Vincent B Éhani

A STUDY OF BISHOP OPERATORS FROM THE POINT OF VIEW OF LINEAR DYNAMICS

Keywords: . 2020 Mathematics Subject Classification. 47A16, 47A10, 11J70, 54E52, 47A15, 46E30 Bishop-type operators, Cyclic vectors, Non-trivial closed invariant subspaces, Approximation by rational numbers, Co-meager sets

In this paper, we study the so-called Bishop operators Tα on L p ([0, 1]), with α ∈ (0, 1) and 1 < p < +∞, from the point of view of linear dynamics. We show that they are never hypercyclic nor supercyclic, and investigate extensions of these results to the case of weighted translation operators. We then investigate the cyclicity of the Bishop operators Tα. Building on results by Chalendar and Partington in the case where α is rational, we show that Tα is cyclic for a dense G δ -set of irrational α's, discuss cyclic functions and provide conditions in terms of convergents of α ∈ R\Q implying that certain functions are cyclic.

Introduction

One of the most famous open problems in modern operator theory is the Invariant Subspace Problem in the Hilbertian setting, which can be stated as follows:

Let H be a separable infinite-dimensional complex Hilbert space. For every bounded linear operator T acting on H, does there exist a non-trivial closed invariant subspace for T ?

Bishop suggested in the fifties a family of operators as counterexamples to the Invariant Subspace Problem on the complex Hilbert space L 2 ([0, 1]), defined for every α ∈ [0, 1] and every f ∈ L 2 ([0, 1]) by T α f (x) = xf ({x + α}) a.e. on [0, 1],

where {•} denotes the fractional part of a real number. These Bishop operators can be seen as weighted translation operators defined for every α ∈ [0, 1], every ϕ ∈ L ∞ ([0, 1]) and every f ∈ L 2 ([0, 1]) by T ϕ,α f (x) = ϕ(x)f ({x + α}) a.e. on [0, 1], that is to say the composition of the multplication operator M ϕ and of the translation operator U α defined on L 2 ([0, 1]) by

M ϕ f = ϕf and U α f = f ({• + α}) for every f ∈ L 2 ([0, 1]).
Even if these two types of operators are separately very well understood, the behaviour of the Bishop operators or of the weighted translation operators are still rather mysterious in the general case of irrational numbers α.

In 1965, Parrott first studied in [START_REF] Parrott | Weighted translation operators[END_REF] these weighted translation operators and succeeded to compute the spectrum of the Bishop operators, proving that σ(T α ) = z ∈ C; |z| ≤ e -1 for every irrational α ∈ [0, [START_REF] Bayart | Dynamics of linear operators[END_REF].

By means of a functional calculus approach, Davie proved in [START_REF] Davie | Invariant subspaces for Bishop's operators[END_REF] that whenever α is a non-Liouville number in [0, 1], the operator T α admits a non-trivial hyperinvariant subspace, that is to say a non-trivial subspace which is invariant by every operator which commutes with T α . Recall that an irrational number α is a Liouville number if there exists a sequence (p n /q n ) n≥0 of rational numbers such that α -p n q n < 1 q n n for every n ≥ 0.

These results have been extended for instance by MacDonald in [START_REF] Macdonald | Invariant subspaces for Bishop-type operators[END_REF] to the weighted translation operators T ϕ,α . However the extensions only involve new weights ϕ while the parameter α still must be a non-Liouville number.

The first one who was able to pass this barrier of Liouville numbers was Flattot in 2008, who proved in [START_REF] Flattot | Hyperinvariant subspaces for Bishop-type operators[END_REF] that T α admits non-trivial hyperinvariant subspaces for some Liouville numbers α.

The set of such parameters was recently further extended in [START_REF] Chamizo | Invariant subspaces for Bishop operators and beyond[END_REF] by Chamizo, Gallardo-Gutiérrez, Monsalve-López and Ubis to include more Liouville numbers. However the existence of non-trivial closed invariant subspaces for T α for every irrational number α in [0, 1] is still a tantalizing open problem.

We recall that an operator T does not admit any non-trivial closed invariant subspace if and only if the subspace generated by the T -orbit {T n x; n ≥ 0} is dense in H for every x ∈ H\{0}, that is to say if every x ∈ H\{0} is cyclic for T.

It is thus an interesting question to investigate whether the operators T α are cyclic, that is to say admit a vector whose orbit under the action of T α spans a dense subspace of L 2 ([0, 1]). Surprisingly enough, this natural question seems to have been considered only in the case where α ∈ Q, by Chalendar, Partington and Pozzi, see [START_REF] Chalendar | Modern approaches to the invariant-subspace problem[END_REF] and [START_REF] Chalendar | Multivariable weighted composition operators: lack of point spectrum, and cyclic vectors[END_REF]. It is the main goal of this paper to study cyclicity of T α for α ∈ R\Q. We also consider the related hypercyclicity (existence of a dense orbit) and supercyclicity (existence of a projective dense orbit) notions. Since the Bishop operators and the weighted translation operators are well-defined on the spaces L p ([0, 1]) with 1 < p < +∞, we will carry out our study in this setting.

The paper is organized as follows. Section 2 deals with hypercyclicity properties of Bishop and weighted translation operators. After recalling the main definitions, we observe (Proposition 2.7) that the operators T α cannot be hypercyclic, and that the weighted translation operators do not satisfy the so-called Hypercyclicity Criterion (Theorem 2.9). In Section 3, we will study supercyclicity properties of our operators. We show (Proposition 3.13) that weighted translation operators are not supercyclic for a large class of weights, and that no weighted translation operator satisfies the Supercyclicity Criterion (Theorem 3.14). The proofs of these results rely on the Positive Supercyclicity Criterion from [START_REF] León-Saavedra | Rotations of hypercyclic and supercyclic operators[END_REF] as well as on an observation of the point spectrum of the adjoints of these operators (Theorem 3.12). In Section 4 we study cyclicity of the operators T α acting on L p ([0, 1]). Building on results from [START_REF] Chalendar | Modern approaches to the invariant-subspace problem[END_REF] which characterize cyclic functions for T α when α ∈ Q, and using Baire Category arguments, we show (Theorem 4.9) that T α is cyclic for a co-meager set of parameters α ∈ [0, 1]. We then explicit a set of irrational numbers α ∈ [0, 1] such that T α is cyclic, using the continued fraction expansion of α. More precisely, for a certain class of functions f ∈ L p ([0, 1]), we show that f is cyclic for T α as soon as the sequence (q n ) n≥0 of denominators of the convergents of α ∈ R\Q has infinitely many sufficiently large gaps (Theorem 4.16). We then generalize the results from [START_REF] Chalendar | Modern approaches to the invariant-subspace problem[END_REF] to the case of weighted translation operators with α rational, and generalize our cyclicity results for irrational α to this setting (Theorems 4.24, 4.27 and 4.28). The final Section 5 presents some further remarks and open questions.

Hypercyclicity

The first well known dynamical property we study is called hypercyclicity. We refer the reader to the books [START_REF] Bayart | Dynamics of linear operators[END_REF] and [START_REF] Grosse-Erdmann | Linear chaos[END_REF] for an in-depth study of this notion, as well as further topics in linear dynamics. Let X be a separable infinite-dimensional complex Banach (or Fréchet) space. We will denote by B(X) the set of linear and continuous operators on X. Definition 2.1 ( [START_REF] Bourdon | Cyclic composition operators on H 2[END_REF]Introduction]). An operator T ∈ B(X) is said to be hypercyclic if there exists x ∈ X such that the T -orbit {T n x; n ≥ 0} is dense in X. In this case x is called a hypercyclic vector for T.

This notion is linked to a problem similar to the Invariant Subspace Problem called the Invariant Subset Problem. Indeed, an operator T ∈ B(X) does not admit any non-trivial invariant closed subset if and only if for every x ∈ X\{0} the T -orbit {T n x; n ≥ 0} is dense in X, that is to say, if and only if every x ∈ X\{0} is hypercyclic for T.

The following theorem shows that the set of hypercyclic vectors of an operator T ∈ B(X) is either empty or a dense G δ -set in X. We recall that a G δ -set in X is a countable intersection of open sets of X. Theorem 2.2 (Birkhoff's transitivity theorem, [4, §62]). An operator T ∈ B(X) is hypercyclic if and only if for every pair (U, V ) of non-empty open sets in X, there exists n ≥ 0 such that T n (U ) ∩ V ̸ = ∅. In this case, the set of hypercyclic vectors for T is a dense G δ -set in X.

Hypercyclicity being a well studied notion, we give here some important examples of hypercyclic operators.

• ∥ n,∞ : f ∈ H(C) → sup |z|≤n |f (z)|, n ≥ 1. The operators λB on ℓ 2 (N), D on H(C) and T (a) on H(C), defined for every (x 1 , x 2 , . . .) ∈ ℓ 2 (N) and every f ∈ H(C) by λB(x 1 , x 2 , . . .) = (λx 2 , λx 3 , . . .), D(f ) = f ′ and T (a) f = f (• + a)
and respectively called Rolewicz's operator, MacLane's operator and Birkhoff's operator, are hypercyclic whenever |λ| > 1 and a ̸ = 0.

A useful tool to prove the hypercyclicity of an operator is the following criterion.

Theorem 2.4 (Hypercyclicity Criterion, [START_REF] Bès | Hereditarily hypercyclic operators[END_REF]Theorem 2.3]). Let T ∈ B(X). Suppose that there exist dense subsets X 0 and Y 0 of X, an increasing sequence (n k ) k≥1 of integers and a sequence (S n k : Y 0 → X) k≥1 of maps such that:

(i)

T n k x → 0 as k → +∞ for every x ∈ X 0 ; (ii) S n k y → 0 as k → + ∞ for every y ∈ Y 0 ; (iii) T n k S n k y → y as k → +∞ for every y ∈ Y 0 .
Then the operator T is hypercyclic.

Bès and Peris actually proved in [START_REF] Bès | Hereditarily hypercyclic operators[END_REF]Theorem 2.3] that an operator T on X satisfies the Hypercyclicity Criterion if and only if the operator T ⊕ T : (x, y) ∈ X ⊕ X → (T x, T y) is hypercyclic. It was an open question for a long time to give examples of hypercyclic operators not satisfying the Hypercyclicity Criterion.

It has been proved by Peris in [START_REF] Peris | Hypercyclicity criteria and the Mittag-Leffler theorem[END_REF]Theorem 2.3] that the Hypercyclicity Criterion as stated in Theorem 2.4 is actually equivalent to a similar criterion involving a partial inverse map S and its iterates (S n k ) k≥0 instead of a sequence of maps (S n k ) k≥0 .

Theorem 2.5 (Gethner-Shapiro Criterion, [START_REF] Gethner | Universal vectors for operators on spaces of holomorphic functions[END_REF]Theorem 2.2] and [START_REF] Grosse-Erdmann | Linear chaos[END_REF]Theorem 3.10]). Let T ∈ B(X). Suppose that there exist dense subsets X 0 and Y 0 of X, an increasing sequence (n k ) k≥1 of integers and a map S : Y 0 → Y 0 such that:

(i)

T n k x → 0 as k → +∞ for every x ∈ X 0 ; (ii) S n k y → 0 as k → +∞ for every y ∈ Y 0 ; (iii) T Sy = y for every y ∈ Y 0 .
Then the operator T is hypercyclic.

These two criteria are in fact equivalent, altough the Hypercyclicity Criterion looks more general than the Gethner-Shapiro Criterion.

Theorem 2.6 ([21, Theorem 2.3]). An operator T ∈ B(X) satisfies the Hypercyclicity Criterion if and only if it satisfies the Gethner-Shapiro Criterion.

2.1. Hypercyclicity of Bishop operators. First of all, it is useful to compute the iterates of T α . For every α ∈ [0, 1], every f ∈ L p ([0, 1]) and every n ≥ 0 T n α f (x) = x{x + α} . . . {x + (n -1)α}f ({x + nα}) a.e. on [0, 1]. We will be interested here in the hypercyclicity of the Bishop operators, and later on in the hypercyclicity of the weighted translation operators. Bishop operators are easily seen not to be hypercyclic. Proposition 2.7. For every α ∈ [0, 1], the Bishop operator T α is not hypercyclic.

Proof. For every

f ∈ L p ([0, 1]), ∥T α f ∥ p p = 1 0 x p |f ({x + α})| p dx ≤ 1 0 |f (y)| p dy ≤ ∥f ∥ p p .
Thus ∥T α ∥ ≤ 1, so every T α -orbit is bounded in L p ([0, 1]) and T α is not hypercyclic. □ 2.2. Hypercyclicity of weighted translation operators.

Since ∥T ϕ,α ∥ = ∥ϕ∥ ∞ for every ϕ ∈ L ∞ ([0, 1]) and every α ∈ [0, 1], T ϕ,α cannot be hypercyclic if ∥ϕ∥ ∞ ≤ 1.
We also observe that T ϕ,α cannot be hypercyclic in the case where m({ϕ = 0}) > 0 if m denotes the Lebesgue measure on [0, 1].

Proposition 2.8. Let ϕ ∈ L ∞ ([0, 1]) and let α ∈ [0, 1]. If ∥ϕ∥ ∞ ≤ 1 or if m({ϕ = 0}) > 0,
then the operator T ϕ,α is not hypercyclic.

Proof. Suppose that m({ϕ = 0}) > 0. Since for every f ∈ L p ([0, 1]) and every n ≥ 0,

T n ϕ,α f (x) = ϕ(x)ϕ({x + α}) . . . ϕ({x + (n -1)α})f ({x + nα}) a.e. on [0, 1], then {ϕ = 0} ⊂ {T n ϕ,α f = 0}. Hence the orbit {T n ϕ,α f ; n ≥ 0} cannot be dense in L p ([0, 1]) since for every n ≥ 0 ∥T n ϕ,α f -1 {ϕ=0} ∥ p p ≥ {ϕ=0} 1dx ≥ m({ϕ = 0}) > 0.

