Γ-supercyclicity

Charpentier, Stéphane; Ernst, Romuald; Menet, Quentin

doi:10.1016/j.jfa.2016.03.005

Cited by 8 publications

(29 citation statements)

References 25 publications

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“…The first part of Theorem A gives a complete characterization of hypercyclic subsets among finite dimensional subsets of a separable Hilbert space. It also answers Question 6 from [10] and provides with a wide class of examples of sets which are not hypercyclic. For example, in the Hilbert setting, a segment joining two linearly independent points, a non-trivial sphere, or a non-empty open set of X is never a hypercyclic subset.…”

Section: Question 4 Does There Exist a Countably Hypercyclic Operatomentioning

confidence: 83%

“…Yet, it is worth saying that we cannot apply a Bourdon-Feldman type Theorem to prove the whole Theorem 2.1. Indeed, by [10], the orbit of [1, b]Tx under some non-hypercyclic operator T , on some Banach space X, and for some x ∈ X, may be somewhere dense in X but not everywhere dense (see also Theorem B). Instead we will generalize the original proof of Costakis-Peris' result, as it is given in [25].…”

Section: A Sufficient Condition For a Set In X To Be A Hypercyclic Sumentioning

confidence: 99%

“…The classical Costakis and Peris' result is a consequence of the Bourdon-Feldman Theorem. According to [10,Theorem B], subsets of C of the form [a, b]T -bounded and bounded away from 0, but somewhere dense in C whenever a < b -do not satisfy the property (P'). This result indicates that a positive answer to Question 1 is not likely to be obtained, as for the classical Costakis and Peris' result, as an application of some Bourdon-Feldman type Theorem.…”

Section: Introductionmentioning

confidence: 99%

“…Roughly speaking, they are those multi-dimensional scalar subsets which satisfy a property similar to (P) and (P') respectively. The following question was already mentioned in [10]. Question 2.…”

Section: Introductionmentioning

confidence: 99%

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Hypercyclic subsets

Charpentier

Ernst

2020

JAMA

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We completely characterize the finite dimensional subsets C of any separable Hilbert space for which the notion of C-hypercyclicity coincides with the notion of hypercyclicity, where an operator T on a topological vector space X is said to be Chypercyclic if the set {T n x, n ≥ 0, x ∈ C} is dense in X. We give a partial description for non necessarily finite dimensional subsets. We also characterize the finite dimensional subsets C of any separable Hilbert space H for which the somewhere density in H of {T n x, n ≥ 0, x ∈ C} implies the hypercyclicity of T . We provide a partial description for infinite dimensional subsets. These improve results of Costakis and Peris, Bourdon and Feldman, and Charpentier, Ernst and Menet.2010 Mathematics Subject Classification. 47A16.

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Section: Question 4 Does There Exist a Countably Hypercyclic Operatomentioning

confidence: 83%

Section: A Sufficient Condition For a Set In X To Be A Hypercyclic Sumentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Hypercyclic subsets

Charpentier

Ernst

2020

JAMA

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“…polynomials with an orbit which is dense with respect to the weak topology)? (4) Recently, Charpentier, Ernst and Menet [11] characterized the subsets Γ ⊂ C for which every Γ-supercyclic linear operator (i.e. operators T such that Γ · Orb T (x) is dense for some x ∈ X) is a hypercyclic linear operator.…”

Section: Introductionmentioning

confidence: 99%

Orbits of homogeneous polynomials on Banach spaces

Cardeccia

Muro

2020

Ergod. Th. Dynam. Sys.

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We study the dynamics induced by homogeneous polynomials on Banach spaces. It is known that no homogeneous polynomial defined on a Banach space can have a dense orbit. We show, a simple and natural example of a homogeneous polynomial with an orbit that is at the same time d-dense (the orbit meets every ball of radius d), weakly dense and such that Γ · OrbP (x) is dense for every Γ ⊂ C that is either unbounded or that has 0 as an accumulation point. Moreover we generalize the construction to arbitrary infinite dimensional separable Fréchet spaces. To prove this we study Julia sets of homogeneous polynomials on Banach spaces.However, the behavior of the orbits induced by a homogeneous polynomial can be highly nontrivial and it is far from being understood. For example, in [4] Bernardes showed that the orbits may oscillate between infinity and the boundary of the limit ball. He also proved that every infinite dimensional and separable Banach space supports supercyclic homogeneous polynomials. More recently Peris, Kim and Song [16,17] proved that every separable Banach space of dimension greater than one supports numerically hypercyclic homogeneous polynomials. This means that there are vectors x ∈ S X , x * ∈ S X * for which its numerical orbit, N orb P (x, x * ) := {x * (P n (x)) : n ∈ N 0 } is dense in C.The following result shows that the Julia set of a homogeneous polynomial share some of the properties satisfied by the Julia set of a holomorphic function on the complex plane.Proposition 2.12 (Properties of J P ). Let P be a non linear homogeneous polynomial. The Julia set J P satisfies the following properties:Proof. i) This is clear since J P is the boundary of an open set.ii) First note that y ∈ A P if and only if P (y) ∈ A P . Thus, if x ∈ J P then P (x) is not in A P .On the other hand, there exists (a n ) n ⊆ A P such that a n → x. Since P (a n ) belongs also to A P and P (a n ) → P (x), we have that P (x) ∈ J P .iii) Just note that if x ∈ J P then λx ∈ J P for every |λ| = 1.iv) It suffices to show that A P n = A P . Clearly if P k (x) → 0 then (P n ) k (x) → 0 and therefore A P ⊆ A P n . The converse follows thanks to the existence of the limit ball (Proposition 2.1). If P nk (x) → 0 then Orb P n (x) meets eventually the limit ball of P and therefore P n (x) must tend to zero.In contrast to the one dimensional case, the Julia set J P may be empty. Indeed, if P ∈ P( 2 ℓ 2 ; ℓ 2 ) is defined as P (x) = (x 2 2 , x 2 3 , x 2 4 , . . .), then P n (x) → 0 for every x ∈ ℓ 2 . Thus A P = ℓ 2 and hence J P = R P = ∅.Recall that a set A is completely invariant under P if P (A) ⊆ A and P −1 (A) ⊆ A. The Julia set of a homogeneous polynomials need not to be completely invariant, as we will show in Example 3.19. On the other hand, under certain conditions on P , J P results completely invariant.have that P n (x) → 0 and hence P (x) ∈ A P . If R P = ∅ this implies that x ∈ J P . If R P = ∅ and x ∈ R P , there is some ǫ > 0 so that P n (y) → ∞ for every y ∈ B ǫ (x).Since P is open, P (B ǫ (x)) is an open neighborhood of...

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$$\Gamma $$-Supercyclicity for Strongly Continuous Semigroups

Abbar

2019

Complex Anal. Oper. Theory

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Γ-supercyclicity

Cited by 8 publications

References 25 publications

Hypercyclic subsets

Hypercyclic subsets

Orbits of homogeneous polynomials on Banach spaces

$$\Gamma $$-Supercyclicity for Strongly Continuous Semigroups

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