Yanni Zhou

Florin Hutu

Guillaume Villemaud

Analysis of a Spatial Modulation System over Time-varying Rician Fading Channel with a CSI Detector

Keywords: Spatial Modulation, Rician channels, Channel Estimation, Channel State Information

Spatial Modulation (SM) is proposed in the literature as a new Multiple-Input Multiple-Output (MIMO) transmission scheme. The advantage of the SM consists in using only one RF (Radio Frequency) chain in the transmitting end which helps to increase the energy efficiency and the performance of the transmission. The related work in the literature is mostly based on the study of the SM transmission under the simple channel models. Moreover, it is assumed that the perfect Channel State Information (CSI) is known by the receiving end. This article discusses a CSI detector for the channel estimation and analyzes the performance of the proposed system over the time-varying Rician fading channel. The ADS-Matlab co-simulation results demonstrate that the bit error rate (BER) can be kept at an acceptable level with a realistic CSI detector, and in this case the system offers better performance with a small Rician factor.

I. INTRODUCTION

MIMO technology plays a significant role in wireless communications. However, when multiple antennas transmit data simultaneously, high inter-channel interference is generated, which leads to a decrease in performance. It is well known that in the case of MIMO systems, the system's complexity increases with the number of antennas since each antenna is associated with an RF chain, which is increasing the total energy consumption of the transceiver. In order to overcome this issue, one solution is to use an unique RF transmission chain together with an RF switch in order to convey each emitted symbol to a particular antenna. This method is called Spatial Modulation and was proposed for the first time in [START_REF] Mesleh | Spatial Modulation-OFDM[END_REF].

SM uses the active state of antennas as a support for the transmission of information, which effectively simplifies the problems of inter-channel interference and spectrum congestion in the traditional MIMO scheme. It was explained in [START_REF] Renzo | Spatial modulation for multiple-antenna wireless systems: a survey[END_REF] that, compared to the V-BLAST (Vertical Bell Laboratories Layered Space-Time) system, the SM receiver does not require a continuous interference cancellation step in order to handle inter-channel interference, and has low system complexity. Moreover, the use of a unique RF link greatly reduces the cost of a SM system. SM can work normally when the number of receiving antennas is smaller than the number of transmitting antennas, which avoids the limitation of the number of receiving antennas compared to other MIMO technologies.

In the literature, [START_REF] Phan-Huy | Single-Carrier Spatial Modulation for the Internet of Things: Design and Performance Evaluation by Using Real Compact and Reconfigurable Antennas[END_REF] highlights the benefits of SM and the potential in 5G. Moreover, the design guidelines are discussed in [START_REF] Yang | Design Guidelines for Spatial Modulation[END_REF]. The related techniques of SM, such as Space Shift Keying (SSK) [START_REF] Jeganathan | Space shift keying modulation for MIMO channels[END_REF], Space-Time Shift Keying (STSK) [START_REF] Sugiura | Coherent and differential space-time shift keying: A dispersion matrix approach[END_REF] and Generalized Spatial Modulation (GSM) [START_REF] Younis | Generalised spatial modulation[END_REF] have also been proposed. Additionally, various lowcomplexity or high-accuracy detectors are also proposed. In [START_REF] Jeganathan | Spatial Modulation : Optimal Detection and Performance Analysis[END_REF], the authors present the performance analysis of a SM scheme in the case of a ML (Maximum Likelihood) detector (i. e. optimal detector). However, in their study, the channel model is a basic Additive White Gaussian Noise (AWGN) one.

Furthermore, the performance analysis over the more complex channel models has been presented in [START_REF] Renzo | On the Performance of Space Shift Keying MIMO Systems Over Correlated Rician Fading Channels[END_REF] and [START_REF] Alwakeel | Quadrature Spatial Modulation Performance Analysis over Rician Fading Channels[END_REF]. However, most of the previous studies assume that perfect CSI is available at the receiver, which may be difficult to implement in real time. The difficulty is even greater in scenarios with mobility, where there is high relative speed between the emitter and the receiver. Hence, this assumption is unrealistic. Furthermore, several authors [START_REF] Afana | On the Joint Impact of Hardware and Channel Imperfections on Cognitive Spatial Modulation MIMO Systems : Cramer -Rao Bound Approach[END_REF], [START_REF] Sugiura | Effects of Channel Estimation on Spatial Modulation[END_REF] have studied the effects of the channel estimation errors on the SM performances. In [START_REF] Basar | Performance of Spatial Modulation in the Presence of Channel Estimation Errors[END_REF], it analyzes the performance of SM in the presence of channel estimation errors by introducing different variances of the estimation errors. However, to the best of the author's knowledge, studies on the design of the CSI detector are still lacking.

In this paper, we propose a SM system with a CSI detector. It is no longer assumed that a perfect CSI can be obtained at the receiving end. The CSI is considered to be degraded by the AWGN noise. In order to study the performance of the proposed system, Advanced Design System (ADS) -Matlab system co-simulations will be performed. More precisely, the performance of the SM over the timevarying Rician fading channel will be analysed.

The remainder of this paper is structured as follows. In Section II, the system and channel models will be discussed, and the channel model together with the channel estimation strategy will be pointed out. The ADS-Matlab co-simulation results are presented in Section III. Section IV summarizes the results of this work and gives the future directions.

A. SM modulator

The input bit stream is divided into N groups. Each group has log 2 (N t ) + log 2 (M ) bits, where M is the number of the distinct symbols that the transmitter can send. The transmitter activates the corresponding antenna according to the first log 2 (N t ) bits of the input bitstream. The rest log 2 (M ) bits are used in order to form the M-ary signal constellation. Besides, in the SM system, there is a limit on the number of the transmitting antennas, which is required to be 2 n , so at least 2 transmitting antennas are required to ensure the normal operation of the system. This paper deals with the particular case of the quadrature phase-shift keying (QPSK) modulation, consequently, log 2 (M )=2. A 4×4 SM mapping table is shown in TABLE I.

B. Channel Model

As stated, the symbols are transmitted over a Rician fading channel H, where H ∈ C Nt×Nr . Moreover, an AWGN noise η, is considered.

