Ismael Charlet

Yohan Desieres

Delphine Marris-Morini
email: delphine.morini@u-psud.fr

Frederic Boeuf
email: frederic.boeuf@st.com

Ismaël Charlet

Yohan Désières

Frédéric Boeuf

Capacitive Modulator Design Optimization Using Si and Strained-SiGe for Datacom Applications

Keywords: Capacitive modulator, electro-refractive modulator, silicon photonics, strained-SiGe

published or not. The documents may come

I. INTRODUCTION

ILICON photonics is a low-cost technology for optical interconnects, by using CMOS facilities. The strong index contrast between Si and SiO 2 enables propagation within compact waveguides. Currently the main commercial application concerns datacom operating at 1310nm wavelength within datacenters. Among the required components used on photonic chips, modulators play a key role. The 2 main standards for datacom are transceivers at 25Gb/s with 4 parallel lines to achieve 100Gb/s, and 4x53GBd using PAM-4 for 400Gb/s [START_REF] Boeuf | A Silicon Photonics Technology for 400 Gbit / s Applications[END_REF].

Commercial modulators used for datacom are mainly based on phase-shifter sections within an interferometer structure. Since Si exhibits low Kerr and Pockels effects, modulation is achieved using the free plasma dispersion effect within a Mach-Zehnder Interferometer (MZI). Typically, modulators are based on a PN-junction working on reverse voltage within the optical waveguide. This solution offers bandwidth above 50 GHz and low insertion penalty [START_REF] Patel | Silicon Photonic Segmented Modulator-Based Electro-Optic DAC for 100 Gb/s PAM-4 Generation[END_REF]. However, PN-junction suffers from a relative lack of efficiency, as a consequence the modulator footprint is few millimeters-long. This turns out to be a problem when considering the Electronic Integrated Circuit (EIC). For long phase shifters, the device cannot be driven using a single-Manuscript received August 1, 2020. This work was supported in part by the IRT Nanoelec I. Charlet is with STMicroelectronis, 850 rue Jean Monnet, 38920 Crolles, France (phone: +33438923981; e-mail: ismael.charlet@st.com).

Y. Desières is with Commissariat à l'énergie atomique et aux énergies alternatives, LETI, Minatec Campus, University Grenoble Alpes, F-38054 Grenoble, France (e-mail: yohan.desieres@cea.fr). stage electrode. Instead Travelling Wave Electrode (TWE) or Multistage Electrode are required, which represent an extraenergy cost [START_REF] Ackerman | The Effect of a Mach-Zehnder Modulator's Traveling-Wave Electrode Loss on a Photonic Link's Noise Figur[END_REF] and lead to a more complex EIC design.

It has been estimated that modulators can be driven with single-stage electrodes for lengths below 1mm at 25GBd and 500µm at 53GBd [START_REF] Temporiti | Insights Into Silicon Photonics Transmitter Architectures[END_REF]. Si-based PN-modulators are not efficient enough to achieve these footprint targets. Changing electronic structure or materials is required to improve electro-refractive modulator efficiency.

Capacitive modulators are based on a Semiconductor Insulator Semiconductor (SIS) electrical junction. In accumulation regime, the modulator is highly efficient since the free carrier concentration varies linearly with the applied voltage. Several demonstrations have been performed, showing modulation up to 28GHz and VπLπ below 1 V.cm [START_REF] Liu | A high-speed silicon optical modulator based on a metal-oxide-semiconductor capacitor[END_REF][6] [START_REF] Byers | 3D Fin Waveguide on 10nm Gate Oxide Bonded Double-SOI for Low VπL Accumulation Modulator[END_REF]. Thus capacitive modulators constitute an appealing solution to reduce modulator footprints. However, the capacitive modulator presents several technological challenges. First it requires a more complex integration than PN-modulators, even though CMOS-compatible. Indeed, few steps dedicated to the capacitive modulator fabrication must be added to the conventional photonic flow. Moreover, unless advanced processes based on recrystallization are used [START_REF] Debnath | Fabrication of Arbitrarily Narrow Vertical Dielectric Slots in Silicon Waveguides[END_REF], in most of the case polycrystalline silicon (polySi) is necessary to form the SIS junction. This has a huge impact on the device performances. Optical losses about 10 dB/cm have been reported within polycrystalline silicon waveguides [START_REF] Douix | Low loss poly-silicon for high performance capacitive silicon modulators[END_REF] [START_REF] Zhu | Influence of RTA and LTA on the Optical Propagation Loss in Polycrystalline Silicon Wire Waveguides[END_REF]. III-V semiconductor alloys can be used to improve dramatically the efficiency but at the cost of a complex integration process [START_REF] Takenaka | High-efficiency , Low-loss Optical Phase Modulator based on III-V / Si Hybrid MOS Capacitor[END_REF].

Alternatively, strained SiGe is CMOS-compatible material which can be used to improve electro-refractive effect. Strained-SiGe takes benefit from a lower effective mass than Si [START_REF] Manku | Drift hole mobility in strained and unstrained doped Si(1-x)Gex alloys[END_REF], leading to higher efficiency for holes than Si. Several PNmodulators using strained-SiGe have been reported and have shown high efficiency [START_REF] Fujikata | High-performance Si optical modulator with strained p-SiGe layer and its application to 25 Gbps optical transceiver[END_REF] but the impact on insertion losses remains unclear. Indeed, the ratio Δn/Δα is lower for strained SiGe than for Si at 𝜆=1310nm This means that optical losses could increase more than the phase shifter efficiency. Therefore, it is not straightforward to know if higher Optical Modulation Amplitude (OMA) can be reached using SiGe alloys, and to determine the optimum operating geometry and Ge concentration for a given operating frequency.

This work focuses therefore on the optimization of Si-based capacitive modulator using an optional strained-SiGe, for datacom applications by maximizing OMA instead of efficiency. The objective is to achieve a footprint small enough such as the phase shifter can be driven by a single-stage electrode. Previous work on modeling SIS modulator were relying on fully analytic model [START_REF] Fujikata | High-performance Si optical modulator with strained p-SiGe layer and its application to 25 Gbps optical transceiver[END_REF] [START_REF] Boeuf | Benchmarking Si, SiGe, and III-V/Si Hybrid SIS Optical Modulators for Datacenter Applications[END_REF] or fully numerical approach [START_REF] Douix | Design of integrated capacitive modulators for 56Gbps operation[END_REF] [START_REF] Dourado | Modeling and trade-off analysis of a capacitive silicon Mach-Zehnder modulator for telecom applications[END_REF]. In [START_REF] Boeuf | Benchmarking Si, SiGe, and III-V/Si Hybrid SIS Optical Modulators for Datacenter Applications[END_REF] optical modes and carrier distribution were determined using fully analytical approach to assess the performance of devices with various materials such as SiGe and InGaAsP, but no device optimization was carried out. In [START_REF] Douix | Design of integrated capacitive modulators for 56Gbps operation[END_REF][16] a full simulation of Si-based SIS was proposed, including the RF propagation line but no systematic optimization of geometry nor materials was done. In this work we propose to address this point by using a combination of numerical and analytical models allowing to run the exploration of a wide space of parameters. In part I, a numerical model coupled with an optimization algorithm is developed to assess ultimate modulator performances. In part II, optimization is performed for full-Si and Si/SiGe capacitive modulators for 100Gbs/s and 400Gb/s applications. Finally, in part III, we discuss the obtained results.

I. MODELLING CAPACITIVE MODULATOR WITH STRAINED-SIGE

For Datacom applications, dynamic Optical Amplitude Modulation (OMA) is the figure of merit (FoM) that should be considered, which takes into account for both phase shifter efficiency and losses. Fig. 1(a) shows a MZI set in quadrature. Considering ON-OFF Keying (OOK) operation, the intensity differences between the different symbols should be maximized. This FoM is OMA, which corresponds to the optical intensity difference between 0 and 1 bits as shown in Fig. 1(b). In the case of PAM-4 application, 4 different OMA values are involved as indicated in Fig. 1(c). For an electrorefractive modulator within a MZI in quadrature working in push-pull operation, analytic calculations of OMA have been performed in [START_REF] Boeuf | Benchmarking Si, SiGe, and III-V/Si Hybrid SIS Optical Modulators for Datacenter Applications[END_REF]. For OOK-NRZ operation (2 different symbols) with an input power I0 = 1mW, OMA in mW can expressed as:

OMA = exp [-(α prop + α max + α min 2 ) . L] . sin(|ΔΦ. L|) (1)
With L the modulator length (mm), ΔΦ the phase-shift (rad/mm), α prop the scattering losses of an undoped waveguide (mm -1 ), α Vmin the linear losses due to material & free carrier absorption at the minimal voltage (mm -1 ) and α Vmax the linear losses due to material & free carrier absorption at the maximal voltage (mm -1 ). In the ideal case OMA should be equal to 0dBm, meaning perfect lossless constructive and destructive interferences. PN-modulators exhibiting OMA equal to -2.25dBm at 50GHz have been reported [START_REF] Monfray | Optimization of deep rib high speed phase modulators on 300mm industrial Si-photonics platform[END_REF]. From this, the optimum length to maximize OMA can be deduced:

L opt = 1 ΔΦ . atan ( Δϕ α prop + α max +α min 2 ) (2)
For PAM-4 operation, 4 different OMA values are defined. OMA outer is the maximal amplitude difference, whereas OMAs top, central and bottom concern inner amplitude differences between intermediate levels. In the ideal case, OMA outer is 0dB, while OMA top, central and bottom are -4.77dB. In the case where the PAM-4 signal is generated optically using LSB and MSB modulator sections, it is demonstrated that these inner OMA are all equivalent and equal to 1/3 of the outer OMA if the LSB modulator length is around:

L = 2 3 L opt outer (3)
With 𝐿 𝑜𝑝𝑡 𝑜𝑢𝑡𝑒𝑟 the optimum length to maximize OMA outer.

A. Capacitive Structure

Studied capacitive modulators are based on stacking a pdoped SOI layer, a dielectric layer (SiO 2 ) and a n-doped polySi layer. An extra p-doped SiGe-layer can be added between the Si and the capacitor dielectric as shown on Fig. 2.

Waveguide thickness is either 220 or 300nm [START_REF] Thomson | Roadmap on silicon photonics[END_REF]. Because our Si-Platform relies on 300nm thick SOI waveguides, we

(a) (b) ) ( a ) (c) ) ( a )
focus the later work on 150nm-thick Si and polySi layers while maximizing the overlap between the guided mode and the accumulation layers. For results presented here, the waveguide width is taken as 400nm as it has been seen as a good compromise to avoid leakage in doped arms and field dissymmetry. Trends are however similar for single-mode waveguide widths in the range 300-500nm. The bandgap of strained SiGe (eV) as a function of the Ge-content is taken from [START_REF] Lang | Measurement of the band gap of GexSi1-x/Si strained-layer heterostructures[END_REF]:

Eg(x) = 1.12 -0.96x + 0.43x² (4)
Strained-SiGe leads to higher optical losses due to bandgap shrinking [START_REF] Zollner | Optical constants and ellipsometric thickness determination of strained Si 1 -x Ge x : C layers on Si ( 100 ) and related heterostructures[END_REF] so thickness should be minimized. A thickness larger than the Debye length (~10nm) is detrimental in term of the n/ ratio. Thus, we set the SiGe thickness to 15nm as show in Fig. 2 and take the Ge-content in the SiGe layer as a parameter. The oxide thickness 𝑡𝑜𝑥 has a major impact on both efficiency and electrical bandwidth and is a second parameter. Its maximum value is set to 15nm to achieve higher efficiency than PN-modulator [START_REF] Douix | Design of integrated capacitive modulators for 56Gbps operation[END_REF].

Finally, when designing capacitive modulators for frequencies > 1GHz, minimizing the access resistance is mandatory. Therefore, a special emphasis is put on the access resistance optimization, with 3 sections of homogeneous doping for each arm of the modulator (𝑁 ++ , 𝑁 + , 𝑁 𝑑 for the polySi doping levels, 𝑃 ++, 𝑃 + , 𝑁 𝑎 for the SOI doping levels, 𝑥𝑛 + , 𝑥𝑝 + , 𝑙𝑛 + , 𝑙𝑝 + for the doping positions). We consider that the strained-SiGe layer is p-doped with a concentration equal to 𝑁 𝑑 due to fabrication constraints [START_REF] Douix | Capacitive modulator integration in a 300 mm silicon photonics platform for low power and high efficiency[END_REF]. This constitutes a model with 12 tunable parameters shown in Fig. 2.

In order to optimize capacitive modulators design, a model is developed to be able to compute several devices allowing deep optimization. The model is based on a perturbative approach [START_REF] Weber | Perturbation Theory Applied to Gain or Loss in an Optical Waveguide[END_REF]. The problem can be split into 3 parts: a computation of the mode profile in the undoped waveguide, a carrier profile distribution computation and finally an electro-refractive computation to estimate the resulting impact on the mode's effective index.

B. Optical Mode calculation

First the optical mode is calculated. Since the oxide thickness lies only between 5 and 15nm, we can approximate that the oxide thickness variation has a little impact on the optical mode. This means only one optical computation is required for Si-based capacitive modulator, with an oxide thickness set in the middle to 10nm.