□

In the next section, we will extend the set of weights ϕ such that T ϕ,α is not hypercyclic, but we don't know if T ϕ,α can ever be hypercyclic. However we will prove that it cannot satisfy the Gethner-Shapiro Criterion, and hence cannot satisfy the Hypercyclicity Criterion either. Theorem 2.9. Let ϕ ∈ L ∞ ([0, 1]) and let α ∈ [0, 1]. The operator T ϕ,α cannot satisfy the Hypercyclicity Criterion.

Proof. Suppose that T ϕ,α satisfies the Hypercyclicity Criterion. Then by Theorem 2.6, it satisfies the Gethner-Shapiro Criterion. Suppose also that m({ϕ = 0}) = 0. One can remark that for every f, h

∈ L p ([0, 1]), we have T ϕ,α f = h if and only if f (x) = h({x -α})/ϕ({x -α}) a.e. on [0, 1].
Let X 0 and Y 0 be the two dense sets given by the Gethner-Shapiro Criterion, and let S : Y 0 → Y 0 be the associated inverse map necessarily given by Sh(x) = h({x-α})/ϕ({x-α}) a.e. on [0, 1] for every h ∈ Y 0 . Since X 0 and Y 0 are dense subsets of L p ([0, 1]), there exist f ∈ X 0 and g ∈ Y 0 such that m({|f | > 1}) ≥ 3/4 and m({|g| > 1}) ≥ 3/4. Otherwise one would have for every

f ∈ X 0 ∥f -2∥ p p ≥ {|f |≤1} |2 -|f (x)|| p dx ≥ m({|f | ≤ 1}) ≥ 1 4 ,
contradicting the density of X 0 in L p ([0, 1]) (a similar argument holds for g ∈ Y 0 ). Then by the Cauchy-Schwarz inequality, we have for every k ≥ 0

1 0 |f ({x + n k α})g(x)| p/2 dx 2 ≤ 1 0 |ϕ(x) . . . ϕ({x + (n k -1)α})f ({x + n k α})| p dx × 1 0 |g(x)| p |ϕ(x) . . . ϕ({x + (n k -1)α})| p dx ≤ ∥T n k ϕ,α f ∥ p p 1 0 |g({y -n k α})| p |ϕ({y -n k α}) . . . ϕ({y -α})| p dy ≤ ∥T n k ϕ,α f ∥ p p ∥S n k g∥ p p ----→ k→+∞ 0.
Hence there exists a subsequence (m k ) k≥0 of (n k ) k≥0 such that f ({x + m k α})g(x) → 0 as k → +∞ a.e. on [0, 1]. We consider now the sets ω = {|f | > 1}, Ω = {|g| > 1} and

Ω ′ = Ω ∩ {x ∈ [0, 1]; f ({x + m k α})g(x) → 0 as k → +∞}. We have m(Ω ′ ) = m(Ω) ≥ 3/4. Since Ω ′ ⊂ K≥0 k≥K {x ∈ [0, 1]; {x + m k α} / ∈ ω} and since the sequence (∩ k≥K {x ∈ [0, 1]; {x + m k α} / ∈ ω}) K≥0 is increasing, it follows that the sequence (m(∩ k≥K {x ∈ [0, 1]; {x + m k α} / ∈ ω})) K≥0 admits a limit satisfying m(Ω ′ ) ≤ lim K→+∞ m   k≥K {x ∈ [0, 1]; {x + m k α} / ∈ ω}   ≤ lim inf K→+∞ m(R -m K α ([0, 1]\ω))
where R α is the translation

R α : x ∈ [0, 1] → {x+α} which preserves the Lebesgue measure m. So 3 4 ≤ m(Ω ′ ) ≤ lim inf K→+∞ m(R -m k α ([0, 1]\ω)) ≤ lim inf K→+∞ m([0, 1]\ω) ≤ 1 4 ,
which is a contradiction. □

Supercyclicity

We now consider a less restrictive dynamical property, called supercyclicity, which has been much studied in the literature. ). An operator T ∈ B(X) is said to be supercyclic if there exists x ∈ X such that the subset {λT n x; λ ∈ C, n ≥ 0} is dense in X. In this case x is called a supercyclic vector for T. The Birkhoff's Theorem admits a supercyclic version, which runs at follows. Theorem 1.12]). An operator T ∈ B(X) is supercyclic if and only if for every pair (U, V ) of non-empty open sets in X, there exist n ≥ 0 and λ ∈ C such that λT n (U ) ∩ V ̸ = ∅. In this case, the set of supercyclic vectors for T is a dense G δ -set in X.

Theorem 3.2 ([1,
The so-called Supercyclicity Criterion is the most useful tool to prove that an operator is supercyclic. It is patterned after the Hypercyclicity Criterion. Theorem 3.3 (Supercyclicity Criterion, [START_REF] Bermúdez | On hypercyclicity and supercyclicity criteria[END_REF]Lemma 3.1]). Let T ∈ B(X). Suppose that there exist dense subsets X 0 and Y 0 of X, an increasing sequence (n k ) k≥1 of integers, a sequence (λ n k ) k≥1 of non-zero complex numbers and a sequence (S n k : Y 0 → X) k≥1 of maps such that:

(i) λ n k T n k x → 0 as k → +∞ for every x ∈ X 0 ; (ii) λ -1 n k S n k y → 0 as k → +∞ for every y ∈ Y 0 ; (iii) T n k S n k y → y as k → +∞ for every y ∈ Y 0 .
Then the operator T is supercyclic.

We are looking for a supercyclic version of Theorem 2.9. To do so, we will prove that the Supercyclicity Criterion is equivalent to a Gethner-Shapiro-type Supercyclicity Criterion. For this we will need the notion of universality which is a generalization of hypercyclicity applied to a family of operators (T n ) n≥0 instead of the iterates of a unique operator T. Definition 3.4 ([14, Definition 1]). Let Y be a metric space. A sequence (T n : X → Y ) n≥0 of continuous maps is said to be universal if there exists x ∈ X such that its orbit {T n x; n ≥ 0} under (T n ) n≥0 is dense in Y. In this case x is called a universal vector for (T n ) n≥0 . Theorem 3.5 (Universality Criterion, [START_REF] Erdmann | Universal families and hypercyclic operators[END_REF]Theorem 1]). Let Y be a separable metric space and (T n : X → Y ) n≥0 a family of continuous maps. The set of universal vectors for (T n ) n≥0 is dense in X if and only if for every non-empty open sets U and V of X and Y, there exists n ≥ 0 such that T n (U ) ∩ V ̸ = ∅. In this case the set of universal vectors for

(T n ) n≥0 is a dense G δ -set in X.
With some adjustements in the proof of the equivalence between the Hypercyclicity Criterion and the Gethner-Shapiro Criteron, given in [START_REF] Grosse-Erdmann | Linear chaos[END_REF]Theorem 3.22], one can prove that the Supercyclicity Criterion is equivalent to the following Gethner-Shapiro-type Supercyclicity Criterion.

Theorem 3.6. An operator T ∈ B(X) satisfies the Supercyclicity Criterion if and only if the following holds true: there exist dense subsets X 0 and Y 0 of X, an increasing sequence (n k ) k≥1 of integers, a sequence (λ n k ) k≥1 of non-zero complex numbers and a map S :

Y 0 → Y 0 such that: (i) λ n k T n k x → 0 as k → +∞ for every x ∈ X 0 ; (ii) λ -1 n k S n k y → 0 as k → +∞ for every y ∈ Y 0 ; (iii) T Sy = y for every y ∈ Y 0 .
Proof. We only have to prove that the Supercyclicity Criterion implies the criterion given by Theorem 3.6. So let us suppose that T satisfies the Supercyclicity Criterion. Since T has dense range the Mittag-Leffler's Theorem, as stated in [START_REF] Grosse-Erdmann | Linear chaos[END_REF]Theorem 3.21] with X n = X and f n = T : X → X for every n ∈ N, gives that the subspace Y = x ∈ X; there exists (x n ) n∈N ∈ X N , x = x 1 and T x n+1 = x n for every n ∈ N is dense in X. Consider X N endowed with the product topology, and its closed subspace

X = (x n ) n∈N ∈ X N ; T x n+1 = x n for every n ∈ N ,
which is a separable Fréchet space endowed with the induced topology. We consider the operator T : X → X defined by T (x 1 , x 2 , . . .) = (T x 1 , T x 2 , . . .) for every (x 1 , x 2 , . . .) ∈ X and its inverse B : X → X defined by B(x 1 , x 2 , . . .) = (x 2 , x 3 , . . .) for every (x 1 , x 2 , . . .) ∈ X .

Let us now prove that the family (λ n k T n k ) k≥1 is topologically transitive on X , that is to say that for every pair (U 0 , V 0 ) of non-empty open sets in X , there exists

k ≥ 1 such that λ n k T n k (U 0 ) ∩ V 0 ̸ = ∅. Given U 0 and V 0 , there exist N ≥ 1 and non-empty open sets U 1 , . . . , U N , V 1 , . . . , V N of X such that U = {(x n ) n∈N ∈ X ; x j ∈ U j for every j ∈ {1, . . . , N }} ⊂ U 0 and V = {(y n ) n∈N ∈ X ; y j ∈ V j for every j ∈ {1, . . . , N }} ⊂ V 0 . Let x = (x n ) n∈N ∈ U and y = (y n ) n∈N ∈ V. Since T j x N = x N -j ∈ U N -j and T j y N = y N -j ∈ V N -j whenever j ∈ {1, . . . , N }, there exist non-empty open neighborhoods U ′ N ⊂ U N and V ′ N ⊂ V N of x N and y N such that T j (U ′ N ) ⊂ U N -j and T j (V ′ N ) ⊂ V N -j whenever j ∈ {1, . . . , N }. Since T satisfies the Supercyclicity Criterion, there exists k ≥ 1 such that λ n k T n k (U ′ N ) ∩ V ′ N ̸ = ∅,
U ′′ N ⊂ U ′ N such that λ n k T n k (U ′′ N ) ⊂ V ′ N . Let u N ∈ Y ∩ U ′′ N , which exists by density of Y. Since u N ∈ Y , there exists (u n ) n≥N such that T u n+1 = u n whenever n ≥ N. Moreover we consider u j = T N -j u N ∈ U j whenever j ∈ {1, . . . , N -1} since u N ∈ U ′′ N ⊂ U ′ N . So u = (u n ) n∈N ∈ U and besides λ n k T n k u ∈ V. Indeed λ n k T n k u = (λ n k T n k u 1 , λ n k T n k u 2 , . . .) and λ n k T n k u j = λ n k T n k T N -j u N = T N -j (λ n k T n k u N ) ∈ T N -j (V ′ N ) ⊂ V j whenever j ∈ {1, . . . , N } because λ n k T n k u N ∈ λ n k T n k (U ′′ N ) ⊂ V ′ N . Thus λ n k T n k u ∈ λ n k T n k (U) ∩ V, which is hence non-empty.
Let us prove now that there exist a dense subset Y ′ 0 of X, a map S :

Y ′ 0 → Y ′ 0 and a subsequence (m k ) k≥1 of (n k ) k≥1 such that T Sy = y and λ -1 m k S m k y → 0 as k → +∞ whenever y ∈ Y ′ 0 . Since (λ n k T n k ) k≥1 is topologically transitive, so is (λ -1 n k B n k ) k≥1 and the sets {x ∈ X ; {λ n k T n k x; k ≥ 1} is dense in X } and x ∈ X ; λ -1 n k B n k x; k ≥ 1 is dense in X are G δ -
subsets of X by Theorem 3.5. So there exists y = (y n ) n∈N ∈ X such that the sets

{λ n k T n k y; k ≥ 1} = {(λ n k T n k y 1 , λ n k T n k y 2 , . . .); k ≥ 1} and λ -1 n k B n k y; k ≥ 1 = (λ -1 n k y 1+n k , λ -1 n k y 2+n k , . . .); k ≥ 1 are dense in X .
We now consider the subset Y ′ 0 = {λy n ; λ ∈ C\{0} and n ∈ N} of X, which is dense in X by density of the projection onto the first coordinate of the dense subset

{λ -1 n k B n k y; k ≥ 1} in X . Let us consider the map S : Y ′ 0 → Y ′ 0 defined by S(λy n ) = λy n+1
for every λy n ∈ Y ′ 0 . It is well defined because if λy n = µy m with n < m, then y n = (µ/λ)y m and T m-n y m = y n = (µ/λ)y m since y ∈ X . Hence the subset

{λ n k T n k y m ; k ≥ 1} ⊂ span[T k y m ; k ∈ {0, . . . , m -n -1}]
could not be dense in X, which would contradict the density of the projection onto the m-th coordinate of the dense subset {λ n k T n k y; k ≥ 1} of X .

Moreover T Sλy n = λT y n+1 = λy n whenever λy n ∈ Y ′ 0 because y ∈ X . Since the subset

λ -1 n k B n k y; k ≥ 1 = λ -1 n k y 1+n k , λ -1 n k y 2+n k , . . . is dense in X , there exists a subsequence (m k ) k≥1 of (n k ) k≥1 such that λ -1 m k B m k y = λ -1 m k y 1+m k , λ -1 m k y 2+m k , . . . → 0 as k → +∞ and hence λ -1 m k S m k λy n = λλ -1 m k y n+m k → 0 as k → +∞ whenever λy n ∈ Y ′ 0 .
Eventually since T satisfies the Supercyclicity Criterion, we consider the dense subset X ′ 0 = X 0 of X which has the property that λ n k T n k x → 0 as k → +∞ whenever x ∈ X ′ 0 . We have thus shown that T satisfies the following three properties:

(i) λ m k T m k x → 0 as k → +∞ for every x ∈ X ′ 0 ; (ii) λ -1 m k S m k y → 0 as k → +∞ for every y ∈ Y ′ 0 ; (iii) T Sy = y for every y ∈ Y ′
0 , and this terminates the proof. □ 3.1. Supercyclicity of Bishop operators. In order to prove that the Bishop operator cannot be supercyclic we will need the so-called Positive Supercyclicity Theorem, proved by León-Saavedra and Müller in [18, Corollary 4], see also [START_REF] Bayart | Dynamics of linear operators[END_REF]Corollary 3.4]. We will denote by σ p (T ) the set of eigenvalues of an operator T ∈ B(X).

