Chaos and frequent hypercyclicity for composition operators
aa r X i v : . [ m a t h . D S ] S e p CHAOS AND FREQUENT HYPERCYCLICITY FOR COMPOSITIONOPERATORS
UDAYAN B. DARJI AND BENITO PIRES
Abstract.
The notions of chaos and frequent hypercyclicity enjoy an intimate relationship in lineardynamics. Indeed, after a series of partial results, it was shown by Bayart and Rusza in 2015 thatfor weighted bilateral shifts on ℓ p -spaces, the notions chaos and frequent hypercyclicity coincide. Itis with some effort that one shows that these two notions are distinct. Bayart and Grivaux in 2007constructed a non-chaotic frequently hypercyclic weighted shift on c . It was only in 2017 that Menetsettled negatively whether every chaotic operator is frequently hypercylic. In this article, we show thatfor a large class of composition operators on L p -spaces the notions of chaos and frequent hypercyclicitycoincide. Moreover, in this particular class an invertible operator is frequently hypercyclic if and onlyif its inverse is frequently hypercyclic. This is in contrast to a very recent result of Menet wherean invertible frequently hypercyclic operator on ℓ whose inverse is not frequently hypercyclic isconstructed. Introduction
Let H be a separable Banach space and T : H → H a continuous linear operator. T is chaotic if T has a dense orbit and the set of periodic points of T is dense in X . Chaotic operators arewell studied in linear dynamics and well understood. For example, Grosse-Erdmann in [12] gave acomplete characterization of backward weighted shifts which are chaotic.The notion of frequent hypercyclicity is a quantitative version of hypercyclicity. It was introducedby Bayart and Grivaux in [3] in the linear setting, but it also makes sense for Polish dynamicalsystems. Although such notion is purely topological, it is related to measure-theoretical features ofthe topological dynamical system ( H , T ). More specifically by Birkhoff’s Pointwise Ergodic Theoremit follows that if T admits an invariant ergodic Borel probability measure with full support, then T is frequently hypercyclic. Conversely, Grivaux and Matheron showed in [11] that if H is a reflexiveBanach space, then any frequently hypercyclic operator T on H admits a continuous invariant Borelprobability measure with full support. The notion of frequent hypercyclicity is tricky: in fact it tooka span of 10 years between the pioneer partial works [3, 13] and the full characterization of frequentlyhypercyclic backward weighted shifts obtained in [5].An important problem in linear dynamics is distinguishing between chaotic operators and fre-quently hypercyclic operators. Bayart and Grivaux constructed in [4] a frequently hypercyclic con-tinuous linear operator that is not chaotic while Menet showed in [16] that there exist chaotic con-tinuous linear operators that are not frequently hypercyclic. However, for many natural classes ofoperators such as the backward weighted shifts, these two notions coincide. Mathematics Subject Classification.
Primary 47A16, 47B33; Secondary 37D45.
Key words and phrases.
Frequently hypercyclic, chaotic operator, composition operator, L p -space.The first author was supported by grant Another important problem concerning frequently hypercyclic operators is whether the inverseof every invertible frequently hypercyclic operator is frequently hypercyclic. This was solved inthe negative very recently by Menet [17]. However, again for natural classes of operators such asbackward weighted shifts the inverse of an invertible frequently hypercyclic operator is frequentlyhypercyclic.Our aim in this article is to give a very large class of composition operators for which the notionsof chaos and frequent hypercyclicity coincide (see Theorem 3.7 and Corollary 3.9). Moreover, as theinverse of an invertible chaotic operator is chaotic, we also obtain that in our class the inverse of afrequently hypercyclic operator is also frequently hypercyclic (See Corollary 3.8).A powerful method for constructing frequently hypercyclic operator is to apply the Frequent Hyper-cyclicity Criterion (FHC). This criterion was introduced by Bayart and Grivaux in [3] and strength-ened by Bonilla and Grosse-Erdmann in [12]. We also explore the relationship between operatorssatisfying (FHC) and chaotic operators. Theorem 3.1 and Theorem 3.2 imply that in our settingchaotic operators satisfy (FHC). Moreover, we prove the partial converse for composition operatorson L p ( X ), p ≥ L p ( X ) where ( X, B , µ ) is a σ -finite measure spaceand f : X → X is a bijective, bimeasurable, nonsingular transformation satisfying µ (cid:0) f − ( B ) (cid:1) ≤ cµ ( B ) for all B ∈ B and some c >
0. The continuous linear operator T f : L p ( X ) → L p ( X ) definedby ϕ ϕ ◦ f is called the composition operator induced by f . In such a general setting, hypercycliccomposition operators and topologically mixing composition operators on L p -spaces were completelycharacterized in a joint work of the authors with Bayart in [2]. Li-Yorke chaotic composition operatorson L p -spaces were completely characterized in a joint work of the authors with Bernardes Jr. in[6]. Recently, generalized hyperbolicity among composition operators was characterized by the firstauthor, D’Aniello and Maiuriello in [10].We would also like to point out that in a very different direction, Charpentier, Grosse-Erdmannand Menet [9] give conditions under which backward weighted shifts on K¨othe sequence spaces havethe property that the notions of chaos and frequent hypercyclicity coincide.The article is organized as follows: In Section 2 we give definitions and background results. InSection 3 we state the main results and their consequences. Section 4 consists of examples, Section 5of proofs and Section 6 of open problems.2. Definitions and Background Results
Topological dynamics of linear operators
Let T : H → H be a continuous linear operator acting on a separable Banach space H . Definition 2.1.
The operator T is topologically transitive (or hypercyclic) if for any pair of non-empty open sets U, V ⊆ H , there is k > such that T k ( U ) ∩ V = ∅ . If, in addition, the set of periodicpoints of T is dense in H , then T is said to be chaotic. We recall that in the setting of Banach spaces, T is topologically transitive if and only if T admitsa hypercyclic vector ϕ , i.e., ϕ ∈ H such that the orbit { ϕ, T ϕ, T ϕ . . . } is dense in H . Definition 2.2.
The operator T is topologically mixing if for any pair of non-empty open sets U, V ⊆H , there exists k ≥ such that T k ( U ) ∩ V = ∅ for all k ≥ k . HAOS AND FREQUENT HYPERCYCLICITY FOR COMPOSITION OPERATORS 3
Definition 2.3.
