Abstract
In this paper, recurrent -semigroups are introduced and investigated. It is proved that, despite hypercyclic -semigroups, recurrent -semigroups can be found on finite-dimensional Banach spaces. Some criteria are stated for recurrence, which is based on open sets, neighborhoods of zero, and special eigenvectors. It is established that having a dense set of recurrent vectors is a sufficient and necessary condition for a -semigroup to be recurrent. Moreover, the direct sum of recurrent -semigroups is investigated.
1. Introduction
The study of the dynamical system is a notable branch in mathematics. Hypercyclicity, recurrency, chaoticity, and mixing are investigated in this branch. For a Banach space , an operator is named hypercyclic if for some or equivalently for some integer , where and are arbitrary open sets in . If for any open set in , a positive integer can be chosen such that ; then, is called a recurrent operator. Recurrence is a remarkable case of hypercyclicity. In fact, in this case, the inverse image of any open set under the operator intersects with itself. More information are accessible in [1, 2].
A hypercyclic operator on , with a dense set of periodic points is named a chaotic operator. Also, is called mixing if for any open set and of , for any greater or equal a natural number . References [3, 4] contain valuable information about the above notions and results.
One of the significant structures that are considered by mathematicians is -semigroups. Suppose that be a family of operators. is called a -semigroup if for any , , , and , for any .
Similar to operators, hypercyclic -semigroups, chaotic -semigroups, and mixing -semigroups are defined.
A -semigroup on is named hypercyclic if it has a dense orbit or equivalently for any open sets and of , there is such that . One can see [5] either. Moreover, there are remarkable criteria in this matter that can be found in [6]. Also, new points about constructive approximation of semigroups can be observed in [7]. A hypercyclic -semigroup is called chaotic if it has a dense set of periodic points. It means there are points like with this property that for some . Also, a -semigroup on is called mixing if for any open sets and of , can be chosen such that, for any , .
In the following, we mention hypercyclicity criterion and a recurrent hypercyclicity criterion for -semigroups.
Definition 1. (see [8])(HCC). A semigroup on fulfills the hypercyclicity criterion if and only if can be found such thatwhere and are arbitrary nonempty and open sets and is an arbitrary neighborhood of zero in .
Definition 2. (see [8])(RHCC). A semigroup on fulfills the recurrent hypercyclicity criterion if and only if for any nonempty and open sets and and any neighborhood of zero in , , and can be found such that, for any , and can be found such thatDesch and Schappacher [8] proved that if a semigroup is RHCC, then it is HCC. In [9, 10], more matters can be found about properties of -semigroups.
By various types of -semigroups and their notable properties, it sounds interesting to investigate -semigroups with this property that returns back an open set to itself. If we name this new concept as recurrent -semigroup, the question arises as to whether we can find interesting properties for them or whether we can get some relations between them and concepts such as hypercyclicity, chaoticity, and mixing for -semigroups?
In this paper, we introduce the concept of recurrent -semigroups and look into their properties. In Section 2, we state some preliminaries about this concept. We show that the recurrence of any operator in a -semigroup implies the -semigroup recurrence. Also, if satisfies HCC, then the recurrence of , can be concluded for any .
In Section 3, recurrent vectors for -semigroups are defined. It is proved that a -semigroup on is recurrent if and only if . It is proved that recurrence preserves under conjugacy. We establish that, despite hypercyclic -semigroups, recurrent -semigroups can be constructed on finite-dimensional spaces. Also, we make various examples of recurrent -semigroups.
In Section 4, various criteria for recurrence of -semigroups are presented. The conditions of these criteria are based on open sets, neighborhoods of zero, dense sets, and special eigenvectors.
In Section 5, we investigate the recurrence of the direct sum of two -semigroups. We establish that recurrence of the direct sum of two semigroups implies recurrence of each of them. Moreover, if at least one of them be mixing, the converse is also true. Additionally, it is proved that if one of the -semigroups is RHCC and the other -semigroup is HCC, then their direct sum is recurrent.