T x1 1 +  0001 T x1 -1 +  0010 T x1 1 - 0011 T x1 -1 - 0100 T x2 1 +  0101 T x2 -1 +  0110 T x2 1 - 0111 T x2 -1 - 1000 T x3 1 +  1001 T x3 -1 +  1010 T x3 1 - 1011 T x3 -1 - 1100 T x4 1 +  1101 T x4 -1 +  1110 T x4 1 - 1111 T x4 -1 -
The Rician fading channel model H takes into account the presence of the line of sight (LOS) path and the Rice distribution is described by the Rician factor K. The factor K is defined as the ratio between the main signal power (usually the LOS path signal) and the sum of the powers of the remaining multipaths. In this particular case, H has the following structure:

H = K K + 1 • A + 1 K + 1 • V (1)
where A is a deterministic matrix and V is a stochastic one.

We assume that the channel parameters change every τ H seconds, and the output time step is T bit (τ H T bit ). More precisely, the same H matrix function is used for a set of τ H /T bit symbols.

The relation between the output signal y and the input one x can be written as follows:

y = H i • x + η, i ∈ [1, ..., N H ] (2)
N H = N • [log 2 (N t ) + log 2 (M )] • T bit τ H (3)
where H i is a particular realization of H, N H is the maximum number of H matrix.

C. CSI detector and demodulator

Since the transmission information bits are respectively mapped in the antenna number and the modulation symbol, it is necessary to detect both the emitting antenna and the transmission symbol. In the detection process of the receiving end, if error occurs in the process of the antenna number detection, the estimated modulation symbol is meaningless. Therefore, the detection of the Tx antenna's number directly impacts the performance of the system, so the receiving end of the SM system needs to be mastered with more complete CSI to better detect the antenna number.

The principle of the CSI detector is based on the Least-Squares (LS) algorithm [START_REF] Neumann | Channel estimation in massive MIMO systems[END_REF]. The detector calculates the channel matrix elements based on the estimated sampling co-variance matrix. More precisely, we establish N t sets of the incoming data. Each set is a repeating loop of log 2 (N t ) + log 2 (M ) bits. The first log 2 (N t ) bits are fixed and known by the receiver in order to analyze the corresponding channel state. LS algorithm is applied for the CSI detection depending on calculating the linear relationship between the input and the output. In this situation, each set of the incoming data can get N r channel states, the total N t sets can get the real-time channel matrix H, where H ∈ C Nr×Nt .

The current CSI is used for the correction at the receiving end. The correction matrix element w ij is as follows:

w ij = 1 N 2 • x t i • x t i H -1 • y t ij • x t i T (4)
where x t i is the modulated symbols sent by the i th transmitting antenna, and y t ij is the symbols received by j th receiving antenna which is transmitted by the i th transmitting antenna. The operator [ • ] H denotes the Hermitian transpose and [ • ] denotes the complex conjugate.

Because of using the QPSK modulation, in order to detect the transmitting antenna, only four possible symbols exist. The average value of the corrected demodulated signal will be compared to the four possible symbols to find the minimum difference. In other words, the most likely transmitting antenna is the antenna giving the smallest difference between the received symbol and one of the four QPSK symbols.

Finally, the binary values corresponding to the received QPSK symbol are extracted. These values are concatenated with the binary values obtained by identifying the emitting antenna.

III. SIMULATION RESULTS

A simulator of the SM scheme (Fig. 1) was built in order to extract the performance in terms of BER variation with respect of Eb/N o, which is the energy per bit over the noise power spectral density. We simulate the transmitting and receiving ends on Keysight's ADS and implement digital signal processing and part of the detection algorithm on Matlab. The simulation parameters are gathered in From Fig. 2, one can remark that the increase in the number of emitting antennas will increase the BER. More precisely, the BER performance of the 2×2 SM system is better than that of the 4×4 SM system. This is because there are only two possible transmitting antennas in the 2×2 SM system compared with four possibilities in the 4×4 SM system. In other words, the detection will be relatively easier. On the other side, the proposed channel estimation slightly increases the BER compared to the case where the perfect CSI is considered. However, it can be considered that the difference of BER performance between the system with perfect CSI and the system with imperfect estimated CSI is at acceptable level.

From Fig. 3, one can remark that the SM system over a Rician fading channel offers better performance for the smaller values of K. This can be explained because the difference between the N t × N r values in the channel matrix H is minimal if the K factor is significant, which leads to imprecision in the antenna detection. In other words, if the CSI difference between each channel is Fig. 2. BER performance of the 2×2 and 4×4 SM system with the perfect and realistic CSI detectors over time-varying Rician fading channel Fig. 3. BER performance of the 4x4 SM system with the realistic CSI detector and for different Rician factor K assumed to be negligible, all N t × N r elements in the channel matrix H are equal, then the BER of detecting the transmitting antenna will remain at 0.5. Hence, the more multipaths in the channel, the better we can detect the emitting antenna.

IV. CONCLUSION

In this paper, a SM system over the time-varying Rician fading channel was presented. The modulation module as well as the channel model were introduced. In particular, a time-varying CSI detector for the channel estimation was designed instead of assuming the availability of the perfect CSI at the receiver. System simulation results indicate that the detector error is acceptable when the Rician Factor K is small. It is envisaged to improve the simulation framework by taking into account the receiver non-idealities (local oscillator's phase noise, presence of nonlinear effects, etc.) and extending the result to more complex modulations. Moreover, an experimental framework based on National Instruments' VST (Vector Signal Transceiver) platform is envisaged.

Fig. 1 .

 1 Fig. 1. Nt × Nr SM system model with a CSI Detector

Fig. 4 .