For SiGe/Si modulator, the optical mode depends on the Gecontent as well. The refractive index of strained SiGe has been studied experimentally [START_REF] Zollner | Optical constants and ellipsometric thickness determination of strained Si 1 -x Ge x : C layers on Si ( 100 ) and related heterostructures[END_REF]. It has been measured that the refractive index varies slowly with Ge-content, starting at 3.5 for Si and reaching 3.59 for 27% Ge-content at 1310nm. Moreover, the SiGe-thickness is only 15nm, so we can expect a small impact of the Ge-content on the calculated optical mode for Ge-content below 30%. Finally, one field distribution is enough again for SiGe/Si modulator, with a Ge-content set to 20% and an oxide thickness set to 10nm.

These 2 optical mode calculations are performed using Lumerical software as shown on Fig. 3 [START_REF]Lumerical Mode Solutions[END_REF]. Concerning the SiGe device, only 4% of the optical power is within the SiGe layer.

C. Carrier Distribution Modeling

Additionally, the electrical behavior of the SIS junction must be calculated. The carrier distribution variations are mainly located at the oxide interfaces. In the following, the problem is reduced to a 1D-problem, perpendicular to the junction surface as shown in Fig. 4.

The calculation consists on solving the carrier distribution on the SIS junction. This is performed by computing numerically 1D-Poisson equation on Matlab [START_REF]Matlab[END_REF]. In the output results plotted in Fig. 5, the depletion and accumulation regimes are

clearly visible. Moreover, the band offset between Si and SiGe implies a clear and sharp concentration discontinuity at the SiGe/Si interface [START_REF] Shaffler | High-mobility Si and Ge structures[END_REF].

Because we are looking for short modulators, the modulator 3dB-bandwidth is given by the low-pass filter formula:

fc = 1 2πRC (5)
With R the access resistance and C the junction capacitance.

The capacitance of the capacitive modulator (F/mm) is given by the dual plate capacitance:

C = ϵ SiO 2 . w capa /t ox (6)
With 𝜖 𝑆𝑖𝑂 2 the relative permittivity of SiO 2 , 𝑤 𝑐𝑎𝑝𝑎 the waveguide width (set to 400nm) and 𝑡 𝑜𝑥 the grid oxide thickness.

The access resistance is calculated by integrating the doping profiles of the bottom and top slabs. The integration is performed assuming 2.5 width-slab and stops at the waveguide edges as shown in Fig. 4. Here we assume the same resistivity for Si and polySi, given by Masetti model [START_REF] Masetti | Modeling of Carrier Mobility Against Carrier Concentration in Arsenic-, Phosphorus-, and Boron-Doped Silicon[END_REF]:

R = R n + R p (7)
R p = ∫(q. hslab. N 𝑆𝑖 (x). µ p ) -1 . dx With ℎ𝑠𝑙𝑎𝑏 the Si and polySi slab thicknesses set to 150nm, 𝑞 the elementary electric charge, 𝑁 𝑆𝑖 (𝑥) the doping concentration in the SOI as a function of the position, 𝑁 𝑝𝑜𝑙𝑦𝑆𝑖 (𝑥) the doping concentration in the polySi as a function of the position and µ 𝑛,𝑝 respectively the electrons and holes mobility in Si.

R n = ∫(q. hslab. N polySi (x). µ n ) -1 . dx (8)

D. Electro-refractive Perturbation

Finally, the optical and electrical calculations are combined using the perturbation theory. The electro-refractive coefficients of Si are estimated to be at 1310nm [START_REF] Nedeljkovic | Free-carrier electrorefraction and electroabsorption modulation predictions for silicon over the 1-14-μm infrared wavelength range[END_REF]: ΔnSi = 2,98.10 -22 ΔNe 1.016 + 1,25.10 -18 ΔNh 0.835 [START_REF] Zhu | Influence of RTA and LTA on the Optical Propagation Loss in Polycrystalline Silicon Wire Waveguides[END_REF] αSi = 3,48.10 -22 ΔNe 1.229 + 1,02.10 -19 ΔNh 1.089 [START_REF] Takenaka | High-efficiency , Low-loss Optical Phase Modulator based on III-V / Si Hybrid MOS Capacitor[END_REF] For strained-SiGe, the theoretical coefficients for Ge-content between 10% and 50% are [START_REF] Takenaka | Strain Engineering of Plasma Dispersion Effect for SiGe Optical Modulators[END_REF]: ΔnSiGe(x) = (1,2 + 4.x).1,25.10 -18 ΔNh 0.835 [START_REF] Manku | Drift hole mobility in strained and unstrained doped Si(1-x)Gex alloys[END_REF] αSiGe(x) = (1,4 + 6.x).1,02.10 -19 ΔNh 1.089 [START_REF] Fujikata | High-performance Si optical modulator with strained p-SiGe layer and its application to 25 Gbps optical transceiver[END_REF] This leads to the computation of the phase-shift (°/m) and losses (dB/m) thanks to [START_REF] Iizuka | Elements of Photonics, Volume II For Fiber and Integrated Optics[END_REF] With Δn(x, y) the local index variations given by the electrorefractive coefficients, E(x, y) the optical field amplitude from the Lumerical simulation, n eff the effective index and k(x, y) the imaginary part of the refractive index. Optical absorption of intrinsic materials and electro-refractive coefficients are considered for 𝑘(𝑥, 𝑦). For silicon at 1310nm, intrinsic absorption is assumed to be 0 at 1310nm. Theoretical values have been used for strained-SiGe taken from [START_REF] Polleux | Optical Absorption Coefficient Determjnation and Physical Modelling of Strained Sigeki Photodetectors[END_REF].

E. OMA Optimization

Once the optical simulation is performed, the modulator simulation on Matlab takes approximatively 6 s. This is short enough to allow OMA optimization for the 12 optimization parameters. This is performed with a genetic algorithm with elitist selection [START_REF] Davis | The Practical Handbook of GENETIC ALGORITHMS: Applications[END_REF]. Genetic algorithms are suitable to solve this problem since they perform a random search in the parameters space and exhibit a weak dependency on initial values. Fig. 6. shows the algorithm convergence. The algorithm is found to converge after 10000 modulators simulations, which corresponds to 17h of calculations using 8 cores working at 3GHz. Fig. 6. Optimization of the capacitive modulator. The FoM to maximize is the OMA. The algorithm has converged after 10000 simulations.

II. OPTIMIZED OMA RESULTS

The model coupled with the genetic algorithm is used to optimize OMA. This is done at 25Gb/s OOK for 100G applications and at 53GBd PAM-4 for 400G applications. The minimum voltage is set to 0V, and the maximum voltage to 1.8V.

A. Results for 25Gb/s OOK

As seen previously, 25Gb/s operation can be driven with a single-stage electrode if the modulator footprint is below 1mm. Besides it is shown that modulators with a 3dB-bandwidth of 20GHz are sufficient for 25Gb/s NRZ operation [START_REF] Boeuf | Silicon Photonics R&D and Manufacturing on 300mm Wafer Platform[END_REF]. Thus the optimization is performed to maximize OMA for 1mm-long device, with a cut-off frequency of 20GHz at least. This is done for full-Si modulators and for SiGe/Si modulators. Once the optimized modulator design is found, the model result is compared to full numerical simulation with Lumerical. Optimized OMAs calculated are reported in Fig. 7, the scattering losses 𝛼 𝑝𝑟𝑜𝑝 (depending on the fabrication platform) is set on the x-axis and decreases the OMA by exp(-𝛼 𝑝𝑟𝑜𝑝 ).

Concerning the full-Si optimized modulator, the maximum achievable OMA is found to be -1.38dB (for scattering loss = 0). This is obtained with a capacitor oxide thickness equal to 5.5nm, leading to V π L π =0.47V.cm. At 0V, the optical losses are about 0.72dB/mm and increase to 1.43dB/mm at 1.8V. The optimized modulator parameters are given in Table 1.

In addition, using strained-SiGe improves the optimized OMA. The optimized maximum OMA (i.e. for scattering losses equal to 0) is equal to -0.78dB, using 20% Ge in the SiGe alloy and a 9nm-thick capacitor oxide. Despite a thicker grid oxide, this configuration is more efficient thanks to the strained-SiGe with V π L π = 0.4V. cm. Free carrier absorption losses vary from 0.32dB/mm at 0V to 1.09dB/mm at 1.8V. Further, one can notice that the SiGe/Si modulator exhibits higher efficiency than full-Si modulator (71°/mm for full-Si and 78°/mm for SiGe/Si), whereas absorption losses remain lower (1.44dB/mm at 1.8V for full-Si against 1.08dB/mm for SiGe/Si). This is not completely intuitive considering n/ is lower for SiGe than for Si. This results will be discussed in more details later.

B. Results for 53GBaud PAM-4

Here we focus on the possibility to achieve 400G operation, using 53GBd PAM-4 format, with a single-stage electrode for a device length below 500µm. A 3dB-bandwith up to 37GHz is sufficient to achieve 53GBd operation [START_REF] Van Campenhout | Silicon Photonics for 56G NRZ Optical Interconnects[END_REF]. The optimized results for full-Si and SiGe/Si modulators are plotted in Fig. 8.

For the full-Si modulator, the optimized outer OMA is found to be -4.11dB, meaning inner OMA equal to -8.89dB. Here a 8.4nm-thick capacitor oxide is used leading to V π L π =0.59V.cm. Free carrier absorption varies from 1.27dB/mm to 1.78dB/mm between 0V and 1.8V.

Concerning the SiGe/Si modulators, the best achievable outer OMA shifts to -2.5dB (for scattering losses =0). This corresponds to inner OMA equal to -7.3dB. This result is obtained with a 10nm-thick capacitor oxide and a 25% Gecontent, leading to V π L π =0.38V.cm. Free carrier absorption increases from 1.13dB/mm at 0V to 1.98dB/mm at 1.8V.

III. DISCUSSION Fig. 7 and Fig. 8 show that the model results are in good agreement with numerical simulations, validating the initial assumption of reducing the electronic simulation into a 1Dproblem. Assuming approximatively 10min computation time to simulate a capacitive modulator with Lumerical, a single optimization process would require 70 days using Lumerical with the same hardware configuration (8 cores operating at 3GHz).

A. 25Gb/s OOK result discussion

From the 25Gb/s operation results, capacitive modulators seem attractive. Optimum OMA for full-Si modulators is found to be -1.38dB for lossless propagation. This result shifts to -2dB for 7dB/cm propagation losses. Thus, 1mm-long full-Si capacitive modulators with relatively high propagation losses exhibit OMA equivalent to PN-modulators [START_REF] Monfray | Optimization of deep rib high speed phase modulators on 300mm industrial Si-photonics platform[END_REF]. This indicates that it should be possible to reach high OMAs with capacitive modulators for 25GB/s, even with high scattering losses.

Furthermore, using strained-SiGe improves considerably the OMA. For the optimized SiGe/Si modulator, the OMA increases to -0.78dB. Compared to the optimized full-Si modulator, the optimized strained-SiGe modulator exhibits a higher efficiency. With strained-SiGe, the optimum modulator has a thicker interfacial oxide, leading to a smaller device capacitance. This allows to have a slightly higher resistance and thus lower doping levels in Si (𝑁 𝑎 =1.7e17at/cm 3 with SiGe against 𝑁 𝑎 =3.7e17at/cm 3 without SiGe). As a result, and even if the ratio Δn/Δ𝛼 is lower for SiGe alloys, the SiGe/Si configuration shows higher efficiency while exhibiting lower free carrier absorption losses. In particular, the OMA remains below -2dB for propagation losses up to 10dB/cm. This means that SiGe/Si modulators are even more suitable for 25Gb/s. High OMA is achieved, whereas 1mm-device footprint allows a single-stage electrode driving.

B. 53Gbaud PAM-4 result discussion

The obtained results for 53Gbaud show how challenging it is to design 500µm-long modulators operating 37GHz. Indeed, the length of the modulator has to be reduced to remain in lumped mode. Compared to the 25Gbd OOK operation, the modulation length is halved while the bandwidth is almost doubled. It means that the capacitance must increase, while at the same time the RC constant must be reduced to increase the bandwidth. Logically, after the genetic optimization process, the bandwidth target is reached by increasing all doping levels compared to the 25Gb/s results (for both full-Si and SiGe/Si modulators). Besides the doping levels get closer to the waveguide center. Both changes lead to higher free carrier absorption. As a consequence, the optimized full-Si modulator exhibits a low outer OMA of -4.11dB even without considering any scattering losses. This value is lower than values already reported for PN-modulators [START_REF] Monfray | Optimization of deep rib high speed phase modulators on 300mm industrial Si-photonics platform[END_REF]. Thus, using a full-Si capacitive modulator with a single-stage electrode is expected to be less performant than PN junctions for 53GBd operation.

Using strained-SiGe improves significantly the results again. The ideal outer OMA for lossless propagation reaches -2.5dB, which is close to 2mm-long PN-modulators. For scattering losses below 10dB/cm, the outer OMA remains higher than -3dB. Here the optimal Ge-content in the SiGe layer reaches 25%, while it is only 20% for 25GBd OOK operation. Nevertheless, the capacitor oxide thickness for 25GBd OOK and 53GBd PAM-4 remains close (9nm and 10nm). From this, it is clear that the bandwidth is increased by reducing the access resistance rather than the capacitance. Furthermore, the efficiency is improved slightly (78°/mm for 25GBd against 84°/mm for 50GBd) compared to the optimized design for 25Gb/s since the Ge-content increases by 5%. Fig. 9. Influence of the polySi resistivity on the optimal OMA. polySi resistivity is assumed to be proportional to the SOI resistivity.

C. Model limitation

The previous results must be interpreted with caution since several assumptions are made in the model. Here we would like to give an insight on assumptions/approximations made in our model, in order to interpret properly the results and give realistic limits for their validity.

