# Compositional frequent hypercyclicity on weighted Dirichlet spaces.

Dedicated to Professor Jose Rodriguez on the occasion of his 60th birthday

1 Introduction and terminology

The general context containing this paper is the dynamics of operators, while our specifical setting will be the composition operators acting on weighted Dirichlet spaces.

As usual, N denotes the set of positive integers, while [T.sup.n] (n [member of] N) stand for the successive iterates of an operator T. We recall that a (continuous and linear) operator T on a topological vector space X is said to be hypercyclic if there exists a vector x [member of] X, also called hypercyclic, whose orbit {[T.sup.n] x : n [member of] N} is dense in X. Thus, x is a hypercyclic vector if its orbit meets every non-empty open subset U of X. Recently, F. Bayart and S. Grivaux (, ) have introduced the following new, stronger, quantified notion in the theory of hypercyclic operators.

Definition 1.1. Let X be a topological vector space and T : X [right arrow] X an operator. Then a vector x [member of] X is called frequently hypercyclic for T if, for every non-empty open subset U of X, the set

{n [member of] N : [T.sup.n] x [member of] U}

has positive lower density. The operator T is called frequently hypercyclic if it possesses a frequently hypercyclic vector.

We recall that the lower density of a subset A of N is defined as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

The following statement, that is due to Bayart and Grivaux , furnishes a sufficient condition for frequent hypercyclity. Recall that an F-space is a metrizable complete topological vector space.

Theorem 1.2 (Frequent Hypercyclicity Criterion). Let X be a separable F-space and [parallel] x [parallel] a complete F-norm on X defining its topology. Assume that T is an operator on a complete F-norm on X defining its topology. Assume that T is an operator on X satisfying the following property: There exists a dense subset [X.sub.0] of X and a mapping S : [X.sub.0] - [X.sub.0] such that

(i) [[summation].sup.[infinity].sub.n=1] [parallel][T.sup.n] x [parallel] converges for all x [member of] [X.sub.0] ,

(ii) [[summation].sup.[infinity].sub.n=1] [parallel][s.sup.n] x [parallel] converges for all x [member of] [X.sub.0],

(iii) TSx = x for all x [member of] [X.sub.0].

Then T is frequently hypercyclic.

An operator T on an F-space X is said to satisfy the Frequent Hypercyclicity Criterion (in short, FHCC) provided that it possesses the property assumed in the last theorem. We point out that a weaker sufficient condition for frequent hypercyclicity has been recently obtained by Grosse-Erdmann and the second author in . Such a weaker condition will not be used in this paper.

We have that an operator T on an F-space X is hypercyclic if and only if it is topologically transitive, that is, if for any pair of non-empty open subsets U, V of X there exists some n [member of] N such that [T.sup.n](U) [intersection] V [not equal to] [empty set], see . Moreover, T is said to be topologically mixing if for any pair of non-empty open subsets U, V of X there exists some N [member of] N such that [T.sup.n](U) [intersection] V [not equal to] 0 for all n [greater than or equal to] N. Thus, every topologically mixing operator is hypercyclic, but the converse is not true, see . On the other hand, T is called chaotic if it is hypercyclic and it has a dense set of periodic points, that is, vectors x [member of] X such that [T.sup.n] x = x for some n [member of] N, see . Inspired by an approach due to Taniguchi (see the next paragraph), Grosse-Erdmann and the second author have shown [7, Remark 2.2(b)] that if T satisfies the FHCC then it is topologically mixing and chaotic. Nevertheless, Bayart and Grivaux [5, Corollary 5.2] have constructed a frequently hypercyclic operator that is not chaotic, while Badea and Grivaux [1, Corollary 4.4] have proved the existence of a frequently hypercyclic, chaotic but not mixing operator.

