitivity. These relationships are equivalent in the special case that the space is compact and the map 𝑓 is continuous.

Theorem 31. Let 𝑓∶𝑋→𝑋 be a map, where (𝑋,𝑑) is a metric space. If 𝑓^𝑚 is sensitive or syndetically sensitive or ergodically sensitive or multisensitive in 𝑋 for some 𝑚≥2, then so is 𝑓. Moreover, the converses of all the above conclusions are true if 𝑓 is uniformly continuous in 𝑋. In addition, if 𝑓 is cofinitely sensitive, then so is 𝑓^𝑚 for any 𝑚≥2; and if 𝑓 is uniformly continuous in 𝑋 and 𝑓^𝑚 is cofinitely sensitive for some 𝑚≥2, then 𝑓 is cofinitely sensitive.

Proof. The proof is divided into four parts.(1)It is evident that for any nonempty open set 𝑈⊂𝑋 and for any constant 𝛾>0, 𝑚𝑁_𝑓^𝑚𝑈,𝛾⊂𝑁_𝑓𝑈,𝛾, (49)which implies that if 𝑓^𝑚 is sensitive or syndetically sensitive or ergodically sensitive or multisensitive in 𝑋, then so is 𝑓 by the related definitions.(2)Suppose that 𝑓 is uniformly continuous in 𝑋 and fix any integer 𝑚≥2. Then 𝑓^𝑖, 1≤𝑖≤𝑚−1, are uniformly continuous in 𝑋. Let 𝛿>0 be a constant of sensitivity with respect to one of the five types of sensitivity. Then there exists a positive constant 𝜀<𝛿 such that whenever 𝑑(𝑥,𝑦)≤𝜀 for 𝑥,𝑦∈𝑋, one has 𝑑𝑓^𝑖(𝑥),𝑓^𝑖𝑦≤𝛿,0≤𝑖≤𝑚−1. (50)This claim will be often used in the sequent discussion, which is divided into four steps.
Step  1. If 𝑓 is sensitive in 𝑋, then so is 𝑓^𝑚.
Let 𝑈⊂𝑋 be any nonempty open set. Since 𝑓 is sensitive in 𝑋, 𝑁_𝑓(𝑈,𝛿) is not empty. Fix any 𝑛∈𝑁_𝑓(𝑈,𝛿) and then there exist 𝑥,𝑦∈𝑈 such that 𝑑(𝑓^𝑛(𝑥),𝑓^𝑛(𝑦))>𝛿. Let 𝑛=𝑘𝑚+𝑟 with 0≤𝑟≤𝑚−1 and 𝑘∈𝐙⁺. Then 𝑑(𝑓^𝑘𝑚(𝑥),𝑓^𝑘𝑚(𝑦))>𝜀. Otherwise, if 𝑑(𝑓^𝑘𝑚(𝑥),𝑓^𝑘𝑚(𝑦))≤𝜀, then it follows from (14) that 𝑑(𝑓^𝑛(𝑥),𝑓^𝑛(𝑦))≤𝛿, which is a contradiction. Hence, 𝑘∈𝑁_𝑓^𝑚(𝑈,𝜀) and so 𝑁_𝑓^𝑚(𝑈,𝜀)≠∅. Therefore, 𝑓^𝑚 is sensitive in 𝑋.
Step  2. If 𝑓 is syndetically sensitive in 𝑋, then so is 𝑓^𝑚.
Let 𝑈⊂𝑋 be any nonempty open set. Since 𝑓 is syndetically sensitive in 𝑋, 𝑁_𝑓(𝑈,𝛿) is syndetic in 𝐙⁺. Write 𝑁_𝑓(𝑈,𝛿) as an increasing sequence {𝑛_𝑖}^∞_𝑖=1. There exists an integer 𝑀≥1 such that 𝑛_𝑖+1−𝑛_𝑖<𝑀 for all 𝑖≥1. Let 𝑛_𝑖=𝑚𝑘_𝑖+𝑟_𝑖,𝑖≥1, (51) with 0≤𝑟_𝑖≤𝑚−1 and 𝑘_𝑖∈𝐙⁺. Then 𝑛_𝑖+1−𝑛_𝑖=𝑚𝑘_𝑖+1−𝑘_𝑖+𝑟_𝑖+1−𝑟_𝑖<𝑀, (52) which implies that 𝑚𝑘_𝑖+1−𝑘_𝑖<𝑀+𝑟_𝑖−𝑟_𝑖+1<𝑀+𝑚,𝑖≥1. (53) Thus, {𝑘_𝑖}^∞_𝑖=1 is syndetic in 𝐙⁺. In addition, As is shown in Step  1, 𝑘_𝑖∈𝑁_𝑓^𝑚(𝑈,𝜀) by the definition of 𝜀. Hence, 𝑁_𝑓^𝑚(𝑈,𝜀) is syndetic, and consequently 𝑓^𝑚 is syndetically sensitive in 𝑋.
Step  3. If 𝑓 is ergodically sensitive in 𝑋, then so is 𝑓^𝑚.