A first limitation concerns the strained-SiGe layer properties. Electro-refractive coefficients and intrinsic optical losses have been estimated theoretically. Experimental data may give different results, depending on epitaxial recipes for instance.

Besides, no relaxation is considered in the SiGe model. If relaxation occurs during the process, electronic and optical properties of SiGe change and affect the overall result. In particular, it has been shown that strained-SiGe relaxes for high thermal budget [START_REF] Misra | Strain relaxation in SiGe due to process induced defects and their subsequent annealing behavior[END_REF].

In the model we assume the same resistivity for polySi and monocrystalline-Si. Nevertheless, it has been shown that polySi can be more resistive than monocrystalline-Si [START_REF] Kamins | [END_REF], depending on the polysilicon grain size. The cut-off frequency is affected directly. Moreover, the depletion regime should be less efficient than estimated as shown experimentally [START_REF] Douix | SiGe-enhanced Si capacitive modulator integration in a 300 mm silicon photonics platform for low power consumption[END_REF], decreasing the OMA as well. The polySi resistivity influence on the OMA for 25GBd OOK operation with full-Si modulators is shown on Fig. 9. Here we assume for simplicity that polySi resistivity is proportional to the monocrystalline-Si one. For each curve, optimization is performed to find the best OMA with 𝑓 𝑐 >20GHz. For 𝜌 𝑝𝑜𝑙𝑦 = 𝜌 𝑚𝑜𝑛𝑜-𝑆𝑖 , the result is the same than Fig. 7. It is clear that resistivity affects the optimum OMA. For resistivity 7 times higher than monocrystalline-Si, OMA drops by 1.8dB compared to 𝜌 𝑝𝑜𝑙𝑦 = 𝜌 𝑚𝑜𝑛𝑜-𝑆𝑖 .

On the opposite side, this shows that capacitive modulators are relevant only if a good quality polySi, or monocrystalline-Si, is integrated for the upper slab. Several experimental works are on-going & encouraging using Si recrystallization process [START_REF] Debnath | Fabrication of Arbitrarily Narrow Vertical Dielectric Slots in Silicon Waveguides[END_REF] or even III-V bonding [START_REF] Takenaka | High-efficiency , Low-loss Optical Phase Modulator based on III-V / Si Hybrid MOS Capacitor[END_REF].

From these model assumptions, the obtained results should be then seen as an upper limit and a guideline for technological developments.

CONCLUSION

In this paper, we showed an accurate simplified electrooptical model of capacitive optical modulators. Optimization of modulator parameters is performed to estimate ultimate OMA that can be expected from a modulator short enough to be driven by a single stage electrode. It is demonstrated that full-Si modulators are suitable for 25GBd OOK applications even when using waveguides with scattering losses up to 1dB/mm. Furthermore, strained-SiGe enhances the modulator performances considerably. In particular, using strained-SiGe seems mandatory to reach 53GBd PAM-4 applications with 500µm-long modulators with an OMA close to PN-modulators one. This work shows that high quality polycrystalline silicon and strained-SiGe are necessary to do so, giving targets for technological & material developments.

Fig. 1 .

 1 Fig. 1. (a) MZI in quadrature for OOK-NRZ. (b) Eye-diagram for OOK. (c) Eye-diagram for PAM-4 modulation.

Fig. 2 .

 2 Fig. 2. Studied capacitive modulator using a horizontal capacitor oxide and strained-SiGe

Fig. 3 .

 3 Fig. 3. TE-mode in the capacitive structure. (a) full-Si modulator. (b) SiGe/Si modulator with 15nm-thick Si 0.8 Ge0 0.2 computed on Lumerical Mode

Fig. 4 .

 4 Fig. 4. 1D-cut used for solving 1D-Poisson equation and capacitance and resistance values used for bandwidth computation

Fig. 5 .

 5 Fig. 5. Free carrier concentration computed by solving 1D Poisson equation for a 15nm-thick grid oxide and Na = Nd = 5e17at/cm 3 . (a) in depletion at 0V. (b) in accumulation at 1.8V

Fig. 7 .

 7 Fig. 7. Optimized OMA for full-Si and SiGe/Si capacitive modulators for 25GBd/s NRZ operation. Optimization is performed for 1mm-long modulator.

Fig. 8 .

 8 Fig. 8. Outer OMA for optimized full-Si and SiGe/Si capacitive modulators for 53GBd PAM-4 operation. Optimization is performed for 500µm-long modulator.

  :

	ΔΦ =	360 λ	.	∫ ∫ n(x, y). Δn(x, y). |E(x, y)| 2 . dx. dy ∫ ∫ n eff . |E(x, y)| 2 . dx. dy	(14)
	IL = -	20 ln(10)	.	2π λ	.	∫ ∫ n(x, y). k(x, y). |E(x, y)| 2 . dx. dy ∫ ∫ n eff . |E(x, y)| 2 . dx. dy	(15)

TABLE I

 I

	OPTIMIZED MODULATOR PARAMETERS FOR 25GBD AND 53GBD OPERATIONS
	Name	Si 20GHz	SiGe 20GHz	Si 37GHz	SiGe 37GHz
	𝑡 𝑜𝑥 (nm)	5.5	9	8.4	10
	%Ge		--	20%	--	25%
	𝑁 𝑑 (at/cm3)	3.8e17	2e17	6.4e17	4.7e17
	𝑁 𝑎 (at/cm3)	3.7e17	1.7e17	4.5e17	3.2e17
	𝑁 + (at/cm3)	3.2e18	1.2e18	5.3e18	4.5e18
	𝑃 + (at/cm3)	3.6e18	1.2e18	3.6e18	2.8e18
	𝑁 ++ (at/cm3)	7.9e19	5.1e19	9.6e19	8.7e19
	𝑃 ++ (at/cm3)	8.2e19	4.3e19	9.5e19	9.6e19
	Length (mm)	1	1	0.5	0.5


Elsi Ahmadieh
email: elsiahmadieh@gmail.com

Nour El Madhoun
email: nour.el-madhoun@isep.fr@lip6.fr

NFTs for Online Trading of Artworks

Keywords: Blockchain, NFT, Smart-Contract

Smart contracts are an important key component of blockchain technology. They allow the creation of unique digital assets (such as artwork, real estate, collectibles, etc.) by representing them as NFTs. These NFTs can then be sold and purchased transparently and securely on a decentralized NFT marketplace. In this paper, we develop a smart contracts application that allows the creation of an NFT, which represents an artwork, that can be sold on an NFT marketplace while transferring the ownership from the seller to the buyer.

I. INTRODUCTION

In the past, artists would go to art exhibitions to show their works so that people and businessmen would buy them. With blockchain technology, the same process can take place in an online marketplace specifically using NFTs (Non-Fungible Tokens). In this section, we present a basic smart-contracts application that we have developed with the solidity language on the Ethereum blockchain [1] [2]. This application allows to create, from a URI (Uniform Resource Identifier), an NFT that represents an artwork and sell this NFT on an NFT marketplace by transferring its ownership from the seller to the buyer. Before proceeding to create an NFT that represents a particular artwork, it is first required to generate the metadata file of that artwork as shown in Fig. 4. Subsequently, it is required to use the InterPlanetary File System (IPFS) to store the artwork's visual through a token URI. More details are presented in [START_REF] Jain | A Brief Introduction to Web3: Decentralized Web Fundamentals for App Development[END_REF] on this procedure. In the following paragraphs, we illustrate the role of the different functions we have developed in our smart-contacts application by dividing them into several steps: -First Step "Creating an NFT using the token URI": the function shown in Fig. 5 is developed in the contract NFTCreation. It takes as parameter the token URI in order to create the corresponding NFT and returns the NFT identifier (initialized to 0 at the beginning and incremented by 1 each time an NFT is created). As illustrated in Fig. 6, the returned identifier of the NFT is 1 because we have created only one NFT and the owner inventor is the artist who has the address 0x5B...ddC4. If the same artist or another artist will create another NFT for another artwork (and thus another token URI), the function will then return an identifier which is equal to 2.

-Second

Step "Adding the created NFT to the NFT marketplace": the data structure shown in Fig. 7 and the function shown in Fig. 8 are developed in the contract NFTMarketplace. The objective of the function shown in Fig. 8 and Fig. 9 is to add an NFT to the marketplace with a price proposed by the seller. This function uses the data structure MarketItem. The objective of the latter is to reference all NFTs added to the marketplace with item identifiers in the marketplace: "each NFT added to the marketplace for trading (selling and buying) will have its identifier itemMarketId in the marketplace". Moreover, this data structure indicates for each NFT added to the marketplace the address of the inventor owner and the address of the seller owner. Let's take an example (see Fig. 9) of a new NFT which was created by the artist 0x5B...ddC4 and was added (itemMarketId = 1) for the first time to the marketplace for trading by its inventor owner who is this artist 0x5B...ddC4. In this case, the address of the inventor owner and the address of the seller owner are the same and lastItemMarketId will be equal to 0. Let's suppose that this NFT has been bought by the buyer 0xAb....5cb2 (see Fig. 13) and that he later gave it to the marketplace (itemMarketId = 2) to sell it with a new price. In this case, the address of the inventor owner will always remain 0x5B...ddC4, the address of the seller owner will be 0xAb....5cb2 and the lastItemMarketId will be equal to 1 "the identifier of the NFT in the marketplace when it was added by the inventor owner" (see Fig. 10 for more details). 12 and Fig. 13 is to sell an NFT added to the marketplace to another participant (another address) by transferring the ownership of the NFT from the seller to the buyer. This function uses the data structure SoldItem. The objective of the latter is to reference all NFTs sold on the marketplace with identifiers of items sold on the marketplace: "each NFT sold on the marketplace will have its identifier itemSoldId on the marketplace". Moreover, this data structure indicates for each NFT sold on the marketplace the address of the inventor owner and the address of the buyer owner (see Fig. 13 for more details).

III. CONCLUSION

In this paper, we have developed a basic smart-contracts application that allows the creation of an NFT, which represents an artwork, that can then be sold on an NFT marketplace while transferring the ownership from the seller to the buyer.

Fig. 1 .

 1 Fig. 1. Code in solidity of NFTCreation Contract

Fig. 2 .

 2 Fig. 2. Code in solidity of NFTMarketplace Contract

Fig. 3 .

 3 Fig. 3. Smart-Contracts of the Application II. SMART-CONTRACTS APPLICATION FOR NFTS Our application is composed of two smart-contracts (see Fig. 1, Fig. 2 and Fig. 3): (1) the NFTCreation contract which allows the creation of one or more NFTs for the same artist or several artists, (2) the NFTMarketplace contract which

Fig. 4 .

 4 Fig. 4. Creation Token URI

Fig. 7 .

 7 Fig. 7. MarketItem Data Structure

Fig. 10 .

 10 Fig. 10. Function addTokenToMarket (3)


Augustin Mouze
email: augustin.mouze@univ-lille.fr

V Munnier

GROWTH OF HYPERCYCLIC FUNCTIONS: A CONTINUOUS PATH BETWEEN U -FREQUENT HYPERCYCLICITY AND HYPERCYCLICITY

Keywords: 2010 Mathematics Subject Classification. 47A16, 30E25, 47B37, 47B38 hypercyclic vectors, weighted shifts, rate of growth, boundary behavior

Growth of hypercyclic functions: a continuous path between U -frequent hypercyclicity and hypercyclicity

Augustin Mouze, Vincent Munnier

Introduction

A linear operator on a Fréchet space X is said to be hypercyclic if there is a vector x ∈ X such that for every non-empty open set U ⊂ X the set N (x, U ) := {n ∈ N : T n x ∈ U } is infinite, where (T n ) is the sequence of iterates of T . In this situation, x is called a hypercyclic vector. Further there are more precise and stringent notions that allow to quantify how often a hypercyclic vector visits a non-empty open set. A linear operator on a Fréchet space X is said to be frequently hypercyclic (resp. U -frequently hypercyclic) if there is a vector x ∈ X such that for every non-empty open set U ⊂ X the set N (x, U ) has positive lower (resp. upper) density, where the lower and upper densities of a subset A ⊂ N are defined respectively as follows These notions were introduced by Bayart and Grivaux [START_REF] Bayart | Frequently hypercyclic operators[END_REF] and Shkarin [START_REF] Shkarin | On the spectrum of frequently hypercyclic operators[END_REF]. The dynamics of linear operators is a very active branch of research: we refer the reader to [START_REF] Bayart | Dynamics of linear operators[END_REF][START_REF] Grosse-Erdmann | Linear chaos, Universitext[END_REF] and the references therein for background in linear dynamics. Clearly a frequently hypercyclic vector is U -frequently hypercyclic and a U -frequently hypercyclic vector is hypercyclic. Classical examples of frequently or U -frequently hypercyclic operators are given by suitable weighted shifts. As usual we denote by D the open unit disc {z ∈ C : |z| < 1} of the complex plane and by H(D) the set of analytic functions in D. It is well known that H(D) endowed with the topology of uniform convergence on compact subsets is a Fréchet space. For α ∈ R, let w(α) = (w n (α)) be the weighted sequence of nonzero complex numbers given by, for all n ≥ 1,

w n (α) = 1 + 1 n α .
In the present paper, we consider the associated weighted Taylor shift:

T α : H(D) → H(D) given by T α ( k≥0 a k z k ) = k≥0 a k+1 w k+1 (α)z k .
For α = 0, T 0 is the classical Taylor shift operator. It is easy to check that for every real number α, T α is a frequently hypercyclic operator. For instance, we refer the reader to [START_REF] Bernal-González | Universal functions for Taylor shifts[END_REF][START_REF] Grosse-Erdmann | Hypercyclic and chaotic weighted shifts[END_REF][START_REF] Mouze | Growth of frequently hypercyclic functions for some weighted Taylor shifts on the unit disc[END_REF]. The problem of determining possible rates of growth of frequently hypercyclic functions for T α in terms of L p averages was studied in [START_REF] Mouze | Growth of frequently hypercyclic functions for some weighted Taylor shifts on the unit disc[END_REF] (see [START_REF] Mouze | Frequent hypercyclicity of random holomorphic functions for Taylor shifts and optimal growth[END_REF] for the case α = 0 too). For 0 < r < 1 and f ∈ H(D), we consider the classical integral means

M p (f, r) = 1 2π 2π 0 |f (re iθ )| p dθ 1/p (1 ≤ p < ∞) and M ∞ (f, r) = sup 0≤t≤2π |f (re it )|.
In the same way, for any holomorphic polynomial P let us define, for p ≥ 1, In the following, for all p > 1 q will stand for the exponent conjugate to p, i.e. 1 p + 1 q = 1 and we will adopt the convention q = ∞ if p = 1. For 1 ≤ p ≤ ∞, the authors recently highlighted a critical exponent, i.e. a value of the parameter α from which the L p -growth of a frequently hypercyclic function for T α no longer has the same behavior. In the case of frequent hypercyclicity for T α , the critical exponent is equal to α = 1 max (2,q) . Indeed the authors obtained the following statements. First for p > 1 they proved the following result. (1) For any 1 < p < +∞ there is a frequently hypercyclic function f in H(D) for T α satisfying the following estimates: there exists C > 0 such that for every 0 < r < 1 (2,q) . These estimates are optimal: every frequently hypercyclic function f in H(D) for T α is bounded from below by the corresponding previous estimate depending on α.