A vector ϕ ∈ H is called frequently hypercyclic if for each non-empty open set U ⊆ H , the set of integers N ( ϕ, U ) = { n ∈ N : T n ϕ ∈ U } has positive lower density, that is, lim inf N →∞ N { ≤ n ≤ N : T n ϕ ∈ U } > · The operator T is called frequently hypercyclic if it admits a frequently hypercyclic vector. The following Frequent Hypercyclicity Criterion (FHC) was provided by Bonilla and Grosse-Erdmann in [8, Theorem 2.1]. It is a strengthened version of the original criterion obtained byBayart and Grivaux in [3, Theorem 2.1]. Its simplified reformulation is stated below in the contextthat we use.
Theorem 2.4 (Frequent Hypercyclicity Criterion (FHC)) . Let ( H , k · k ) be a separable Banach spaceand let T : H → H be a continuous linear operator. Assume there exists a dense subset H of H anda map S : H → H such that, for any ϕ ∈ H , ( a ) The series P n ≥ T n ( ϕ ) converges unconditionally; ( b ) The series P n ≥ S n ( ϕ ) converges unconditionally ; ( c ) T S ( ϕ ) = ϕ .Then, T is frequently hypercyclic, chaotic and topologically mixing. Measurable dynamicsDefinition 2.5.
A transformation f : X → X on the measure space ( X, B , µ ) is(a) bimeasurable if f ( B ) ∈ B and f − ( B ) ∈ B for all B ∈ B ;(b) nonsingular if µ (cid:0) f − ( B ) (cid:1) = 0 if and only if µ ( B ) = 0 . Definition 2.6.
A measurable system is a tuple ( X, B , µ, f ) , where(1) ( X, B , µ ) is a σ -finite measure space with µ ( X ) > ;(2) f : X → X is a bijective bimeasurable nonsingular transformation;(3) there is c > such that ( ⋆ ) µ ( f − ( B )) ≤ cµ ( B ) for every B ∈ B . If both f and f − satisfy ( ∗ ) , then we say that the measurable system is invertible. Definition 2.7.
Let p ≥ . The composition operator T f induced by a measurable system ( X, B , µ, f ) is the map T f : L p ( X ) → L p ( X ) defined by T f : ϕ → ϕ ◦ f. It is well-known that ( ⋆ ) guarantees that T f is a continuous linear operator. We refer the readerto [19] for a detailed exposition on compositions operators. Definition 2.8.
A measurable transformation f : X → X on the measure space ( X, B , µ ) is(a) conservative if for each measurable set B of positive µ -measure, there is n ≥ such that µ (cid:0) B ∩ f − n ( B ) (cid:1) > ;(b) dissipative if ∃ W ∈ B such that f n ( W ) , n ∈ Z , are pairwise disjoint and X = ∪ n ∈ Z f n ( W ) .The measurable system ( X, B , µ, f ) is called conservative (respectively, dissipative) if f is conservative(respectively, dissipative). UDAYAN B. DARJI AND BENITO PIRES
Theorem 2.9 (Hopf [1, 15]) . Let ( X, B , µ, f ) be a measurable system. Then, X is the union oftwo disjoint f -invariant sets C ( f ) and D ( f ) , called the conservative and the dissipative parts of f ,respectively, such that f | C ( f ) is conservative and f | D ( f ) is dissipative. Definition 2.10.
Let ( X, B , µ, f ) be a measurable system. A measurable set W ⊂ X is a wanderingset if the sets f n ( W ) , n ∈ Z , are pairwise disjoint. The system ( X, B , µ, f ) is said to be generatedby a wandering set W if X = S n ∈ Z f n ( W ) . In the sequel, we let B ( W ) = { B ∩ W : B ∈ B} . Definition 2.11.
We say that a dissipative system ( X, B , µ, f ) is of bounded distortion if there exista wandering set W of finite positive µ -measure and K > such that(i) W generates ( X, B , µ, f ) , i.e, X = S n ∈ Z f n ( W ) ;(ii) For all n ∈ Z and C ∈ B ( W ) with positive µ -measure, K µ ( f n ( W )) µ ( W ) ≤ µ ( f n ( C )) µ ( C ) ≤ K µ ( f n ( W )) µ ( W ) . . Notice that in the definition of dissipative system (see Definition 2.8), we do not require that W have finite measure. 3. Statement of the Main Results
Our main result is a condition called
Summability Condition or simply Condition (SC) that isuseful for constructing composition operators that are simultaneously chaotic, topologically mixingand frequently hypercyclic. We split our results into two subsections. In the first subsection, weintroduce Condition (SC) and we explore its relation to the Frequent Hypercyclicity Criterion (FHC).In particular, we show that if we add the hypothesis that f is dissipative, then we obtain thatCondition (SC) is equivalent to T f being chaotic. In the second subsection we show that if f isa dissipative transformation of bounded distortion, then Condition (SC) is equivalent to T f beingfrequently hypercyclic.3.1. The Summability Condition and the Frequent Hypercyclicity Criterion
Let ( X, B , µ, f ) be a measurable system. We say that f satisfies the Summability Condition (SC)if for each ǫ > B ∈ B with µ ( B ) < ∞ , there exists B ′ ⊆ B such that(SC) µ (cid:0) B \ B ′ (cid:1) < ǫ and X n ∈ Z µ (cid:0) f n ( B ′ ) (cid:1) < ∞ . The following result shows that the Summability Condition (SC) is the natural translation of theFrequent Hypercyclicity Criterion (FHC) to the composition operator framework.
Theorem 3.1.