Denote by D the open unit disk {z : [absolute value of z] < 1} of the complex plane C, and by H(D) the class of holomorphic functions on D. If [phi] : D [right arrow] D is a holomorphic self-map of D, the composition operator Cp generated by [phi] is defined by [C.sub.[phi]]f = f o [phi] (f [member of] H(D)). It is well known (see ) that [C.sub.[phi]] is a well-defined operator on each Hardy space [H.sup.p](D). Taniguchi [21, Proposition 1] has shown there that under conditions that are stronger than those in Theorem 1.2 an operator on a separable Banach space is chaotic. He applies his criterion to deduce that for any hyperbolic (for the notions of hyperbolic and parabolic see below) automorphism [phi] of D the composition operator [C.sub.[phi]] is chaotic on the Hardy space [H.sup.p](D) for 1 [less than or equal to] p < [infinity], while for any parabolic automorphism [phi] the operator [C.sub.[phi]] is chaotic on the Hardy space [H.sub.p](D) for 1 [less than or equal to] p < 2 (see also ). It follows from Theorem 1.2 that these operators are also frequently hypercyclic.

We add that Taniguchi [21, Theorem 3] has shown that if [phi] is a hyperbolic or parabolic automorphism of D then [C.sub.[phi]] is chaotic on [H.sup.p](D) for any p [member of] (0, + [infinity]). Moreover, based on a clever eigenvalue criterion (see Lemma 2.4 below), Bayart and Grivaux [4, Corollary 3.7] have shown that every composition operator generated by any hyperbolic or parabolic automorphism of D is frequently hypercyclic on the Hilbert space [H.sup.2](D).

Weaker cyclicity properties of composition operators on weighted or non-weighted Hardy spaces had been intensively investigated by Bourdon and Shapiro , Zorboska  and Gallardo and Montes .

For each sequence of positive numbers [beta] = [{[[beta].sub.n]}.sup.[infinity].sub.0] with lim [sup.sub.n [right arrow] [infinity]] [[beta].sup.-1.sub.n] [less than or equal to] 1, the weighted Hardy space H2(f) is defined as the Hilbert space of functions f(z) = [[summation].sup.[infinity].sub.n=0] [a.sub.n] [z.sup.n] analytic on D for which the norm [[summation].sup.[infinity].sub.n=0] [[absolute value of [a.sub.n]].sup.2] [[beta].sup.2.sub.n] is finite (see [10, p. 16] or [12, p. 1]). This norm is induced by the inner product

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Notice that the set of monomials [{[z.sup.n]/[[beta].sup.n]}.sup.[infinity].sub.0] forms a complete orthonormal system. In particular, the polynomials are dense in [H.sup.2]([beta]). The weighted Hardy spaces are natural spaces in the sense that the norm convergence in [H.sup.2]([beta]) implies uniform convergence on compact subsets of D.

In the case in which the weights [[beta].sub.n] = [(n + 1).sup.v], v [member of] R, we obtain the so-called weighted Dirichlet spaces or [S.sub.v] spaces. That is,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

For instance, if v = 0, -1/2,1/2, then [S.sub.v] is, respectively, the classical Hardy space [H.sup.2](D), the Bergman space [A.sup.2](D), and the Dirichlet space D.

Let LFT denote the family of linear fractional transformations [phi](z) = az+b/cz+d on the extended complex plane [C.sub.[infinity]]. A member of LFT different from the identity can have either one or two fixed points in [C.sub.[infinity]]. A map [phi] [member of] LFT is said to be parabolic if it has a unique fixed point in Co or, equivalently, if it is conjugate to a translation z [??] z + a. If [phi] is neither the identity nor parabolic, then it is called hyperbolic (elliptic, resp.) whenever it is conjugate to a positive dilation z [??] [alpha]z, [alpha] > 0 (to a rotation z [??] [e.sup.i[theta]]z, resp.). The remainder of maps in LFT are called loxodromic. By LFT(D) we denote the subfamily {[phi] [member of] LFT : [phi](D) [subset] D}. In particular, the derivative [phi]'(z) exists and is finite for every [phi] [member of] LFT(D) and every z in the unit circle T. Let [phi] [member of] LFT(D). If [phi] is parabolic then its fixed point [eta] is in T, and it satisfies [phi]' ([eta]) = 1. If [phi] is a hyperbolic automorphism then its two fixed points are in T, and one of them, say [eta], is attractive. If [phi] is a hyperbolic non-automorphism then it has a fixed attractive point [eta] in T, the other fixed point lying in {[absolute value of z] > 1} [union] {[infinity]}. In the last two cases, we have 0 < [phi]' ([eta]) < 1. By using the Cayley transform z [??] 1+z/1-z from D onto the right half plane [PI], one can easily visualize to which translation (dilation, resp.) a parabolic (hyperbolic, resp.) transformation [phi] is conjugate, assuming that 1 is a fixed point for [phi]. Finally, if [phi] is elliptic (in this case [phi] is always an automorphism of D) or loxodromic, then one fixed point is in D and the other one lies on {[absolute value of z] > 1} (see ).