P p = 1 2π
M p (f, r) ≤        C(1 -r) α- 1 max(2,q) if α < 1 max(2,q) , C| log(1 -r)| 1 p if α = 1 max(2,q) , C if α > 1 max
(2) There is a frequently hypercyclic function f in H(D) for T α satisfying the following estimates: there exists C > 0 such that for every 0 < r < 1

M ∞ (f, r) ≤    C(1 -r) α-1 2 if α < 1/2, C| log(1 -r)| if α = 1/2, C if α > 1/2.
For α = 1/2, these estimates are optimal: every frequently hypercyclic function f in H(D) for T α is bounded from below by the corresponding previous estimate depending on α.

For p = 1, the following result holds. For any positive integer ℓ ≥ 1, log ℓ stands for log ) For any ℓ ≥ 1, there is a frequently hypercyclic function f in H(D) for T α satisfying the following estimates: there exists C > 0 such that for every 0 < r < 1 sufficiently large

M 1 (f, r) ≤    C(1 -r) α log ℓ (-log(1 -r)) if α < 0, C| log(1 -r)| log ℓ (-log(1 -r)) if α = 0, C if α > 0.
Moreover every frequently hypercyclic function f in H(D) for T α satisfies the following estimates:

lim inf r→1 - M 1 (f, r)(1 -r) -α > 0 if α < 0, lim inf r→1 - M 1 (f, r) -log(1 -r) > 0 if α = 0, lim inf r→1 - [M 1 (f, r)] > 0 if α > 0.
It should be noted that the study of the growth of hypercyclic or frequently hypercyclic functions started with those related to the differentiation operator on H(C) (see for instance [START_REF] Blasco | Rate of growth of frequently hypercyclic functions[END_REF][START_REF] Drasin | Optimal growth of frequently hypercyclic entire functions[END_REF][START_REF] Grosse-Erdmann | On the universal functions of G. R. MacLane[END_REF][START_REF] Grosse-Erdmann | Rate of growth of hypercyclic entire functions[END_REF]) but was also recently extended to the partial differentiation operator [START_REF] Gilmore | Optimal growth of harmonic functions frequently hypercyclic for the partial differentiation operator[END_REF] or the Dunkl operator [START_REF] Bernal-González | Rate of growth of hypercyclic and frequently hypercyclic functions for the Dunkl operator[END_REF]. Here, as a first step, we obtain sharp results on the permissible rates of L p -growth of hypercyclic and U -frequently hypercyclic functions for T α . On one hand, for hypercyclicity, for any 1 ≤ p ≤ ∞ we find that the rate of growth (1 -r) min(α,0) turns out to be critical and hence, for any 1 ≤ p ≤ ∞, α = 0 is the critical exponent. Observe that in this case the critical exponent does not depend on p. We refer to Theorem 2.2. In particular this result states that for 1 ≤ p ≤ ∞ there is no hypercyclic function f for T α satisfying lim sup r→1 -((1 -r) -α M p (f, r)) < +∞ if α ≤ 0 while if α > 0 there exist hypercyclic functions f for T α such that the average M p (f, r) is bounded. In passing Theorem 2.2 gives a negative answer to a question of [START_REF] Mouze | Growth of frequently hypercyclic functions for some weighted Taylor shifts on the unit disc[END_REF] which asked if for α < 0 there is a frequently hypercyclic function g α for T α such that lim sup r→1 -((1 -r) -α M 1 (g α , r)) < +∞. On the other hand, for U -frequent hypercyclicity, we find the same critical exponent α = 1 max (2,q) as for frequent hypercyclicity. Therefore contrary to the previous case this exponent depends on p. Moreover we show that the U -frequently hypercyclic functions and the frequently hypercyclic functions for T α share the same admissible (and optimal) L p -growth when α is different from the critical exponent, i.e. α = 1 max (2,q) . Concerning the case α = 1 max(2,q) with 1 ≤ p ≤ ∞, we prove that every U -frequently hypercyclic vector f for T α satisfies lim sup r→1 -M p (f, r) = +∞, without a priori additional information on the growth of the function. We refer to Theorems 3.3 and 3.4. Nevertheless several questions remain and need to be addressed. The first question that comes to mind is the following: what is the optimal boundary growth of U -frequently hypercyclic functions for T α when α is the critical exponent? Further if we go back to what was just said, we see that for p = 1 the critical exponent is always equal to 0 for the hypercyclic case, the U -frequently case and the frequently hypercyclic case. But surprisingly for p > 1 this critical exponent is equal to

1 max(2,q)
for the U -frequently or frequently hypercyclic cases and is equal to zero for the hypercyclic case. Thus, as a second question, we can wonder about what happens between U -frequent hypercyclicity and hypercyclicity. Why does the critical exponent go from 1 max(2,q) to zero? In order to understand this phenomenon, we introduce intermediate notions of linear dynamics between the U -frequent hypercyclicity and the hypercyclicity: the U β γ -frequent hypercyclicity related to notions of upper weighted densities d β γ , with 0 ≤ γ ≤ 1 a continuous parameter, where we replace in the definition of U -frequent hypercyclicity the natural upper density d by d β γ . Moreover for γ = 0 the U β 0 -frequent hypercyclicity will be the frequent hypercyclicity and for γ = 1 the U β 1 -frequent hypercyclicity will be the hypercyclicity. Further for any 0 ≤ γ ≤ γ ′ ≤ 1 and for any subset E ⊂ N, the following chain of inequalities

d(E) ≤ d β γ (E) ≤ d β γ ′ (E) ≤ d β 1 (E)
will show that the U β γ -frequent hypercyclicity for γ ∈ (0, 1) furnish refined notions of linear dynamics between the U -frequent hypercyclicity and the hypercyclicity. We refer the reader to the beginning of Section 4 for the main definitions and properties. Similar notions of weaker densities have been recently studied in the context of linear dynamics (see for instance [START_REF] Bès | Recurrence properties of hypercyclic operators[END_REF][START_REF] Bonilla | Upper frequent hypercyclicity and related notions[END_REF][START_REF] Ernst | U-frequent hypercyclicity notions and related weighted densities[END_REF][START_REF] Ernst | A quantitative interpretation of the frequent hypercyclicity criterion[END_REF][START_REF] Ernst | Frequent universality criterion and densities, Ergodic Theory Dynam[END_REF][START_REF] Menet | A bridge between U-frequent hypercyclicity and frequent hypercyclicity[END_REF] and the references therein). In the present paper, we investigate the growth in terms of L p -averages of U β γ -frequently hypercyclic functions for T α . In particular, for 0 < γ < 1, and for p > 1 we find that the critical exponent is given by α = 1-γ max(2,q) . Hence let us observe that this critical exponent:

• tends to 1 max(2,q) as γ tends to zero, i.e. tends to the critical exponent for the U -frequent hypercyclicity case;

• tends to 0 as γ tends to 1, i.e. tends to the critical exponent for the hypercyclicity case. These estimates thus allow to highlight a continuous path between the rate of growth of hypercyclic and U -frequently hypercyclic functions: the growth (in terms of L p -averages) of a hypercyclic function for T α continuously depends on the frequency of visits (measured by the densities d β γ , 0 ≤ γ ≤ 1) of non-empty open subsets by its orbit under the action of T α . We also show that the estimates on the growth of U β γ -frequently hypercyclic functions that we obtained are optimal. To do this, we apply a method based on the use of Rudin-Shapiro polynomials and inspired by a construction of frequently hypercyclic functions with optimal growth for differentiation operator on H(C) due to Drasin and Saksman [START_REF] Drasin | Optimal growth of frequently hypercyclic entire functions[END_REF] and that has also been adapted for the proofs of Theorems 1.1 and 1.2 in [START_REF] Mouze | Frequent hypercyclicity of random holomorphic functions for Taylor shifts and optimal growth[END_REF][START_REF] Mouze | Growth of frequently hypercyclic functions for some weighted Taylor shifts on the unit disc[END_REF]. For all these results, we refer the reader to Theorems 4.5, 4.10 and 4.12. Finally let us return to the first question mentioned above. In the last section, we answer it by showing that the optimal growth of U -frequently and U β γ -frequently hypercyclic functions for T α coincides whenever α is the critical exponent: actually the L p -growth can be arbitrarily slow as in the hypercyclic case. We refer to Theorem 5.11.

The paper is organized as follows. In Sections 2 and 3 we establish the boundary behavior of hypercyclic functions and U -frequently hypercyclic functions for T α respectively. In Section 4 we deal with the U β γ -frequently hypercyclic functions for T α . In Section 5 we turn our attention to the specific case of critical exponent.

Throughout the paper, whenever A and B depend on some parameters, we will use the notation A B (resp. A B) to mean A ≤ CB (resp. A ≥ CB) for some constant C > 0 that does not depend on the involved parameters.

Growth of hypercyclic functions

In this section, we are going to establish the rate of growth of hypercyclic functions with respect to the weighted Taylor shift operator T α . To do this, inspired by the proofs of [18, Theorem (A)] and [START_REF] Bernal-González | Rate of growth of hypercyclic and frequently hypercyclic functions for the Dunkl operator[END_REF]Theorem 3], where the authors are interested in the rate of growth of hypercyclic functions with respect to the Mac-Lane operator or the Dunkl operator respectively, we need an important tool in linear dynamics: the Universality Criterion. Indeed a natural extension of the notion of hypercyclicity is the concept of universality. A sequence of continuous linear mappings L n : X → Y between topological vector spaces X, Y is said to be universal whenever there exists a vector x ∈ X such that the set {L n x ; n ∈ N} is dense in Y . Such a vector x is called a universal vector for (L n ). Observe that an operator T : X → X is hypercyclic if and only if the sequence (T n ) is universal. The following result which is known as the Universality Criterion furnishes a sufficient condition for universality [START_REF] Grosse-Erdmann | Holomorphe Monster und universelle Funktionen[END_REF]. It is a refined version of Hypercyclicity Criterion [START_REF] Gethner | Universal vectors for operators on spaces of holomorphic functions[END_REF][START_REF] Kitai | Invariant closed sets for linear operators[END_REF].

Theorem 2.1. (Universality Criterion) Assume that X and Y are topological vector spaces, such that X is a Baire space and Y is separable and metrizable. Let L j : X → Y be a sequence of continuous linear mappings. Suppose that there are dense subsets X 0 of X and Y 0 of Y and mappings S j : Y 0 → X such that (i) for every x ∈ X 0 , L j x → 0, (ii) for every y ∈ Y 0 , S j y → 0, (iii) for every y ∈ Y 0 , (L j S j )y → y.

Then (L j ) is universal and the set of universal vectors for (L j ) is residual in X.

Now we are ready to obtain the critical rate of growth for hypercyclic functions with respect to the weighted Taylor shift operator T α . The following statement holds.

Theorem 2.2. Let 1 ≤ p ≤ ∞.
(1) Let α ≤ 0.

(a) For any function ϕ

: [0, 1) → R + with ϕ(r) → ∞ as r → 1 -there is a hypercyclic function f for T α with M p (f, r) ϕ(r)(1 -r) α for 0 < r < 1 sufficiently close to 1.
(b) There is no hypercyclic function f for T α that satisfies, for 0 < r < 1

M p (f, r) ≤ C(1 -r) α , where C > 0. (2) Let α > 0.
(a) There is a hypercyclic function f for T α with

M p (f, r) ≤ C for some C > 0. (b) For any function ϕ : [0, 1) → R + with ϕ(r) → 0 as r → 1 -, there is no hypercyclic function f for T α that satisfies, for 0 < r < 1 M p (f, r) ≤ ϕ(r).
Proof. Since we have for 1

≤ p ≤ l M p (f, r) ≤ M l (f, r), for 0 < r < 1,
it suffices to prove assertions (1a) and (2a) for M ∞ (f, r) and assertions (1b) and (2b) for M 1 (f, r).