Theorem 3.7 (Positive Supercyclicity Theorem, [1, Corollary 3.4]). Let T ∈ B(X). If σ p (T * ) = ∅, then x ∈ X is supercyclic for T if and only if the set {aT n x; a ∈ (0, +∞), n ≥ 0} is dense in X.

We will also need the following straightforward lemma.

Lemma 3.8. If a subset A is dense in L p ([0, 1]), then the set of the Lebesgue measures {m({ℜ(f

) > 0}); f ∈ A} is dense in [0, 1].
Davie proved in [10, Theorem 2] that the Bishop operator T α has no eigenvalue whatever the value of α ∈ [0, 1]. The same ideas are recalled and generalized by Flattot in [START_REF] Flattot | Hyperinvariant subspaces for Bishop-type operators[END_REF]Theorem 2.1]. In fact, the approach proposed by Flattot in [START_REF] Flattot | Hyperinvariant subspaces for Bishop-type operators[END_REF] allows us to prove that the adjoint of the Bishop operator

T * α ∈ B(L p ′ ([0, 1])), defined for every f ∈ L p ′ ([0, 1]) by T * α f (x) = {x -α}f ({x -α})
a.e. on [0, 1] with 1/p + 1/p ′ = 1, has no eigenvalue. Since the arguments are extremely similar to those of Flattot in [START_REF] Flattot | Hyperinvariant subspaces for Bishop-type operators[END_REF], we omit the proof. Theorem 3.9. For every α ∈ [0, 1], the operators

T α ∈ B(L p ([0, 1])) and T * α ∈ B(L p ′ ([0, 1])) have no eigenvalue.
We can now deduce from the Positive Supercyclicity Theorem that T α is not supercyclic. Theorem 3.10. For every α ∈ [0, 1], the Bishop operator T α is not supercyclic.

Proof. Let f ∈ L p ([0, 1]). By Theorem 3.7, f is supercyclic for T α if and only if the set {aT n α f ; a > 0, n ≥ 0} is dense in L p ([0, 1]).
f ) > 0}) = m({ℜ(f )({• + nα}) > 0}) = m(R -n α ({ℜ(f ) > 0})) = m({ℜ(f ) > 0})
since R α preserves the measure m. Then the set of the Lebesgue measures, which is equal to the singleton {m({ℜ(f

) > 0})}, is not dense in [0, 1]. Hence f is not supercyclic for T α . □ 3.2. Supercyclicity of weighted translation operators. Let ϕ ∈ L ∞ ([0, 1]
), one can remark already that T ϕ,α cannot be supercyclic if the measure m({ϕ = 0}) is positive.

Proposition 3.11. Let ϕ be a function in L ∞ ([0, 1]) satisfying m({ϕ = 0}) > 0 and let α ∈ [0, 1]
. The operator T ϕ,α is not supercyclic.

Proof. For every λ ∈ C, every n ≥ 0 and every f ∈ L p ([0, 1]), we have the inequality

0 < m({ϕ = 0}) ≤ m({λT n ϕ,α f = 0}), so the set {λT n ϕ,α f ; λ ∈ C, n ≥ 0} cannot be dense in L p ([0, 1]). □
In order to generalize the result of Theorem 3.10 about the non-supercyclicity to the weighted translation operators T ϕ,α , one would need the adjoint operator T * ϕ,α , defined on

L p ′ ([0, 1]) by T * ϕ,α f (x) = ϕ({x -α})f ({x -α}) for every f ∈ L p ′ ([0, 1]
) and a.e. on [0, 1], to have no eigenvalue whenever α ∈ [0, 1].

The argument of Flattot in [11, Theorem 2.1] proving that weighted translation operators T ϕ,α have no eigenvalue can be adjusted to show the following result. As in Theorem 3.9, we omit the proof because it follows the proof of Theorem 2.1 in [START_REF] Flattot | Hyperinvariant subspaces for Bishop-type operators[END_REF]. Theorem 3.12. Let ϕ ∈ L ∞ ([0, 1]) be an increasing convex function such that ϕ(0) = 0. For every α ∈ [0, 1], the operators T ϕ,α ∈ B(L p ([0, 1])) and

T * ϕ,α ∈ B(L p ′ ([0, 1])) have no eigenvalue.
As in the case of Bishop operators (Theorem 3.10), the non-negativity of the weight gives the following result. Proposition 3.13. Let ϕ ∈ L ∞ ([0, 1]) be an increasing convex function such that ϕ(0) = 0. For every α ∈ [0, 1], the weighted translation operator T ϕ,α cannot be supercyclic.

Proof. Let f ∈ L p ([0, 1]). Since σ p (T * ϕ,α ) = ∅, by Theorem 3.7, f is supercyclic for T ϕ,α if and only if the set {aT n ϕ,α f ; a > 0, n ≥ 0} is dense in L p ([0, 1]). Then by Lemma 3.8, the set {m({ℜ(aT n ϕ,α f ) > 0}); a > 0, n ≥ 0} is dense in [0, 1]. Since aT n ϕ,α f (x) = ϕ(x)ϕ({x + α}) . . . ϕ({x + (n -1)α})f ({x + nα})
a.e. on [0, 1], for every a > 0 and every n ≥ 0, then

ℜ(aT n ϕ,α f )(x) > 0 if and only if ϕ(x)ϕ({x + α}) . . . ϕ({x + (n -1)α})ℜ(f )({x + nα}) > 0. So m({ℜ(aT n ϕ,α f ) > 0}) = m(R -n α ({ℜ(f ) > 0})) = m({ℜ(f ) > 0})
since ϕ is positive on (0, 1] and R α preserves the measure m. Then the set of the Lebesgue measures, which is equal to the singleton {m({ℜ(f )(x) > 0})}, is not dense in [0, 1], and f is not supercyclic for T ϕ,α . □ Using Theorem 3.6, we can prove that the Supercyclicity Criterion cannot be satisfied by any weighted translation operator. The proof proceeds in the same way as in the hypercyclic case. Theorem 3.14. For every ϕ ∈ L ∞ ([0, 1]) and every α ∈ [0, 1], the operator T ϕ,α does not satisfy the Supercyclicity Criterion.

Proof. Suppose that T ϕ,α satisfies the Supercyclicity Criterion. Then by Theorem 3.6, it satisfies the Gethner-Shapiro-type Supercyclicity Criterion with data X 0 , Y 0 and S. Suppose also that m({ϕ = 0}) = 0. Since X 0 and Y 0 are dense subsets of L p ([0, 1]), let f ∈ X 0 and g ∈ Y 0 be functions such that m({|f | > 1}) ≥ 3/4 and m({|g| > 1}) ≥ 3/4. Then by the Cauchy-Schwarz inequality, for every k ≥ 0

1 0 |f ({x + n k α})g(x)| p/2 dx 2 ≤ 1 0 |λ n k ϕ(x) . . . ϕ({x + (n k -1)α})f ({x + n k α})| p dx × 1 0 |g(x)| p |λ n k ϕ(x) . . . ϕ({x + (n k -1)α})| p dx ≤ ∥λ n k T n k ϕ,α f ∥ p p 1 0 |g({y -n k α})| p |λ n k ϕ({y -n k α}) . . . ϕ({y -α})| p dy ≤ ∥λ n k T n k ϕ,α f ∥ p p λ -1 n k S n k g p p ----→ k→+∞ 0 with the map S : Y 0 → Y 0 defined by Sg(x) = g({x -α})/ϕ({x -α}) a.e. on [0, 1] for every g ∈ Y 0 . As in the hypercyclic case if (m k ) k≥0 is a subsequence of (n k ) k≥0 such that f ({x + m k α}g(x) → 0 as k → +∞ for every x in a full measure set A in [0, 1], then the sets ω = {|f | > 1} and Ω ′ = {|g| > 1} ∩ A satisfy 3 4 ≤ m(Ω ′ ) ≤ m   K≥0 k≥K {x ∈ [0, 1]; {x + m k α} / ∈ ω}   ≤ m([0, 1]\ω) ≤ 1 4 ,
which is impossible. □

Cyclicity

Our aim is now to investigate the cyclicity properties of the Bishop and weighted translation operators. Cyclicity is the less restrictive dynamical property of operators which we consider in this paper. We will denote by C[ξ] the set of complex polynomials.

Definition 4.1 ([13, Introduction]

). An operator T ∈ B(X) is said to be cyclic if there exists x ∈ X such that the linear subspace span[T n x; n ≥ 0] = {P (T )x; P ∈ C[ξ]} is dense in X. In this case x is called a cyclic vector for T.

As in the case of hypercyclicity and supercyclicity, there exists a well known necessary and sufficient condition for the cyclicity of an operator in the case where T * has no eigenvalue, proved by Baire Category arguments. Proposition 4.2. Let T ∈ B(X) such that σ p (T * ) = ∅. The following assertions are equivalent:

(i) T is cyclic;

(ii) For every pair (U, V ) of non-empty open sets in X, there exists P ∈ C[ξ] such that P (T )(U ) ∩ V ̸ = ∅. In this case, the set of cyclic vectors for T is a dense G δ -set in X.

Proof. Suppose that the operator T is cyclic. Let x be a cyclic vector for T and let U and V be two non-empty open sets in X. There exists P ∈ C[ξ] such that P (T )x ∈ U. Since σ p (T * ) = ∅, the operator P (T ) has dense range. Hence one can find an open set W of X such that P (T )(W ) ⊂ V. Once again by cyclicity there exists

Q ∈ C[ξ] such that Q(T )x ∈ W and then P (T )Q(T )x = Q(T )P (T )x ∈ Q(T )(U ) ∩ V ̸ = ∅.
Conversely, let us suppose that the condition (ii) holds and let (U k ) k≥1 be a basis of open sets in X. A vector x ∈ X is cyclic for T if and only if for every k ≥ 1, there exists

P ∈ C[ξ] such that P (T )x ∈ U k , that is to say if and only if x ∈ k≥1 P ∈C[ξ] P (T ) -1 (U k ). The union ∪ P ∈C[ξ] P (T ) -1 (U k ) is a dense open set in X for every k ≥ 1. Indeed for every non-empty open set V in X there exists P ∈ C[ξ] such that P (T )(V ) ∩ U k ̸ = ∅, that is to say that P (T ) -1 (U k ) ∩ V ̸ = ∅. So ∪ P ∈C[ξ] P (T ) -1 (U k ) ∩ V ̸ = ∅.
By the Baire Category Theorem, the set of cyclic vectors is a dense G δ -set in X and thus a non-empty set. ) and let α = r/q be a rational number such that 0 < r < q and r and q are coprime. We define ∆(f, r/q) a.e. on [0, 1] by

f (t) T r/q f (t) • • • T q-1 r/q f (t) f ({t + r/q}) T r/q f ({t + r/q}) • • • T q-1 r/q f ({t + r/q}) . . . . . . . . .
f ({t + (q -1)r/q}) T r/q f ({t + (q -1)r/q}) • • • T q-1 r/q f ({t + (q -1)r/q}) .

This function is used in [START_REF] Chalendar | Modern approaches to the invariant-subspace problem[END_REF]Theorem 5.4.4] to give a necessary and sufficient condition for a function f ∈ L p ([0, 1]) to be cyclic for T α , where α ∈ (0, 1) ∩ Q. Theorem 4.4 ([7, Theorem 5.4.4]). Let α = r/q be a rational number such that 0 < r < q and r and q are coprime. A function f ∈ L p ([0, 1]) is cyclic for T r/q if and only if the function ∆(f, r/q) satisfies m({t ∈ [0, 1]; ∆(f, r/q)(t) = 0}) = 0.

One can deduce from the previous theorem a set of common cyclic vectors for the family of operators T α , α ∈ (0, 1) ∩ Q. Theorem 4.5. Any holomorphic function f on a open neighborhood of [0, 1] such that f (0) ̸ = 0 is cyclic for T α for every α ∈ (0, 1) ∩ Q.

Proof. Let α = r/q be a rational number such that 0 < r < q and r and q are coprime. One can remark that the function |∆(f, r/q)| is a 1/q-periodic function by permutation of the rows. Besides the function ∆(f, r/q) is holomorphic on (0, 1/q) and right-continuous at 0 since ∆(f, r/q)(t) is equal for every t ∈ [0, 1/q) to the determinant t + ir q . . . t + (i+j-1)r q f t + (i+j)r q 0≤i,j≤q-1

.

Suppose that f is not cyclic for T r/q . Thus m({t ∈ [0, 1]; ∆(f, r/q)(t) = 0}) > 0 by Theorem 4.4. Since |∆(f, r/q)| is 1/q-periodic, m({t ∈ [0, 1/q); |∆(f, r/q)(t)| = 0}) > 0. Hence ∆(f, r/q)(t) = 0 for every t ∈ [0, 1/q) because ∆(f, r/q) is holomorphic on (0, 1/q) and right-continuous at 0. However ∆(f, r/q)(0) is equal to the determinant

f (0) 0 • • • • • • 0 f ({r/q}) {r/q}f ({2r/q}) • • • • • • {r/q} . . . {(q -1)
f ({(q -1)r/q}) {(q -1)r/q}f (0) 0 • • • 0 which is in turn equal to (-1) (q-2)(q-1)/2 f (0) q q-2 i=0 {(q -1)r/q} • • • {(q -1 -i)r/q} ̸ = 0.
This contradiction shows that f is cyclic for T α . □ Remark 4.6. In particular, for every α ∈ (0, 1) ∩ Q the Bishop operator T α is cyclic and the constant function 1 is a common cyclic vector for these operators.