Let ( X, B , µ, f ) be a measurable system. For all p ≥ , (SC) implies (FHC). More-over, (SC) and (FHC) are equivalent for all p ≥ . Now we will provide a list of results that show how Condition (SC) is useful to characterize whena transformation f is dissipative and when the composition operator T f is frequently hypercyclic orchaotic. The results below are true for all p ≥ Theorem 3.2 ((SC) Characterization, General Case) . Let ( X, B , µ, f ) be a measurable system and T f : L p ( X ) → L p ( X ) be the associated composition operator. The following statements are equivalent. HAOS AND FREQUENT HYPERCYCLICITY FOR COMPOSITION OPERATORS 5 (a) f satisfies Condition (SC);(b) f is dissipative and T f has a dense set of periodic points.Moreover, any of the above implies that T f is chaotic, topologically mixing and frequently hypercyclic. In the sequel, we will need the following definition. Let X be a metric space and f : X → X bea map. We say that x ∈ X is recurrent (for f ) if for every open set U containing x , there exists aninteger n ≥ f n ( x ) ∈ U . Theorem 3.3 ((SC) Characterization, µ ( X ) < ∞ ) . Let ( X, B , µ, f ) be a measurable system with µ ( X ) < ∞ and T f : L p ( X ) → L p ( X ) be the associated composition operator. Then, the followingstatements are equivalent.(a) f satisfies Condition (SC);(b) f is dissipative;(c) If X is a metric space and µ is a Borel measure, then the set of recurrent points of f has µ -measure zero.Moreover, any of the above implies that T f is chaotic, topologically mixing and frequently hypercyclic. Corollary 3.4 ((SC) Characterization, f dissipative) . Let ( X, B , µ, f ) be a dissipative system and T f : L p ( X ) → L p ( X ) be the associated composition operator. Then, the following statements areequivalent.(a) f satisfies Condition (SC);(b) T f is chaotic;(c) T f has dense set of periodic points.Proof. That (a) implies (b) follows from Theorem 3.2. That (b) implies (c) is simply the definition.That (c) implies (a) follows from Theorem 3.2. (cid:3)
Corollary 3.5 ( f dissipative, µ ( X ) < ∞ ) . Let ( X, B , µ, f ) be a dissipative system with µ ( X ) < ∞ and let T f : L p ( X ) → L p ( X ) be the associated composition operator. Then, f satisfies Condition(SC). Hence, T f is chaotic, topologically mixing and frequently hypercyclic.Proof. This follows directly from Theorem 3.3. (cid:3)
Bounded Distortion and Frequent HypercyclicityTheorem 3.6 (Necessary condition for frequent hypercyclicity) . Let ( X, B , µ, f ) be a measurablesystem with associated composition operator T f frequently hypercyclic. Then for every wandering set W with positive finite µ -measure, the following inequality holds X n ∈ Z (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) d µ d (cid:0) µ ◦ f n (cid:1) (cid:12)(cid:12)(cid:12)(cid:12) W (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) − ∞ < ∞· Theorem 3.7 (Frequent Hypercyclicity Characterization) . Let ( X, B , µ, f ) be a dissipative systemof bounded distortion and T f : L p ( X ) → L p ( X ) be the associated composition operator. Then, thefollowing statements are equivalent.(a) f satisfies Condition (SC);(b) T f is frequently hypercyclic;(c) T f is chaotic. UDAYAN B. DARJI AND BENITO PIRES
Corollary 3.8.
Let ( X, B , µ, f ) be an invertible dissipative system of bounded distortion and T f , T f − : L p ( X ) → L p ( X ) be the associated composition operators. Then, T f is frequently hypercyclic (respec-tively, chaotic) if and only if ( T f ) − is.Proof. This follows from the fact that ( T f ) − = T f − . (cid:3) We say that a set A ⊆ X is f -invariant if f − ( A ) = A . We say that f is ergodic if every f -invariantset A ∈ B satisfies µ ( A ) = 0 or µ ( X \ A ) = 0. Corollary 3.9.
Let ( X, B , µ, f ) be a dissipative system with a purely atomic measure µ . Furthermore,assume that f is ergodic. Then the following statements are equivalent.(a) T f is frequently hypercyclic;(b) T f is chaotic;(c) µ is finite.Proof. ( a ) = ⇒ ( c ) Assume that T f is frequently hypercyclic. Let x ∈ X be an atom of µ . Since f is dissipative and ergodic, we have that W = { x } is a wandering set that generates X , thus µ ( X ) = P n ∈ Z µ (cid:0) f n ( W ) (cid:1) . Applying Theorem 3.6 and using the fact that d µ/ d (cid:0) µ ◦ f n (cid:1) on W is equalto the constant µ ( W ) /µ (cid:0) f n ( W ) (cid:1) , we obtain that µ ( X ) µ ( W ) = X n ∈ Z µ (cid:0) f n ( W ) (cid:1) µ ( W ) = X n ∈ Z (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) d µ d (cid:0) µ ◦ f n (cid:1) (cid:12)(cid:12)(cid:12)(cid:12) W (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) − ∞ < ∞ , implying that µ is finite.( c ) = ⇒ ( b ) It follows from Corollary 3.5 that f satisfies Condition (SC) and hence it is chaotic.( b ) = ⇒ ( a ) As f is dissipative and T f chaotic, by Corollary 3.4, we have that f satisfies Condition(SC) and hence it is frequently hypercyclic. (cid:3) Applications and Examples
In this section we give some applications of our main theorems. We also give some examples whichshow that our theorems are sharp.The first result is an application of Theorem 3.3. It shows that a large class of natural, simplemaps f on R d yields complex behavior of T f . We recall that a linear isomorphism L : R d → R d is hyperbolic if L has no eigenvalue of modulus 1. Theorem 4.1.
Consider a measurable system ( R d , B , µ, f ) where µ is a Borel measure on R d , µ ( R d ) < ∞ , µ ( { } ) = 0 and f is a hyperbolic linear isomorphism. Then, T f is chaotic, topolog-ically mixing and frequently hypercyclic.Proof. By [18, Propositions 2.9, 2.10], there exist f -invariant subspaces E s and E u of R d with R d = E s L E u and an adapted norm k · k on R d with respect to which the map f s = f | E s is a contractionand the map f u = f | E u is a dilation (thus its inverse is a contraction). In this way, if x ∈ R d theneither lim n →∞ k f n ( x ) k = 0 or lim n →∞ k f n ( x ) k = ∞ (with respect to any norm k · k ). Hence, the setof recurrent points of f equals { } , which has µ -measure 0 by hypothesis. Hence, by Theorem 3.3,the proof follows. (cid:3) The following is a concrete application of Theorem 3.3.
HAOS AND FREQUENT HYPERCYCLICITY FOR COMPOSITION OPERATORS 7
Example 4.2.