 4 Fig. 4. Nt × Nr SM system model with a CSI Detector

TABLE II :

 II


Tarik Lassouaoui

Florin Hutu

Yvan Duroc

Guillaume Villemaud

Theoretical BER Evaluation of Passive RFID Tag-To-Tag Communications

Keywords: RFID tags, UHF antennas, Bit error rate, Amplitude shift keying, Demodulation

The concept of UHF RFID passive tag-totag communication has been introduced and opens new promising perspectives in the field of Internet-of-Things. This concept enables two powerless tags to exchange information using an external source. Moreover, a simulation framework has been proposed as a tool allowing the performance evaluation of tag-to-tag radio links in terms of Bit Error Rate (BER). The aim of this paper is to provide a theoretical framework for BER estimation. The theoretical framework takes into account various parameters such as modulation rate, receivers topology and the electromagnetic coupling of the tags. Furthermore, in order to validate the obtained results, they are compared with the ones obtained by realistic simulations.

I. INTRODUCTION

Radio frequency identification (RFID) is still a fast emerging technology with numberless applications especially in the areas of security, logistics or traceability [START_REF] Finkenzeller | RFID handbook: fundamentals and applications in contact-less smart cards and identification[END_REF]. RFID is also an evolving technology since new capabilities such as sensing, energy harvesting, spatial identification are added [START_REF] Nikitin | Phase based spatial identification of uhf rfid tags[END_REF] [START_REF] Tedjini | Augmented rfid tags[END_REF]. Another example of this tendency is the development of inter-tag communication concept for applications where data rates can be low and the receiver tag uses passive RF part (i.e. envelope detector) [START_REF] Liu | Ambient backscatter: Wireless communication out of thin air[END_REF].

The concept of tag-to-tag (T2T) communication between two passive UHF RFID tags has already been proposed and experimentally validated in [START_REF] Nikitin | Passive tag-to-tag communication[END_REF] for short distances. The main idea is to establish communication between two passive tags placed in close proximity. Energy feeding of these tags is achieved by an external source which can be a classical RFID reader or any other source providing a power level large enough in order to activate the tags. The communication between the tags is performed by backscattering modulation of the incoming RF sources such as WiFi signals, TV transmissions [START_REF] Liu | Ambient backscatter: Wireless communication out of thin air[END_REF] and Bluetooth 4.0 Low Energy standard [START_REF] Ensworth | Every smart phone is a backscatter reader: Modulated backscatter compatibility with bluetooth 4.0 low energy (ble) devices[END_REF].

However, since the first studies few works have been done about this topic. In [START_REF] Karimi | Design of a backscatter-based tag-to-tag system[END_REF], the design of a power efficient demodulator is investigated to enable the tag-to-tag link to operate at long distances with a modulation depth as low as 75%. In [START_REF] Ryoo | Design and evaluation of bttn: A backscattering tag-to-tag network[END_REF], a novel multiphase backscattering technique is proposed to overcome phase cancellation and demonstrating the capability of tag-to-tag link to operate at long distances (above 3m) under high excitation signal.

All things considered, these efforts mostly focus to deal with the performance limitations of the passive UHF RFID systems, like tag characteristics in terms of antenna gain polarisation and impedance matching, sensitivity of the receiver which defines the minimum level of the tag signal and propagation environment [START_REF] Nikitin | Performance limitations of passive uhf rfid systems[END_REF]. This paper focuses on the evaluation of the passive T2T communication in terms of bit error rate (BER) as traditionally done for evaluating wireless transmission systems. The performance depend on several parameters: i) electromagnetic coupling (i.e., the mutual coupling between the tags or the orientation of the tags with respect to the source); ii) modulation depth; iii) receivers topology (coherent or non-coherent demodulation). The general objective is to achieve a global optimization of the communication link between the tags. More precisely, antenna or radio front-end considerations may be extracted by such kind of analysis.

The proposed approach is based on the simulation framework discussed in [START_REF] Zhou | Simulation framework for performance evaluation of passive rfid tag-to-tag communications[END_REF]. The main idea is to consider the antennas of the two tags (LT -listener tag and RTreader tag) as an array of two mutually coupled radiating elements. With respect of the orientation of the illuminating source relative to the tags plane, the far field gain of the antenna array is varying. Moreover, the impedance provided by the LT antenna varies with respect of the distance between the two tags. The gain and impedance modification has an impact on the modulation depth and consequently on the quality of the communication. As stated, the present work gives for ASK modulation the analytical formula for the probability of error taking into account the particular case.

The rest of the paper is organized as follows. Section II describes more in detail the concept of tag-to-tag (T2T) communication with versatile modeling tools are required to understand and master the design of interacting-oriented tags. The theoretical investigation aimed to rewrite bit error probability according to crucial parameters that improve the range is detailed in section III. Finally conclusion and perspectives are given in section IV.

II. TAG-TO-TAG SIMULATION FRAMEWORK

A T2T communication requires an external electromagnetic source in the employed frequency band that delivers a signal with a power high enough to ensure the tags activation. The information exchange is performed based on the same source by using the retro-modulation principle. As mentioned previously, the RT will modulate the incoming signal to send a message towards the LT (Fig. 1). From the electromagnetic field perspective, the two tags may be described as an array of two mutually coupled antennas. The resulting antenna gains and the impedance variation of the LT antenna with respect of the RT variation are extracted from an electromagnetic field simulation performed with CST microwave studio software.

Then, system-level simulations are performed using Keysights Advanced Design System (ADS) software. The synoptic of the simulation framework is depicted in Fig. 2. As can be remarked, the external source (here a CW) is modulated in amplitude (ASK) by switching between the two gains and impedance of the antennas (G on , Z on and G of f , Z of f respectively). The switch is performed in the rhythm of the data sent from the RT towards the LT.

III. ANALYTICAL FORMULATION OF BER

The variation in terms of antenna array gain and the impedance presented at the input of the LT, can be associated to a variation of the amplitude of the CW signal, at the rhythm of the data sent by the RT. In order to receive the data, on the LT, a demodulation stage is employed. The demodulation may be coherent or non-coherent. For the two cases, analytical expression of the probability of error with respect of signal to noise ratio can be developed [START_REF] Kabara | The Physical Layer of Communications Systems[END_REF] [START_REF] Peebles | Digital Communication Systems[END_REF].