2. Preliminaries
We begin this section by defining the concept of recurrent -semigroup.
Definition 3. We say a -semigroup is recurrent if for any open and nonempty set , some can be found such that .
By Definition 3, it is not complicated to see that if is a hypercyclic -semigroup or a chaotic -semigroup, then is recurrent. Moreover, we establish that hypercyclicity (chaoticity) of an operator in a -semigroup implies its hypercyclicity (chaoticity) as follows.
Corollary 1. Suppose that be a -semigroup on . If be hypercyclic or chaotic for some , then is recurrent.
Proof. Hypercyclicity (chaoticity) of for some indicates that is hypercyclic (chaotic). Accordingly, is recurrent.
The next theorem shows that the recurrence of any operator in a -semigroup implies its recurrence.
Theorem 1. Consume is a -semigroup on . Then, recurrence of is derived from recurrence of any of ’s.
Proof. Let be recurrent for some . Consider be an open and nonempty set. Hence, there is such that . Suppose . Then, and . But is a -semigroup. Hence, . So, . Therefore, . Thus, is recurrent.
Now, this question arises that if the converse of Theorem 1 is true?
The next theorem manifests that the answer is affirmative if fulfills the hypercyclicity criterion.
Theorem 2. If fulfills the hypercyclicity criterion, then is a recurrent operator for any .
Proof. Let fulfills the hypercyclicity criterion. Thus, is hypercyclic (Theorem 7.27 in [4]). Hence, is hypercyclic, and so, it is recurrent for any (Theorem 2.3 in [11]).
The following lemma states that, for a -semigroup and any open and nonempty set , for infinitely many .
Lemma 1. Let be a recurrent -semigroup. Then,which is an infinite subset of .
Proof. Assume, on the contrary, there is an open and nonempty set such thatSuppose that be the greatest number in the right set of (4). Now, is nonempty and open. So, there is such thatHence, . That means belongs to the left set of (4). But this is a contradiction since .
3. Recurrent Vectors, Frequently Recurrent Vectors, and Finite Dimensions
A vector is named a recurrent vector for an operator if , where is an increasing sequence. By this notion, we can state the next lemma.
Lemma 2. Suppose that is a -semigroup on . Then, is recurrent if for some , be dense in .
Proof. Suppose that for some . Then, is recurrent (Proposition 2.1 in [1]). Now, by Theorem 1, is recurrent.
The concept of the recurrent vector can be defined for -semigroups as follows.
Definition 4. A vector is named a recurrent vector for if for some increasing sequence . We denote the set of recurrent vectors of by .
It is affirmed in the next theorem that having a dense set of recurrent vectors is an equivalent condition for a -semigroup to be recurrent.
Theorem 3. Let be a -semigroup on . Then, is recurrent if and only if .
Proof. Let . Suppose that be an open and nonempty subset of . So, can be found. Hence, there is an increasing sequence such that . Therefore, can be chosen such that . Hence, .
Now, suppose that . Consider , where . By recurrence of , can be found such that . So, there is .
Let be such thatAnother by recurrence, can be found such that . Therefore, and can be chosen such that andInductively, an increasing sequence can be found such that, for any ,So, for any ,Now, the Cantor theorem implies that for some . Also, by (9). Hence, is a recurrent vector. Moreover, .
Assume that and be two -semigroups on spaces and , respectively. If a continuous map can be found with this property that for any , then and are named quasi-conjugate. We state in the following theorem that quasi-conjugacy preserves recurrence.
Theorem 4. Recurrence of -semigroups preserves under quasi-conjugacy.
Proof. Let be a recurrent -semigroup, and let and be quasi-conjugate. Let be an open and nonempty set. Now, is open by continuity of . Since has a dense set of recurrent vectors, there exists such that . Hence, can be found such that . So, there is such that . Now, by conjugacy, , and thus, we can conclude that . Hence, . Therefore, is recurrent.