Let 𝑈⊂𝑋 be any nonempty open set. Then 𝑑(N_𝑓(𝑈,𝛿))>0. For any given 𝑘∈𝑁_𝑓(𝑈,𝛿), let 𝑘=𝑚𝑖_𝑘+𝑟_𝑘 with 0≤𝑟_𝑘≤𝑚−1 and 𝑖_𝑘∈𝐙⁺. Then 𝑖_𝑘∈𝑁_𝑓^𝑚(𝑈,𝜀) as is shown in Step  1. Consequently, we have 𝑚limsup_𝑘→∞𝑁_𝑓^𝑚(𝑈,𝜀)⋂𝐍_𝑖𝑘𝑖_𝑘≥limsup_{𝑁𝑓(𝑈,𝛿)∋𝑘→∞}𝑁_𝑓(𝑈,𝛿)⋂𝐍_𝑘+1=𝑘+1𝑑𝑁_𝑓(𝑈,𝛿)>0, (54) which implies that 𝑑𝑁_𝑓^𝑚(𝑈,𝜀)≥limsup_𝑘→∞𝑁_𝑓^𝑚(𝑈,𝜀)⋂𝐍_𝑖𝑘𝑖_𝑘>0. (55) Therefore, 𝑓^𝑚 is ergodically sensitive in 𝑋.
Step  4. If 𝑓 is multisensitive in 𝑋, then so is 𝑓^𝑚.
Since 𝑓 is multisensitive in 𝑋, for each integer 𝑘≥1 and any 𝑘 nonempty open sets 𝑉_𝑖⊂𝑋, 1≤𝑖≤𝑘, we have ^𝑘_𝑖=1𝑁_𝑓𝑉_𝑖,𝛿≠∅. (56) Let 𝑛∈⋂^𝑘_𝑖=1𝑁_𝑓(𝑉_𝑖,𝛿) and 𝑛=𝑙𝑚+𝑟 with 0≤𝑟≤𝑚−1 and 𝑙∈𝐙⁺. It can be easily shown that 𝑙∈⋂^𝑘_𝑖=1𝑁_𝑓^𝑚(𝑉_𝑖,𝜀) by (14), and consequently ^𝑘_𝑖=1𝑁_𝑓^𝑚𝑉_𝑖,𝜀≠∅. (57) Therefore, 𝑓^𝑚 is multisensitive in 𝑋.(3)If 𝑓 is cofinitely sensitive in 𝑋, then so is 𝑓^𝑚 for any 𝑚≥2.
Let 𝛿>0 be a constant of cofinite sensitivity, 𝑈⊂𝑋 any nonempty open set, and 𝑚≥2 any integer. Then there exists an integer 𝑀≥1 such that 𝑁_𝑓(𝑈,𝛿)⊃[𝑀,+∞)𝐍. (58) For any integer 𝑛≥𝑀, it is clear that 𝑛𝑚∈𝑁_𝑓(𝑈,𝛿) and so there exist two points 𝑥,𝑦∈𝑈 such that 𝑑𝑓^𝑛𝑚(𝑥),𝑓^𝑛𝑚𝑦>𝛿, (59) which yields that 𝑛∈𝑁_𝑓^𝑚(𝑈,𝛿), and consequently we get that 𝑁_𝑓^𝑚(𝑈,𝛿)⊃[𝑀,+∞)𝐍. (60) Hence, 𝑓^𝑚 is cofinitely sensitive in 𝑋.(4)If 𝑓 is uniformly continuous in 𝑋 and 𝑓^𝑚 is cofinitely sensitive for some 𝑚≥2, then 𝑓 is cofinitely sensitive.
It is clear that 𝑓^𝑖 is uniformly continuous for 1≤𝑖≤𝑚. For any constant 𝛾>0, there exists 0<𝛾^′≤𝛾 such that 𝑑(𝑥,𝑦)≤𝛾^′ implies that 𝑑(𝑓^𝑖(𝑥),𝑓^𝑖(𝑦))≤𝛾 for 1≤𝑖≤𝑚. This implies that if 𝑛∈𝑁_𝑓^𝑚(𝑈,𝛾), then 𝑚(𝑛−1),𝑚(𝑛−1)+1,…,𝑚𝑛∈𝑁_𝑓(𝑈,𝛾^′). This means that {𝑚(𝑛−1)+𝑖∶𝑛∈𝑁_𝑓^𝑚(𝑈,𝛾), 0≤𝑖≤𝑚} ⊂ 𝑁_𝑓(𝑈,𝛾^′). Hence, 𝑓 is cofinitely sensitive.
The entire proof is complete.

Theorem 32. Let 𝑓∶𝑋→𝑋 be a map, where (𝑋,𝑑) is a nontrivial metric space. For any given 𝑚≥2, if 𝑓^𝑚 is 𝑛-sensitive in 𝑋, then so is 𝑓. Moreover, the converses of the above conclusion are true if 𝑓 is uniformly continuous in 𝑋.

Proof. The proof is similar to that of Theorem 31 and is omitted.

Remark 33. In the study of topological dynamical systems, it is the most important case that the space 𝑋 is compact and the map 𝑓 continuously transforms the space into itself. Clearly, 𝑓 is uniformly continuous in 𝑋, and consequently 𝑓 is sensitive, syndetically sensitive, ergodically sensitive, cofinitely sensitive, 𝑛-sensitive, and multisensitive in 𝑋 if and only if so is 𝑓^𝑚, respectively, for any given integer 𝑚≥2 in this case.

Conflict of Interests

The authors have declared that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors sincerely thank the referees for their careful reading and useful remarks, which helped us improve the paper. This research was supported by the NNSF of Shandong Province (Grant ZR2011AM002).