Let f : R → R be a non-identity affine map of the form x ax + b , < | a | ≤ .Let µ be the probability measure on R defined by µ ( J ) = R J e −| t | d t for all interval J ⊆ R . Then, T f is chaotic, topologically mixing and frequently hypercyclic.Proof. We first show that ( X, B , µ, f ) is a measurable system, i.e., that Condition ( ⋆ ) in Definition2.6 holds. Indeed, such is the case: µ (cid:0) f ( J ) (cid:1) µ ( J ) = R f ( J ) e −| t | d t R J e −| t | d t = a R J e −| at + b | d t R J e −| t | d t ≥ ae −| b | R J e −| at | d t R J e −| t | d t ≥ ae −| b | , where the last inequality follows from the fact 0 < | a | ≤ e −| at | ≥ e −| t | for all t .The transformation f has no recurrent points when a = 1 as, in this case, b = 0 and f n ( x ) = x + nb for all x ∈ R and n ∈ N . For a = 1, some elementary computation shows that f has exactly onerecurrent point, namely the fixed point of f , x = b − a . In either case, as µ is non-atomic, the set ofrecurrent points has measure zero. By Theorem 3.3, we have that T f is chaotic, topologically mixingand frequently hypercyclic. (cid:3) The next example shows that Corollary 3.4 is sharp in the sense that the hypothesis of dissipativitycannot be removed.
Example 4.3.
There exists a measurable system ( X, B , µ, f ) such that T f : L p ( X ) → L p ( X ) ischaotic but f does not satisfy Condition (SC).Proof. We use an example from our earlier work [2]. In particular, our X is an odometer and f isthe +1-map.For i ≥
1, we let Z i = { , . . . , i − } be integers modulo i . We let X = Π ∞ i =1 A i , where A i = Z for i even and A i = Z i for i odd. We put the discrete topology on A i and the associated producttopology on X . Endowed with this topology, X is homeomorphic to the Cantor space.We let B be the collection of Borel subsets of X . We define a product measure µ on X by defininga probability measure µ i on A i as follows: µ i (0) = µ i (1) = 12 , if i is evenand µ i ( j ) = − − i i for j ∈ { , . . . , i − } − i i for j ∈ { i, . . . , i − } , if i is odd.Hence, X = ( X, B , µ ) becomes a topological Borel probability space.The map f : X → X is simply the +1-map with carryover. It was shown in [2] that T f is a well-defined continuous linear operator which is topologically transitive but not topologically mixing.We will show that T f has a dense set of periodic points, implying that T f is chaotic. As T f is nottopologically mixing, by [2, Corollary 2.2] we have that f does satisfy Condition (SC).We recall that the open subsets of X are countable unions of disjoint basic cylinders, i.e., sets ofthe form [ a , . . . , a i ] := { ( x , . . . , x i , x i +1 , . . . ) ∈ X : x = a , ..., x i = a i } . We claim that χ [ a ,...,a i ] is a periodic point of T f . In fact, if N = | A | · | A | · · · | A i | , where | A j | denotesthe cardinality of A j , and x = ( x , x , . . . ) ∈ [ a , . . . , a i ], then f N ( x ) = ( x , x , . . . , x i , y i +1 , . . . ), that UDAYAN B. DARJI AND BENITO PIRES is , f N ( x ) ∈ [ a , . . . , a i ]. Hence, f − N (cid:0) [ a , . . . , a i ] (cid:1) = [ a , . . . , a i ]. In this way, T Nf χ [ a ,...,a i ] = χ f − N (cid:0) [ a ,...,a i ] (cid:1) = χ [ a ,...,a i ] , implying that χ [ a ,...,a i ] is a periodic point of T f .Now it is easy to verify that the characteristic function of the finite union of cylinders is alsoa periodic point. From this one can easily show that collection of simple functions of the form P mi =1 a i · χ C i , where C i is the finite union of cylinders, is dense in L p ( X ), completing the proof. (cid:3) The following example shows that there are simple situations where the full strength of Theorem 3.2is realized. It also shows that the hypothesis of ergodicity is necessary in Corollary 3.9.
Example 4.4.
There exists a dissipative system ( X, B , µ, f ) with a purely atomic measure µ suchthat µ ( X ) = ∞ and T f satisfies Condition (SC).Proof. Let X = Z × Z and f : X → X defined by f (cid:0) ( i, j ) (cid:1) = ( i, j + 1), i, j ∈ Z . Let B = 2 X be thediscrete σ -algebra and µ : B → [0 , ∞ ] be the σ -finite measure defined by µ (cid:0) { ( i, j ) } (cid:1) = 2 −| j | . Clearly, µ ( X ) = ∞ .Now let us verify Condition (SC). Let B ∈ B and ǫ > µ ( B ) < ∞ . As µ ( B ) < ∞ , thereis L ≥ µ ( B \ ([ − L, L ] × [ − L, L ])) < ǫ . Let B ′ = B ∩ ([ − L, L ] × [ − L, L ]). Clearly, µ ( B \ B ′ ) < ǫ . Next we observe that for any ( i, j ) ∈ X , we have that P n ∈ Z µ ( f n ( i, j )) = 3. As B ′ ⊆ [ − L, L ] × [ − L, L ], we have that P n ∈ Z µ ( f n ( B ′ )) ≤ · (2 L + 1) . (cid:3) Proofs of the Main Results
Lemma 5.1 (Orlicz [14, Theorem 4.2.1]) . Let ( X, B , µ ) be a σ -finite measure space. Let the series P n ∈ N ϕ n of elements of L p ( X ) converge unconditionally. Then for each ≤ p ≤ , the series P n ∈ N k ϕ n k p converges, and for each ≤ p < ∞ , the series P n ∈ N k ϕ n k pp converges. Lemma 5.2.