The probability of error Pe in the case of a coherent and non-coherent reception is given below. The coherent or synchronous demodulation requires a carrier recovery circuit to transpose the received signal to the low frequencies. A block diagram of coherent detection is shown in Fig. 3. In this circuit, a local oscillator is tuned to oscillate at the same frequency as the received signal, the received RF signal multiplies with the local oscillator through an RF mixer generating the sum and the difference frequencies between them. The lowpass filter is used to recover the bandpass signal by eliminating the sum frequency component .

A. Coherent detection

P e,ASKC = 1 2 .erf c   d 2 4N 0   (1)
The erf c function is given as: The asynchronous ASK detector consists of a bandpass filter to reduce noise outside the frequency band, an envelope detector as a half-wave rectifier used to recover the baseband signal and a quantifier which delivers a digital output. The non-coherent detection does not require a synchronization mechanism, in the presence of noise that disturbs the signal, a system using non-coherent detection has a lower BER than coherent detection.

erf c (x) = 2 √ π x 0 e -t 2 dt (2)

B. Non coherent detection

P e,ASKN C = 1 2 • exp - d 2 4N 0 (3)
Where d corresponds to the difference between the amplitudes of the high and low bit states, and N 0 is the noise power spectral density.

E b = 1 2 • A 2 1 2 • T s + A 2 0 2 • T s (4)
The average energy per bit E b is defined by the amplitudes of the two states A 0 , A 1 of the binary flow and bit time T s .

A 1/0 = A.G on/of f 1 - Z on/of f -Z 0 Z on/of f + Z 0 (5)
It is worth noticing that the modulation depth M depends on the gain and switching impedance together with the received power coming from the external source.

M = A 1 -A 0 A 1 (6)
For binary modulation schemes, we have two signals S 1 (t) and S 0 (t).

S 1 (t) = 2E b β • ϕ(t) (7)
S 0 (t) = α • S 1 (t) (8)
With :

β = G on 1 - Z on -Z 0 Z on + Z 0 +G of f 1 - Z of f -Z 0 Z of f + Z 0 (9) ϕ(t) = 2 T s • cos (2πf c t) (10)
d = 2E b β • (1 -α) (11)
The parameter d is strictly related to the average energy per bit E b which is defined according to the amplitudes of the two states A 0 , A 1 of the binary flow and modulation rate α= A0 A1 . By replacing the term d in ( 1) and ( 3) with [START_REF] Kabara | The Physical Layer of Communications Systems[END_REF], we obtain:

P e,ASKC = 1 2 • erf c   (1 -α) 2 β • E b 2N 0   (12)
The erf c function is defined in [START_REF] Nikitin | Phase based spatial identification of uhf rfid tags[END_REF].

P e,ASKN C = 1 2 • exp - (1 -α) 2 β • E b 2N 0 (13) E b
N0 is the signal to noise ratio (SNR) for a digital signal, its a normalized version of SNR for a defined bandwidth and bit rate, used as a figure of merit.

IV. SIMULATION RESULTS By using the simulation framework previously described, BER variation has been calculated. The simulations were performed considering the LT and RT illuminated by a CW at 868 MHz and with a power level of -50 dBm.

In order to have good accuracy for the obtained BER values, the number of transmitted bits was 10 6 . The bit time was set to 50 nsec. The gain and the coupling impedance value are defined to have several modulation depths ranging from 10% to 100%. Without loss of generality, the simulation considered the impedance matching case (where Z on = Z of f = 50 Ω) together with a missmatch case.

The BER value is estimated by comparing the base band signal of RT with the recovered signal of LT using synchronous or asynchronous demodulation, the digital version is recovered with a decision circuit after the demodulation.

As can be seen from Fig. 5 and Fig. 6, for a coherent and non-coherent detection, by assuming that the signal collected by the listener tag is degraded by AWGN. The performance of the T2T is proportional to the modulation depth that must be maximized with a proper impedance matching strategy accounting for the electromagnetic coupling among tags. We focus on the non coherent detection results that is used in passive T2T system for simplified circuitry and very low power cost.

In Fig. 6, for a bit error rate of 10 -5 , ASK N C100% needs a signal to noise ratio of approximately 13 dB while ASK N C50% is around 19 dB. The reason is ASK N C50% expends energy in both symbols used for transmission, but the symbols are alike and distance between them in signal space is small called modulation depth, the receiver is not able to make the difference between two binary levels in low signal to noise ratio. Fig. 6.

BER variation for non-coherent demodulation for different modulation depths in matched case. The difference between the theoretical results and the simulation in Fig. 7 comes from the way in which the BER and SNR is determined in simulation (statistical computation), while in MATLAB we define an interval for the SNR in which the theoretical BER is ploted.

V. CONCLUSION

This paper presents a theoretical background for the analysis of a passive tag-to-tag communication. BER evaluation demonstrates the importance of modulation depth, which is defined by the gain, coupling impedance and received power from the external source. Concluding that, the distance between the tags, their orientation and the position of the source play a crucial role in performance limitations of passive tag-to-tag systems and that the theoretical formulation will help to design an efficient T2T system.

A further study will be performed by completing the theoretical study, the aim is to make simulation framework as realistic as possible by considering several parameters like electromagnetic coupling (i.g. complex coupling impedance) and different tags configuration where the antennas are spaced, staggered and arbitrarily oriented. Also, by adding scenarios where the external source is a complex RF waveform from, using a more realistic radio channel and receivers with passive circuits. This will leads us to measure the maximum RT-to-LT tag communication distance in a variety of controlled scenarios.

Fig. 1 .

 1 Fig. 1. General principle of tag-to-tag communication.

Fig. 2 .

 2 Fig. 2. Synoptic of the tag-to-tag system-level simulations.

Fig. 3 .

 3 Fig. 3. Coherent ASK receiver.

Fig. 4 .

 4 Fig. 4. Non-coherent ASK reciever.

Fig. 5 .

 5 Fig. 5. Coherent demodulation: BER variation for different modulation depths in matched case.

Fig. 7 .

 7 Fig. 7. BER variation for non-coherent demodulation in the miss-match case.

VI. ACKNOWLEDGEMENT

This work was supported by the SPIE ICS-INSA Lyon IoT chair.