Definition 4.7. For every subset A of [0, 1], we define the set

Cycl A = α∈A {f ∈ L p ([0, 1]); f is cyclic for T α }.
of common cyclic vectors for all operators T α , α ∈ A.

In particular Cycl Q∩(0,1) contains any holomorphic function f on an open neighborhood of [0, 1] such that f (0) ̸ = 0. 4.2. Cyclicity of Bishop operators in the irrational case. We are now going to use the cyclicity of the Bishop operator T α for all rational numbers α ∈ (0, 1) to deduce the cyclicity of T α for some irrational numbers α ∈ (0, 1). To do so we will first need a notion of large sets in Baire spaces, which are called co-meager sets. We refer the reader to [START_REF] Kechris | Classical descriptive set theory[END_REF]Section 8.A] for more on this notion. Definition 4.8 ([17, Section 8.A]). Let X be a Polish space, that is to say a separable completely metrizable topological space. A subset A of X is said to be:

(i) a co-meager set in X if it contains a dense G δ -set in X;

(ii) a meager set in X if the complement X\A is a co-meager set in X.

(iii) satisfying the Baire property if there exists an open set U of X such that A∆U = A\U ∪ U \A is a meager set in X.

Our aim is now to prove the following theorem:

Theorem 4.9. The set {α ∈ [0, 1]; T α is cyclic} is a co-meager set in [0, 1].
To do so, we recall the Kuratowski-Ulam Theorem.

Theorem 4.10 (Kuratowski-Ulam, [17, Theorem 8.41]). Let X and Y be two Polish spaces and let A be a subset of X × Y satisfying the Baire property. Then the following assertions are equivalent:

(i) A is a co-meager set in X × Y ; (ii) {x ∈ X; {y ∈ Y ; (x, y) ∈ A} is a co-meager set in Y } is a co-meager set in X; (iii) {y ∈ Y ; {x ∈ X; (x, y) ∈ A} is a co-meager set in X} is a co-meager set in Y.
One can remark that a co-meager set satisfies the Baire property, the chosen open set being the full space. Then we deduce from the Kuratowski-Ulam's Theorem the following corollary. To prove Theorem 4.9, our first strategy is to directly apply Corollary 4.11 to the set

A = {(α, f ) ∈ [0, 1] × L p ([0, 1]); f is cyclic for T α },
which does not give any information about the irrationals α such that T α is cyclic. A second more explicit strategy would be to find a G δ -set in {α ∈ [0, 1]; T α is cyclic} (see Remark 4.17).

We will now show that A is a dense G δ -set in [0, 1] × L p ([0, 1]). To do so, we will need the following lemma. Lemma 4.12. For every Q ∈ C[ξ], the map

ϕ Q : ([0, 1], |•|) × (L p ([0, 1]), ∥ • ∥ p ) → (L p ([0, 1]), ∥ • ∥ p ) (α, f ) → Q(T α )f is continuous.
Proof. For any M > 0 we define B M (L p ([0, 1])) to be the set of the operators T of L p ([0, 1]) such that ∥T ∥ ≤ M. We denote by SOT the Strong Operator Topology on L p ([0, 1]), i.e the topology generated by the seminorms T → ∥T f ∥ p for every f ∈ L p ([0, 1]). Then one can write

ϕ Q = φ 3,Q • ((φ 2,Q • φ 1 ) ⊕ id)
where id :

(L p ([0, 1]), ∥ • ∥ p ) → (L p ([0, 1]), ∥ • ∥ p ) f → f, φ 1 : ([0, 1], |•|) → (B 1 (L p ([0, 1])), SOT ) α → T α , φ 2,Q : (B 1 (L p ([0, 1])), SOT ) → B ∥Q∥ 1 (L p ([0, 1])), SOT T → Q(T ), φ 3,Q : (B ∥Q∥ 1 (L p ([0, 1])) × SOT ) × (L p ([0, 1]), ∥ • ∥ p ) → (L p ([0, 1]), ∥ • ∥ p ) (T, f ) → T f.
The continuity of the maps φ 2,Q and φ 3,Q being easily proved, we only need to prove that φ 1 is a continuous map. Let α ∈ [0, 1], and let (α n ) n≥1 to be a sequence of elements of [0, 1] such that α n → α as n → +∞. Given f ∈ L p ([0, 1]), we are going to show that ∥T αn f -T α f ∥ p → 0 as n → +∞.

Let ε > 0. By density there exists g ∈ C([0, 1]) such that ∥f -g∥ p < ε. For every n ≥ 1,

∥T αn f -T α f ∥ p ≤ ∥T αn f -T αn g∥ p + ∥T αn g -T α g∥ p + ∥T α g -T α f ∥ p .
On the one hand, we have for every β ∈ [0, 1] that

∥T β f -T β g∥ p p = 1 0 |tf ({t + β}) -tg({t + β})| p dt ≤ 1 0 |f (s) -g(s)| p ds ≤ ∥f -g∥ p p , so that ∥T αn f -T αn g∥ p + ∥T α g -T α f ∥ p ≤ 2∥f -g∥ p < 2ε.
On the other hand

∥T αn g -T α g∥ p p = 1 0 |tg({t + α n }) -tg({t + α})| p dt ≤ 1 0 |g({t + α n }) -g({t + α})| p dt ≤ 1 0 |g(y) -g({y + α -α n })| p dy ≤ 1-{α-αn} 0 |g(y) -g(y + {α -α n })| p dy + 1 1-{α-αn} |g(y) -g(y + {α -α n } -1)| p dy.
Since g is uniformly continuous on [0, 1], there exists δ > 0 such that |g(x) -g(y)| < ε whenever x, y ∈ [0, 1] satisfy |x -y| < δ. Moreover since α n → α as n → +∞, there exists

n 0 ≥ 1 such that |α -α n | < min(δ, ε p /(2∥g∥ ∞ ) p , 1) for every n ≥ n 0 . Fix n ≥ n 0 . First, we suppose that 0 ≤ α -α n . Then {α -α n } = α -α n = |α -α n | and ∥T αn g -T α g∥ p p ≤ (1 -{α -α n })ε p + 2 p ∥g∥ p ∞ {α -α n } ≤ (1 -|α -α n |)ε p + 2 p ∥g∥ p ∞ |α -α n | ≤ 2ε p . Then, we suppose that 0 > α -α n . So {α -α n } = α -α n + 1 = 1 -|α -α n | and ∥T αn g -T α g∥ p p ≤ (1 -{α -α n })2 p ∥g∥ p ∞ + {α -α n }ε p ≤ |α -α n | 2 p ∥g∥ p ∞ + (1 -|α -α n |)ε p ≤ 2ε p .
Eventually we get that ∥T αn f -T α f ∥ p ≤ (2 + 2 1/p )ε for every n ≥ n 0 , and thus φ 1 is continuous at α. □

We can now show Theorem 4.9.

Proof of Theorem 4.9. First, we prove that the set previously defined as

A = {(α, f ) ∈ [0, 1] × L p ([0, 1]); f is cyclic for T α } is a G δ -set in [0, 1] × L p ([0, 1]). Let (U n ) n≥1 be a basis of open sets in L p ([0, 1]). For every (α, f ) ∈ [0, 1] × L p ([0, 1]), f is cyclic for T α if and only if for every n ≥ 1, there exists Q ∈ C[ξ] such that Q(T α )f ∈ U n . Thus A = n≥1 Q∈C[ξ] ϕ -1 Q (U n ) where the map ϕ Q is defined by ϕ Q (α, f ) = Q(T α )f for every (α, f ) ∈ [0, 1] × L p ([0, 1]) and is continuous by Lemma 4.12. So A is a G δ -set in [0, 1] × L p ([0, 1]). We next prove that the set A is dense in [0, 1]×L p ([0, 1]). Let g ∈ L p ([0, 1]), let β ∈ [0, 1]
, and ε > 0. There exists α ∈ (0, 1) ∩ Q such that |β -α| < ε. Since T α is cyclic by Theorem 4.5 and since σ p (T * α ) = ∅ by Theorem 3.9, the set of cyclic vectors for T α is a dense G δ -set by Proposition 4.2. Thus there exists f ∈ L p ([0, 1]) cyclic for T α such that ∥g -f ∥ p < ε. So (α, f ) ∈ A and satisfies |β -α| < ε and ∥g -

f ∥ p < ε. Hence A is a dense G δ -set in [0, 1] × L p ([0, 1]).
Then the set

B = {α ∈ [0, 1]; {f ∈ L p ([0, 1]); f is cyclic for T α } is a co-meager set in L p ([0, 1])}
is a co-meager set in [0, 1] by Corollary 4.11. Moreover, B ⊂ {α ∈ [0, 1]; T α is cyclic}, and Theorem 4.9 follows. □

We want now to explicit a set of irrational numbers α in [0, 1] such that T α is a cyclic operator. More precisely, we will give a sufficient condition on α expressed in terms of rational approximations implying that T α is cyclic. Some results on approximations of irrationals by rational are recalled here and can be found in [START_REF] Bugeaud | Approximation by algebraic numbers[END_REF]. Definition 4.13 ([6, Section 1.2]). Let n ≥ 0 and let (a k ) 0≤k≤n be such that a 0 ∈ Z and a k ∈ N for every k ∈ {1, . . . , n}. The rational number

a 0 + 1 a 1 + 1 . . .+ 1
an is called a finite continued fraction and is written [a 0 ; a 1 , . . . , a n ].

Let us now recall the continued fraction expansion of an irrational number. Definition 4.14 ([6, Definition 1.2]). Let x ∈ [0, 1]\Q. Let a 0 ∈ Z and ξ 0 ∈ (0, 1) such that x = a 0 + ξ 0 . Let (a k ) k≥1 be a sequence of positive integers and (ξ k ) k≥1 be a sequence of elements in (0, 1) such that 1/ξ k = a k+1 + ξ k+1 for every k ≥ 0.

The rational numbers of the sequence (p n /q n ) n≥0 = ([a 0 ; a 1 , . . . , a n ]) n≥0 are called the convergents of x.

The following result shows that these convergents give an approximation by rational numbers of an irrational number, which turns out to be optimal. Proposition 4.15 ([6, Theorem 1.3, Corollary 1.4 and Theorem 1.4]). Let x be an irrational number and (p n /q n ) n≥0 its convergents. If p -1 = 1, q -1 = 0, p 0 = a 0 and q 0 = 1, then p n = a n p n-1 + p n-2 , q n = a n q n-1 + q n-2 and p n and q n are coprime for every n ≥ 1.

Moreover the convergents (p n /q n ) n≥0 converge to x and satisfy

x -p n q n < 1 q n q n+1 for every n ≥ 0.

We are now able to prove the following theorem, which gives a sufficient condition on the convergents of an irrational number α in [0, 1] implying the cyclicity of T α . We recall that the set Cycl Q∩(0,1) has been defined in Definition 4.7 as the set of common cyclic vectors for all T α , α ∈ Q ∩ (0, 1). Theorem 4.16. Let f ∈ Cycl Q∩(0,1) . There exists a function ψ f : N → R + with the following property: if (p n /q n ) n≥0 are the convergents of an irrational number α in [0, 1] and if for every n ≥ 0 there exists n 0 ≥ n such that q n 0 +1 > ψ f (q n 0 ), then f is cyclic for T α .

Proof. Let (g j ) j≥1 be a dense set of functions in L p ([0, 1]). Let q ≥ 2 be an integer. We consider the sets R q = r q ; 0 < r < q and gcd(r, q) = 1 ⊂ [0, 1] and G q = {g 1 , . . . , g q } ⊂ L p ([0, 1]).

Since T r/q is cyclic for every r/q ∈ R q , for every j ∈ {1, . . . , q} there exists Q r/q,j ∈ C[ξ] such that ∥Q r/q,j (T r/q )f -g j ∥ p < 1/2 q . Then the finite set P q = Q r/q,j ; (r/q, j) ∈ R q × {1, . . . , q} is such that for every j ∈ {1, . . . , q} and every p/q ∈ R q , there exists Q ∈ P q such that ∥Q(T r/q )f -g j ∥ p < 1/2 q . Since the maps ϕ Q,f : α → Q(T α )f are continuous for every polynomial Q in P q by Lemma 4.12, there exists δ(q) > 0 such that for every Q ∈ P q and every r/q ∈ R q , ∥Q(T β )f -Q(T r/q )f ∥ p < 1/2 q whenever β ∈ [0, 1] satisfies |β -r/q| < δ(q). Let us consider the function ψ f : N → R + defined by ψ f (q) = 1/(qδ(q)) and an irrational number α ∈ (0, 1) whose convergents (p n /q n ) n≥0 are such that for every n ≥ 0, there exists n 0 ≥ n such that q n 0 +1 > ψ f (q n 0 ). Since (q n ) n≥0 is an increasing sequence of positive integers, q n ≥ n for every n ≥ 0. Fix n ≥ 0 and n 0 ≥ n such that q n 0 +1 > ψ f (q n 0 ). Then g n ∈ G qn ⊂ G qn 0 and p n 0 /q n 0 ∈ R qn 0 , so there exists

Q n 0 ∈ P qn 0 such that ∥Q n 0 (T pn 0 /qn 0 )f -g n ∥ p < 2 -qn 0 . Since |α -p n 0 /q n 0 | < 1/(q n 0 q n 0 +1 ) < δ(q n 0 ), we have ∥Q n 0 (T α )f -Q n 0 (T pn 0 /qn 0 )f ∥ p < 2 -qn 0 . So ∥Q n 0 (T α )f -g n ∥ p ≤ ∥Q n 0 (T α )f -Q n 0 (T pn 0 /qn 0 )f ∥ p + ∥Q n 0 (T pn 0 /qn 0 )f -g n ∥ p < 2 -qn 0 + 2 -qn 0 ≤ 2 -(qn 0 -1) ≤ 2 -(n 0 -1) ≤ 2 -(n-1) .
Hence f is cyclic for T α . □ Remark 4.17. As in Theorem 4.9, Theorem 4.16 implies that the set of parameters α such that T α is cyclic is a co-meager set in [0, 1]. Indeed, if q k (α) denotes the denominator of the k-th convergent of α, then for every f ∈ Cycl Q∩(0,1) the set n≥0 n 0 ≥n {α ∈ R\Q ∩ (0, 1);

q n 0 +1 (α) > ψ f (q n 0 (α))} is a G δ -set in [0, 1].
Remark 4.18. The proof of the previous theorem does not require the function f to be cyclic for T α for every rational number α, but only for α ∈ Q ∩ (0, 1)\F , where F is a finite set.