According to a result by P.R. Hurst , the composition operator [C.sub.[phi]] : [S.sub.v] [right arrow] [S.sub.v] is bounded for any v [member of] R and any [phi] [member of] LFT(D). E. Gallardo and A. Montes  have furnished a complete characterization of the hypercyclicity of [lambda][C.sub.[phi]] on [S.sub.v] in terms of [lambda], v, [phi]. This characterization can be summarized as follows.

Theorem 1.3. Let [lambda] [member of] C, v [member of] R and [phi] [member of] LFT(D). Let [C.sub.[phi]] : [S.sub.v] - [S.sub.v] be the composition operator generated by Cp. We have:

(a) If [phi] is a hyperbolic automorphism and [eta] is its attractive fixed point, then [lambda][C.sub.[phi]] is hypercyclic if and only if v < 1/2 and [phi]'{[absolute value of [lambda]] < [phi]] ([eta]).sup.2v-1/2].

(b) If [phi] is a parabolic automorphism, then [lambda][C.sub.[phi]] is hypercyclic if and only if v < 1/2 and [absolute value of [lambda]] = 1.

(c) If [phi] is a hyperbolic non-automorphism and [eta] is its boundary fixed point, then [lambda][C.sub.[phi]] is hypercyclic if and only if v [less than or equal to] and [phi]' [([eta]).sup.1-2v/2] < [absolute value of [lambda]].

(d) If [phi] is either an elliptic automorphism, or a loxodromic map, or a parabolic nonautomorphism, or the identity, then [lambda][C.sub.[phi]] is never hypercyclic.

In view of Theorem 1.3, and encouraged by Corollary 3.7 in , it is natural to pose the following question: For which triples (v, [lambda], [phi]) [member of] R x C x LFT(D) is it true that [lambda][C.sub.[phi]] is frequent hypercyclicity on [S.sub.v]?

In this paper, we provide an a partial answer to the last question, namely, for all triples except perhaps for triples satisfying v [member of] [1/4,1/2), [absolute value of [lambda]] = 1, [phi] a parabolic automorphism. As a byproduct, Taniguchi's chaoticity results are extended to the weighted Dirichlet spaces.

2 Frequently hypercyclic composition operators

Here the result announced at the end of the previous section will be formally stated.

We need the following five auxiliary results. The first two of them are density results and can be found respectively in [12, Lemma 2.13 and Lemma 4.7]. The third one asserts that the all the dynamical properties considered in Section 1 are invariant under conjugation. Its proof is elementary, so it will be omitted. The fourth lemma contains an eigenvalue criterion for frequent hypercyclicity, that was developed by Bayart and Grivaux (see , ,  and ) and inspired by Flytzanis . This lemma adopts an heuristic idea due to Godefroy and Shapiro  (see also ), namely, rich supplies of eigenvectors associated to eigenvalues [lambda] with [absolute value of [lambda]] < 1 and to eigenvalues [lambda] with [absolute value of [lambda]] > 1 imply hypercyclicity. The fifth lemma can be found in [12, Lemma 1.2] and furnishes a useful renorming of the weighted Dirichlet spaces.

Lemma 2.1. Assume that v [less than or equal to] 1/2, that [m.sub.1] and [m.sub.2] are any positive integers and that [[alpha].sub.1], [[alpha].sub.2] are complex numbers with [absolute value of [alpha].sub.i]] [greater than or equal to] 1 for i = 1,2. Then the set of all polynomials that vanish at least [m.sub.1] times at a1 and at least [m.sub.2] times at [[alpha].sub.2] is dense in the space [S.sub.[??]].

For t [greater than or equal to] 0, let [e.sub.t] be the function

[e.sub.t](z) = exp (t z+1/z - 1).