Let ( X, B , µ, f ) be a measurable system. If f satisfies Condition (SC), then f isdissipative.Proof. By the Hopf Decomposition Theorem, we may write X = C ( f ) ∪ D ( f ) as the union of theconservative and the dissipative parts of f , respectively. We will prove that µ (cid:0) C ( f ) (cid:1) = 0. By way ofcontradiction, suppose that B ⊆ C ( f ) is a measurable set with 0 < µ ( B ) < ∞ .We claim that there exist W ⊆ B and N ∈ N such that µ ( W ) > f n ( W ) ∩ W = ∅ for all n ∈ N satisfying | n | ≥ N . By Condition (SC), there exist a measurable set B ′ ⊆ B and N ∈ N suchthat µ ( B \ B ′ ) < µ ( B )4 and X | n |≥ N µ (cid:0) f n ( B ′ ) (cid:1) < µ ( B )4 . Let W = B ′ (cid:15) S | n |≥ N f n ( B ′ ). Notice that µ ( B ′ \ W ) ≤ X | n |≥ N µ (cid:0) f n ( B ′ ) (cid:1) < µ ( B )4 . In this way, µ ( B \ W ) < µ ( B ) /
2, which yields µ ( W ) >
0. Moreover, f n ( W ) ∩ W = ∅ for all n ∈ Z satisfying | n | ≥ N . This proves the claim.Now let A = { x ∈ W : f n ( x ) W, ∀ n ≥ } . Since A ⊂ C ( f ), we have that µ ( A ) = 0. Hence, theset W ′ = W \ S n ∈ Z f n ( A ) has positive µ -measure. Moreover, if x ∈ W ′ , then f n ( x ) ∈ W for infinitelymany positive n , contradicting the claim. Therefore, µ (cid:0) C ( f ) (cid:1) = 0 and f is dissipative. (cid:3) HAOS AND FREQUENT HYPERCYCLICITY FOR COMPOSITION OPERATORS 9
Proof of Theorem 3.1
Proof of ( SC ) = ⇒ ( F HC ) . Assume that f satisfies (SC). We will show that T f satisfies (FHC). ByLemma 5.2, f is dissipative. Let W be a wandering set such that X = S n ∈ Z f n ( W ). Let H = L p ( X )and H ⊆ H be defined as follows: ϕ ∈ H if and only if there exist m ≥ , a , . . . , a m ∈ R , k , . . . , k m ∈ Z , and pairwise disjoint measurable sets B , . . . , B m such that ϕ = P mi =1 a i χ B i and, foreach 1 ≤ i ≤ m , B i ⊆ f k i ( W ) and P n ∈ Z µ (cid:0) f n ( B i ) (cid:1) < ∞ .We will now show that H is dense in H . Given ǫ > ψ ∈ H , let ϕ = P rj =1 a j χ D j ∈ H\{ } be such that D , . . . , D r are pairwise disjoint measurable sets with finite positive µ -measures and k ϕ − ψ k p < ǫ/
2. Set M = max {| a j | : 1 ≤ j ≤ r } , then M >
0. As f satisfies Condition (SC) and X = S n ∈ Z f n ( W ), there exist an integer N > C , . . . , C r such that, for each1 ≤ j ≤ r , C j ⊆ D j ∩ S | n |≤ N f n ( W ), µ ( D j \ C j ) < (cid:16) ǫ rM (cid:17) p and X n ∈ Z µ (cid:0) f n ( C j ) (cid:1) < ∞ . Let ϕ ′ = P rj =1 a j χ C j . Since C j ⊆ S | n |≤ N f n ( W ), we have that each C j is the union of finitely manypairwise disjoint sets B i satisfying B i ⊆ f k i ( W ) for some − N ≤ k i ≤ N . In this way, ϕ ′ ∈ H .Moreover, k ϕ − ϕ ′ k p = (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) r X j =1 a j (cid:0) χ D j − χ C j (cid:1)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) p = (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) r X j =1 a j χ D j \ C j (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) p ≤ M r X j =1 [ µ ( D j \ C j )] p < ǫ . In this way, k ϕ ′ − ψ k p < ǫ , completing the proof of the denseness of H .We next show that P n ≥ T nf ϕ is unconditionally convergent for all ϕ = P mi =1 a i χ B i ∈ H . Since B i ⊆ f k i ( W ), we have that the sets f n ( B i ), n ∈ Z , are pairwise disjoint. Hence, for each sequence ofintegers 1 ≤ n < n < . . . , we have that(1) X j ≥ T n j f ϕ = m X i =1 a i X j ≥ χ f − nj ( B i ) = m X i =1 a i χ ∪ j ≥ f − nj ( B i ) . Since P n ∈ Z µ (cid:0) f n ( B i ) (cid:1) < ∞ for all 1 ≤ i ≤ m , we have that χ ∪ j ≥ f − nj ( B i ) ∈ L p ( X ). Thus P j ≥ T n j f ϕ converges for each sequence of integers 1 ≤ n < n < . . . . Therefore, P n ≥ T nf ϕ isunconditionally convergent and Condition (a) in Theorem 2.4 is true.Given ϕ = P mi =1 a i χ B i ∈ H , let Sϕ = P mi =1 a i χ f ( B i ) . Since f is bijective, bimeasurable andnon-singular, we have that Sϕ ∈ H and S : ϕ → Sϕ is a self-map on H . Moreover, T Sϕ = P mi =1 a i χ f − ( f ( B i )) = ϕ , showing that Condition (c) in Theorem 2.4 is true. By proceeding as in (1),we can show that P n ≥ S n ϕ is unconditionally convergent for all ϕ ∈ H . We have proved that T f satisfies (FHC). Proof of ( F HC ) = ⇒ ( SC ) when p ≥ . Assume the hypothesis, i.e., T f satisfies the FrequentHypercyclicity Criterion (FHC). Let H ⊆ L p ( X ) be as in the statement of (FHC). Let 0 < ǫ < and B ∈ B with µ ( B ) < ∞ . By the denseness of H , there is ϕ ∈ H such that k ϕ − χ B k p < ǫ ( p ) . Let B ′ = { x ∈ B : | ϕ ( x ) − | ≤ ǫ } . Then, ǫ [ µ ( B \ B ′ )] p ≤ (cid:18)Z B \ B ′ | ϕ − | p d µ (cid:19) p = (cid:18)Z B \ B ′ | ϕ − χ B | p d µ (cid:19) p ≤ k ϕ − χ B k p < ǫ ( p ) , showing that µ ( B \ B ′ ) < ǫ .We next show that P n ≥ µ ( f − n ( B ′ )) < ∞ . Proceeding as earlier we have that for any n ≥ − ǫ ) (cid:2) µ (cid:0) f − n ( B ′ ) (cid:1)(cid:3) p ≤ (cid:18)Z f − n ( B ′ ) | ϕ ◦ f n | p d µ (cid:19) p implying that X n ≥ µ (cid:0) f − n ( B ′ ) (cid:1) < p X n ≥ (cid:18)Z f − n ( B ′ ) | ϕ ◦ f n | p d µ (cid:19) ≤ p X n ≥ k T nf ϕ k pp . Since P n ≥ T nf ϕ converges unconditionally and p ≥