(1) We begin by the case α ≤ 0. First we can assume without loss of generality that the function ϕ is increasing and continuous with ϕ(0) > 0. Let us consider the space X of all functions f in H(D) with f (z) = k≥0 a k z k satisfying for every n ≥ 0 ρ n (f ) < +∞ and ρ n (f ) → 0 as n → +∞, where

ρ n (f ) = sup |z|<1 +∞ k=n a k z k (1 -|z|) -α [ϕ(|z|)] -1 .
It is easy to check that X endowed with the norm . = sup n ρ n (.) is a Banach space. Therefore (X, . ) is a Baire space. For all integer j, let L j : X → H(D) be the operator given by L j f = T j f . Clearly (L j ) is a sequence of continuous linear operators. We choose X 0 = Y 0 = P the set of polynomials. The set P is dense in H(D). Moreover, setting the polynomial s

N (f ) = N k=0 a k z k we get ρ n (f -s N (f )) = ρ n (f ) for n ≥ N + 1 ρ N +1 (f ) for n ≤ N which implies f -s N (f ) = sup n≥N +1 ρ n (f ) → 0 as N tends to infinity. Hence P is dense in X.
Then we define the operators S j as follows

S j : P → X, S j ( n k=0 a k z k ) = n k=0 a k (k + 1) α (k + j + 1) α z k+j .
Clearly we have, for all P ∈ P, L j (P ) → 0, as j → +∞, and L j S j (P ) = P. Now we prove that, for all P ∈ P, S j (P ) → 0, as j → +∞. Since S j (z k ) = (k + 1) α S j+k [START_REF] Bayart | Frequently hypercyclic operators[END_REF], it suffices to show that S j (1) → 0, as j → +∞. To do this, observe that

S j (1) = sup 0<r<1 r j (1 -r) -α (j + 1) α ϕ(r) .
Let us define h j : [0, 1) → R + given by h j (r) = r j (1-r) -α ϕ(r)

. We have h j (0) = 0 = lim

r→1 - h j (r).
Let 0 < r j < 1 with h j (r j ) = sup 0<r<1

r j (1-r) -α ϕ(r)
. If r j+1 < r j , we get

h j+1 (r j+1 ) = r j+1 h j (r j+1 ) < r j h j (r j+1 ) ≤ r j h j (r j ) = h j+1 (r j )
which gives a contradiction. Hence the sequence (r j ) is increasing. If r j → γ with γ < 1, then S j (1) = (j + 1) -α h j (r j ) ≤ (j + 1) -α γ j ϕ(0) → 0, as j → +∞.

Otherwise r j → 1 and

S j (1) ≤ (j + 1) -α ϕ(r j ) j j -α j 1 - j j -α -α → 0, as j → +∞.
Thus we have S j (1) → 0 as j tends to infinity. We apply the universality criterion to obtain universal elements for the sequence (L j ) that are hypercyclic functions for T α satisfying the growth condition required.

For assertion (1b), assume that f = k≥0 a k z k is a function in H(D) with, for all 0 < r < 1,

M 1 (f, r) ≤ C(1 -r) α ,
for some C > 0. By Cauchy estimates we get

|a n | ≤ M 1 (f, r) r n .
We obtain, for all n ≥ 0 and all 0 < r < 1,

|a n w 1 (α) . . . w n (α)| ≤ C |w 1 (α) . . . w n (α)| r n (1 -r) α .
Hence we get for all n ≥ 0,

|a n (n + 1) α | ≤ C (n + 1) α e -n/(n+1) (1 -e -1/(n+1) ) α
which is bounded. Hence f cannot be hypercyclic for T α .

(2) Now let us consider the case α > 0.

First we can assume without loss of generality that the function ϕ is increasing and continuous with ϕ(0) > 0. Let us consider the Banach space (H ∞ (D), . )

H ∞ (D) = f ∈ H(D) ; f := sup 0<r<1 M ∞ (f, r) < ∞
which is continuously embedded in H(D). We set X the closure of the polynomials in H ∞ (D). Let us define the sequences (L j ) and (S j ) of linear operators as in the previous case. Clearly we have, for all P ∈ P, where P is the set of polynomials, L j (P ) → 0, as j → +∞, and L j S j (P ) = P. Now we prove that, for all P ∈ P, S j (P ) → 0, as j → +∞. To do this, it suffices to show that S j (1) → 0, as j → +∞. Since

S j (1) ≤ 1 (j + 1) α
and α > 0 we have S j (1) → 0 as j tends to infinity. We apply the universality criterion to obtain universal elements for the sequence (L j ). These universal vectors are clearly hypercyclic functions for T α satisfying the growth condition required.

For assertion (2b), assume that f = k≥0 a k z k is a function in H(D) with, for all 0 < r < 1, M 1 (f, r) ≤ ϕ(r), where ϕ : [0, 1) → R + is a function such that ϕ(r) → 0 as r → 1 -. We obviously may assume that ϕ is continuous and decreasing. By Cauchy estimates we get

|a n | ≤ ϕ(r) r n .
We obtain, for all n ≥ 0 and all 0 < r < 1,

|a n (n + 1) α | ≤ (n + 1) α ϕ(r) r n .
Let us choose a sequence (r n ) such that r n ≥ max(1 -1/n, ϕ -1 ((n + 1) -α )). Hence we get for all n ∈ N,

|a n (n + 1) α | ≤ (n + 1) α ϕ(r n ) r n n ≤ e -n log(1-1/n)
which is bounded. Hence f cannot be hypercyclic for T α .

Remark 2.3. For α < 0, Theorem 2.2 ensures that every hypercyclic function f for T α satisfies lim sup

r→1 - (1 -r) -α M 1 (f, r) = +∞.
Since a frequent hypercyclic function is necessarily hypercyclic, this observation gives a negative answer to the first part of the question from Remark 4.5 of [START_REF] Mouze | Growth of frequently hypercyclic functions for some weighted Taylor shifts on the unit disc[END_REF] which asked if for α < 0 there is a frequently hypercyclic function g α for T α such that lim sup

r→1 - ((1 -r) -α M 1 (g α , r)) < +∞.

Growth of U -frequently hypercyclic functions

In this section we are interested in the growth of U -frequently hypercyclic functions for T α . First of all, in the sequel we will need the following easy lemmas. Lemma 3.1. Let N ∈ N. Let A N be a subset of {1, . . . , N }. For all γ ∈ R \ {1} the following estimate holds

k∈A N (k + 1) γ ≥          (#A N + 1) γ+1 -1 γ + 1 if γ ≥ 0, (N + 2) γ+1 γ + 1 1 -1 - #A N N + 2 γ+1 if γ < 0.
Proof. We begin by the case γ ≥ 0. We write

k∈A N (k + 1) γ ≥ #A N k=1 (k + 1) γ ≥ #A N 0 (t + 1) γ dt,
which gives the announced result. For γ < 0 with γ = -1, we obtain in an analogue way

k∈A N (k + 1) γ ≥ N k=N -#A N +1 (k + 1) γ ≥ N +1 N -#A N +1 (t + 1) γ dt,
which allows to finish the proof.

Lemma 3.2. Let (u k ) and (v k ) be two sequences of non-negative real numbers. Assume that (v k ) is decreasing. For any increasing sub-sequence (N j ) ⊂ N, the following inequality holds, for all l ≥ 1:

N l k=1+N 0 u k v k ≥ S N l v N l -S N 0 v N 0 + l j=1 S N j-1 (v N j-1 -v N j ), with S N = N k=1 u k .
Proof. Observe that (S N ) is increasing and (v k ) is decreasing. Thus by using a summation by parts we derive

N l k=1+N 0 u k v k = l j=1 N j 1+N j-1 u k v k = l j=1   S N j v N j -S N j-1 v N j-1 + N j -1 k=N j-1 S k (v k -v k+1 )   = S N l v N l -S N 0 v N 0 + l j=1 N j -1 k=N j-1 S k (v k -v k+1 ) ≥ S N l v N l -S N 0 v N 0 + l j=1 S N j-1 (v N j-1 -v N j ).
We are ready to establish the boundary behavior of U -frequently hypercyclic functions for T α . Actually we are going to prove that these functions share the same optimal growth as frequently hypercyclic functions except in the case where α is the critical exponent, for which we will show in Section 5 that the growth can be arbitrarily slow. We begin by the case p > 1.

Theorem 3.3. Let f be a U -frequently hypercyclic function for the operator T α and 1 < p ≤ ∞. Then the following estimates hold (2, q) .

lim sup r→1 - (1 -r) 1 max(2,q) -α M p (f, r) > 0, if α < 1 max(2, q) , lim sup r→1 - (M p (f, r)) = +∞, if α = 1 max(2, q) , lim sup r→1 - M p (f, r) > 0, if α > 1 max
For α = 1 max(2,q) , these results are optimal in the following sense: for all p > 1 there exists a U -frequently hypercyclic function for T α such that, for every 0 < r < 1,

M p (f, r) (1 -r) α- 1 max(2,q) if α < 1 max(2,q) 1 if α > 1 max(2,q) .
Proof. We write f = k≥0 a k (k+1) α z k . Since f is U -frequently hypercyclic there exists an increasing sub-sequence (n k ) ⊂ N with positive upper density such that for all k ≥ 1

|T n k α f (0) -3/2| = |a n k -3/2| < 1/2. We get, for all k ≥ 1, |a n k | ≥ 1. Set I = ((n k )
) and for all N ≥ 1, I N = I ∩ {1, . . . , N }. The hypothesis d(I) > 0 ensures that there exist 0 < C < 1 and an increasing sequence (N l ) of positive integers such that

(1) #I N l ≥ CN l .
Up to take a sub-sequence, we can also assume that ( 2)

C(N l+1 + 1) ≥ N l + 1.
Let us consider, for all l ≥ 1, 1

-1 N l -1 ≤ r l < 1 -1 N l . Thus we derive (3) N l -1 ≤ 1 1 -r l < N l .
(1) Case 2 ≤ p ≤ ∞: Jensen'inequality and Parseval's Theorem give

[M p (f, r l )] 2 ≥ [M 2 (f, r l )] 2 = k≥0 |a k | 2 (k + 1) 2α r 2k l ≥ N l k=1 |a k | 2 (k + 1) 2α r 2k l .
Thus we deduce

(4) [M p (f, r l )] 2 ≥ 1 - 1 N l -1 2N l N l k=1 |a k | 2 (k + 1) 2α N l k=1 |a k | 2 (k + 1) 2α
and using the inequality

|a k | ≥ 1 for k ∈ I N l (5) [M p (f, r l )] 2 k∈I N l 1 (k + 1) 2α .
(a) Case α ≤ 0: Combining Lemma 3.1 with (1), ( 3) and [START_REF] Bès | Recurrence properties of hypercyclic operators[END_REF] we get

[M p (f, r l )] 2 N -2α+1 l ≥ (1 -r l ) 2α-1 .
(b) Case 0 < α < 1/2: using (5) and Lemma 3.1 again, we get

[M p (f, r l )] 2 ≥ (2 + N l ) -2α+1 -2α + 1 1 -1 - #I N l 2 + N l -2α+1
.

The inequality (1) ensures 1 -

#I N l 2+N l -2α+1 ≤ 1 -CN l 2+N l -2α+1
. We deduce by using (1) and ( 3) again

[M p (f, r l )] 2 N -2α+1 l ≥ (1 -r l ) 2α-1 .
Hence we conclude lim sup

r→1 - (1 -r) 1 2 -α M p (f, r) > 0.
(c) Case α = 1 2 : using (4), we know

[M p (f, r l )] 2 l j=1 N j k=1+N j-1 |a k | 2 k + 1 .
Hence applying Lemma 3.2 with u k = |a k | 2 and v k = 1/(k + 1), we get, for all l ≥ 1,

[M p (f, r l )] 2 S N l N l + 1 - S N 0 N 0 + 1 + l j=1 S N j-1 1 N j-1 + 1 - 1 N j + 1
.

By construction S N k-1 N k-1 + 1.
Thus taking into account (2) we derive, for all l ≥ 1, the inequality

[M p (f, r l )] 2 l j=1 1 - N j-1 + 1 N j + 1 l,
that allows to obtain lim sup

r→1 - M p (f, r) = +∞.
(2) Case 1 < p < 2: From Hausdorff-Young inequality (see [START_REF] Duren | Theory of H p spaces[END_REF]) we get

[M p (f, r l )] q ≥ k≥0 |a k | q (k + 1) qα r qk l ≥ N l k=1 |a k | q (k + 1) qα r qk l .
Thus we deduce

[M p (f, r l )] q ≥ 1 - 1 N l -1 qN l N l k=1 |a k | q (k + 1) qα N l k=1 |a k | q (k + 1) qα .
Using the same strategy as in the case 2 ≤ p ≤ ∞, we obtain, lim sup

r→1 - (1 -r) 1 q -α M p (f, r) > 0, for α < 1/q, lim sup r→1 - (M p (f, r)) = +∞, for α = 1/q.
Moreover, since a U -frequently hypercyclic function is necessarily hypercyclic, the assertion for the case α > 1 max(2,q) of the statement is given by the assertion (2b) of Theorem 2.2. Finally, since a frequently hypercyclic function is necessarily U -frequently hypercyclic, Theorem 1.1 ensures that the estimates we have proved are optimal when α = 1 max(2,q) . Now we deal with the case p = 1.