Pay attention to the fact that periodic points of a -semigroup are recurrent. In fact, if be a periodic point for -semigroup , then for some . Hence, for any , we have . Therefore, , and hence, is a recurrent vector for .
By this fact, interesting examples can be constructed as follows.
Example 1. Suppose that be a -semigroup on so that, for any , is defined by . Let . So, , and hence, every point of is a periodic point for . Hence, the set of periodic points of is dense, and hence, is recurrent. So, by Theorem 3, is recurrent.
Bermúdez et al. in [12] proved that there are hypercyclic -semigroups that are not chaotic. Hence, there exist recurrent -semigroups that do not have a dense set of periodic points.
By Theorem 2.4 in [12], hypercyclic -semigroups can be found on any infinite-dimensional and separable Banach spaces. So, we can deduce that recurrent -semigroups exist on these spaces since hypercyclic -semigroups are recurrent. By Example 1, we can deduce the following theorem.
Theorem 5. There are finite-dimensional Banach spaces that support recurrent -semigroups.
Proof. In Example 1, we make a recurrent -semigroup on a finite-dimensional Banach space. So, recurrent -semigroups can be found on finite-dimensional spaces.
As it is established in Theorem 7.15 in [4], hypercyclic -semigroups cannot be built on finite-dimensional spaces. So, we can state the following corollary.
Corollary 2. The set of hypercyclic -semigroups is a proper subset of the set of recurrent -semigroups.
Proof. The proof is evident by Theorem 5 and this fact that hypercyclic -semigroups do not exist on finite-dimensional spaces.
Frequently recurrent vectors for operators are defined in [13]. Similarly, frequently recurrent vectors for -semigroups can be defined as follows.
Definition 5. A vector is named a frequently recurrent vector for a -semigroup on ifhas positive lower density for any open set that contains . We denote the set of frequently recurrent vectors of by .
By Definition 5, it is not complicated to see thatBy Theorem 3 and (11), we can conclude the following corollary.
Corollary 3. If a -semigroup has a dense set of frequently recurrent vectors, then it is recurrent.
In the next example, a -semigroup is constructed with a dense set of frequently recurrent vectors.
Example 2. Let , whereConsider . Let . If for any and we definethen is a chaotic -semigroup (Example 7.10 in [4]). Hence, it has a dense set of periodic points in . Therefore, has a dense set of frequently recurrent vectors.
4. Some Criteria for Recurrence of -Semigroups
Some criteria for recurrency of -semigroups are presented in this section. The conditions of the first theorem are based on open sets and neighborhoods of zero. Moreover, it has weaker conditions than conditions in HCC.
Theorem 6. Assume be a -semigroup on . If for any open and nonempty set and any neighborhood of zero, there is some such thatand then, is recurrent.
Proof. Suppose that be an open and nonempty set. So, there is an open set and a neighborhood of zero so that (Lemma 2.36 in [4]). By hypothesis, there is such thatHence, there are and such thatNow, by (16),Therefore, and . Hence, can be selected such that . So, is recurrent.
We can rewrite Theorem 6 as follows.
Theorem 7. Consider be a -semigroup on . If there is such that, for any open and nonempty set and any neighborhood of zero, can be found so thatand then, is recurrent.
Proof. Assume that is a neighborhood of zero. Consider . Then, is a nonempty neighborhood of zero. By hypothesis, can be selected such thatHence,Therefore,So,Now, by Theorem 6, is recurrent.
In Theorem 7, is a positive and arbitrary scalar. So, by considering , we get the following corollary.
Corollary 4. Suppose that is a -semigroup on . If for any open and nonempty set and any neighborhood of zero, can be found so thatthen is recurrent.
Proof. It is enough to consider in Theorem 7
Like Theorem 4.1 in [14], we can define another recurrent criterion that is based on dense subsets.
Theorem 8. Assume that be a -semigroup on . Suppose that there is a dense subset of and there is a sequence of positive real numbers such that(i)For any , (ii)For any , there is in such that and Then, is recurrent.