Éphane St

Seuret

A SURVEY ON PRESCRIPTION OF MULTIFRACTAL BEHAVIOR

published or not. The documents may come   L'archive ouverte pluridisciplinaire

Multifractality between pure and applied mathematics

The notion of multifractal functions and measures can be traced back to the interest of physicists in the Hölder singularities structure in fully developed turbulence, which is described in terms of large deviations for the distribution at small scales of Mandelbrot random multiplicative cascades in [START_REF] Mandelbrot | Multiplications aléatoires itérées et distributions invariantes par moyennes pondérées[END_REF], and in a geometric setting in the version of the socalled multifractal formalism for functions proposed by Frisch and Parisi [START_REF] Frisch | Fully developed turbulence and intermittency in turbulence, and predictability in geophysical fluid dynamics and climate dynamics[END_REF], see Section 4. Another source leading to multifractal ideas is provided by the works of Henschel & Procaccia [START_REF] Hentschel | The infinite number of generalized dimensions of fractals and strange attractors[END_REF] and Halsey & al. [25]. Since then, multifractal analysis was further developed in dynamical systems theory and geometric measure theory, and has become a standard tool to describe the fine geometric structure of objects possessing nice invariance properties, such as self-similar and self-affine measures and functions, many classes of stochastic processes such as Lévy processes and more general Markov processes, as well as random measures emerging from multiplicative chaos theory.

Let us recall the notion of singularity spectrum of a function, leading to multifractals. Let d ≥ 1 be an integer. Given a real function f ∈ L ∞ loc (R d ) and x 0 ∈ R d , f is said to belong to C H (x 0 ), for some H ≥ 0, if there exists a polynomial P of degree at most H and a constant C > 0 such that for x close to x 0 , |f (x) -P (x

-x 0 )| ≤ C|x -x 0 | H . Definition 1. The pointwise Hölder exponent of f ∈ L ∞ loc (R d ) at x 0 is h f (x 0 ) = sup H ≥ 0 : f ∈ C H (x 0 ) ,
and f is said to have a Hölder singularity of order h f (x 0 ) at x 0 . The singularity spectrum D f of f is the map:

D f : H ∈ [0, ∞] -→ dim E f (H), where E f (H) := {x 0 ∈ R d : h F (x 0 ) = H}.
The notation dim stands for the Hausdorff dimension, and by convention dim ∅ = -∞. The multifractal spectrum D f encapsulates key information on a given function f , in particular it carries a description of the distribution of the singularities of f . But the computation of D f often raises deep mathematical questions (for instance, it took almost 130 years to find the multifractal spectrum of the famous Riemann series

+∞ n=1 sin(n 2 πx) n 2 ),
and in most cases the exact value of D f happens to be not directly accessible, neither theoretically nor numerically. Fortunately, the notion of multifractal formalism furnishes a clever way to circumvent this difficulty and to compute the explicit value of the spectrum of large classes of measures and functions. Also, multifractal formalism provides ideas to develop numerical algorithms able to estimate D f on real-life data. The main idea is that for very large classes of functions f (and also for other mathematical objects like measures, stochastic processes -such examples will be given in this paper), D f is equal to the Legendre transform of the so-called L qspectrum τ f of f : this L q -spectrum is computed directly using the values of f , and is numerically accessible. When these two quantities (D f and the Legendre transform of τ f ) coincide, it is said that f satisfies the multifractal formalism. Examples of L q -spectra for functions (and measures) based on increments, wavelet coefficients or wavelet leaders, are given in the upcoming sections (see [START_REF] Abry | When Van Gogh meets Mandelbrot: Multifractal classification of painting's texture[END_REF], (3), [START_REF] Barral | From multifractal measures to multifractal wavelet series[END_REF] or [START_REF] Barral | Besov spaces in multifractal environement, and the Frisch-Parisi conjecture[END_REF]). The intuition that a multifractal formalism should hold is due to U. Frisch and G. Parisi, we refer the reader to Section 4 for an account on the ideas leading to this formula.

The multifractal formalism, and its validity for many mathematical models, explains the success of the multifractal approach used as classification tool in signal and image processing. Indeed, algorithms have been developed (mainly based on wavelet theory, see [START_REF] Muzy | Multifractal formalism for fractal signals: The structure-function approach versus the wavelet-transform modulus-maxima method[END_REF] for the original WTMM method and more recently [START_REF] Abry | Irregularities and scaling in signal and image processing: Multifractal analysis[END_REF] for a mathematical study of the wavelet leaders algorithm and the latest developments and algorithms based on wavelet leaders) to estimate numerically L q -scaling functions, the stability and efficiency of these algorithms being mathematically grounded. Using these algorithms, it is now established that many data coming from Geophysics, turbulence, Physics, Biology, exhibit non-linear L q -scaling functions, which for a given function f is interpreted thanks to the Frisch-Parisi heuristics Resuming the above, we have on one side many mathematical objects f with non-linear L q -scaling functions and a non-trivial singularity spectrum D f , and on the other side an impressive quantity of signals, images and multivariate, multi-dimensional data whose estimated L q -spectra and singularity spectra are non-trivial. It is worth asking which mathematical objects are indeed the most relevant to model the observed data, and how to create models with any reasonable multifractal behavior.

This general problematics can be understood in various ways, and raises several theoretical questions, most of them still being open: {-∞}, is there a functional space in which Baire typical functions have σ as singularity spectrum? Do typical functions satisfy a multifractal formalism? (iv) Given an admissible (homogeneous or not) singularity spectrum σ : R + → [0, d] ∪ {-∞}, is there a differential equation, a PDE or a stochastic (P)DE whose solution has σ as singularity spectrum?

These problems have their counterpart in terms of L q -spectra: replacing everywhere σ : R + → [0, d] ∪ {-∞} by τ : R → R, one may ask for the admissible τ that can be the L q -spectrum of a function (homogeneous or not), and if such an L q -spectrum is typical in some functional space.

The same questions arise when considering probability measures instead of functions. The main difference with the function setting is that there are additional constraints when dealing with measures, see Sections 2 and 4.1.

