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

and Applied Analysis 3 (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:Nφ(U, V) ̸ = 0 andNφ(U, V) ⊃ [L, +∞) for some constant L > 0. (ii) A semiflowφ is said to be topologically weaklymixing onX if φ×φ is topologically transitive on the product spaceX × X. Let S be a subset of Z (resp., a Lebesgue measurable subset of R). Its upper and lower densities are defined, respectively, by


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][2][3][4][5][6][7][8][9][10][11][12][13][14]).
In [1], Abraham et al. proved that if a measure-preserving map  on a metric probability space (, , B(), ) with supp  =  is either topologically mixing or weak mixing and satisfies that for any nonempty open set  ⊂  and there exists a subsequence {  } with positive upper density such that then  is sensitive.In the same paper, they proved that if  is strong mixing and sup  = , then it is sensitive; and if  is an exact endomorphism and sup  = , then it is cofinitely sensitive.He et al. [8] showed that if a measurepreserving map  (resp., a measure-preserving semiflow ) on (, , B(), ) with supp  =  is weak mixing, then it is sensitive.In addition, if  is a nontrivial metric space (i.e., a metric space is not reduced to a single point) and a map  on  is topologically mixing, then  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 -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  there is a pair (; ) 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   (, ).
Previously, the third way was considered by several scholars.More recently, Moothathu [12] studied continuous self-maps Abstract and Applied Analysis 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 sensitivity for continuous self-maps of a complete metric space.Later, Shao et al. [13] investigated some properties of -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 -sensitivity, -sensitivity for , -complexity, and -equicontinuity for a measure-preserving and continuous map on a metric probability space (, , B(), ) and presented a sufficient condition for -sensitivity for , where  is a compact metric space.They proved that -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 -sensitivity) of a map  and its iterations   for  ≥ 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.

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.
We refer to [12,19,20] for the following basic concepts.Let (, ) be a metric space, B() the sigma-algebra of Borel subsets of , and  a probability measure on (, B()).Then the space  is called to be a metric probability space, denoted by the quadruple (, , B(), ), which is often briefly denoted by the triple (, B(), ).
The following concepts are about mixing properties of maps and semiflows in the measure-theoretical sense.Definition 1. (i) A measure-preserving map  on (, B(), ) is called weak mixing and strong mixing if, for any ,  ∈ B(), the following two equalities hold, respectively: (ii) A measure-preserving semiflow  on (, B(), ) is called weak mixing and strong mixing if, for any ,  ∈ B(), the following two equalities hold, respectively: The following concepts describe three different forms of transitivity of a map  :  →  and a semiflow  : R + × →  in the topological sense.For convenience, denote for any sets ,  ⊂ .
Definition 2. Let  :  →  be a map and  a metric space.
(ii) The map  is said to be topologically weakly mixing on  if × is topologically transitive on the product space  × .
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  : R + ×  →  be a semiflow and  a metric space.
(ii) A semiflow  is said to be topologically weakly mixing on  if  ×  is topologically transitive on the product space  × .
For convenience, such a constant  in the above definitions is called a sensitive constant of  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 cofinitely sensitive ⇒ syndetically sensitive ⇒ ergodically sensitive ⇒ sensitive.
So, cofinite sensitivity is the strongest one among the above five different forms of sensitivity.
Definition 5. Let (, ) be a nontrivial metrics space and  :  →  a map.For a given integer  ≥ 2, the system (, ) or the map  is said to be -sensitive if there exists a constant  > 0 such that, for any nonempty and open set , there are distinct points  1 ,  2 , . . .,   ∈  and some  ∈  + satisfying that (  (  ),   (  )) ≥  for 1 ≤  <  ≤ .Such a constant  is called an -sensitive constant of .
It is clear that 2-sensitivity is just sensitivity.For any given  ≥ 2, there exists a minimal system, that is, -sensitive, but not ( + 1)-sensitive (see [13]).Remark 6.In the case that  is a locally connected and compact nontrivial metric space, Shao et al. [13] showed that if a continuous and surjective map  :  →  is sensitive, then it is -sensitive for all  ≥ 2. Note that if the assumptions that  is compact and  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  is multisensitive, then it is -sensitive for each  ≥ 2 in this case.