2, by Theorem 5.1 we have that X n ≥ µ (cid:0) f − n ( B ′ ) (cid:1) < ∞ . We next show that P n ≥ µ ( f n ( B ′ )) < ∞ . As { ϕ, Sϕ, . . . , S n ϕ } ⊂ H and T f S is the identity mapon H , we have that T nf S n ( ϕ ) = ϕ for all n ≥
1. Since f is bijective, bimeasurable and non-singular,we have that (cid:2) T nf S n ( ϕ ) | B = ϕ | B (cid:3) = ⇒ [ S n ( ϕ ) ◦ f n | B = ϕ | B ] = ⇒ (cid:2) S n ( ϕ ) | f n ( B ) = ϕ ◦ f − n | f n ( B ) (cid:3) . Using this we have that(1 − ǫ ) (cid:2) µ (cid:0) f n ( B ′ ) (cid:1)(cid:3) p ≤ (cid:18)Z f n ( B ′ ) (cid:12)(cid:12) ϕ ◦ f − n (cid:12)(cid:12) p d µ (cid:19) p = (cid:18)Z f n ( B ′ ) | S n ( ϕ ) | p d µ (cid:19) p ≤ k S n ϕ k p . Since P n ≥ S nf ϕ converges unconditionally and p ≥
2, by Lemma 5.1 we have that X n ≥ µ (cid:0) f n ( B ′ ) (cid:1) ≤ p X n ≥ k S n ( ϕ ) k pp < ∞ , which completes the proof.5.2. Proof of Theorem 3.2
Proof of ( a ) = ⇒ ( b ). Suppose that f satisfies Condition (SC). Then, by Lemma 5.2, f is dissi-pative. By Theorem 3.1, T f satisfies (FHC). Hence, by Theorem 2.4, T f is frequently hypercyclic,topologically mixing and chaotic. In particular, T f has a dense set of periodic points. Proof of ( b ) = ⇒ ( a ) . Since f is dissipative, there exist a wandering set W ∈ B of positive µ -measuresuch that X = S n ∈ Z f n ( W ). Let ǫ > B ∈ B with µ ( B ) < ∞ . Let n ∈ N be such that if B = B ∩ [ | k |≤ n f k ( W ) , then µ ( B \ B ) < ǫ . By hypothesis, there exists a periodic point ϕ ∈ L p ( X ) of T f such that k ϕ − χ B k pp ≤ p · ǫ . HAOS AND FREQUENT HYPERCYCLICITY FOR COMPOSITION OPERATORS 11
For B ′ = (cid:26) x ∈ B : | ϕ ( x ) | > (cid:27) , we have that µ ( B \ B ′ ) < ǫ p µ ( B \ B ′ ) ≤ Z B \ B ′ | ϕ − | p d µ ≤ k ϕ − χ B k pp ≤ p · ǫ , implying µ (cid:0) B \ B ′ (cid:1) ≤ ǫ . In this way, µ ( B \ B ′ ) = µ ( B \ B ) + µ ( B \ B ′ ) < ǫ. Let N ≥ n + 1 be such that T Nf ϕ = ϕ ◦ f N = ϕ µ -a.e. Hence, since f is bijective, we have that ϕ ◦ f kN = ϕ for all k ∈ Z . Because B ′ ⊆ B and N ≥ n + 1, we have that the sets in the family (cid:8) f kN ( B ′ ) : k ∈ Z (cid:9) are pairwise disjoint. Moreover,34 X k ∈ Z µ (cid:0) f kN ( B ′ ) (cid:1) ≤ X k ∈ Z Z f kN ( B ′ ) | ϕ ◦ f − kN | d µ = X k ∈ Z Z f kN ( B ′ ) | ϕ | d µ == Z S k ∈ Z f kN ( B ′ ) | ϕ | d µ ≤ Z | ϕ | d µ, showing that P k ∈ Z µ (cid:0) f kN ( B ′ ) (cid:1) < ∞ . By ( ⋆ ), we have that X n ∈ Z µ ( f n ( B ′ )) = N − X i =0 X k ∈ Z µ ( f kN − i ( B ′ )) ≤ N max { , c, . . . , c ( N − } · X k ∈ Z µ (cid:0) f kN ( B ′ ) (cid:1) < ∞ . This proves that Condition (SC) is true, which concludes the proof of ( b ) = ⇒ ( a ).5.3. Proof of Theorem 3.3
Proof of ( a ) = ⇒ ( b ). This is Lemma 5.2. Proof of ( b ) = ⇒ ( a ). Assume f is dissipative and µ ( X ) < ∞ . Then, there exists a wandering set W such that X = S n ∈ Z f n ( W ). Let ǫ > B ∈ B with µ ( B ) < ∞ be given. Let us verify Condition(SC). Let N ∈ N be such that µ (cid:16) B \ S | n |≤ N f n ( W ) (cid:17) < ǫ . Then B ′ = B ∩ S | n |≤ N f n ( W ) verifiesCondition (SC) because µ ( B \ B ′ ) < ǫ and by the dissipativity of f , we have that X n ∈ Z µ (cid:0) f n ( B ′ ) (cid:1) ≤ (2 N + 1) µ ( X ) < ∞ . Concerning the next two proofs, we assume that X is a metric space and µ is a Borel measure. Proof of ( a ) = ⇒ ( c ). By Lemma 5.2, f is dissipative, thus the set of recurrent points has µ -measurezero. Proof of ( c ) = ⇒ ( a ). The proof consists in verifying Condition (SC). Hence, to begin with, let ǫ > B ∈ B with 0 < µ ( B ) < ∞ be given. We may assume that B has no recurrent points. For each n ≥
1, set A n = (cid:26) x ∈ B : { f ( x ) , f ( x ) , . . . } ∩ B (cid:18) x, n (cid:19) = ∅ (cid:27) , where B (cid:0) x, n (cid:1) = { y ∈ X : d ( y, x ) < n } , where d is the metric on X . Notice that A ⊆ A ⊆ . . . and B = S ∞ n =1 A n . In this way, there exist N large enough so that µ ( B \ A N ) < ǫ . Let K be acompact subset of A N such that µ ( A N \ K ) < ǫ . Let U , U , . . . , U r be a finite collection of ballsof radius N such that K ⊂ S ri =1 U i . Since U i ∩ K ⊆ A N , we have that if x ∈ U i ∩ K , then { f ( x ) , f ( x ) , . . . } ∩ U i = ∅ . In particular, if K i = U i ∩ K , then the following statements are true:(a) K = S ri =1 K i ;(b) f n ( K i ) ∩ K i = ∅ for all n ≥ The bijectivity of f and (b) imply that W is a forward wandering set (i.e. K i , f ( K i ) , f ( K i ),. . . arepairwise disjoint) and a backward wandering set (i.e. K i , f − ( K i ), f − ( K i ), . . . , are pairwise disjoint).Hence, X n ∈ Z µ (cid:0) f n ( K i ) (cid:1) ≤ µ ( X ) . In this way, µ ( B \ K ) < ǫ and X n ∈ Z µ (cid:0) f n ( K ) (cid:1) ≤ rµ ( X ) < ∞ . Thus, Condition (SC) is true, which concludes the proof.5.4.