Although the tools used in the two contexts (functions and measures) are of different nature, a connection between the two situations is provided by the following theorem from [START_REF] Barral | From multifractal measures to multifractal wavelet series[END_REF], based on wavelet analysis. Theorem 1. Let µ be a probability measure on R d such that there exist α, C > 0 satisfying that for every x ∈ R d and 0 ≤ r ≤ 1, µ(B(x, r)) ≤ Cr α .

Consider the function F µ : R d → R whose wavelet coefficients are given by d λ = µ(λ) for every dyadic cube λ ∈ Λ (see Section 4.2 for definitions).

Then the multifractal spectra of µ and F µ coincide.

Our purpose here is to provide a survey on recent results and on some open problems related to these various research directions, which combine many ideas coming from (and having applications to) geometric measure theory, functional and harmonic analysis, and real analysis, as well as ergodic theory and dynamical systems.

Prescription of exponents and local dimensions

For a given mapping f : R d → R belonging to L ∞ loc (R d ), its associated pointwise Hölder exponent mapping h f : x → h f (x) may be very erratic, changing violently from one point to the other. Nevertheless h f (viewed as a function) is quite well understood, as confirmed by the following theorem by S. Jaffard which provides a full characterization of h f [START_REF] Jaffard | Exposants de Hölder en des points donnés et coefficients d'ondelettes[END_REF][START_REF] Jaffard | Functions with prescribed Hölder exponent[END_REF]. Recall that C log (R d ) is the space of those functions f : R d → R satisfying that there exists C > 0 such that for every

x, y ∈ R d with |x -y| ≤ 1/2, |f (x) -f (y)| ≤ C| log |x -y|| -1 . Theorem 2. When f ∈ C log (R d ), the mapping h f is a liminf of a sequence of continuous functions.
Conversely, let H : R d → R + ∪ {+∞} be a liminf of a sequence of continuous functions. There exists a function f :

R d → R, f ∈ C log (R d ), such that for every x ∈ R d , h f (x) = H(x).
Let us also mention that in [START_REF] Ayache | Wavelet construction of generalized multifractional processes[END_REF] the authors build a continuous nowhere differentiable stochastic process (M x ) x≥0 whose pointwise Hölder exponents have the most general form, i.e. the mapping x → h M (x) ∈ (0, 1) can be any liminf of a sequence of continuous functions.

It is a natural question to investigate the same issues for local dimensions for measures. Definition 2. Let M(K) be the set of Borel probability measures on a Borel set K ⊂ R d .

For µ ∈ M(R d ), the support of µ is the set Supp(µ) = {x ∈ R d : µ(B(x, r)) > 0 for every r > 0}.

The (lower) local dimension of µ at x ∈ Supp(µ) is

(1) h µ (x) = lim inf r→0 + log µ(B(x, r)) log r
and the singularity spectrum of µ is defined for

H ∈ R ∪ {+∞} by D µ (H) = dim E µ (H) where E µ (H) = {x ∈ Supp(µ) : h µ (x) = H}.
It is common (and in many situations, relevant and important) to look at points x at which (1) turns out to be a limit (and not only a liminf). Nevertheless, in this article only lower local dimensions are considered (we will forget the term "lower" in the following), since we are interested in quantities defined for all x ∈ Supp(µ). Definition 3. A function f (resp. a measure µ) on R d is called homogeneous (in short: HM) if the restriction of f (resp. µ) on any finite subcube I ⊂ R d has the same singularity spectrum as f (resp. µ).

The same definition applies to a function or measure when R d is replaced by [0, 1] d .

One could expect that an analog of Theorem 2 should hold for local dimensions of measures. Unfortunately, the situation is not as clear, as proved by the next lemma [START_REF] Buczolich | Measures and functions with prescribed singularity spectrum[END_REF]. Lemma 3. Let µ ∈ M(R d ) with a support containing a cube U ⊂ R d . If the mapping x → h µ (x) is continuous on U , then h µ is locally constant and equal to d on U .

Last lemma leads to the two following open problems: What are the admissible mappings H : R d → R + satisfying H = h µ for some probability measure µ? Given an admissible mapping H, can one explicitly build a measure µ ∈ M(R d ) such that h µ = H? Even if all these questions are mathematically relevant and raise delicate questions (in geometric measure theory for instance), in many situations it is even more important to construct functions with prescribed singularity spectrum. This is the case in particular when trying to model real-life data, for which essentially only global quantities (like the L q -spectrum) are accessible.

Prescription of multifractal behavior

As expected, the prescription of singularity spectrum for functions or measures is more involved than that of exponents. Indeed, there is no obvious characterization for the admissible singularity spectrum for functions. Yet, using wavelet techniques, S. Jaffard was able to prove the following theorem [START_REF] Jaffard | Construction de fonctions multifractales ayant un spectre de singularités prescrit[END_REF]. Let

R = σ : R + → [0, d] ∪ {-∞} : ∃ bounded interval I ⊂ R + and α ∈ [0, d] such that σ = α1 1 I + (-∞)1 1 R + \I . Theorem 4. Let σ : R + → [0, d] ∪ {-∞} be the supremum of a countable sequence of functions (σ n ) n≥1 ∈ R.
Then there exists a continuous function f :

R d → R such that D f = σ.
Although probably not optimal, this theorem already covers a large class of singularity spectra, certainly sufficient to mimic precisely all the singularity spectra that can be estimated on real data.