Theorem 3.4. Let f be a U -frequently hypercyclic function for the operator T α . Then, the following estimates hold

lim sup r→1 - (1 -r) -α M 1 (f, r) = +∞, if α ≤ 0, lim sup r→1 - M 1 (f, r) > 0, if α > 0.
For α = 0, these results are optimal in the following sense: for any positive integer l ≥ 1, there exists a U -frequently hypercyclic function for the operator T α such that for every 0 < r < 1 sufficiently large

M 1 (f, r) (1 -r) α log l (-log(1 -r)) if α < 0 1 if α > 0.
Proof. First since a U -frequently hypercyclic function is necessarily hypercyclic, the assertions (1a) and (2b) of Theorem 2.2 ensures that, lim sup

r→1 - (1 -r) -α M 1 (f, r) = +∞, if α ≤ 0, and lim sup r→1 - M 1 (f, r) > 0, if α > 0.
Moreover, since a frequently hypercyclic function is necessarily U -frequently hypercyclic, Theorem 1.2 shows that the previous estimates are optimal when α = 0.

Between U -frequent hypercyclicity and hypercyclicity

Let 1 ≤ p ≤ ∞. In view of Theorems 1.1, 1.2, 3.3 and 3.4, the critical exponent related to the L p growth of frequently hypercyclic functions for T α is the same as that related to the L p growth of U -frequently hypercyclic functions. It is equal to 1 max (2,q) . Nevertheless this critical exponent is always equal to 0 in the case of hypercyclic functions and, hence it does not depend on p. In this section we are interested in what happens between U -frequent hypercyclicity and hypercyclicity. In particular, when p > 1, we will try to understand why and how the critical exponent goes from 1 max(2,q) in the case of L p -norm of U -frequent hypercyclic functions to 0 for the L p -norm of hypercyclic functions. To do this, we introduce intermediate notions of linear dynamics that link U -frequent hypercyclicity and hypercyclicity. First of all, we need some definitions and results. These quantities enjoy all the classical properties of densities (see [START_REF] Ernst | A quantitative interpretation of the frequent hypercyclicity criterion[END_REF][START_REF] Freedman | Densities and summability[END_REF]) and allow to define dynamical notions of the same nature as hypercyclicity or U -frequent hypercyclicity. Definition 4.2. Let β = (β n ) be a non-decreasing sequence of positive real numbers tending to infinity and let E be a subset of N. An operator T : X → X, where X is a Fréchet space, is said to be U β -frequently hypercyclic if there exists x ∈ X such that for every non-empty open subset

U ⊂ X, d β ({n ∈ N : T n x ∈ U }) > 0.
In the sequel, we are interested in densities given by the weighted sequence denoted by β γ and defined by β γ = (e n γ ), where γ is a parameter with 0 ≤ γ ≤ 1. First of all, let us notice that:

(i) the density d β 0 coincides with the upper natural density d, (ii) for any subset E ⊂ N, d β 1 (E) > 0 if and only if E is infinite. Moreover for 0 < γ < 1 an integral comparison test leads to the estimate

n k=1 e k γ ∼ n 1-γ γ e n γ ,
as n tends to infinity, that we will use regularly in the rest of the paper. In addition, according to Lemma 2.8 of [START_REF] Ernst | A quantitative interpretation of the frequent hypercyclicity criterion[END_REF] the following inequalities hold.

Lemma 4.3. For any 0 ≤ γ 1 ≤ γ 2 ≤ 1 and for any subset E of N, we have

d(E) ≤ d β γ 1 (E) ≤ d β γ 2 (E) ≤ d β 1 (E).
Therefore the densities d β γ can give very different notions of dynamics that are intermediate between U -frequent hypercyclicity and hypercyclicity. In particular the following lemma holds. Lemma 4.4. Let 0 < γ ≤ 1. There exists a subset E γ ⊂ N such that, for any 0 ≤ γ ′ < γ ≤ 1,

d β γ (E) > 0 and d β γ ′ (E) = 0.
Proof. First observe that, for all 0 < t < 1, (6)

2 n-1 k=1 e k t 2 n k=1 e k t ∼ 2 -(1-t) e -2 nt (1-2 -t ) → 0 as n → +∞. Let γ = 1. Set E γ = N n≥⌊ 1 γ ⌋+1 2 n -⌊2 n(1-γ) ⌋; 2 n .
Clearly, for all n large enough, we have (7)

2 n k=1;k∈Eγ e k γ ≥ 2 n k=2 n -⌊2 n(1-γ) ⌋+1 e k γ .

Moreover we get

2 n k=2 n -⌊2 n(1-γ) ⌋+1 e k γ 2 n k=1 e k γ = 1 - 2 n -⌊2 n(1-γ) ⌋ k=1 e k γ 2 n k=1 e k γ .
But we compute

2 n -⌊2 n(1-γ) ⌋ k=1 e k γ 2 n k=1 e k γ ∼ 1 - ⌊2 n(1-γ) ⌋ 2 n 1-γ e 2 nγ ((1-2 -n ⌊2 n(1-γ) ⌋) γ -1
) → e -γ as n → +∞.

Taking into account ( 6) and ( 7), we deduce

d β γ (E γ ) > 0. Now let 0 ≤ γ ′ < γ ≤ 1.
Clearly keeping in mind that

2 n k=1;k∈Eγ e k γ ′ ≤ 2 n-1 k=1 e k γ ′ + 2 n k=2 n -⌊2 n(1-γ) ⌋ e k γ ′
and by using both γ ′ -γ < 0, the estimate

2 n -⌊2 n(1-γ) ⌋ k=1 e k γ ′ 2 n k=1 e k γ ′ ∼ 1 - ⌊2 n(1-γ) ⌋ 2 n 1-γ ′ e 2 nγ ′ ((1-2 -n ⌊2 n(1-γ) ⌋) γ ′ -1) → 1 as n → +∞,
and (6) we derive

d β γ ′ (E γ ) = 0.
Finally, for γ = 1, the lemma is easy to establish since a subset E ⊂ N satisfies d β 1 (E) > 0 if and only if E is infinite. This finishes the proof.

In some sense, the densities d β γ , 0 ≤ γ ≤ 1, will us allow to interpolate the behavior of hypercyclic vectors between the U -frequent hypercyclicity and the hypercyclicity. 4.2. Rate of growth of U β γ -frequently hypercyclic functions. First we deal with the case p > 1. We will discuss the case p = 1 at the end of the section. We are ready to state the result that highlights both the continuous variation of the critical exponent and that of the growth (in term of L p averages) of a hypercyclic function for T α according to the frequency of visits of non-empty open subsets by its orbit under the action of T α . Theorem 4.5. Let 0 < γ < 1 and 1 < p ≤ ∞. Let f be a U β γ -frequently hypercyclic function for the operator T α . Then, the following hold

if α < 1 -γ max(2, q) , lim sup r→1 - [1 -r] 1-γ max(2,q) -α M p (f, r) > 0, if α = 1 -γ max(2, q) , lim sup r→1 - (M p (f, r)) = +∞, if α > 1 -γ max(2, q)
, lim sup

r→1 - (M p (f, r)) > 0.
Proof. Let f be a U β γ -frequently hypercyclic function for T α . We write f = k≥0 a k (k+1) α z k . Since f is U β γ -frequently hypercyclic there exists an increasing sub-sequence (n k ) ⊂ N with positive upper β γ -density such that, for all k ≥ 1,

|T n k α f (0) -3/2| = |a n k -3/2| < 1/2. We get, for all k ≥ 1, |a n k | ≥ 1. Set I = ((n k )
) and for all N ≥ 1, I N = I ∩ {1, . . . , N }. The hypothesis d β γ (I) > 0 ensures that there exist 0 < C < 1 and an increasing sequence (N l ) of positive integers such that [START_REF] Drasin | Optimal growth of frequently hypercyclic entire functions[END_REF]

k∈I N l e k γ ≥ C N 1-γ l γ e N γ l .
Up to take a sub-sequence, we can suppose that ( 9)

C(N k+1 + 1) ≥ N k + 1.
Let us consider, for all l ≥ 1, a sequence (r l ) with 1

-1 N l -1 ≤ r l < 1 -1 N l . Observe that (10) N l -1 ≤ 1 1 -r l < N l .
(1) Case 2 ≤ p ≤ ∞: Jensen'inequality and Parseval's Theorem give

(11) [M p (f, r l )] 2 ≥ [M 2 (f, r l )] 2 = k≥0 |a k | 2 (k + 1) 2α r 2k l ≥ N l k=1 |a k | 2 (k + 1) 2α r 2k l N l k=1 |a k | 2 (k + 1) 2α .
Let us choose j 0 ∈ N such that the function t → t -2α e -t γ is decreasing for t ≥ N j 0 . Thus we can write, for all l ≥ j 0 + 1,

[M p (f, r l )] 2 l j=1+j 0 N j k=1+N j-1 |a k | 2 (k + 1) 2α e k γ e -k γ .
Then applying Lemma 3.2 with u k = |a k | 2 e k γ , we get [START_REF] Ernst | A quantitative interpretation of the frequent hypercyclicity criterion[END_REF]

[M p (f, r l )] 2 S N l (N l + 1) -2α e -N γ l -S N j 0 (N j 0 + 1) -2α e -N γ j 0 + l j=1+j 0 S N j-1 (N j-1 + 1) -2α e -N γ j-1 -(N j + 1) -2α e -N γ j . Since S N i = k≤N i |a k | 2 e k γ ,
by construction and by ( 8), we get, for all i ≥ 1, ( 13)

S N i ≥ k∈I N i e k γ N 1-γ i e N γ i (N 1-γ i + 1)e N γ i .
From ( 12) and ( 13) we deduce ( 14)

[M p (f, r l )] 2 (N l + 1) (1-γ)-2α + l j=1+j 0 (N j-1 + 1) 1-γ e N γ j-1 (N j-1 + 1) -2α e -N γ j-1 -(N j + 1) -2α e -N γ j .
(a) Case α < 1-γ 2 : From ( 14) we get, for l large enough [START_REF] Gethner | Universal vectors for operators on spaces of holomorphic functions[END_REF] [M p (f,

r l )] 2 (N l + 1) (1-γ)-2α
Thanks to [START_REF] Edwards | Littlewood-Paley and Multiplier Theory[END_REF] and [START_REF] Gethner | Universal vectors for operators on spaces of holomorphic functions[END_REF] we deduce

[M p (f, r l )] 2 (1 -r l ) 2α-(1-γ) .
Hence we conclude lim sup

r→1 - (1 -r) -α+ 1-γ 2 M p (f, r) > 0.
(b) Case α = 1-γ 2 : taking into consideration ( 14), we can write, for all l ≥ 1 + j 0 ,

[M p (f, r l )] 2 l j=1 1 - N j-1 + 1 N j + 1 1-γ e N γ j-1 -N γ j .
Thus taking into account [START_REF] Duren | Theory of H p spaces[END_REF] we derive, for all l ≥ 1 + j 0 , [M p (f, r l )] 2 l, which allows to obtain lim sup

r→1 - M p (f, r) = +∞. (c) Case α > 1-γ 2 : since f is hypercyclic, the conclusion is given by Theorem 2.2. (2) Case 1 < p < 2:
It suffices to combine the arguments of the proof of the preceding case with those of the proof of (2) of Theorem 3.3 to obtain the desired conclusions.

4.3.

Optimal growth of U β γ -frequently hypercyclic functions: a constructive proof. In this subsection, we intend to prove that the estimates given by Theorem 4.5 whenever α is different from the critical exponent, i.e. α = 1-γ max(2,q) , are optimal. The case α = 1-γ max(2,q) will be treated separately in Section 5. Thus for all 0 < γ < 1 and for α = 1-γ max(2,q) , we propose to build U β γfrequently hypercyclic functions for T α that have the required L p growth and no more. To do this, we follow the construction of frequently hypercyclic functions for T α given in [START_REF] Mouze | Growth of frequently hypercyclic functions for some weighted Taylor shifts on the unit disc[END_REF] which itself was partly inspired by [START_REF] Drasin | Optimal growth of frequently hypercyclic entire functions[END_REF]. In particular we will need the so-called Rudin-Shapiro polynomials (combined with the de la Vallée-Poussin polynomials), which have coefficients ±1 (or bounded by 1) and an optimal growth of L p -norm. Let us recall the associated result in the form of Lemma 2.1 of [START_REF] Drasin | Optimal growth of frequently hypercyclic entire functions[END_REF] that summarized the result of Rudin-Shapiro [START_REF] Rudin | Some theorems on Fourier coefficients[END_REF].

Lemma 4.6.

(1) For each N ≥ 1, there is a trigonometric polynomial p N = N -1 k=0 ε N,k e ikθ where ε N,k = ±1 for all 0 ≤ k ≤ N -1 with at least half of the coefficients being +1 and with

p N p ≤ 5 √ N for p ∈ [2, +∞].
(2) For each N ≥ 1, there is a trigonometric polynomial p * N = N -1 k=0 a N,k e ikθ where |a N,k | ≤ 1 for all 0 ≤ k ≤ N -1 with at least ⌊ N 4 ⌋ coefficients being +1 and with p * N p ≤ 3N 1/q for p ∈ [START_REF] Bayart | Frequently hypercyclic operators[END_REF][START_REF] Bayart | Dynamics of linear operators[END_REF].