We now move over to the study of cyclicity properties of weighted translation operators.

4.3.

Cyclicity of weighted translation operators. Let ϕ be a function in L ∞ ([0, 1]). Again, we observe that T ϕ,α cannot be cyclic in the case where m({ϕ = 0}) > 0.

Proposition 4.19. Let ϕ ∈ L ∞ ([0, 1]) satisfying m({ϕ = 0}) > 0. For every α ∈ [0, 1], the operator T ϕ,α is not cyclic. Proof. Let A = {ϕ = 0}. Then for every f ∈ L p ([0, 1]) and every P = 0≤k≤d a k ξ k ∈ C[ξ] P (T ϕ,α )f (x) = d k=0 a k ϕ(x) . . . ϕ({x + (k -1)α})f ({x + kα}) = a 0 f (x) a.e. on A. Suppose that f is cyclic for T ϕ,α . Then the set {P (T ϕ,α )f |A ; P ∈ C[ξ]} = span[f |A ] is dense in L p (A), which is impossible. □
Our aim is now to extend the result of Chalendar and Partington characterizing cyclic functions for rational Bishop operators to the case of weighted translation operators T ϕ,α , with α ∈ Q and an increasing function

ϕ ∈ C([0, 1], R). Definition 4.20. Let ϕ ∈ L ∞ ([0, 1]), let f ∈ L p ([0, 1]
) and let α = r/q be a rational number such that 0 < p < q and that r and q are coprime. We define ∆ ϕ (f, r/q)(t) a.e. on [0, 1] by

f (t) T ϕ,r/q f (t) • • • T q-1 ϕ,r/q f (t) f ({t + r/q}) T ϕ,r/q f ({t + r/q}) • • • T q-1 ϕ,r/q f ({t + r/q}) . . . . . . . . . f ({t + (q -1)r/q}) T ϕ,r/q f ({t + (q -1)r/q}) • • • T q-1 ϕ,r/q f ({t + (q -1)r/q})
.

We will prove the following theorem:

Theorem 4.21. Let α = r/q be a rational number in (0, 1) such that 0 < r < q and r and

q are coprime. Let ϕ ∈ C([0, 1], R) be an increasing function. A function f ∈ L p ([0, 1]) is cyclic for T ϕ,α if and only if the function ∆ ϕ (f, r/q) satisfies m({t ∈ [0, 1]; ∆ ϕ (f, r/q)(t) = 0}) = 0.
The proof of Theorem 4.21 follows the lines of the proof of Theorem 5.4.4 in [START_REF] Chalendar | Modern approaches to the invariant-subspace problem[END_REF]. The first step in the proof of Theorem 4.21 is the following lemma: Lemma 4.22. Let ϕ ∈ C([0, 1], R) be an increasing function and let α = r/q be a rational number such that 0 < r < q and that r and q are coprime. Let

f ∈ L p ([0, 1]), let n ≥ 1 and let h ∈ L ∞ ([0, 1]) vanishing on the set Ω n,f,ϕ ={t ∈ [0, 1]; |∆ ϕ (f, r/q)(t)| < 1/n}∪ 0≤i,j≤q-1 {t ∈ [0, 1]; T j ϕ,r/q f ({t + ir/q}) > n}.
There exist 1/q-periodic functions h 0 , . . . , h q-1 ∈ L ∞ ([0, 1]) such that h = q-1 j=0 h j T j ϕ,r/q f. Proof. The proof of Lemma 4.22 is patterned after the proof of Lemma 5.4.2 in [START_REF] Chalendar | Modern approaches to the invariant-subspace problem[END_REF]. A decomposition of h of the form h = h 0 f + . . . + h q-1 f is by 1/q-periodicity equivalent to

           h(t) = h 0 (t)f (t) + . . . + h q-1 (t)T q-1 ϕ,r/q f (t) h(t + 1/q) =
h 0 (t)f (t + 1/q) + . . . + h q-1 (t)T q-1 ϕ,r/q f (t) . . . h(t + (q -1)/q) = h 0 (t)f (t + (q -1)/q) + . . . + h q-1 (t)T q-1 ϕ,r/q f (t + (q -1)/q) a.e. on [0, 1/q), that is to say to equivalent to the system

   h(t)
. . .

h(t + (q -1)/q)    =     f (t)
. . . T q-1 ϕ,r/q f (t) . . . . . . f (t + (q -1)/q) . . . T q-1 ϕ,r/q f (t + (q -1)/q)

       h 0 (t) . . . h q-1 (t)   
whose determinant is ±∆ ϕ (f, r/q)(t) by permutation of the rows since r and q are coprime. On the one hand there exist such solutions if t ∈ [0, 1]\Ω n,f,ϕ because |∆ ϕ (f, r/q)(t)| ≥ 1/n. On the other hand it suffices to set h 0 (t) = . . . = h q-1 (t) = 0 whenever t ∈ Ω n,f,ϕ . Such functions h 0 , . . . , h q-1 are then bounded by the definition of the set Ω n,f,ϕ . □

We will also need the following density lemma, which is an analogue of Lemma 5.4.3 in [START_REF] Chalendar | Modern approaches to the invariant-subspace problem[END_REF]. Lemma 4.23. Let ϕ ∈ C([0, 1], R) be an increasing function and let F be a function in L p ([0, 1/q]) such that m({t ∈ [0, 1/q]; F (t) = 0}) = 0. The set {Q(w)F ; Q ∈ C[ξ]} is dense in L p ([0, 1/q]) where w is the function defined on [0, 1/q) by w(x) = ϕ(x)ϕ(x + 1/q) . . . ϕ(x + (q -1)/q). Proof. Suppose that 1/p + 1/p ′ = 1 and that G is a function in L p ′ ([0, 1/q]) such that

1/q 0 Q(w(t))F (t)G(t)dt = 0 for every Q ∈ C[ξ].
Since ϕ is continuous on [0, 1], the function w admits a continuous extension on [0, 1/q], denoted by w 0 . Then

1/q 0 Q(w 0 (t))F (t)G(t)dt = 0 for every Q ∈ C[ξ]. Since ϕ ∈ C([0, 1], R) is an increasing function, the algebra {Q(w 0 ); Q ∈ C[ξ]} in C([0, 1/q])
separates points, contains the constant functions and is closed under complex conjugation. Then this algebra is dense in (C([0, 1/q]), ∥ • ∥ ∞ ) by the Stone-Weierstrass's Theorem. Therefore

1/q 0 f (t)F (t)G(t)dt = 0 for every f ∈ C([0, 1/q]).
Thus the function F G in L 1 ([0, 1/q]) is the constant function equal to zero and so is the function G in L p ′ ([0, 1/q]). □

With these tools at hand, we can now prove Theorem 4.21.

Proof of Theorem 4.21. Assume that f satisfies m({t ∈ [0, 1]; ∆ ϕ (f, r/q)(t) = 0}) = 0. The union

A = n∈N {h ∈ L ∞ ([0, 1]); h = 0 on Ω n,f,ϕ } is dense in L p ([0, 1]
) since m(∩ n∈N Ω n,f,ϕ ) = 0. Fix ε > 0 and h = q-1 j=0 h j T j ϕ,α f ∈ A. In order to prove that there exists Q ∈ C[ξ] such that ∥Q(T ϕ,α )f -h∥ p < ε, one can remark that

Q(T ϕ,α )f = d k=0 a k T k ϕ,α f = q-1 j=0 m n=0 a nq+j T nq+j ϕ,α f = q-1 j=0 m n=0 a nq+j w n T j ϕ,α f = q-1 j=0 Q j (w)T j ϕ,α f for every Q = d k=0 a k ξ k ∈ C[ξ]
since T q ϕ,α f = wf and by 1/q-periodicity of w. Therefore, in order to prove that there exists Q ∈ C[ξ] such that ∥Q(T ϕ,α )f -q-1 j=0 h j T j ϕ,α f ∥ p < ε, it suffices to show that for every j ∈ {0, . . . , q -1}, there exists Q j ∈ C[ξ] such that ∥Q j (w)(T ϕ,α )-h j T j ϕ,α f ∥ p < ε/q and then the polynomial Q defined by Q = q-1 j=0 Q j (ξ q )ξ j will satisfy ∥Q(T ϕ,α )f -h∥ p < ε. Let j ∈ {0, . . . , q -1} and consider the function

F j : t ∈ [0, 1/q] → T j ϕ,α f (t) + T j ϕ,α f ({t + 1/q}) + . . . + T j ϕ,α f ({t + (q -1)/q}) .
For every t ∈ [0, 1/q], if F j (t) = 0 then ∆ ϕ (f, r/q)(t) = 0 because its j-th column would be zero. So m({t ∈ [0, 1/q]; F j (t) = 0}) = 0 since m({t ∈ [0, 1]; ∆ ϕ (f, r/q)(t) = 0}) = 0.

By Lemma 4.23 there exists Q j ∈ C[ξ] such that ∥Q j (w)F j -h j F j ∥ [0,1/q],p < ε/q. However since p > 1 and by 1/q-periodicity of

Q j (w) -h j ∥(Q j (w) -h j )F j ∥ p [0,1/q], p = 1/q 0 |Q j (w(t)) -h j (t)| p q-1 k=0 T j ϕ,α f (t + k/q) p dt ≥ q-1 k=0 1/q 0 |Q j (w(t)) -h j (t)| p T j ϕ,α f (t + k/q)) p dt ≥ 1 0 (Q j (w(t)) -h j (t))T j ϕ,α f (t) p dt ≥ ∥Q j (w)T j ϕ,α f -h j T j ϕ,α f ∥ p p .
Then ∥Q(T ϕ,α )f -h∥ p ≤ q-1 j=0 ∥Q j (w)T j ϕ,α f -h j T j ϕ,α f ∥ p < q-1 j=0 ε/q ≤ ε, so f is cyclic for T ϕ,α .

Conversely, let us now assume that the set A = {t ∈ [0, 1]; ∆ ϕ (f, r/q)(t) = 0} satisfies m(A) > 0. Then there exist functions a 0 , . . . , a q-1 on [0, 1], not all zero, such that

     f (t)
. . . f ({t + (q -1)r/q}) T ϕ,α f (t) . . . T ϕ,α f ({t + (q -1)r/q}) . . . . . .

T q-1 ϕ,α f (t) . . . T q-1 ϕ,α f ({t + (q -1)r/q})           a 0 (t) a 1 (t) . . . a q-1 (t)      =      0 0 . . . 0     
a.e. on A, that is to say q-1 j=0 a j (t)T i ϕ,α f ({t + jr/q}) = 0 for every i ∈ {0, . . . , q -1}. If

φ = Q(T ϕ,α )f with Q = d k=0 b k ξ k ∈ C[ξ], then q-1 j=0 a j (t)φ({t + jr/q}) = d k=0 b k q-1 j=0 a j (t)T k ϕ,α f ({t + jr/q}) = m n=0 q-1 i=0 b nq+i q-1 j=0 a j (t)T nq+i ϕ,α f ({t + jr/q}) = m n=0 q-1 i=0 b nq+i w n (t) q-1 j=0 a j (t)T i ϕ,α f ({t + jr/q}) = 0
a.e. on A since T nq+i ϕ,α f = w n T i ϕ,α f and by 1/q-periodicity of w. Therefore q-1 j=0 a j (t)φ({t+ jr/q}) = 0 whenever φ is a function in the closed subspace generated by

{T n ϕ,α f ; n ≥ 0}. So the subspace {Q(T ϕ,α )f ; Q ∈ C[ξ]} is not dense in L p ([0, 1]), that is to say, f is not cyclic for T ϕ,α . □
Hence if ϕ is a holomorphic function, Theorem 4.5 still holds for the weighted translation operators. Proof. Let α = r/q be a rational number such that 0 < r < q and r and q are coprime. It suffices to show that m({t ∈ [0, 1]; ∆ ϕ (f, r/q)(t) = 0}) = 0 by Theorem 4.21. The function |∆ ϕ (f, r/q)| is 1/q-periodic on [0, 1] and ∆ ϕ (f, r/q) is holomorphic on (0, 1/q), since ϕ is holomorphic, and right-continuous at 0. Indeed ∆ ϕ (f, r/q)(t) is equal to the determinant ϕ t + ir q . . . ϕ t + (i+j-1)r q f t + (i+j)r q 0≤i,j≤q-1 whenever t ∈ [0, 1/q). Then as in the case of the Bishop operators, ∆ ϕ (f, r/q)(t) = 0 for every t ∈ [0, 1/q) if f is not cyclic for T ϕ,α . However ∆ ϕ (f, r/q)(0) is equal to the determinant f f ({(q -1)r/q}) ϕ({(q -1)r/q})f (0) 0 . . . 0 which is in turn equal to (-1) (q-2)(q-1)/2 f (0

) q q-2 i=0 ϕ({(q -1)r/q}) . . . ϕ({(q -1 -i)r/q}) ̸ = 0. So f is cyclic for T ϕ,α . □ Definition 4.25. Let ϕ ∈ L ∞ ([0, 1]). For every subset A of [0, 1], we define the set Cycl (ϕ) A = α∈A {f ∈ L p ([0, 1]); f is cyclic for T ϕ,α }
of common cyclic vectors for all operators T ϕ,α , α ∈ A.