It is shown in [12, Proposition 3.10] (see also ) that [e.sub.t] [member of] [S.sub.v] if and only v < 1/4.

Lemma 2.2. Suppose that v < 1/4. Then

[bar.span] {[e.sub.t] : t [greater than or equal to] 0} = [S.sub.v].

Lemma 2.3. Let X be a separable F-space. Assume that T is a hypercyclic (mixing, chaotic, frequently hypercyclic) operator on X satisfying the FHCC, and that R is an invertible operator on X. Then the operator [RTR.sup.-1] is also hypercyclic (mixing, chaotic, frequently hypercyclic, respectively).

Recall that a measure a defined on the Borel [sigma]-algebra generated by a topological space is said to be continuous if [sigma]({a}) = 0 for each singleton {a}.

Lemma 2.4. Let T be an operator on a separable complex Banach space X. Assume that T has perfectly spanning set ofeigenvectors associated to unimodular eigenvalues, that is, there is a continuous probability measure [sigma] on T = {z : [absolute value of z] = 1} such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

for every Borel set A [subset] T with [sigma](A) = 1. Then T is hypercyclic. Moreover, we have:

(a) If X is a Hilbert space then T is even frequently hypercyclic.

(b) If [sigma] can be chosen to be absolutely continuous with respect to the Lebesgue measure on T, then T is even mixing.

By d A(z) we denote the normalized Lebesgue measure dxdy/[pi] (z = x + iy) on D.

Lemma 2.5. If v [member of] (-[infinity], 1) then the expression

[parallel]f[[parallel].sup.2] = [[absolute value of f(0)].sup.2] + [[integral].sub.D] [[absolute value of f'(z)].sup.2] [(1 - [[absolute value of z].sup.2].sup.1-2v] d A(z) (f [member of] [S.sub.v])

defines an equivalent norm on [S.sub.v].

We are now ready to establish and prove our theorem.

Theorem 2.6. Let v [member of] R, [lambda] [member of] C and [phi] [member of] LFT(D), with (v, [lambda], [phi]) [not member of] [1/4,1/2) x T x {parabolic automorphisms of D}.

Let [C.sub.[phi] : [S.sub.v] [right arrow] [S.sub.v] be the composition operator generated by p. Then the following statements are equivalent:

(a) [lambda][C.sub.[phi]] is frequently hypercyclic.

(b) [lambda][C.sub.[phi] is topologically mixing.

(c) [lambda][C.sub.[phi] is chaotic.

(d) [lambda][C.sub.[phi] is hypercyclic.

Proof. The implications (a) [??] (d), (b) [??] (d) and (c) [??] (d) are trivial. Now, let [lambda][C.sub.[phi] : [S.sub.v] [right arrow] [S.sub.v] be hypercyclic. By Theorem 1.3, [phi] is either a hyperbolic map or a parabolic automorphism. At this point we distinguish three cases. In the first two cases we will prove that [lambda][C.sub.[phi] satisfies the FHCC. Then it is frequently hypercyclic and, according to [7, Remark 2.2(b)], it is also topologically mixing and chaotic, so (b) are (c) are fulfilled. In the third case we will use the eigenvalue criterion to prove the frequent hypercyclicity as well as the mixing property, while the chaoticity will be demonstrated directly. We denote T = [lambda][C.sub.[phi].

Case 1: [phi] is a hyperbolic automorphism. In this case, [phi] has its two fixed points [eta], [eta]' on T. Let [eta] be the attractive one. Take any automorphism [sigma] of D satisfying [sigma]([eta]) = 1, [sigma]([eta]') = -1. Then [[phi].sub.0] := [sigma] o [phi] o [[sigma].sup.-1] is a hyperbolic automorphism of D with fixed points at 1, -1, such the point 1 is the attractive one. Moreover, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. An application of Lemma 2.3 yields that it is enough to prove that ACp0 satisfies the FHCC. Consequently, we can assume without loss of generality that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] are the fixed points of [phi], the point 1 being attractive.