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.

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 -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  :  →  and  :  →  be maps, where  and  are metric spaces.If there exists a continuous and surjective map  :  →  such that  ∘  =  ∘ , then (, ) is said to be a factor of the system (, ), and (, ) is said to be an extension of (, ), while  is said to be a factor map between (, ) and (, ).Proposition 9. Let (, ) and (, ) be nontrivial metric spaces, let  :  →  and  :  →  be maps, and let  :  →  be a semiopen factor map between (, ) and (, g).If  is multisensitive, then so is .
Proof.Suppose that  is multisensitive with sensitive constant  > 0. By the continuity of , there exists a constant Proof.As  is not reduced to a single point, there exists a constant  > 0 such that, for every  ∈ , there is  ∈  satisfying (, ) > 3.We will remark that this claim will be repeatedly used in this section.
Remark 11.It is known that a continuous map on a compact space is topologically weak mixing if and only if  ×  × ⋅ ⋅ ⋅ ×  ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟  is topologically transitive for each  ≥ 2.
(1) In [22, Theorem 3.1], it was shown that if  is continuous and topologically mixing on a compact metric space , 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  is topologically weak mixing on a compact metric space, then it is multisensitive in .So, Lemma 10 relaxes the conditions of this result.
Proof.The proof is similar to that of Theorem 1.17 in [20] and then is omitted.

Lemma 15.
The following are equivalent: Proof.The proof is similar to that of Theorem 1.20 in [20] and then is omitted here.

Lemma 16.
Let  be a measure-preserving semiflow on (, , B(), ).Then the following are equivalent: (i)  is weak mixing; (ii) for every pair of sets ,  ∈ B(), there is a subset (, ) ⊂ R + of density zero such that lim which yield that lim So, by Lemma 15 one has lim Since the measurable rectangles form a semialgebra that generates B() × B().Therefore,  ×  is weak mixing by Lemma 14.
Next, we consider the sufficiency.Suppose that  ×  is weak mixing.Fix any sets ,  ∈ B().It is evident that which, together with the assumption that  ×  is weak mixing, imply that lim Hence,  is weak mixing.The entire proof is complete.
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.

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  and  be nontrivial metric spaces, let  :  →  and  :  →  be maps, and let  :  →  be a semiopen factor map between (, ) and (, ).If  is cofinitely sensitive, then so is .
Proof.The proof is similar to that of Proposition 9 and so its details are omitted.
Remark 21.Proposition 2 in [12] shows that if a map  is topologically mixing and continuous in a compact metric space , then  is cofinitely sensitive in .Note that the compactness of the space  and the continuity of the map  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 (, ) be a nontrivial metric space.If a semiflow  : R + ×  →  is topologically mixing, then it is cofinitely sensitive in .
Proof.The proof is similar to that of Lemma 20 and so it is omitted.
Theorem 23.Let (, ) be a nontrivial metric space and let  be a measure-preserving map on (, , B(), ).If  is strong mixing and supp  = , then   is cofinitely sensitive in  for each integer  ≥ 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  is strong mixing if and only if   is too for each  ≥ 1.So it suffices to show that  is cofinitely sensitive in .Because of supp  = , every nonempty open set in  has a positive measure.So  is topologically mixing.Consequently, by Lemma 20,  is cofinitely sensitive in .Thus, the proof is complete.
Theorem 24.Let (, ) be a nontrivial metric space and let  be a measure-preserving semiflow on (, , B(), ).If  is strong mixing and supp  = , then  is cofinitely sensitive in .
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.