In particular Cycl

4.26. Let ϕ ∈ L ∞ ([0, 1]). For every Q ∈ C[ξ], the map φ Q : ([0, 1], |•|) × (L p ([0, 1]), ∥ • ∥ p ) → (L p ([0, 1]), ∥ • ∥ p ) (α, f ) → Q(T ϕ,α )f is continuous. Proof. Let Q = d k=0 a k ξ k ∈ C[ξ].
As in the case of the Bishop operators, one can write

φ Q = φ 3,Q • ((φ 2,Q • φ 1 ) ⊕ id) where id : (L p ([0, 1]), ∥ • ∥ p ) → (L p ([0, 1]), ∥ • ∥ p ) f → f, φ 1 : ([0, 1], |•|) → (B ∥ϕ∥∞ (L p ([0, 1])), SOT ) α → T ϕ,α , φ 2,Q : (B ∥ϕ∥∞ (L p ([0, 1])), SOT ) → (B M Q (L p ([0, 1])), SOT ) T → Q(T ), φ 3,Q : (B M Q (L p ([0, 1])), SOT ) × (L p ([0, 1]), ∥ • ∥ p ) → (L p ([0, 1]), ∥ • ∥ p ) (T, f ) → T f with M Q = d k=0 |a k | ∥ϕ∥ k ∞ .
We will only prove the continuity of φ 1 since the other are easily proved. Let α ∈ [0, 1], and let (α n ) n≥1 te be a sequence of elements of [0, 1] such that α n → α as n → +∞. Given f ∈ L p ([0, 1]), we are going to show that ∥T ϕ,αn f -T ϕ,α f ∥ p → 0 as n → + ∞. Let ε > 0. By density there exists g ∈ C([0, 1]) such that ∥f -g∥ p < ε. For every n ≥ 1, ∥T ϕ,αn f -T ϕ,α f ∥ p ≤ ∥T ϕ,αn f -T ϕ,αn g∥ p +∥T ϕ,αn g-T ϕ,α g∥ p +∥T ϕ,α g-T ϕ,α f ∥ p . On the one hand for every β ∈ [0, 1]

∥T ϕ,β f -T ϕ,β g∥ p p = 1 0 |ϕ(t)f ({t + β}) -ϕ(t)g({t + β})| p dt ≤ ∥ϕ∥ p ∞ ∥f -g∥ p p , so that ∥T ϕ,αn f -T ϕ,αn g∥ p + ∥T ϕ,α g -T ϕ,α f ∥ p ≤ 2∥ϕ∥ ∞ ∥f -g∥ p .
On the other hand as in the proof of Lemma 4.12

∥T ϕ,αn g -T ϕ,α g∥ p p = 1 0 |ϕ(t)g({t + α n }) -ϕ(t)g({t + α})| p dt ≤ ∥ϕ∥ p ∞ 1 0 |g({t + α n }) -g({t + α})| p dt
≤ 2∥ϕ∥ p ∞ ε p whenever n is sufficiently large since g is uniformly continuous and since α n → α as n → +∞. Eventually if n is sufficiently large, we get that ∥T ϕ,αn f -T ϕ,α f ∥ p ≤ (2 + 2 1/p )∥ϕ∥ ∞ ε and thus φ 1 is continuous at α. □ ]). There exists α ∈ (0, 1) ∩ Q such that |β -α| < ε. Since T ϕ,α is cyclic by Theorem 4.24 and since σ p (T * ϕ,α ) = ∅ by Theorem 3.12, the set of cyclic vectors for T ϕ,α is a dense G δ -set in L p ([0, 1]). Then there exists a cyclic vector f for T ϕ,α such that ∥g -f ∥ p < ε. So A is a dense G δ -set in [0, 1] × L p ([0, 1]).

[0, 1]; T ϕ,α is cyclic} is a co-meager set in [0, 1]. Proof. Let us set A = {(α, f ) ∈ [0, 1] × L p ([0, 1]); f is cyclic for T ϕ,α }. We will prove that this is a dense G δ -set in [0, 1] × L p ([0, 1]). Let (U n ) n≥1 be a basis of open sets in L p ([0, 1]). For every (α, f ) ∈ [0, 1] × L p ([0, 1]) f is cyclic for T ϕ,α if and only if for every n ≥ 1, there exists Q ∈ C[ξ] such that Q(T ϕ,α ) ∈ U n . Thus A = n≥1 Q∈C[ξ] φ -1 Q (U n ) where the map φ Q is defined on [0, 1]×L p ([0, 1]) by φ Q (α, f ) = Q(T ϕ,α )f and is continuous by Lemma 4.26 since ϕ is bounded on [0, 1]. So A is a G δ -set in [0, 1] × L p ([0, 1]). Let now ε > 0 and (β, g) ∈ [0, 1] × L p ([0, 1
By the Kuratowski-Ulam's Theorem, the set B = {α ∈ [0, 1]; {f ∈ L p ([0, 1]); f is cyclic for T ϕ,α } is co-meager set in L p ([0, 1])} is a co-meager set in [0, 1] and B ⊂ {α ∈ [0, 1]; T ϕ,α is cyclic}. □

The function φ P,f : α → P (T ϕ,α )f still being continuous, one can extend Theorem 4.16 for a weighted translation operator T ϕ,α whenever α is an irrational number in [0, 1] sufficiently well approached by its convergents. Q∩(0,1) be a common cyclic vector for T ϕ,α for every α ∈ (0, 1) ∩ Q. There exists a function ψ ϕ,f : N → R + with the following property: if (p n /q n ) n≥0 are the convergents of an irrational number α in [0, 1] and if for every n ≥ 0 there exists n 0 ≥ n such that q n 0 +1 > ψ ϕ,f (q n 0 ), then f is cyclic for T ϕ,α .

Proof. Since the weighted translation operator T ϕ,α is cyclic whenever α ∈ (0, 1) ∩ Q and since the map φ Q,f : α → Q(T ϕ,α )f is continuous whenever Q ∈ C[ξ], the result follows from the proof of Theorem 4.16. □ Remark 4.29. Since we don't assume in Theorem 4.28 that ϕ is a convex function, the operator T * ϕ,α may have eigenvalues according to Theorem 3.12. Then we don't know if the set of cyclic functions for T ϕ,α is a dense set in L p ([0, 1]) whenever T ϕ,α is cyclic.

Open questions

We present in this last section some further comments and open questions.

To begin with, although we know that the weighted translation operators T ϕ,α do not satisfy the Hypercyclicity Criterion, we are unable to prove that they are not hypercyclic. Question 5.1. For every ϕ ∈ L ∞ ([0, 1]) and every α ∈ [0, 1], is it true that T ϕ,α is not hypercyclic on L p ([0, 1])?

A negative answer would lead to a natural example of a hypercyclic but not weakly mixing operator on a Banach space.

As in the hypercyclic case, we do not know if the weighted translation operators T ϕ,α are not supercyclic in general, even if we do know that they cannot satisfy the Supercyclicity Criterion.

Question 5.2. For every ϕ ∈ L ∞ ([0, 1]) and every α ∈ [0, 1], is it true that T ϕ,α is not supercyclic on L p ([0, 1])?

Our results on cyclicity of operators T α for α ∈ R\Q give rise to many questions. Question 5.3. Given f ∈ L p ([0, 1])\{0}, does there exist a finite set F ⊂ Q such that f is cyclic for T α for every α ∈ Q\F ? Thanks to Theorem 4.16, this set could help to explicit a set of common cyclic vectors for T α for a large set of α ∈ [0, 1].

It would also be interesting to take a closer look at the function ψ appearing in the statement of Theorem 4.16.

Question 5.4. Can the function ψ appearing in the statement of Theorem 4.16 be made explicit, for instance in the case where f = 1?

We know that f = 1 is a common cyclic vector for T α whenever α ∈ (0, 1) ∩ Q. Let ε > 0 and let α = r/q be a rational number in (0, 1). In order to obtain an explicit form for ψ, we would need for every g ∈ L ∞ ([0, 1]) vanishing on a set Ω n,f to explicit a polynomial Q ∈ C[ξ] such that ∥Q(T α )1 -g∥ p < ε. To do so, writing g = g 0 T 0 α f + . . . + g q-1 T q-1 α f , we would have to take a look at polynomials Q j ∈ C[ξ] such that ∥Q j -g j ∥ ∞ < ε whenever j ∈ {0, . . . , q -1}. In the study of such polynomials, an obstacle is the study of the determinant ∆(1, r/q)(t) = 1 t . . . t{t + r/q} . . . {t + (q -2)r/q} 1 {t + r/q} . . . {t + r/q} . . . {t + (q -1)r/q} . . . . . . . . . 1 {t + (q -1)r/q} . . . {t + (q -1)r/q}t . . . {t + (q -3)r/q} . This determinant seems to never vanish on [0, 1/q) and to be a monotone function on [0, 1/q). Proving such properties should lead to estimates allowing to obtain an explicit form of the function ψ.

Getting back to the study of invariant subspaces, we recall that in [START_REF] Chamizo | Invariant subspaces for Bishop operators and beyond[END_REF], the authors have proved that T α admits a non-trivial closed hyperinvariant subspace in L p ([0, 1]) as soon as the convergents (p n /q n ) n≥0 of α satisfy log(q n+1 ) = n→+∞ O q n log(q n ) 3 .

This condition gives a bound on the growth of the sequence (q n ) n≥0 of denominators of the convergents of α. On the other hand, the results we proved in Theorem 4.16 state that T α will be cyclic as soon as the sequence (q n ) n≥0 has infinitely many sufficiently large gaps. This is not surprising. Indeed cyclicity and admiting non-trivial closed invariant subspaces can be seen as opposed properties of an operator T since T does not admit such a subspace if and only if every non-zero vector is cyclic for T. Nevertheless, an optimization of the function in the statement of Theorem 4.16 ψ could possibly lead to a positive answer to the following question. Question 5.5. Is it possible to explicit real numbers α such that T α is cyclic and admits a non-trivial (hyper)invariant subspace?

Example 2 . 3 (

 23 [START_REF] Grosse-Erdmann | Linear chaos[END_REF] Examples 2.20, 2.21 and 2.22]). Let us consider the Hilbert space ℓ 2 (N) and the Fréchet space H(C) of entire functions endowed with the seminorms ∥

Definition 3 . 1 (

 31 [START_REF] Hilden | Some cyclic and non-cyclic vectors of certain operators[END_REF] Definition §4]

  and hence there exists a non-empty open set

□ 4 . 1 .

 41 Cyclicity of Bishop operators in the rational case. The cyclicity of the Bishop operator T α has already been studied in the case α ∈ Q by Chalendar and Partington in [7, Section 5.4] and generalized to multivariable Bishop operators by Chalendar, Partington and Pozzi in [8, Section 4].Definition 4.3 ([7, Definition 5.4.1]). Let f ∈ L p ([0, 1]

Corollary 4 . 11 .

 411 Let X and Y be two Polish spaces and let A be a co-meager set in X ×Y. Then the sets {x ∈ X; {y ∈ Y ; (x, y) ∈ A} is a co-meager set in Y } and {y ∈ Y ; {x ∈ X; (x, y) ∈ A} is a co-meager set in X} are co-meager sets in X and Y respectively.

Theorem 4 .

 4 24. Let ϕ ∈ C([0, 1], R) be an increasing function satisfying ϕ(0) = 0 and which is holomorphic on an open neighborhood of [0, 1]. Any holomorphic function f on an open neighborhood of [0, 1] such that f (0) ̸ = 0 is cyclic for T ϕ,α for every α ∈ (0, 1) ∩ Q.

1 )

 1 contains any holomorphic function f on an open neighborhood of [0, 1] such that f (0) ̸ = 0 if ϕ is an increasing function on [0, 1] that is holomorphic on an open neighborhood of [0, 1] satisfying ϕ(0) = 0.It follows that Theorem 4.9 can be generalized to the weighted translation operators.

Lemma

Theorem 4 .

 4 27. Let ϕ ∈ C([0, 1], R) be an increasing convex function satisfying ϕ(0) = 0 and which is holomorphic on an open neighborhood of [0, 1]. The set of parameters {α ∈

Theorem 4 .

 4 28. Let ϕ ∈ C([0, 1], R) be an increasing function satisfying ϕ(0) = 0 and which is holomorphic on an open neighborhood of [0, 1]. Let f ∈ Cycl

This work was supported in part by the project FRONT of the French National Research Agency (grant ANR-17-CE40-0021) and by the Labex CEMPI (ANR-11-LABX-0007-01).

I am grateful to Sophie Grivaux for our interesting discussions and her helpful proofreadings.


Eve F Fabre

Patrick Braca

Vsevolod Peysakhovich

Frédéric Dehais

A New Interface to Cope With Unreliable Airspeed Indications A Behavioral and Eye-Tracking Study

Keywords: Sycopaero, unreliable airspeed event, aviation, workload, pilot flying

When unreliable airspeed events occur, the pilot flying (PF) is required to fly the aircraft using the thrust and the pitch parameters that are displayed in two distanced locations of the flight deck. The Sycopaero interface was designed to limit the PF's workload by automatically displaying thrust and pitch values specific to aircraft configuration on the primary flight display. Participants performed a simulated flight scenario in which they lost airspeed information during take-off with and without the Sycopaero interface. Both behavioral and ocular results demonstrate that the Sycopaero interface significantly lowers the mental workload of PFs and improves their monitoring performance. Taken together, these results suggest that the Sycopaero interface may be a suitable solution to safely handle airspeed failures.

Over the past few decades, the automatization and complexity of aircraft have increased drastically [START_REF] Lee | Human factors and ergonomics in automation design[END_REF].