According to Theorem 1.3, we have that v < 1/2 and ([phi]'[(1).sup.1-2v/2] < [absolute value of [lambda]] < [phi][(1).sup.2v-1/2]. We follow the proof of Theorem 3.5 in . The explicit expression of [phi] is

[phi](z) = 1 + [mu] z + 1 - [mu]/(1 - [mu]) z + 1 + [mu]'

where [mu] [member of] (0,1) and, in fact, [phi]' (1) = [mu]. Therefore

[mu].sup.1-2v/2] < [absolute value of [lambda]] < [mu].sup.2v-1/2]. (1)

Choose m [member of] N with m > 2 - 2v and

m > - log [absolute value of [lambda]]/log [mu]. (2)

Let X be the set of all holomorphic functions on a neighborhood of the closed disk [bar.D] that vanish at least m times at 1. Fix f [member of] X. It is proved in  that

[parallel][T.sup.n][parallel] [less than or equal to] ([[absolute value of [lambda]].sup.2n] [[mu].sup.2nm] + [[absolute value of [lambda]].sup.2n][[mu].sup.n(1-2v)]) (n [member of] N),

where C is a constant independent of n. From (1) and (2), we obtain that [absolute value of [lambda]][[mu].sup.m] and [[absolute value of [lambda]].sup.2] [[mu].sup.1-2v] are less that 1, so

[[infinity].summation over (n=1)] [paralle][T.sup.n]f[parallel] < + [infinity] for all f [member of] X. (3)

Now, take S := [T.sup.-1] = [[lambda].sup.-1] [C.sup.-1.sub.[phi]] = [[lambda].sup.-1] [C.sub.[phi]-1] and consider the set [gamma] of all holomorphic functions on a neighborhood of [bar.D] that vanish at least m times at -1. Observe that -1 is the attractive fixed point of [[phi].sup.-1] with ([[phi].sup.-1])'(-1) = 1/p'(-1) = [mu] and that [[mu].sup.1-2v/2] < [absolute value of [lambda]] < [[mu].sup.2v-1/2]. Therefore, a similar argument leads to

[[infinity].summation over (n=1)] [paralle][S.sup.n]f[parallel] < + [infinity] for all f [member of] X. (4)

If we set [X.sub.0] := X [intersertion] [gamma], then we have [X.sub.0] [contains] {polynomials vanishing at least m times at 1 and -1}, so [X.sub.0] is dense in [S.sub.v] by Lemma 2.1. Clearly, (3) and (4) hold for all f [member of] [X.sub.0]. In addition, TS is the identity and [X.sub.0] is S-invariant, because [[phi].sup.-1] is conformal and fixes the points 1, -1. Consequently, T satisfies the FHCC.

Case 2: [phi] is a hyperbolic non-automorphism. This time [phi] has two fixed points, one on T and the other one outside [bar.D]. Choose an automorphism [sigma] of D sending those points, respectively, to 1 and to certain [alpha] [member of] (-[infinity], -1) (see [20, p. 114 and Exercise 10 on p. 125]). By using Lemma 2.3 as in the first part of Case 1, one can suppose without loss of generality that the fixed points of [phi] are 1, [alpha]. We follow the proof of Theorem 2.11 in . The explicit expression of [phi] is

[phi](z) = ([mu][alpha] - 1)z + [alpha] (1 - [mu])/([mu] -1) z + [alpha] - [mu],

where [alpha] [member of] (-[infinity], -1), [mu] [member of] (0,1) and, in fact, [phi] (1) = [mu]. By Theorem 1.3, we must have v [less than or equal to] 1/2 and

[[mu].sup.1-2v/2] < [absolute value of [lambda]]. (5)

Choose m [member of] N satisfying m > ( 1 - 2v) /2 and (2). Denote by X ([gamma], resp.) the set of all polynomials that vanish at least m times at 1 (at a, resp.). This time, the inverse map S = [[lambda].sup.-1] C[[phi].sub.-1] is not bounded on [S.sub.v] but it is well defined on the polynomials. It is proved in  that

[parallel][T.sup.n]f[parallel] [less than or equal to] M [[absolute value of [lambda][[mu].sup.m].sup.n] for all f [member of] X

and

[parallel][S.sup.n]f[parallel] [less than or equal to] C [[absolute value of [lambda].sup.-n] [mu].sup.n(1-2v)/2] for all f [member of] Y,

where the constants M, C are independent of n. By (2) and (5), both numbers [absolute value of [lambda][[mu].sup.m]], [[absolute value of [lambda]].sup.-1] [mu].sup.1-2v/2] are less than 1. Therefore, we obtain