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 measurepreserving maps and semiflows on a probability space and give a sufficient condition for each of them.
Proposition 25.Let  and  be nontrivial metric spaces, let  :  →  and  :  →  be maps, and let  :  →  be a semiopen factor map between (, ) and (, ).If  is ergodically sensitive, then so is .
Proof.The proof is similar to that of Proposition 9 and so its details are omitted.
Lemma 26.Let (, ) be a nontrivial metric space and  a measure-preserving map on (, , B(), ).If  is not ergodically sensitive in  and supp  = , then there exist a constant  > 0 and two disjoint and nonempty open sets  and  in  such that  ( ⋂ (Z + \   (, ))) > 0 (22) for some nonempty open set  ⊂ , where Proof.As is shown in the proof of Lemma 10, there exists a constant  > 0 such that, for every  ∈ , there is  ∈  with (, ) > 3.Since  is not ergodically sensitive in , there exists a nonempty open set  ⊂  such that (  (, )) = 0 and so It is clear that Fix a point  ∈  and take a constant 0 <  <  with the open ball (, ) ⊂ .Then ((, )) > 0 because of supp  = .By Lemma 7, the set is relatively dense in Z + .Now, for any  ∈  1 , take which implies that   (  ) ∈  (, ) ⋂   ( (, )) .
Set  = (, ) and  =  \ (, 2).Then  and  are disjoint and nonempty open sets, and   () ⋂  = 0 for any  ∈  1 ⋂(Z + \   (, )), which implies that and consequently As the lower density of  1 is positive which implies that the upper density of is positive, since the upper density of 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 (, ) be a nontrivial metric space and let  be a measure-preserving map on (, , B(), ).If  is weak mixing and supp  = , then   is ergodically sensitive in  for each integer  ≥ 1.
Proof.As is shown in the proof of Theorem 12,  is weak mixing if and only if   is too for each  ≥ 1.So it suffices to show that  is ergodically sensitive in .
Suppose on the contrary that  is not ergodically sensitive in .Then, by Lemma 26 there exist a constant  > 0 and two disjoint and nonempty open sets  and  in  such that the set  ⋂(Z + \   (, )) has a positive upper density for some nonempty open set  ⊂ , where  = { ∈ Z + :   () ⋂  = 0}.Then which implies that ( ⋂  − ()) = 0 for each  ∈ .Thus, one has that  (36) This is a contradiction since  is weak mixing.Therefore,  is ergodically sensitive in .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].
Theorem 30.Let (, ) be a nontrivial metric space, whose bounded and closed subsets are compact, and let  be a continuous measure-preserving semiflow on (, , B(), ).
If  is weak mixing and supp  = , then  is ergodically sensitive in .
Proof.On the contrary,  is not ergodically sensitive in .
Then, by Lemma 29, there exist two disjoint and nonempty open sets ,  in  such that the set (48) This contradicts the assumption that  is weak mixing.Therefore,  is ergodically sensitive in .This completes the proof.