Proof of Theorem 3.6
Throughout this section, ( X, B , µ, f ) will denote a measurable system. Given a measurable set W ⊂ X with finite positive µ -measure, let (cid:0) d n ( W ) (cid:1) n ∈ Z be the sequence defined by(2) d n ( W ) = (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) d µ d (cid:0) µ ◦ f n (cid:1) (cid:12)(cid:12)(cid:12)(cid:12) W (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) − ∞ , where we set 1 / ∞ = 0 so that d n ( W ) is well-defined even in the case in which (cid:13)(cid:13)(cid:13)(cid:13) d µ d (cid:0) µ ◦ f n (cid:1) (cid:12)(cid:12)(cid:12)(cid:12) W (cid:13)(cid:13)(cid:13)(cid:13) ∞ = ∞ . Lemma 5.3.
Let c > be as in ( ⋆ ) and let W be a measurable set with finite positive µ -measure.Then, d n +1 ( W ) ≥ c − d n ( W ) for all n ∈ Z .Proof. By ( ⋆ ), µ (cid:0) f n ( B ) (cid:1) ≤ cµ (cid:0) f n +1 ( B ) (cid:1) for all B ∈ B and n ∈ Z . Hence, the Radon-Nikodymderivatives d (cid:0) µ ◦ f n (cid:1) / d (cid:0) µ ◦ f n +1 (cid:1) , n ∈ Z , are bounded by c at µ ◦ f n +1 -almost every point, andtherefore, µ -a.e. In this way, for all n ∈ Z ,(3) d µ d (cid:0) µ ◦ f n +1 (cid:1) = d µ d (cid:0) µ ◦ f n (cid:1) · d (cid:0) µ ◦ f n (cid:1) d (cid:0) µ ◦ f n +1 (cid:1) ≤ c d µ d (cid:0) µ ◦ f n (cid:1) µ -a.e.Therefore, for all n ∈ Z , we have that(4) 1 d n +1 ( W ) = (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) d µ d (cid:0) µ ◦ f n +1 (cid:1) (cid:12)(cid:12)(cid:12)(cid:12) W (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∞ ≤ c (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) d µ d (cid:0) µ ◦ f n (cid:1) (cid:12)(cid:12)(cid:12)(cid:12) W (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∞ = c d n ( W ) , showing that d n +1 ( W ) ≥ c − d n ( W ) for all n ∈ Z . (cid:3) Lemma 5.4.
Suppose that T f is frequently hypercyclic. If W is a wandering set with finite positive µ -measure, then there exists a set A ⊂ N with positive lower density such that for each n ∈ A , X m ∈ A d m − n ( W ) < . Proof.
Let ϕ ∈ L p ( X ) be a frequently hypercyclic vector for T f and let W be a wandering set withfinite positive µ -measure. Let(5) A = n n ∈ N : (cid:13)(cid:13) T nf ϕ − χ W (cid:13)(cid:13) pp < p µ ( W ) o , then A has positive lower density. HAOS AND FREQUENT HYPERCYCLICITY FOR COMPOSITION OPERATORS 13
By combining the definition of A with the fact that W is a wandering set, we obtain for each n ∈ A ,(6) Z W | ϕ ◦ f n − | p d µ + X m =0 Z f m ( W ) | ϕ ◦ f n | p d µ ≤ (cid:13)(cid:13) T nf ϕ − χ W (cid:13)(cid:13) pp < p µ ( W ) . In particular, the second term of (6) satisfies the following inequality for each n ∈ A ,(7) X m =0 Z f m ( W ) | ϕ ◦ f n | p d µ < p µ ( W ) . Applying the triangle inequality to the first term of (6) yields for each n ∈ A ,(8) [ µ ( W )] p − (cid:18)Z W | ϕ ◦ f n | p d µ (cid:19) p ≤ (cid:18)Z W | ϕ ◦ f n − | p d µ (cid:19) p <
12 [ µ ( W )] p . Therefore, renaming n by k in (8) yields(9) Z W (cid:12)(cid:12) ϕ ◦ f k (cid:12)(cid:12) p d µ > p µ ( W ) , ∀ k ∈ A. In what follows, keep n ∈ A fixed and let m ∈ A − n be arbitrary. Setting k = n + m in (9) yields R W | ϕ ◦ f n + m | p d µ > p µ ( W ). By combining this with (7), we reach,(10) X m ∈ A − nm =0 R f m ( W ) | ϕ ◦ f n | p d µ R W | ϕ ◦ f n + m | p d µ < . By the Change of Variables Formula, for all m ∈ A − n ,(11) Z f m ( W ) | ϕ ◦ f n | p d µ = Z f m ( W ) (cid:12)(cid:12) ϕ ◦ f n + m ◦ f − m (cid:12)(cid:12) p d µ = Z W (cid:12)(cid:12) ϕ ◦ f n + m (cid:12)(cid:12) p d (cid:0) µ ◦ f m (cid:1) . Replacing (11) in (10) yields(12) X m ∈ A − nm =0 R W | ϕ ◦ f n + m | p d (cid:0) µ ◦ f m (cid:1) R W | ϕ ◦ f n + m | p d µ d (cid:0) µ ◦ f m (cid:1) d (cid:0) µ ◦ f m (cid:1) < . For all m ∈ A − n ,(13) Z W (cid:12)(cid:12) ϕ ◦ f n + m (cid:12)(cid:12) p d µ d (cid:0) µ ◦ f m (cid:1) d (cid:0) µ ◦ f m (cid:1) ≤ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) d µ d (cid:0) µ ◦ f m (cid:1) (cid:12)(cid:12)(cid:12)(cid:12) W (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∞ · Z W (cid:12)(cid:12) ϕ ◦ f n + m (cid:12)(cid:12) p d (cid:0) µ ◦ f m (cid:1) . Replacing (13) in (12) and canceling out the nonzero term R W | ϕ ◦ f n + m | p d (cid:0) µ ◦ f m (cid:1) yields X m ∈ A − nm =0 (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) d µ d (cid:0) µ ◦ f m (cid:1) (cid:12)(cid:12)(cid:12)(cid:12) W (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) − ∞ < . Therefore, after renaming variables, we obtain X m ∈ A d m − n ( W ) = X m ∈ A (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) d µ d (cid:0) µ ◦ f m − n (cid:1) (cid:12)(cid:12)(cid:12)(cid:12) W (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) − ∞ = 1 + X m ∈ A − nm =0 (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) d µ d (cid:0) µ ◦ f m (cid:1) (cid:12)(cid:12)(cid:12)(cid:12) W (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) − ∞ < . (cid:3) Lemma 5.5 (Bayart-Ruzsa [5, Corollary 9]) . Let ( α n ) n ∈ Z be a sequence of non-negative real numberssuch there exists C > such that either α n +1 ≥ Cα n for all n ∈ Z , or α n ≥ Cα n +1 for all n ∈ Z . Suppose that for some set A ⊂ Z with positive upper density the sequence ( β n ) n ∈ A defined by β n = P m ∈ A α m − n is bounded. Then P n ∈ Z α n < ∞ . Proof of Theorem 3.6
Set α n = d n ( W ) for all n ∈ Z . By Lemma 5.3, we have that α n +1 ≥ c − α n for all n ∈ Z . Let A be as in Lemma 5.4 and let (cid:0) β n (cid:1) n ∈ A be defined by β n = P m ∈ A α m − n . Then, by Lemma 5.4, for each n ∈ A , β n = X m ∈ A α m − n = X m ∈ A d m − n ( W ) < . Hence, (cid:0) β n (cid:1) n ∈ A is bounded. By Lemma 5.5, X n ∈ Z d n ( W ) = X n ∈ Z α n < ∞ . Proof of Theorem 3.7
Proof of ( a ) = ⇒ ( b ). This follows immediately from Theorem 3.2.Proof of ( b ) = ⇒ ( a ). Let B ∈ B with µ ( B ) < ∞ and ǫ > W be a wandering set of finite positive µ -measure satisfying the bounded distortion Conditions(i) and (ii) in Definition 2.11. For each i ∈ Z , set W i = f i ( W ). By (i), there exists N ∈ N such thatif B ′ = B ∩ S | i |≤ N f i ( W ), then µ ( B \ B ′ ) < ǫ . We claim that there exists K > − N ≤ i ≤ N , each D ∈ B ( W i ) with positive µ -measure and each n ∈ Z , µ ( D ) µ (cid:0) f n ( D ) (cid:1) ≤ K µ ( W i ) µ (cid:0) f n ( W i ) (cid:1) . In fact, since f is bijective, bimeasurable and non-singular, any set D ∈ B ( W i ) has positive µ -measureif and only if D = f i ( C ), where C ∈ B ( W ) has positive µ -measure. By Condition (ii) in Definition2.11, we have that µ (cid:0) W i (cid:1) µ (cid:0) f n ( W i ) (cid:1) = µ (cid:0) f i ( W ) (cid:1) µ (cid:0) f n + i ( W ) (cid:1) = µ (cid:0) f i ( W ) (cid:1) µ ( W ) · µ ( W ) µ (cid:0) f n + i ( W ) (cid:1) ≥ K µ (cid:0) f i ( C ) (cid:1) µ ( C ) µ ( C ) µ (cid:0) f n + i ( C ) (cid:1) = 1 K µ ( D ) µ (cid:0) f n ( D ) (cid:1) . which proves the claim.Hence, since D ∈ B ( W i ) is an arbitrary measurable set of positive µ -measure, we reach µ ( W i ) µ (cid:0) f n ( W i ) (cid:1) ≥ K d µ d (cid:0) µ ◦ f n (cid:1) ( x ) , for µ -almost every x ∈ W i . Therefore, since W i is a wandering set and T f is frequently hypercyclic, by Theorem 3.6 we reach X n ∈ Z µ (cid:0) f n ( W i ) (cid:1) µ ( W i ) ≤ K X n ∈ Z (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) d µ d (cid:0) µ ◦ f n (cid:1) (cid:12)(cid:12)(cid:12)(cid:12) W i (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) − ∞ < ∞ . HAOS AND FREQUENT HYPERCYCLICITY FOR COMPOSITION OPERATORS 15
In this way, X n ∈ Z µ (cid:0) f n ( B ′ ) (cid:1) ≤ X n ∈ Z X | i |≤ N µ (cid:0) f n ( W i ) (cid:1) = X | i |≤ N µ ( W i ) X n ∈ Z µ (cid:0) f n ( W i ) (cid:1) µ ( W i ) < ∞ . Proof of ( a ) ⇐⇒ ( c ). This follows immediately from Corollary 3.4.6. Problems
Below we list some problems related to our work.
Problem 6.1.
In Theorem 3.1, we show that (FHC) implies (SC) for p ≥ . What happens for ≤ p < ? Problem 6.2.
Is the condition of bounded distortion necessary in Theorem 3.7?
Problem 6.3.
In Example 4.3 is T f frequently hypercyclic? If not, this would give an alternativesolution to a result of Menet [16] . Problem 6.4.
We have thoroughly studied the case when the measurable system is dissipative. Whathappens in the case when the measurable system is conservative? In particular, what can be saidwhen X is an odometer endowed with the product measure? Characterizations of hypercyclic andtopologically mixing operators of such type were given in [7] . References [1] Jon Aaronson.
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Department of Mathematics, University of Louisville, Louisville, KY 40292, USA.
E-mail address : [email protected] Departamento de Computac¸˜ao e Matem´atica, Faculdade de Filosofia, Ciˆencias e Letras, Univer-sidade de S˜ao Paulo, Ribeir˜ao Preto, SP, 14040-901, Brazil.
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