[[infinity].summation over (n=1)] [paralle][T.sup.n]f[parallel] < + [infinity] and [[infinity].summation over (n=1)] [paralle][S.sup.n]f[parallel] < + [infinity] for all f [member of] X [intersection] [gamma]. (6)

Now we define [X.sub.0] := [[union].sup.[infinity].sub.n=0] [S.sup.n](X [intersection] [gamma]). Then S is well defined on [X.sub.0], the set [X.sub.0] is dense in [S.sub.v] (by Lemma 2.1, because [X.sub.0] [contains] X [intersection] [gamma]) and S-invariant, TS is the identity on [X.sub.0] and (6) is satisfied for all f [member of] [X.sub.0] (this only carries a translation of the indexes n in both series). Again, conditions (i), (ii) and (iii) of Theorem 1.2 are fulfilled. Thus, T satisfies the FHCC.

Case 3: [phi] is a parabolic automorphism. According to Theorem 1.3 and the hypothesis, we must have v < 1/4 and [absolute value of [lambda]] = 1. Since [phi] is conjugate to a translation, we may suppose, after applying a similarity if necessary, that [phi] is conjugate to a translation z [??] z + ia (a [member of] R \ {0}), which is a self-map of the right half-plane [C.sub.+]. Then an appropriate linear fractional transformation mapping D onto [C.sub.+] shows that we can assume (with a further application of Lemma 2.3) that [phi] has the form

[phi](z) = (2 - ai)z + ai/-aiz + 2 + ai,

with a [member of] R \ {0}. Note that [e.sub.t] (t [greater than or equal to] 0) is an eigenfunction for T associated to the eigenvalue [lambda][e.sup.-iat]. As in [4, Proof of Example 3.6], take [sigma] to be the normalized length measure on TT, and let A be a measurable set of T with [sigma](A) = 1. Then [sigma]([[lambda].sup.-1] A) = 1. If m is the Lebesgue measure on [0, + [infinity]), we have

m({t [greater than or equal to] 0: [lambda][e.sup.-iat] [not member of] A} [less than or equal to] 1/[absolute value of a] m({t [member of] R : [not member of] [[lambda].sup.-1] A}) = 0.

Hence the set B := {t [greater than or equal to] 0 : [lambda][e.sup.-iat] [member of] A} is dense in [0, + [infinity]). Observe that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (7)

Let f [member of] [S.sub.v] with (f, [e.sub.t]) = 0 for all t [member of] B. Since [e.sub.t] depends continuously on t (this will be detailed at the end of the proof) and B is dense, we get (f, [e.sub.t]) = 0 for all t [greater than or equal to] 0, so (f,g) = 0 for all g [member of] span {[e.sub.t] : t [greater than or equal to] 0}. By Lemma 2.2, the last span is dense in [S.sub.v], whence f = 0. Consequently, {[e.sub.t] : t [member of] B} is total in [S.sub.v]. It follows

from (7) that span ([[union].sub.[alpha][member of]A] Ker(T - [alpha]I)) is dense in [S.sub.v]. Then Lemma 2.4 applies yielding that T is frequently hypercyclic and mixing.

Now, we prove that T is chaotic. Since T is already hypercyclic, our task is to demonstrate that the set P of T-periodic functions in [S.sub.v] is dense. For this, observe that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (8)

where Q is the set of rational numbers, C := [0, + [infinity]) n ([a.sup.-1] +([beta] [pi]Q)), with [lambda] = [e.sup.i[beta]]. Note that C is dense in [0, + [infinity]). An argument as in the above paragraph shows that {[e.sub.t] : t [member of] C} spans a dense set in [S.sub.v]. Since P is a linear manifold, it follows from (8) that P is dense in [S.sub.v], as required.