Proof. Assume that be an open and nonempty set. By density of , there is . Hence, can be chosen such that .
By (i), , and by (ii), can be found such that and . So, for sufficiently large , we haveTherefore, if we consider , then by (24),Hence, and . It means that . Thus, is recurrent.
By Theorem 2.3 in [15], the idea of the next theorem comes to mind.
Theorem 9. Assume be a -semigroup on . Suppose that a dense subset of exists such that(i)For any , when (ii)For any and for any , and exist such thatThen, is recurrent.
Proof. Suppose that be an open and nonempty set. Let , and let be such that . By (i), there is such thatBy (ii), there is such thatSo, if we consider , by (27) and (28), similar to the proof of Theorem 8, can be found such that .
It is considerable that if a -semigroup fulfills the conditions of Theorem 9, then it fulfills the conditions of Theorem 8.
The next sufficient condition for recurrence is based on special eigenvectors of operators of a -semigroup.
Theorem 10. Consume is a -semigroup on . Assume that and exist such thatbe dense in . Then, is recurrent.
Proof. Assume that is an open and nonempty subset of . By density of , we can find . So, we can write , where and for . Now,If we consider , then and sinceHence, andSo, for a large enough , and . Accordingly, is recurrent.
By Theorem 10, we can state the following interesting corollary.
Corollary 5. Consume is a -semigroup on . Assume that is a scalar. If there is such thatbe dense in , then is recurrent.
Proof. Suppose that . Hence, , where . But . So, if we consider , by Theorem 10, is recurrent.
5. Direct Sum of -Semigroups
We investigate the direct sum of recurrent -semigroups and their properties in this section. We begin this section by showing that recurrence of the direct sum of two -semigroups implies recurrence of each of them.
Theorem 11. If be a recurrent -semigroup on , then and are recurrent on and , respectively.
Proof. Assume that be an open and nonempty set in . Then, is an open set of . By hypothesis, is recurrent. So, there exists so thatHence, . This indicates that is recurrent. Similarly, is recurrent.
It is proved that the recurrence of implies the recurrence of and . But if the converse is true?
In the following theorem, we state that the answer to the question is positive if at least one of them be mixing.
Theorem 12. Suppose that is a recurrent -semigroup on , and suppose that is a -semigroup on .(i)If be a mixing semigroup, then is recurrent(ii)If there is such that be mixing, then is recurrent
Proof. To prove part (i), consume is an open subset of . By hypothesis, is mixing. So, there is such that, for any ,Also, is recurrent. So, there is such thatBy Lemma 1, we can assume that . Hence, by (35) and (36),For proving part (ii), note that, by Proposition 7.21 in [4], mixing of implies that is mixing. Hence, is recurrent by part (i).
The following corollary can be deduced from Theorem 12.
Corollary 6. Consume and are mixing -semigroups on and , respectively. Then, is recurrent.
Especially, and are recurrent.
So, notable examples can be made by using mixing -semigroups as follows.
Example 3. Consider to be a sequence in with this property that . Consider is an operator on so that and . It is proved that the generated -semigroup by is mixing (Theorem 1.6 in [16]). So, if we denote this -semigroup with , then by Corollary 6, is recurrent.
Also, we can state the following lemmas by using RHCC and HCC.
Lemma 3. If is RHCC and is HCC, then is recurrent.
Proof. It is deduced from Lemma 5.3 in [8] that is HCC. Hence, it is recurrent.
Lemma 4. If is HCC, then is recurrent. Especially, is recurrent.
Proof. If is HCC, then by Theorem 2.5 in [16] and Theorem 7.28 in [4], is hypercyclic, and so, it is recurrent.
Also, we make the following example by Lemma 4.
Example 4. Let , and let denote an admissible weight function on . Assume that . If , then fulfills the hypercyclicity criterion (Proposition 4.4 in [8]). Then, is recurrent.
Data Availability
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Conflicts of Interest
The author declares that there are no conflicts of interest.