In particular, any concave mapping σ : R + → [0, d] ∪ {-∞} can be written as sup n∈N σ n for some well chosen functions σ n ∈ R, hence it is possible to build a function f :

R d → R such that D f = σ.
The same questions were addressed for measures first in [START_REF] Buczolich | Measures and functions with prescribed singularity spectrum[END_REF] and then in [START_REF]Inverse problems in multifractal analysis of measures[END_REF]. The admissible singularity spectra for measures are not characterized either, but when compared to spectra of functions, there are additional constraints: if [START_REF] Brown | On the multifractal analysis of measures[END_REF][START_REF] Olsen | A multifractal formalism[END_REF]).

d µ = σ for some µ ∈ M(R d ), then σ(h) ≤ min(h, d) (see
Another surprising constraint obtained in [START_REF] Buczolich | Measures and functions with prescribed singularity spectrum[END_REF] is that the support of the singularity spectrum of a 1-dimensional HM measure contains an interval. We call Supp(σ) the support of a function σ : R d → R, and by abuse of notation, if

σ : R → R + ∪ {-∞}, Supp(σ) = {H : σ(H) ≥ 0}. Proposition 5. For any non-atomic HM probability measure µ ∈ M(R), Supp(D µ ) ∩ [0, 1] is necessarily an interval of the form [α, 1], where 0 ≤ α ≤ 1.
This proposition leads to the following notation: for σ :

R + → [0, 1] ∪ {-∞}, consider the mapping σ † (H) = max σ(H), 0 • 1 1 [inf(Supp(σ)),sup(Supp(σ))] (H) .
Essentially, σ † fills the gaps in the support of σ by replacing the value -∞ by 0.

The result concerning the prescription of singularity spectrum of measures obtained in [START_REF] Buczolich | Measures and functions with prescribed singularity spectrum[END_REF] is the following. Theorem 6. Let σ : R + → [0, 1] ∪ {-∞} be the supremum of a countable sequence of functions (σ n ) n≥1 ∈ R satisfying in addition that for every n ≥ 1, calling I n the interval on which σ n is not -∞,

•

I n ⊂ [0, 1], • I n is closed, • σ n (x) ≤ x for x ∈ I n .

Then:

(i) There exists µ ∈ M(R) such that D µ = σ.

(ii) There exists a HM measure µ ∈ M(R) with support equal to [0, 1] such that D µ = σ † , and D µ (1) = 1.

Observe that although the class of singularity spectra obtained here is quite large, only local dimensions less than 1 are dealt with, and only the one-dimensional case is covered.

Theorem 6 is completed by the result by Barral [START_REF]Inverse problems in multifractal analysis of measures[END_REF].

Theorem 7. Let σ : R + → [0, d]∪{-∞} be an upper semi-continuous function with support included in [α, β] for some 0 < α < β < +∞, satisfying σ(h) ≤ h for every h ∈ [α, β],
and such that σ(h) = h for some h. Then there exists µ ∈ M(R d ) such that D µ = σ.

In the last theorem, Barral was also able to build measures that were "homogeneous" in the sense that the restriction of µ to any bounded cube I ⊂ R d such that µ(I) = 0 has the same singularity spectrum as µ itself. A comparison between Theorems 6 and 7 yields that (at least) in dimension 1, the measures constructed by Barral are necessarily not supported by a full interval (their support is a Cantor-like set), otherwise σ should be replaced by σ † . Theorems 4, 6 and 7 are not entirely satisfying. Indeed,

• the construction used in Theorem 4 does not guarantee that the corresponding spectrum is homogeneous. Homogeneous spectra are yet very common (for instance, trajectories of stationary processes usually exhibit homogeneous spectra). • in the three previous theorems, even if the prescribed spectrum is concave, the corresponding function or measure a priori does not satisfy a multifractal formalism. • the functions and measures built along the proofs of Theorems 4 and 6 are not "typical" in any sense, and may essentially appear, from the modeling standpoint, as mathematical extreme toy examples. These issues will be addressed in the next sections.

Prescription of multifractal formalisms

Let us very quickly recall the intuition by Frisch & Parisi [START_REF] Frisch | Fully developed turbulence and intermittency in turbulence, and predictability in geophysical fluid dynamics and climate dynamics[END_REF], who studied the velocity v of a turbulent fluid in a bounded domain Ω ⊂ R 3 . More precisely, inspired by the seminal works by Kolmogorov on turbulent fluids and the study of the local fluctuations of their velocity, Frisch and Parisi were interested in the moments of the increments of v defined by [START_REF] Abry | When Van Gogh meets Mandelbrot: Multifractal classification of painting's texture[END_REF] for every q ∈ R, S v (q, l)

= Ω |v(x + l) -v(x)| q dx.
For real data, q being fixed, it has been observed that when |l| becomes small, S v (q, l) obeys a scaling law: S v (q, l) ∼ |l| ζv(q) for some exponent ζ v (q) ∈ R. The mapping q → ζ v (q) is called the scaling function of the velocity of the fluid. It can be seen that if v is modeled at small scales by a fractional Brownian motion of index H 0 (as did Kolmogorov for instance), then ζ v (q) is linear with slope H 0 . However, in the 1980's, numerical experiments for the velocity show that ζ v (q) is non-linear and concave. The seminal idea by Frisch and Parisi consists in interpreting this non-linearity in terms of multifractality of v, via the following heuristic argument.

Replacing Hausdorff by box dimension, and making all kind of rough approximations (i.e. assuming that limits exist, etc), for all points x ∈ R 3 at which

h v (x) = H, one has |v(x + l) -v(x)| ∼ |l| H for small l. Since dim E v (H) = D v (H)
, there should exist approximately |l| -Dv(H) cubes of size l in the domain Ω containing points x which are singularities of order H for the velocity v. All these intuitions lead to the estimates

S(q, l) = Ω |v(x + l) -v(x)| q dx ∼ H |l| qH |l| -Dv(H) |l| 3 ∼ H |l| qH-Dv(H)+3 .
When |l| → 0, the greatest contribution is obtained for the smallest exponent:

ζ v (q) = inf H (qH -D v (H) + 3).
The corresponding mapping q → ζ v (q) is called the L q -spectrum or the scaling function of v -soon we will see more relevant formulas for ζ v (q) and how to define it for measures.

By inverse Legendre transform, one deduces that

D v (H) = inf q∈R (qH -ζ v (q) + 3)
which justifies that D v has a concave shape.

It is striking that despite the series of crude approximations, this intuition has proved to hold true in many (if not most of) situations, after some renormalization and suitable choices for the scaling functions. Definition 4. We call multifractal formalism any formula relating the singularity spectrum of a function (or a measure) to a scaling function via a Legendre transform.