Relationships between Sensitive Properties of a Map and Its Iterations
In the final section, we discuss relationships between sensitive properties of a map  and its iterations   , including sensitivity, syndetic sensitivity, ergodic sensitivity, cofinite sensitivity, multisensitivity, and -sensitivity.These relationships are equivalent in the special case that the space is compact and the map  is continuous.
This claim will be often used in the sequent discussion, which is divided into four steps.
Step 2. If  is syndetically sensitive in , then so is   .
(3) If  is cofinitely sensitive in , then so is   for any  ≥ 2.
Hence,   is cofinitely sensitive in .
(4) If  is uniformly continuous in  and   is cofinitely sensitive for some  ≥ 2, then  is cofinitely sensitive.
The entire proof is complete.
Theorem 32.Let  :  →  be a map, where (, ) is a nontrivial metric space.For any given  ≥ 2, if   is sensitive in , then so is .Moreover, the converses of the above conclusion are true if  is uniformly continuous in .
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  is compact and the map  continuously transforms the space into itself.Clearly,  is uniformly continuous in , and consequently  is sensitive, syndetically sensitive, ergodically sensitive, cofinitely sensitive, -sensitive, and multisensitive in  if and only if so is   , respectively, for any given integer  ≥ 2 in this case.
is syndetically sensitive in  if there is a constant  > 0 such that   (, ) is a syndetic set for any nonempty open set  ⊂ .
Proof.By the definition, one can easily prove that  is weak mixing if and only if   is too for each  ≥ 1.So it suffices to show that  is multisensitive in .Since  is weak mixing,  ×  × ⋅ ⋅ ⋅  ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟In order to extend the above result for measure-preserving maps to measure-preserving semiflows, we first show the following five lemmas.Let (, ) be a nontrivial metric space.If a semiflow  : R + ×  →  satisfies that  ×  × ⋅ ⋅ ⋅ ×  ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ Let  be a measure-preserving semiflow on (, , B(), ) and Ω a semialgebra that generates B().Then  is weak mixing if and only if, for any ,  ∈ Ω, we have [15]rem 12. Let (, ) be a nontrivial metric space and let  be a measure-preserving map on (, , B(), ).If  is weak mixing and supp  = , then   is multisensitive in  for every integer  ≥ 1.2is weak mixing for each  ≥ 1 by Theorem 1.24 in[15].Further, every nonempty open set in  has a positive measure because of supp  = .It follows that  ×  × ⋅ ⋅ ⋅  ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟2is topologically transitive for each  ≥ 1.Therefore,  is multisensitive in  by Lemma 10.This completes the proof.
29.Let (, ) be a nontrivial metric space, whose bounded and closed subsets are compact, and let  be a continuous measure-preserving semiflow on (, , B(), ).If  is not ergodically sensitive in  and supp  = , then there exist two disjoint and nonempty open sets ,  in  such that () > 0, where 1) It is evident that for any nonempty open set  ⊂  and for any constant  > 0,    (, ) ⊂   (, ) , (49) which implies that if   is sensitive or syndetically sensitive or ergodically sensitive or multisensitive in , then so is  by the related definitions.(2) Suppose that  is uniformly continuous in  and fix any integer  ≥ 2. Then   , 1 ≤  ≤  − 1, are uniformly continuous in .Let  > 0 be a constant of sensitivity with respect to one of the five types of sensitivity.Then there exists a positive constant  <  such that whenever (, ) ≤  for ,  ∈ , one has  (  () ,   ()) ≤ , 0 ≤  ≤  − 1.
then so is .Moreover, the converses of all the above conclusions are true if  is uniformly continuous in .In addition, if  is cofinitely sensitive, then so is   for any  ≥ 2; and if  is uniformly continuous in  and   is cofinitely sensitive for some  ≥ 2, then  is cofinitely sensitive.Proof.The proof is divided into four parts.( {  } ∞ =1 is syndetic in Z + .In addition, As is shown in Step 1,   ∈    (, ) by the definition of .Hence,    (, ) is syndetic, and consequently   is syndetically sensitive in .Step 3. If  is ergodically sensitive in , then so is   .Let  ⊂  be any nonempty open set.Then (N  (, )) > 0. For any given  ∈   (, ), let  =   +   with 0 ≤   ≤  − 1 and   ∈ Z + .Then   ∈    (, ) as is shown in Step 1. Consequently, we have         (, ) ⋂ N                (, ) ⋂ N          is ergodically sensitive in .Step 4. If  is multisensitive in , then so is   .Since  is multisensitive in , for each integer  ≥ 1 and any  nonempty open sets   ⊂ , 1 ≤  ≤ , we have  =1   (  , ) and  =  +  with 0 ≤  ≤  − 1 and  ∈ Z + .It can be easily shown that  ∈ ⋂  =1    (  , ) by (14), and consequently