Finally, we demonstrate that the map [member of] : t [member of] [0, + [infinity]) [??] [e.sub.t] [member of] [S.sub.v] is continuous. To this end, we fix u [greater than or equal to] 0. Let t [member of] [u/2, u + 1]. By using the equivalent norm furnished by Lemma 2.5, one obtains

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [sigma](z) := i1+z/1-z is the Cayley transform from D onto the upper half plane [PI]. Therefore

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and the last function does not depend on t, and it is integrable on [(0, + [infinity]).sup.2] because 2(1 - 2v) > 1. From the Lebesgue Dominated Convergence Theorem, it follows that J(t) [right arrow] 0 (t [right arrow] u), hence [parallel]E(t) - E(u)[parallel] [right arrow] 0 (t [right arrow] u), so yielding the continuity of E.

(v, [lambda], [phi]) [member of] [1/4,1/2) x T x {parabolic automorphisms of D}.

Denote T = [lambda][C.sub.[phi]]. The approach of the proof of Theorem 3.3 in  only leads to estimates of the form

[parallel][T.sup.n]f[parallel] [less than or equal to] C/[n.sup.1-2v] (9)

for v < 1/2, that is not sufficient to apply Theorem 1.2. Nevertheless, since the so-called Hypercyclicity Criterion is satisfied for the entire sequence (n) of positive integers (see ), the operator T is in fact mixing for v < 1/2. Moreover, for v [greater than or equal to] 1/4 and t > 0, the functions [e.sub.t] (the "most natural" eigenfunctions of [C.sub.[phi]]) do not belong to [S.sub.v]. These functions seem to be the best candidate to be periodic functions (with appropriate values of t) for the operator [C.sub.[phi]]. This leads us to conjecture that T is not chaotic for v [member of] [1/4,1/2). If this is the case, T would not satisfy the FHCC either.

Remark 2.7. The referee has kindly provided a new way to derive the frequent hypercyclicity of T = [C.sub.[phi]] in the parabolic case, with [absolute value of [lambda]] = 1, v < 1/4. This way does not use the eigenvalue criterion. Namely, following the proof of Theorem 3.3 in  we get estimates like (9) for f [member of] [X.sub.0] := {holomorphic functions in some neighborhood of [bar.D] that vanish at least twice at 1}, where we are assuming without loss of generality that 1 is the fixed point of [phi]. Similar estimates hold for S := [C.sub.[phi]]-1. Then TS = I and both series [[summation].sup.[infinity].sub.n=1][parallel][T.sup.n]f[[parallel].sup.2], [[summation].sup.[infinity].sub.n=1] [parallel][S.sup.n]f[[parallel].sup.2] converge for f in the dense set [X.sup.0]. Since [S.sup.v] is a Hilbert space (so a Banach space with cotype 2), the conditions of the "random" Frequent Hypercyclicity Criterion, see [15, Theorem 2.1 and the subsequent comments] are satisfied, which implies that [C.sub.[phi]] is frequently hypercyclic.

To conclude the paper, we want to pose the problem ar s ng from the comments following the proof of Theorem 2.6.

Question. Assume that [phi] is a parabolic automorphism of D, [absolute value of [lambda]] = 1 and v [member of] [1/4,1/2). Is [lambda][C.sub.[phi]] chaotic? Is [lambda][C.sub.[phi]] frequently hypercyclic?

Acknowledgements. We thank the referee for helpful comments and suggestions, which led to an improvement of this paper.

References

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L. Bernal-Gonzalez Departamento de Anaa lisis Matemaa tico Facultad de Matemaaticas, Apdo. 1160 Avda. Reina Mercedes, s/n 41080 Sevilla, Spain

E-mail: lbernal@us. es

Antonio Bonilla Departamento de Anaa lisis Matemaa tico Universidad de La Laguna C/Astroffsico Fco. Sanchez, s/n 38271 La Laguna, Tenerife, Spain E-mail: abonilla@ull.es

Luis Bernal-Gonzalez Antonio Bonilla, The first author has been partially supported by the Plan Andaluz de Investigacion de la Junta de Andalucla FQM-127 and by MEC Grant MTM2006-13997-C02-01. The second author has been partially supported by MEC and FEDER MTM2005-07347. Both authors have been partially supported by MEC Accion Especial MTM2006-26627-E.

Received by the editors January 2008--In revised form in July 2008. Communicated by F. Bastin.

2000 Mathematics Subject Classification : Primary 47A16. Secondary 30E10, 30H05, 47B33, 47B38.
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