While this evolution helped significantly improve air safety [START_REF] Boeing | Statistical summary of commercial jet airplane 582 accidents[END_REF], it also led to the emergence of new types of accidents linked to breakdowns in human-automation coordination, where pilots lack knowledge and have trouble understanding the current and future state of automation (e.g., [START_REF] Dehais | Automation surprise in aviation: Real-time solu-586 tions[END_REF][START_REF] Parasuraman | Complacency and bias in human use of automation: An attentional integration[END_REF][START_REF] Sarter | Pilots' monitoring strategies and performance on automated flight decks: An empirical study combining behavioral and eyetracking data[END_REF], as it sometimes happens when an unreliable airspeed event (UAE) occurs [START_REF] Silva | Categorization of unreliable airspeed events using Rasmussen's human performance model[END_REF].

Most of the time, UAEs translate into abnormally large

Mach number or indicated airspeed fluctuations and differences between the indications for the two pilots. In the worst-case scenario, all pitot probes are affected at the same time, which makes the detection of the failure more difficult as the information between the sources seems consistent. Abnormal behavior of the autothrust and/or the autopilot can also be observed (e.g., disconnection), which triggers a reversion of the flight control laws resulting in a lower flight envelope protection compared to a nominal flight situation. As a failure of the autopilot and/or the autothrust can trigger a total loss of control of the aircraft, UAEs have to be quickly detected and properly handled [START_REF] Bea | Final report: On the accident on 1st June 2009 to the 578 Airbus A330-203 registered F-GZCP operated by Air France flight 579 AF 447 Rio de Janeiro[END_REF][START_REF] Silva | Categorization of unreliable airspeed events using Rasmussen's human performance model[END_REF].

When the pilots detect a UAE, the pilot flying (PF) is required to disconnect the autothrust, the autopilot, and 48 the flight directors, resulting in a decreased automation 49 level. The PF also has to adjust manually the thrust and 50 the pitch of the aircraft according to predefined settings 51 that can be found in the flight crew operating manual and 52

are usually provided by the pilot monitoring [START_REF] Bea | Final report: On the accident on 1st June 2009 to the 578 Airbus A330-203 registered F-GZCP operated by Air France flight 579 AF 447 Rio de Janeiro[END_REF]. 53

In Airbus 320 aircraft, the thrust information (N1) is dis-54 played in the electronic centralized aircraft monitor 55

(ECAM) located in the middle of the flight desk, while 56 the pitch of the aircraft is displayed in the primary flight dis-57 play (PFD) located in front of the pilots. UAEs were found 58 to significantly increase airline pilots' stress, both because 59 they feel less protected due to the sudden drop in automa-60 tion and because they are not always able to understand 61 what is happening [START_REF] Bea | Final report: On the accident on 1st June 2009 to the 578 Airbus A330-203 registered F-GZCP operated by Air France flight 579 AF 447 Rio de Janeiro[END_REF]. They also trigger a sudden 62 increase in workload, due to (1) the effort associated with 63 the understanding of the failure and (2) the operation of 64 the aircraft in degraded conditions (e.g., the recovery proce-65 dure and the repeated switch of attention between the 66 ECAM and the PFD; [START_REF] Fitts | Eye fixations of aircraft pilots. III. Frequency, duration, and sequence fixations when flying air force ground-controlled approach system (GCA)[END_REF][START_REF] Bea | Final report: On the accident on 1st June 2009 to the 578 Airbus A330-203 registered F-GZCP operated by Air France flight 579 AF 447 Rio de Janeiro[END_REF]. Work-67 load and stress are likely to affect the pilots' cognitive skills, 68

to undermine their reasoning and piloting performances; 69 and therefore to negatively affect flight safety (Goode, 70 2003;[START_REF] Sarter | Automation surprises[END_REF]. Even though the airspeed failure 71 is properly handled by the pilots, the increased workload action selection errors and execution errors [START_REF] Silva | Categorization of unreliable airspeed events using Rasmussen's human performance model[END_REF], which are the main causes of accident in these conditions (e.g., [START_REF] Bea | Final report: On the accident on 1st June 2009 to the 578 Airbus A330-203 registered F-GZCP operated by Air France flight 579 AF 447 Rio de Janeiro[END_REF][START_REF] Dgac | Reporte final accidente aereo birgenair, vuelo alw-590 301[END_REF][START_REF] Dgatp | Final report, Boeing 757-200 Accident[END_REF][START_REF] Ntsb | Aircraft accident report[END_REF]Wiegmann & Shappell, 2017).

While one's first reaction may be to blame the pilots for not always being able to handle UAEs, it is important to ask the question as to the extent to which an appropriate interface could have supported the pilots in these very stressful and hazardous conditions and prevented these terrible accidents [START_REF] Silva | Categorization of unreliable airspeed events using Rasmussen's human performance model[END_REF]. The lesson to be learned from these accidents is that the human operator is not always considered as being part of the overall system when designing automated systems, thus, these problems are the result of a failure to design for a coordinated team effort across human and machine agents as one cooperative system [START_REF] Sarter | Automation surprises[END_REF]. Instead, the human operator is considered as a providential who will take over when automation fails [START_REF] Gateau | Considering human's non-deterministic behavior and his availability state when designing a collaborative human-robots system[END_REF]. Unfortunately, the technological response to these accidents often consists in implementing even more automation rather than improving the integration of the human in the control and decision loop [START_REF] Billings | Human-centered aviation automation: Prin-574 ciples and guidelines[END_REF][START_REF] Chialastri | Automation in aviation[END_REF]. However, in the case of unreliable airspeed indication, human-centered solutions could be designed and based on basic flying procedures that are taught to pilots from their ab initio training.

In reaction to the accident of Air France Flight 447 (BEA, 2012), Airbus developed a system called "Back-Up Speed Scale" (BUSS Ó ) designed to cope with the potential danger of unreliable speed events. The BUSS Ó consists of displaying speed information based on the angle of attack as a function of the slat/flap configuration. While this system was recently found to be useful to handle UAEs (ATSB, 2018), the EASA still advises against its use above flight level 250 and recommends referring to the pitch and thrust tables in case of strong turbulence (EASA, 2018a). Moreover, the BUSS Ó was found to be likely to display erroneous information when two angle-of-attack sensors are affected by icing conditions at the same time, making its use potentially dangerous in these specific conditions (EASA, 2018a).

A simpler and more reliable solution may be to display the thrust and pitch values presented in the flight manual directly into the PFD, as a function of the altitude, the weight, the flight phase, and the configuration of both flaps and slats of the aircraft.

The present study aimed at evaluating the impact of the Sycopaero interface on PFs in the occurrence of an identified UAE. This interface consisted of displaying both the pitch and the thrust indications corresponding to the four flight configurations (i.e., TOGA for Take Off/Go-Around, climb, level off, and descent/final approach) into the PFD to facilitate both the monitoring and the aircraft operation.

Participants had to perform a simulated flight scenario as PF (in which the airspeed information was lost in a collision 128 with birds during take-off, forcing them to return to the 129 departure airport) with (i.e., Sycopaero scenario) and with-130 out (i.e., control scenario). The subjective mental workload 131 of the participants was assessed in each scenario using the 132 NASA-TLX questionnaire (Hart, 2006). The mean speed 133

and the speed variations were measured in both scenarios. 134

Participants' ocular activity was also measured with a 135 remote eye-tracker set up in the flight simulator. This tech-136 nique is particularly suited to investigate the attentional 137 processes of the operators [START_REF] Helleberg | Effects of data-link modality and display redundancy on pilot performance: An attentional perspective[END_REF]138 Peysakhovich et al., 2018), as it provides objective and 139

quantitative evidence of the user's visual and (overt) 140

attentional processes reflecting their mental processes 141 [START_REF] Duchowski | A breadth-first survey of eye-tracking 596 applications[END_REF]. Consequently, it appeared to be the 142 As the latter is quite far from the PFD (see Figure 3), we decided to duplicate the N1 (i.e., N1d) and to place it directly into the PFD in order to minimize the workload associated with monitoring (i.e., only one area to monitor instead of two). Four notches were placed around the half-circles representing the thrust in the N1d (see Figure 1B). The color code of the notches is the same as the pitch value lines: interface that was going to be tested during the experiment.

most
The three remaining interfaces/systems were fictional (see ESM 1) and were presented to the participants in order to prevent them from predicting the failure they would have to handle during the experiment. Participants took the left seat of the PEGASE flight simulator. They were then asked to perform one landing procedure and one take-off procedure to become acquainted with the PEGASE flight simulator. Once they felt at ease, the eye-tracker was calibrated.

Participants were asked to look at the 11 different calibration points located around the flight deck and the exterior. Each participant was calibrated to reach an overall accuracy of at least 1.5°. This level of accuracy ensured a sufficient level of precision to detect the subparts of the PFD (13°Â 17°)

with: the speed and altitude indicators (3.5°Â 10°), the AI (6°Â 10°), and the N1d area (i.e., the lower part of the PFD; 5°Â 13°). Once the calibration was completed, the experiment started. Just after take-off, participants who performed the flight as PF heard a loud noise. The experimenter who served as a "pseudo" pilot monitoring, informed participants that the aircraft had just collided with birds, which resulted in the destruction of all pitot probes (see Figure 2). Then the former provided participants with the thrust and the pitch values to be applied (that could also be retrieved in the flight crew operating manual) and informed them that (1) they had been cleared by the ATC to perform a go-around procedure and to land on the same runway they took off and that (2) no other aircraft was present in the area in which they were operating the aircraft.

They had to perform a go-around procedure at an altitude subjective mental demand (Hart, 2006). At the end of the 282 experiment, participants were debriefed and asked to give 283 their views on the interface and how they thought it could 284 be improved.

285

Results

286

Behavioral Results

287

Speed Variations 288

The speed of the aircraft and the standard deviations of the 289 and perceived performance, T = 62, z = À1.620, p = .105, (see Figure 3).

speed

Postexperimental Debriefing

During the debriefing, all 22 participants reported that the interface was very natural to use, well-designed, and that it made them feel safer in this very stressful situation. All reported that they thought crossing-out the airspeed indicator was a good idea, because it helped them remember that this informationthey are used to monitor continuouslywas no longer valid, which lowered their cognitive charge.

Except for one pilot who declared that he had some difficulty adapting to the N1d, the remaining 21 pilots reported that duplicating the N1 in the PFD made the monitoring easier and lessened their workload. They were pleased that the N1 was not removed from the ECAM when the Sycopaero interface appeared. They found it reassuring that the Sycopaero interface did not upset their routine, allowing them to continue monitoring thrust in the ECAM if they wished to.

Oculometric Results

Dwell Times

In order to compare the ocular activity of the pilots during the two versions of the flight scenario, six areas of interest (AOIs) were created (see Figure 4). for the remaining AOIs between the scenario with the 373

Sycopaero interface and the control scenario (Figure 5A). 374

The analysis also revealed a main effect of AOIs, 375

F(1, 21) = 97.22, p < .001, η p 2 = .82. For the sake of clarity, 376 the details of these results are reported in ESM 1 (see 377

Tables E1 andE2 ECAM/ND ? N1d; AI ? AOIr; AI ? ECAM/ND; AI ? N1d; N1d ? AOIr; N1d ? ECAM/ND; N1d ? AI) -ANOVA was conducted on the data, revealing a significant Scenario  Transition interaction, F(11, 231) = 7.26, p < .001, η p 2 = .26 (see Figure 6), with an increase in AI ?

N1d transitions (M = 82.47%, SD = 9.86, p < .01) and The details of this analysis are presented in ESM 1(see that they perceived their performance as being similar 439 in both scenarios (i.e., performance sub-scale of the

440

NASA-TLX), suggesting that they were able to accurately 441 evaluate their performance. Despite the absence of differ-442 ence in flight performance, participants reported during 443 the postexperimental debriefing that the interface was very 444 natural to use, well-designed and that it made them feel 445 safer in this very stressful situation. The analysis of the 446 raw NASA-TLX data (Hart, 2006) revealed that overall par-447 ticipants estimated that the Sycopaero interface signifi-448 cantly reduced their mental workload. More specifically, 449 participants reported lower mental demand, physical 450 demand, effort, and frustration in the Sycopaero scenario 451 than in the control scenario. Taken together, these results 452 suggest that the Sycopaero interface was very well accepted 453 by the participants operating as PF and lowered their men-454 tal workload, which is a key point when facing a UAE (Silva 455 & Nicholson, 2012). 456

Participants' ocular activity was analyzed in order to 457 assess how the Sycopaero interface impacted the way PFs 458 allocate their attentional resources in the occurrence of a 459 UAE with and without the Sycopaero interface (Duchowski, 460 2002). Proportionally to the duration of the flight scenarios, 461 The present study suggests that this new and easy-toimplement human-machine interface may be a suitable solution for supporting the PF when reliable airspeed information is no longer available and may help prevent disastrous accidents in the future.

Further investigation is, nevertheless, needed before this interface can be implemented in commercial aircraft.

The present experiment aimed at investigating how the Sycopaero interface impacts the mental workload and the performance of PFs when an unreliable speed event occurs and is immediately identified. While we did not investigate its impact on the pilots monitoring, it is very likely that the interface will also benefit them, as the simplification of the aircraft stabilization procedure may (1) lower both their stress and workload and (2) enable them to almost directly focus on the correction of the airspeed failure. Nevertheless, this assumption should be confirmed experimentally.

In the present study, we chose to use a scenario in which the UAE was immediately detected and understood, and we were only able to investigate its impact on the mental workload associated with the aircraft operation. The present study does not allow us to conclude on whether the Sycopaero interface would facilitate the detection and/or understanding of UAEs. Therefore, this interface should be tested on both the PFs and the pilots monitoring using different scenarios where the UAE is hard to detect and/ or understand, as it happened in the AF447 accident for instance. Further research is also needed [author: edit ok?] on the way the Sycopaero interface should be displayed and reversed in the event of, respectively, an airspeed failure and its resolution. We think that the automatic display of the Sycopaero interface when the system detects airspeed inconsistencies (as it was implemented in the present study) would make these failureswhich can sometimes be difficult to detectmore explicit for the pilots and strongly facilitate the detection of the system change of 547 state [START_REF] Silva | Divergence between flight crew mental model and aircraft system state in auto-throttle mode confusion accident and incident cases[END_REF]. It may also be helpful to 548 allow pilots to manually activate the interface in the case 549 of doubt and/or to decide whether they want to cross out 550 the airspeed indicator or not. Most UAEs resolve in some 551 minutes (BEA, 2012), raising the question of whether and 552 how the Sycopaero interface should be reversed. We think 553 that it might be a better option to let the pilots decide 554 whether they want to continue operating the aircraft with 555 the Sycopaero interface or not. These assumptions should 556 now be confirmed experimentally. Finally, UAEs are an 557 important problem in both general and military aviation. 558

It would be interesting to investigate whether the Sycopaero 559 is a suitable solution to this problem in other aviation fields. 560

In closing, the present study advocates for a more wide-561 spread use of the eye-tracking method for the evaluation 562 of human-machine interfaces before they are implemented 563 in cockpits.