For any given polynomial q with q(z) = d j=0 b j z j with b d = 0, we denote d = deg(q) and

q ℓ 1 = d j=0 |b j |. We set 2N = k≥1 A k where for any k ≥ 1, A k = 2 k (2j -1); j ∈ N .
Denote by P the countable set of polynomials with rational coefficients and let us also consider pairs (q, l) with q ∈ P and l ∈ N satisfying q ℓ 1 ≤ l, displayed as a single sequence (q k , l k ). Clearly (q k ) is a dense set in H(D). Hence, for any k ≥ 1, we set d k = deg(q k ) and we have

q k ℓ 1 ≤ l k for every k ≥ 1.
For any α ∈ R, for any positive integer k ≥ 1, we set qk (z) =

d k j=0 (j + 1) α b (k) j z j
Let α be a real number and p ∈ (1, ∞]. For all integer n ≥ 0, we set I n = {2 n , . . . , 2 n+1 -1}. Next, for k ≥ 1, let us define the integers

α k = 1 + max l 2 k (1 + d k ) 2 max(α,0) , d k + max(3, 3 + α)l 2 k + max(α, 0)l k log(1 + d k )
and

α * k = 1 + max l q k (1 + d k ) q max(α,0) , d k + max(3, 3 + α)l 2 k + max(α, 0)l k log(1 + d k ) .
We set f α = n≥0 P n,α where the blocks (P n,α ) are polynomials defined as follows, using Rudin-Shapiro polynomials given by Lemma 4.6, ( 16)

P n,α (z) =    0 if n is odd 0 if n ∈ A k and 2 n-1 < α k z 2 n Q n (z) if n ∈ A k and 2 n-1 ≥ α k with for n ∈ A k , Q n (z) = j∈In (j + 1) -α c (k) j-2 n z j-2 n
where the sequence (c (k) j ) denotes the sequence of the coefficients of the polynomial p

⌊ 2 n(1-γ) α k ⌋ (z α k ) qk (z).
We also set f * α = n≥0 P * n,α where the blocks (P * n,α ) are polynomials defined as follows, using the de la Vallée-Poussin polynomials given by Lemma 4.6,

P * n,α (z) =    0 if n is odd 0 if n ∈ A k and 2 n-1 < α * k z 2 n Q * n (z) if n ∈ A k and 2 n-1 ≥ α * k , with, for n ∈ A k , Q * n (z) = j∈In (j + 1) -α c (k) j-2 n z j-2 n (17)
where the sequence (c

(k) j ) denotes the sequence of the coefficients of the polynomial p * ⌊ 2 n(1-γ) α * k ⌋ (z α * k ) qk (z).
A combination of Lemma 4.7 below with the triangle inequality shows that the function f α (resp. f * α ) belongs to H(D). Observe that, if we denote the polynomial

z → p ⌊ 2 n-1 α k ⌋ (z α k ) (resp. z → p * ⌊ 2 n-1 α k ⌋ (z α * k )) by g k (resp. g * k ), we have, for all 1 ≤ p ≤ +∞, g k p = p ⌊ 2 n-1 α k ⌋ p (resp. g * k p = p * ⌊ 2 n-1 α * k ⌋ p ).
Finally for any integer n, let us denote (φ n (k)) the sequence defined as follows

φ n (k) = (k + 1) -α if k ∈ I n 0 otherwise. Lemma 4.7. Let α ∈ R.
The following estimates hold. (i) For any 2 ≤ p ≤ +∞, any 0 < r < 1 and any n ∈ N, we have

M p (P n,α , r) 2 n( 1-γ 2 -α) r 2 n .
(ii) For any 1 < p < 2, any 0 < r < 1 and any n ∈ N, we have

M p (P * n,α , r) 2 n( 1-γ q -α) r 2 n .
Proof. (i) On one hand, we deal with the case 2 ≤ p < +∞. Let n be a positive integer. Without loss of generality, we can assume that n belongs to the set A k for some k ≥ 1. Let r be in (0, 1). Since r → M p (f, .) is increasing, we get

M p (P n,α , r) ≤ r 2 n Q n p .
Then, the polynomial Q n can be viewed as a trigonometric polynomial obtained by an abstract convolution operator on T, given by (c k ) k≥0 → (φ n (j)c

(k) j-2 n ) j≥0 (where (c (k)
j ) denotes the sequence of the coefficients of the polynomial p ⌊ 2 n-1 α k ⌋ qk ). Now, we are going to apply the Marcinkiewicz Multiplier Theorem [10, Theorem 8.2 p.148]. To do this, observe that we have, for any l ≥ 1, sup

j∈I l |φ n (j)| ≤ sup j∈In |φ n (j)| 2 -nα
and sup

l j∈I l |φ n (j + 1) -φ n (j)| ≤ j∈In |φ n (j + 1) -φ n (j)| 2 -nα .
Hence, taking into account the choice of α k and Lemma 4.6, we get

Q n p 2 -nα p ⌊ 2 n(1-γ) α k ⌋ p qk ∞ 2 -nα 2 n(1-γ) α k l k (1 + d k ) max(α,0) 2 n( 1-γ 2 -α) .
Finally, we obtain the desired estimate

M p (P n,α , r) 2 n( 1-γ 2 -α) r 2 n .
On the other hand we deal with the case p = ∞. Let us recall that P n,α (z) = 0 or z

2 n Q n (z) with Q n (z) = j∈In (j + 1) -α c (k) j-2 n z j-2 n where (c (k)
j ) denotes the sequence of the coefficients of the polynomial p

⌊ 2 n(1-γ) α k ⌋ (z α k ) qk (z).
First, assume that α ≤ 0. We write

M ∞ (P n,α , r) r 2 n Q n ∞ .
Using the form of n , as in the proof of Lemma 3.6 of [START_REF] Mouze | Growth of frequently hypercyclic functions for some weighted Taylor shifts on the unit disc[END_REF] we apply a fractional Bernstein's inequality to obtain, taking into consideration Lemma 4.6,

M ∞ (P n,α , r) r 2 n 2 -nα Q n ∞ 2 -nα p ⌊ 2 n(1-γ) α k ⌋ ∞ qk ∞ 2 -nα ⌊ 2 n(1-γ) α k ⌋l k .
Thanks to the choice of α k , we have, for α ≤ 0,

M ∞ (P n,α , r) 2 n( 1-γ 2 -α) .
To conclude it suffices to mimic the induction of the proof of Lemma 3.7 of [START_REF] Mouze | Growth of frequently hypercyclic functions for some weighted Taylor shifts on the unit disc[END_REF] (ii) The proof is similar as that of the case 2 ≤ p < +∞ by applying Lemma 4.6 for 1 < p < 2. Now we are ready to obtain the rate of growth of the aforementioned functions f α and f * α . We refer to Lemmas 3.4, 3.5 and 3.8 of [START_REF] Mouze | Growth of frequently hypercyclic functions for some weighted Taylor shifts on the unit disc[END_REF] with obvious modifications. Lemma 4.8.

(1) Let 2 ≤ p ≤ +∞. For all 0 < r < 1, the following estimates hold

M p (f α , r) (1 -r) α-1-γ 2 if α < 1-γ 2 , 1 if α > 1-γ 2
(2) Let 1 < p < 2. For all 0 < r < 1, the following estimates hold

M p (f * α , r) (1 -r) α-1-γ q if α < 1-γ q , 1 if α > 1-γ q
Now we are going to prove that the functions f α and f * α are U β γ -frequently hypercyclic for T α .

Proposition 4.9. For p ≥ 2 (resp. 1 < p < 2), the function f α (resp. f * α ) is a U β γ -frequently hypercyclic vector for the operator T α .

Proof. We only prove the frequent hypercyclicity of f α for the operator T α . We do not repeat the details for f * α : it will be enough to make the appropriate modifications. Let k be a large enough integer. Let us consider n ∈ A k such that 2 n-1 ≥ α k . We consider B n the set of s in I n such that the coefficient z s in the polynomial z 2 n p (1-γ) ⌋ and since at least half of the coefficients of p

⌊ 2 n(1-γ) α k ⌋ (z α k ) is equal to 1 and we denote by T k = s : s ∈ B n , n ∈ A k , 2 n-1 ≥ α k . Observe that max(B n ) ≤ 2 n + ⌊2 n
e j γ ≥ 2 n +2 -1 ⌊2 n(1-γ) ⌋ j=2 n e j γ 2 n +⌊2 n(1-γ) ⌋ j=1 e j γ = 2 n +2 -1 ⌊2 n(1-γ) ⌋ j=1 e j γ 2 n +⌊2 n(1-γ) ⌋ j=1 e j γ - 2 n -1 j=1 e j γ 2 n +⌊2 n(1-γ) ⌋ j=1 e j γ .
Clearly we have

2 n + ⌊2 n(1-γ) ⌋ 2 γ -2 n + ⌊2 n(1-γ) ⌋ γ → - γ 2 and (2 n -1) γ -2 n + ⌊2 n(1-γ) ⌋ γ → -γ,
which implies, using similar estimations as those of the proof of Lemma 4.4,

2 n +2 -1 ⌊2 n(1-γ) ⌋ j=1 e j γ 2 n +⌊2 n(1-γ) ⌋ j=1 e j γ → e -γ/2 and 2 n -1 j=1 e j γ 2 n +⌊2 n(1-γ) ⌋ j=1 e j γ → e -γ , as n → +∞.
Hence the inequality [START_REF] Grosse-Erdmann | On the universal functions of G. R. MacLane[END_REF] ensures that

d β γ (T k ) > 0.
Then let α be a real number and let k ∈ N. Let us consider s ∈ B n with n ∈ A k satisfying 2 n-1 ≥ α k . As in the proof of Lemma 3.9 of [START_REF] Mouze | Growth of frequently hypercyclic functions for some weighted Taylor shifts on the unit disc[END_REF] with easy modifications, we can prove that sup

|z|=1-1 l k |T s α (f α )(z) -q k (z)| 1 l k ,
provided that k is chosen large enough. This allows to obtain the frequent hypercyclicity of f α .

In summary, Lemma 4.8 and Proposition 4.9 leads to the following result, which shows that the statement of Theorem 4.5 is optimal whenever α is not the critical exponent. Theorem 4.10. Let 0 < γ < 1 and 1 < p ≤ ∞.

(1) for α < 1-γ max(2,q) there exists a U β γ -frequently hypercyclic function for the operator

T α such that M p (f, r) (1 -r) α-1-γ max(2,q) ;
(2) for α > 1-γ max(2,q) there exists a U β γ -frequently hypercyclic function for the operator T α such that M p (f, r) 1.

Remark 4.11. Let 0 < γ < 1. It seems important to note that the functions constructed for the proof of Theorem 4.10 are U β γ -frequently hypercyclic for T α but neither U β γ ′ -frequently hypercyclic for 0 < γ ′ < γ nor U -frequently hypercyclic, since they don't satisfy the estimates given by Theorem 4.5 or Theorem 3.3.

Finally let us say some words for the case p = 1. As in the proof of Theorem 3.4, observe that a U β γ -frequently hypercyclic function is necessarily hypercyclic and a U -frequently hypercyclic function is necessarily U β γ -frequently hypercyclic. This leads to the following statement. Theorem 4.12. Let 0 < γ < 1. Let f be a U β γ -frequently hypercyclic function for the operator T α . Then, the following assertions hold

lim sup r→1 - (1 -r) -α M 1 (f, r) = +∞, if α ≤ 0, lim sup r→1 - M 1 (f, r) > 0, if α > 0.
These results are optimal in the following sense: for any positive integer l ≥ 1, there exists a U β γ -frequently hypercyclic function for the operator T α such that for every 0 < r < 1 sufficiently large

M 1 (f, r) (1 -r) α log l (-log(1 -r)) if α < 0 1 if α > 0.

Optimal estimates: the case of the critical exponent

In this section, we are going to show that the growth of U -frequently or U β γ -frequently hypercyclic functions for T α can be arbitrarily slow when α is the critical exponent. The situation will therefore be similar to the hypercyclic case for which for all 1 ≤ p ≤ ∞ the critical exponent is α = 0 and, according Theorem 2.2, the two following properties hold: for all hypercyclic function f for T α , lim sup r→1 -M p (f, r) = +∞ and for any function ϕ : [0, 1) → R + tending to infinity as r tends to 1, there is a hypercyclic function f such that M p (f, r) ≤ ϕ(r). For this, we are going to adapt the constructive method used in Section 4.3. Before we start, we establish a lemma that will be useful in the following. Lemma 5.1. Let (w n ) be an increasing sequence of positive integers such that w n+1 wn → +∞ as n tends to infinity. Let (a n ) be a bounded sequence of positive real numbers such that a n = +∞. If we denote by h : R + → R + a continuous increasing function with, for all n ∈ N, h(n) = w n , the following estimate holds

n≥0 a n r wn ∼ (θ a • h -1 ) 1 1 -r as r → 1 -,
where for all x ∈ R + , θ a (x) = n≤x a n .

Proof. By hypothesis the power series S(x) := n≥0 a n x n has radius of convergence 1. For all j ≥ 2, we denote by n j the only positive integer satisfying w n j +1 > j and w n j ≤ j.

Then we write

(19) S 1 - 1 j = n j +1 n=0 a n 1 - 1 j wn + +∞ n=n j +2 a n 1 - 1 j wn .
On one hand, we get, thanks to the choice of j and using the inequality 1 -t ≤ e -t , +∞

n=n j +2 a n 1 - 1 j wn ≤ +∞ n=n j +2
a n e where M = sup |a n |. Moreover, up to take j large enough, since w n+1 wn tends to infinity, we have, for all k ≥ 2,

w n j +k w n j +1 ≥ 2 k-1 , which implies (20) +∞ n=n j +2 a n 1 - 1 j wn ≤ M +∞ k=2 e -2 k-1 .
On the other hand, we have, for all n = 0, . . . , n j -1,

1 - 1 j wn ≥ 1 - 1 j w n j -1 = 1 - 1 j wn j w n j -1 wn j ≥ 1 - 1 j j w n j -1 wn j
.

From this, we derive the following inequality

n j +1 n=0 a n 1 - 1 j wn ≥ 1 - 1 j j w n j -1 wn j n j -1 n=0 a n .
Thus, we obtain

1 - 1 j j w n j -1 wn j θ a (n j -1) θ a (n j ) ≤ n j +1 n=0 a n 1 -1 j wn θ a (n j ) ≤ θ a (n j + 1) θ a (n j ) .
Since an θa(n) = 1 -θa(n-1) θa(n) → 0 and

w n j -1 wn j → 0, we obtain n j +1 n=0 a n 1 - 1 j wn ∼ θ a (n j ), as j → +∞,
which gives, thanks to ( 19) and ( 20),

S 1 - 1 j ∼ θ a (n j ) as j → +∞. (21)
By construction the sequence (n j ) satisfies n j ≤ h -1 (j) < n j + 1. Thus, combining this inequality with [START_REF] Grosse-Erdmann | Linear chaos, Universitext[END_REF] and the hypothesis θa(n+1) θa(n) → 1 again, we get

(22) S 1 - 1 j ∼ θ a (h -1 (j)), as j → +∞. Let 0 < r < 1 large enough with 1 -1 j ≤ r < 1 -1 j+1 , i.e j ≤ 1 1-r < j + 1. Clearly we have (23) S 1 - 1 j ≤ S(r) ≤ S 1 - 1 j + 1 and h -1 (j) ≤ h -1 1 1 -r ≤ h -1 (j + 1).
Let us recall that n j was chosen so that n j ≤ h -1 (j) < n j +1. Assume that h -1 (j)+1 < h -1 (j +1). Therefore

w n j ≤ j < w n j +1 ≤ h(h -1 (j) + 1) < h(h -1 (j + 1)) = j + 1,
which gives a contradiction since w n j +1 is a positive integer. We deduce h -1 (j + 1) ≤ h -1 (j) + 1 and by ( 23)

(24) h -1 (j) ≤ h -1 1 1 -r ≤ h -1 (j) + 1.
Thanks to ( 22), ( 23) and ( 24), we conclude

n≥0 a n r wn ∼ (θ a • h -1 ) 1 1 -r , as r → 1 -.
5.1. The U -frequently hypercyclic case. We keep the definitions and the notations of Subsection 4.3. Let us also consider an increasing function h : R + → R + tending to infinity such that, for any n ∈ N, h(n) := u n ∈ N and u n+1 -u n → +∞ as n tends to infinity. Let α be a real number. For all integer n ≥ 0, we set I For 1 ≤ p ≤ ∞, we denote by α c the critical exponent α c = 1 max(2,q) . Lemma 5.2. We have, for any 0 < r < 1, M p (P (u) n,αc , r) r 2 un if 2 ≤ p ≤ ∞, M p (P * (u) n,αc , r) r 2 un if 1 < p < 2, and M 1 (P * (u) n,0 , r) r 2 un l k . Proof. For p > 1, it suffices to argue along the same lines as the proof of Lemma 4.7 replacing the sequence (2 n ) by (2 un ). Now let us consider the case p = 1 (hence α c = 0). We can write, keeping in mind thatq k = qk for α = 0, Lemma 5.8. Let 1 < p ≤ ∞ and α c = 1-γ max(2,q) . Then, for all 0 < r < 1,

M p (f (u) αc , r) h -1 -log(1 -r) log(2) if 2 ≤ p ≤ ∞, and
M p (f * (u) αc , r) h -1 -log(1 -r) log(2) if 1 < p < 2.
Now we are going to prove that the functions f Proof. We only prove the frequent hypercyclicity of f (u) αc for the operator T αc . We do not repeat the details for f * (u) αc : it will be enough to make the appropriate modifications. Let k be a large enough integer. Let us consider n ∈ A k such that 2 u n-1 ≥ α k . We consider B n the set of s in I (2 un -1) γ -2 un + ⌊2 un(1-γ) ⌋ γ → -γ,

  d(A) = lim inf n→+∞ #A ∩ {1, . . . , n} n and d(A) = lim sup n→+∞ #A ∩ {1, . . . , n} n .

2π 0 |P

 0 (e iθ )| p dθ 1/p and P ∞ = sup 0≤t≤2π |P (e it )|.

Theorem 1 . 1 . ([ 25 ,

 1125 Theorem 1.2]) Let α ∈ R. The following assertions hold

4. 1 .Definition 4 . 1 .

 141 Some weighted densities. First we introduce a refined notion of upper densities. Let β = (β n ) be a non-decreasing sequence of positive real numbers tending to infinity. For a subset E ⊂ N, its upper β-density is given byd β (E) = lim sup

n=j- 2 j- 2

 22 {2 un , . . . , 2 u n+1 -1}. Next, for k ≥ 1, we keep the definition of integers α k and α * k given in Subsection 4.3. We set f (u) α = n≥0 P (u) n,α where the blocks (P (u) n,α ) are polynomials defined as follows, using Rudin-Shapiro polynomials given by Lemma 4n is odd0 if n ∈ A k and 2 u n-1 < α k z 2 un Q n (z) if n ∈ A k and 2 u n-1 ≥ α k with for n ∈ A k , un z j-2 unwhere the sequence (c(k) j ) denotes the sequence of the coefficients of the polynomial p ⌊ 2 un α k ⌋ (z α k ) qk (z). We also set f * (u) α = n≥0 P * (u)n,α where the blocks (P * (u) n,α ) are polynomials defined as follows, using polynomials given by Lemma 4.6,(26)P * (u) n,α (z) =    0 if n is odd 0 if n ∈ A k and 2 u n-1 < α * k z 2 un Q * (u) n (z) if n ∈ A k and 2 u n-1 ≥ α * k , with, for n ∈ A k , un z j-2 unwhere the sequence (c (k) j ) denotes the sequence of the coefficients of the polynomial p * ⌊ 2 un α * k ⌋ (z α * k ) qk (z).

M 1 (⌋j- 2 Lemma 5 . 7 .

 1257 ((re it ) α * k ) qk (re it ) dt ≤ r 2 un p * ⌊ 2 un α * k ⌋ 1 q k ∞ r 2 un l k .From Lemma 5.1 and 5.2, we deduce the rate of growth of the functions f (u) αc and f * (u) αc . We begin by the case p = 1. n is odd0 if n ∈ A k and 2 u n-1 < α * k z 2 un Q * (u) n (z) if n ∈ A k and 2 u n-1 ≥ α * k , with, for n ∈ A k , un z j-2 unwhere the sequence (c (k) j ) denotes the sequence of the coefficients of the polynomial p *⌊ 2 un(1-γ) α * k ⌋ (z α * k ) qk (z). Let 1 < p ≤ ∞. Set α c = 1-γ max(2,q) . We have, for any 0 < r < 1 and all n ∈ N,M p (P (u) n,αc , r) r 2 un if 2 ≤ p ≤ ∞ and M p (P * (u) n,αc , r) r 2 un if 1 < p < 2.Proof. It suffices to argue along the same lines as the proof of Lemma 4.7 replacing the sequence (2 n ) by (2 un ). From Lemma 5.1 and 5.7, we deduce the rate of growth of the functions f (u) αc and f * (u) αc .

  αc and f * (u) αc are U β γ -frequently hypercyclic for T αc . Proposition 5.9. For p ≥ 2 (resp. 1 < p < 2), the function f(u) αc (resp. f * (u)αc ) is a U β γ -frequently hypercyclic vector for the operator T αc .

n⌊ 2 . 2 γ- 2

 222 such that the coefficient z s in the polynomial z 2 un p⌊ 2 un(1-γ) α k ⌋ (z α k ) is equal to 1 and we denote by T k = {s : s ∈ B n , n ∈ A k , 2 u n-1 ≥ α k } . Observe that max(B n ) ≤ 2 un + ⌊2 un(1-γ) ⌋ and since at least half of the coefficients of p Clearly we have2 un + ⌊2 un(1-γ) ⌋ un + ⌊2 un(1-γ) ⌋ γ → -γ 2 ,

The first author was partly supported by the grant ANR-17-CE40-0021 of the French National Research Agency ANR (project Front).

Lemma 5.3. Let 1 < p ≤ ∞. Under the preceding definitions and assumptions, the following estimates hold: for all 0 < r < 1,

Proof. Let 2 ≤ p ≤ ∞. Combining Lemma 5.2 with triangle inequality, we get

By hypothesis 2 u n+1 -un → +∞ as n tends to infinity. We apply Lemma 5.1 with w n = 2 un and a n = 1 and we obtain, for 0 < r < 1,

.

For the case 1 < p < 2, the proof works along the same lines.

Now we are interested in the specific case p = 1.

Lemma 5.4.

There is a function of the form f * (u) 0 such that

Proof. Without loss of generality we can assume that

). Taking into account Lemma 5.2 and the inequality 1 -t ≤ e -t , we get, for any 1

n,0 , r)

) and we get

2 . Now we are going to prove that the functions f (u) αc and f * (u) αc are U -frequently hypercyclic for T αc .

Proposition 5.5. For p ≥ 2 (resp.

αc ) is a U -frequently hypercyclic vector for the operator T αc .

Proof. We only prove the frequent hypercyclicity of f (u) αc for the operator T αc . We do not repeat the details for f * (u) αc : it will be enough to make the appropriate modifications. Let k be a large enough integer. Let us consider n ∈ A k such that 2 u n-1 ≥ α k . We consider B n the set of s in

Observe that max(B n ) ≤ 2 un+1 and since at least half of the coefficients of p ⌊ 2 un α k ⌋ being +1, we get

Then let α be a real number and let k be in N. Now let us consider s ∈ B n with n ∈ A k satisfying 2 u n-1 ≥ α k . As in Proposition 4.9, by construction we get sup

provided that k is chosen large enough. This allows to obtain the frequent hypercyclicity of f (u) αc .

Combining Lemma 5.3 with Proposition 5.5 we obtain the following result.

Theorem 5.6. Let 1 ≤ p ≤ ∞ and α c = 1 max(2,q) . Then, for any function ϕ : [0, 1) → R + with ϕ(r) → +∞ as r → 1 -, there is a function f in H(D) with M p (f, r) ϕ(r), for 0 < r < 1 sufficiently close to 1, that is U -frequently hypercyclic for T αc .

Proof. We begin by the case p > 1. Without loss of generality, we can assume that the function ϕ is a continuous increasing function that can be written, for all 0 < r < 1, ϕ(r) = ψ 1 1-r where ψ is a continuous increasing function with, for all n ∈ N, u n := ψ -1 (n) ∈ N and u n+1 -u n → +∞. Thus Lemma 5.3 and Proposition 5.5 ensure that, for all 1 < p ≤ ∞, there is a function f in H(D) with

that is U -frequently hypercyclic for T αc . Now we deal with the case p = 1. Without loss of generality, we can assume that ϕ is a continuous increasing function that can be written, for all 0 < r < 1, ϕ(r

where ψ is a continuous and increasing function such that, for all n ∈ N, u n := ψ -1 (n) ∈ N and u n+1 -u n → +∞. Applying Lemma 5.4 and Proposition 5.5, we find a function f ∈ H(D) with

that is U -frequently hypercyclic for T 0 . The proof is complete.

5.2.

The U β γ -frequently hypercyclic case. We keep the definitions and the notations of Subsection 5.1. We modify the definitions of polynomials P (u) n,α and P * (u) n,α as follows:

(28)

where the sequence (c (k) j ) denotes the sequence of the coefficients of the polynomial p

which implies, using similar estimations as those of the proof of Lemma 4.4,

Hence the inequality (30) ensures that

Then let α be a real number and let k be in N. Now for s ∈ B n with n ∈ A k satisfying 2 u n-1 ≥ α k , as in Proposition 4.9, by construction we get sup

provided that k is chosen large enough. This allows to obtain the frequent hypercyclicity of f (u) αc .

Combining Lemma 5.8 with Proposition 5.9 we obtain the following result.

Theorem 5.10. Let 0 < γ < 1. Let 1 ≤ p ≤ ∞ and α c = 1-γ max (2,q) . Then, for any function ϕ : [0, 1) → R + , with ϕ(r) → +∞ as r → 1 -, there is a function f in H(D) with M p (f, r) ϕ(r), for 0 < r < 1 sufficiently close to 1, that is U β γ -frequently hypercyclic for T αc .

Proof. For p = 1, we have α c = 0 and the result is given by Theorem 5.6. Now let p > 1.

Without loss of generality, we can assume that the ϕ is a continuous increasing function such that, for all 0 < r < 1, ϕ(r) = ψ 1 1-r where ψ is continuous and increasing with, for all n ∈ N, u n := ψ -1 (n) ∈ N and u n+1 -u n → +∞. Thus Lemma 5.8 and Proposition 5.9 ensure that, for all 1 < p ≤ ∞, there is a function f in H(D) with

that is U β γ -frequently hypercyclic for T αc . The proof is complete.

From Theorems 2.2, 5.6 and 5.10, we can state the following result that unifies what happens in the critical case, given by the critical exponent, for the L p growth of U β γ -frequently hypercyclic functions for T α , when γ belongs to [0, 1]. Theorem 5.11. Let 0 ≤ γ ≤ 1 and 1 ≤ p ≤ ∞. Then, for any function ϕ : [0, 1) → R + , with ϕ(r) → +∞ as r → 1 -, there is a function f in H(D) with M p (f, r) ϕ(r) that is U β γ -frequently hypercyclic for T αc where α c = 1-γ max(2,q) .

Concluding remark. In summary, thanks to Theorems 4.10, 4.12, 5.6 and 5.10, the results given by Theorems 3.3, 3.4 and 4.5 are optimal.