564

Electronic Supplementary Material

  according to the Helsinki Declaration. The experimental protocol was reviewed and approved by a regional ethics committee(CERNI no. 2018-108).Material InterfacesIn this experiment, we compared the performances of the participants in a flight scenario performed with the Sycopaero interface (Figure1B) with the performances in the same flight with the interface that is currently displayed when an unreliable speed event occurs (i.e., control interface; Figure1A). The Sycopaero interface consisted of displaying the pitch and the thrust values for the climb, cruise (i.e., level off), descent, and approach phases as a function of the current aircraft, flap/slat configuration, landing gear configuration, and estimated weight of the aircraft (see Figure1B). The pitch values are symbolized by three colored lines with two arrowheads at their extremities and displayed directly on the PFD: The pitch value of the level-off is a green line with the arrowheads pointing to the center of the line, while the climb and the descent pitch values are yellow lines with the arrowheads pointing up and down, respectively. The pitch value of TOGA was not displayed as pilots know this value by heart. The thrust information (N1) is displayed on the ECAM (see Figure1C).

205

  The notch representing the cruise thrust value is colored in 206 green, the notches representing the thrust values of descent 207 and climb are colored in yellow, and the notch representing 208 the TOGA thrust value is colored in amber. The color code 209 recommended by the EASA (2018b; see Part 25.1321) and 210 the FAA (2010; see Part 25.1322) for flight deck design was 211 applied. Yellow and amber inks were used to indicate 212 marginal conditions or to alert of situations where caution, 213 recheck, or unexpected delay is necessary. Green was used 214 to indicate that monitored equipment/processes are within 215 tolerance or a condition is satisfactory and that it is all right 216 to proceed with an operation or transaction. Finally, the 217 airspeed indicator was crossed out in order to prevent the 218 pilots from consulting it. The design process is described in 219 detail in the Electronic Supplementary Material, ESM 1. 220 Flight Simulator 221 The experiment was conducted in a simplified A320 flight 222 simulator called Pegase at ISAE-Supaero in France. 223 Eye Tracker 224 A Smart Eye Pro (Smart Eye AB, Gothenburg, Sweden) 225 remote eye-tracking system, with a 60-Hz sampling rate 226 and five-camera set-up with two IR flash illuminators was 227 used in the present study. Eleven points were used for 228 calibration. 229 Procedure 230 After they gave their written consent, participants were 231 handed a booklet describing four different new interfaces/ 232 warning systems. Only one of them was the Sycopaero

Figure 1 .

 1 Figure 1. Illustrations of the PFD in (A) the control scenario and (B) the Sycopaero scenario. In the Sycopaero scenario, the airspeed indicator was crossed out to indicate that the airspeed was unreliable and the thrust indicator was duplicated at the bottom of the PDF (N1d) to facilitate the joint monitoring of the thrust and the attitude. (C) Illustration of the ECAM in both scenarios.

of 3 ,

 3 Figure 1B). Participants' ocular activity was recorded in both 279 scenarios. After each flight scenario, participants filled out 280 the NASA-TLX questionnaire aimed at evaluating the 281

  were measured in both scenarios from the moment 290 the aircraft took off to the moment it touched the runway 291 for landing. After ensuring that the data were normally 292 distributed, two two-tailed paired Student t tests were 293 conducted on the mean speeds and the standard deviations 294 of the speed measured in the Sycopaero scenario and the 295 control scenario. The analysis revealed no significant differ-296 ences in mean speed, t(21) = .20, p = .84, d = .05; control 297 scenario: M = 148.93 knots, SD = 9.22; Sycopaero scenario: 298 M = 149.40 knots, SD = 8.24, or in speed standard 299 deviation, t(21) = À.48, p = .63, d = .06; control scenario: 300 M = 9.74 knots, SD = 5.45; Sycopaero scenario: M = 9.21 301 knots, SD = 4.31.

Figure 2 .

 2 Figure 2. Illustration of the flight scenario with r the bird-strike following take-off, s go-around at 3,000 feet, and t touch-down.

  They corresponded to the main flight instruments: the ECAM, the navigator display (ND), the flight control unit (FCU), and the exterior (EXT). As the introduction of the Sycopaero interface had modified the PFD, we divided the PFD into two separate areas: one area corresponding to the Sycopaero interface (SYC) comprising the N1d and the AI in the center of the PFD, and an area comprising the remaining parts of the PFD (PFDr).351 The dwell time percentages to the six AOIs and the area 352 of "non-interest" (i.e., the area regrouping the zones of the 353 cockpit not included in the AOIs) were measured in the 354 control scenario and the Sycopaero scenario from the 355 moment the aircraft took off to the moment it touched 356 the runway for landing. A 2  6 -Scenario (Sycopaero; 357 control)  AOIs (SYC, ECAM, PFDr, ND, FCU, EXT) -358 ANOVA was performed with Tukey-HSD post hoc tests 359 conducted on the dwell times [author: edits ok?]. A 360 significant main effect of scenario, F(1, 21) = 7.56, p < .05, 361 η p 2 = .26, was found with longer dwell times on the AOIs 362 in the Sycopaero scenario (M = 14.40%, SD = 28.55) com-363 pared with the control scenario (M = 13.88%, SD = 26.61). 364The analysis revealed a significant Scenario  AOIs interac-365 tion, F(5, 105) = 17.20, p < .001, η p 2 = .45 (see Figure5A), 366with longer dwell times on the Sycopaero Area (SYC) and 367shorter dwell times on the ECAM in the scenario with the 368Sycopaero interface (SYC: M = 37.25 %, SD = 9.27; ECAM: 369 M = 1.09%, SD = 1.18) than in the control scenario (SYC: 370 M = 29.82%, SD = 10.42, p < .001; ECAM: M = 5.01%, 371 SD = 3.18, p < .01). No significant differences were found 372

  [Author: ok?]). A 2  2 À Scenario 378 (Sycopaero; control)  AOIsyc (N1d; AI) -ANOVA was also 379 performed on the dwell times and revealed a main effect of 380 scenario, F(1, 21) = 39.78, p < .001, η p 2 = .65 (see Figure 5B) 381 with longer dwell times on the two AOIs composing the 382 SYC area (AOIsyc) with the Sycopaero interface (M = 383 18.62%, SD = 15.53) than without it (M = 14.91%, SD = 384 16.25). 385 The analysis also revealed a main effect of AOIsyc, 386 F(1, 21) = 123.95, p < .001, η p 2 = .85, with more time 387 spent on the AI (M = 30.43%, SD = 10.71) than on the 388 N1d (M = 3.11%, SD = 3.94). The Scenario  AOIsyc inter-389 action, F(1, 21) = 1.03, p = .32, η p 2 = .05, did not reach 390 significance.

391

  Transitions Between the Areas 392Complementary analyses were conducted to investigate the 393 transitions between the three AOIs: N1d, the ECAM, and 394 the AI; and an area covering the remaining AOIs in order 395 to investigate how the introduction of the N1d impacted 396 the joint monitoring of the thrust and the attitude. As the 397 ND is located next to the ECAM and the pilots' gaze had 398 to pass through the ND to access the ECAM, this may have 399 interfered with the measurements of the saccades between 400 the ECAM and the remaining AOIs. To prevent this, a 401 unique area merging the ECAM and the ND was created, assuming that introducing the Sycopaero interface would not modify the number of transitions between the ND and the remaining AOIs. A 2 Â 12 -Scenario (Sycopaero; control) Â Transition (AOIr ? ECAM/ND; AOIr ? AI; AOIr ? N1d; ECAM/ND ? AOIr; ECAM/ND ? AI;

Figure 4 .

 4 Figure 4. Illustration of the PEGASE flight deck with the six areas of interest.

Figure 3 .

 3 Figure 3. Illustration of the raw NASA-TLX subscales for the control scenario (dark gray) and the Sycopaero scenario (light gray).

  [Autor: ok?]). The main effect of condition did not reach significance, F(1, 21) = 1.00, p = .32, η p 2 = .05. 427 Discussion 428 The present study aimed at testing a new interface called 429 Sycopaero, designed to support the PFs and lower their 430 workload in the event of an identified airspeed failure. All 431 pilots were able to safely land the aircraft in both scenarios. 432 No differences in mean speed or speed standard deviation 433 were found between the Sycopaero scenario and the control 434 scenario. These results are reassuring as they demonstrate 435 the effectiveness of the intensive training pilots receive to 436 manage this type of failure since the accident of Air France 437 flight 447 (BEA, 2012). Interestingly, participants reported 438

Figure 5 .

 5 Figure 5. Illustrations of the dwell times as a function of the duration of the flight scenario (%): (A) for the six areas of interest and in total and for the control scenario (dark gray) and the Sycopaero scenario (light gray); (B) for the N1d and the AI composing the Sycopaero area in the control scenario (left) and the Sycopaero scenario (right).

Figure 6 .

 6 Figure 6. Transition matrices for the four areas of interest: the ECAM and ND merged area (ECAM/ND), the AI and heading merged area (ATT/ HDG), the N1d and the AOIr in (A) the control scenario and (B) the Sycopaero scenario.

  participants were found to spend significantly more time 462 monitoring the AOI (i.e., main flight instruments) with the 463 Sycopaero interface than with the control interface. In case 464 of a UAE, the thrust and the pitch values corresponding to 465 the current condition of the aircraft (i.e., altitude, weight, 466 and flap/slat configuration) are required to maintain the 467 balance of the aircraft and to prevent it from stalling. This 468 information has to be retrieved from the flight crew operat-469 ing manual by the pilot monitoring[START_REF] Bea | Final report: On the accident on 1st June 2009 to the 578 Airbus A330-203 registered F-GZCP operated by Air France flight 579 AF 447 Rio de Janeiro[END_REF]. The time 470 period in which pilots retrieve this informationbefore 471 being able to stabilize and secure the aircraftis particu-472 larly stressful and associated with an important workload. 473Displaying the thrust and pitch data directly into the PFD 474 via the Sycopaero interface facilitated the retrieval of the 475 relevant flight parameters, eliminating the division of atten-476 tional resources, and enabling almost immediately the PF 477 to stabilize and secure the aircraft[START_REF] Fitts | Eye fixations of aircraft pilots. III. Frequency, duration, and sequence fixations when flying air force ground-controlled approach system (GCA)[END_REF]. Partic-478 ipants were also found to spend proportionally more time 479 monitoring the N1d and less time monitoring the ECAM 480 with than without the Sycopaero interface. Moreover, the 481 analysis of the transition rate matrices between the merged 482 AOI revealed that the transition rates between the ATT/ 483 HDG area (i.e., attitude and heading indicators) and the 484 N1d in the PFD tended to increase (+12%). These results 485 suggest that the participants most often monitored the 486 thrust on the N1d and at least partially abandoned the usual 487 N1 indicator located in the ECAM. The interface appears to 488 have limited the scattering of the participants' attentional 489 resources over different and distant flight instruments, 490 facilitating the monitoring during this specific type of 491 UAE(Wickens et al., 2017). These results are also bolstered 492 by the pilots' comments during the postexperimental debriefing. Except for one pilot who declared that he had some difficulty adapting to the N1d, the remaining 21 pilots reported that duplicating the N1 in the PFD made the monitoring easier and lessened their workload. Moreover, they positively valued the decision not to delete the N1 indicator in the ECAM, indicating that they found it reassuring.More generally, no difference in dwell time percentages and transition rates was found for the remaining AOIs between the two scenarios. This result demonstrates that the introduction of the Sycopaero interface did not affect the monitoring of the remaining flight instruments (i.e., the areas aside from the ECAM and the SYC areas). Taken together, these results suggest that the Sycopaero interface, by locating the critical indicators directly in the PFD, facilitated the retrieval of the thrust and pitch information, increased the time spent monitoring the AOIs (i.e., main flight instruments) and did not distort the way PFs allocated their attentional resources to the main fight instruments.

  546

565

  The electronic supplementary material is available with the 566 online version of the article at https://doi.org/10.1027/ 567 2192-0923/a000221 568 ESM 1. [Author: Please provide brief description]

Table 1 .

 1 Means and standard deviations of the NASA-TLX

		Control	Sycopaero
	NASA-TLX	M	S D	M	S D
	Total	47.00	21.32	32.23	13.07
	Mental demand	10.41	4.11	6.73	3.13
	Physical demand	6.45	3.90	5.09	3.28
	Temporal demand	5.50	4.82	4.81	2.86
	Performance	6.05	2.80	4.95	3.60
	Effort	10.82	4.49	7.55	4.57
	Frustration	5.77	4.23	3.09	2.24

Ó 2022 Hogrefe Publishing Aviation Psychology and Applied Human Factors (2022)

Table E4

 E4

[Author: ok?]

). The analysis also revealed a main effect of transitions, F(11, 231) = 239.46, p < .001, η p 2 = .91.

The details of this result are reported in ESM 1 for sake of clarity (see Table

E3

Ethics Statement

Wickens, C. D., Goh, J., Helleberg, J., Horrey, W. J., & Talleur, D. A. (2017). [Author: editor names?] Attentional models of multitask pilot performance using advanced display technology. In