For almost 30 years now, many efforts have been made to prove the validity of multifractal formalism(s) in various functional spaces, for many mathematical objects (self-similar or selfaffine functions and measures) including random processes (Mandelbrot cascades, Gaussian multiplicative chaos, Lévy processes). This line of research was constantly followed and fostered by applications which gave mathematicians lots of signals and physical phenomena to study and work on, see Figures 1 and2. In particular, stable algorithms to estimate L qspectra of data have been developed, furnishing to the scientific community many robustly analyzed sets of data [START_REF] Abry | Irregularities and scaling in signal and image processing: Multifractal analysis[END_REF].

A remaining question though lies in the existence of a functional setting in which a given multifractal behavior would be "generic". This is known after [START_REF] Jaffard | On the Frisch-Parisi conjecture[END_REF] as the Frisch-Parisi conjecture, which can be formulated as follows:

Conjecture 1. Given any admissible concave mapping σ : R + → [0, d] ∪ {-∞}, is there a functional space in which typical functions have σ as singularity spectrum and satisfy a multifractal formalism? Notice that ideas leading to a multifractal formalism can also be found in thermodynamics (see [START_REF] Halsey | Fractal measures and their singularities: the characterisation of strange sets[END_REF][START_REF] Hentschel | The infinite number of generalized dimensions of fractals and strange attractors[END_REF] and the large literature around thermodynamical formalism). This outlines the universality of the approach consisting in describing local fluctuations via the (Legendre transform of) global statistical quantities computed directly on the object (function, measure, random process) under consideration.

From now on, and without loss of generality, we restrict our statements to measures and functions supported in the cube [0, 1] d .

τ µ (q) q 0 -d 1 Dµ(H) = τ * µ (H) H 0 d Figure 3. Left: L q -spectrum of a measure µ on [0, 1]. Right:
The corresponding singularity spectrum of µ when it satisfies a multifractal formalism. 4.1. Prescription of multifractal formalism for measures. In case of measures µ ∈ M([0, 1] d ), the formula for the L q -spectrum is quite standard and given by (3)

τ µ (q) = lim inf j→+∞ 1 -j log 2 λ∈D j : µ(λ) =0 µ(λ) q ,
where D j stands for the set of dyadic cubes

λ j,k = 2 -j k + [0, 2 -j ] d , k ∈ Z d , of generation j ∈ Z (i.
e. dyadic cubes with side-length equal to 2 -j ). It is easily seen that τ µ is always concave, non-decreasing, and that -d ≤ τ µ (0 + ) ≤ τ µ (1) = 0. In addition, the support of τ µ is equal to R when lim sup r→0 + log(inf{µ(B(x,r)): x∈Supp(µ)}) log r

< +∞, and it is [0, +∞) when the same quantity is infinite [START_REF]Inverse problems in multifractal analysis of measures[END_REF].

Recall that the Legendre transform of a mapping τ : R → R (used in the previous section) is defined for H ≥ 0 as τ * (H) := inf q∈R (qH -τ (q)).

Barral solved in [START_REF]Inverse problems in multifractal analysis of measures[END_REF] the following inverse problem.

Theorem 8. Let τ : R → R be concave, non-decreasing, with -d ≤ τ (0 + ) ≤ τ (1) = 0.

There exists a probability measure µ ∈ M([0, 1] d ) compactly supported, such that τ µ = τ and µ satisfies the multifractal formalism, i.e. D µ = τ * .

See Figure 3 for an illustration. The drawback of this first important step is that the measure constructed by Barral in [START_REF]Inverse problems in multifractal analysis of measures[END_REF] has again a Cantor-like set as support (so it is not fully supported on [0, 1] d ), hence is not suitable to model any real-life signal supported by, say, an interval. The result is reinforced in the upcoming paper [START_REF] Barral | Besov spaces in multifractal environement, and the Frisch-Parisi conjecture[END_REF], in which we build fully supported measures satisfying a prescribed multifractal formalism, which in addition are almost-doubling in the following sense.

A Borel set function is a mapping µ associating with every Borel set B ⊂ [0, 1] d a positive real number µ(B) ∈ [0, +∞]. A Borel set function µ is almost-doubling when there exists a non increasing function θ : (0, 1] → R + \ {0} such that :

• θ(1) = 0 and lim r→0 + θ(r) log(r) = 0 • there is a constant C ≥ 1 such that for all x ∈ [0, 1] d and r ∈ (0, 1] one has (4)

C -1 e -θ(r) µ(B(x, r)) ≤ µ(B(x, 2r)) ≤ Ce θ(r) µ(B(x, r)).

When θ ≡ 0, then µ is said to be doubling. Doubling and almost-doubling measures occupy a special place in geometric measure theory since they are easier to deal with in many situations -such properties guarantee a certain stability of the values of µ in the sense that µ(B) and µ(B ) have comparable values as soon as B and B are two balls of comparable radii that are close to each other. It is thus important to investigate the possible combination of these properties with the multifractal ones, as done in the following theorem proved in [START_REF] Barral | Besov spaces in multifractal environement, and the Frisch-Parisi conjecture[END_REF]. Although Gibbs measures associated with Hölder regular potentials and smooth maps provide examples of doubling measures with non-trivial multifractal behavior, it may seem surprising that the almost doubling property (which, as said above, limits the local variations of a measure) does not constitute a constraint from the multifractal formalism standpoint: every (admissible) concave mapping can be obtained as the singularity spectrum of a compactly supported probability measure satisfying the multifractal formalism.

Theorem 9 leaves open interesting questions in ergodic theory and dynamical systems, and geometric measure theory, which to the best of our knowledge are not completely addressed yet:

(i) Can the almost doubling property be simplified in a "simple" doubling property in Theorem 9? (ii) Given an almost doubling measure µ, is there a doubling measure µ with same multifractal behavior as µ? (iii) Is it possible to find a Hölder potential on a suitable dynamical system such that the associated invariant measure satisfies the multifractal formalism with a L qspectrum given in advance?

Remark 1. In Theorem 9, it is possible to impose additional conditions on the measures µ so that the same result (D µ = τ * ) holds. One useful condition, which will be used later, is the following.

Definition 5. Let Θ be the set of non decreasing functions θ : N → R * + such that: (i) θ(j) = o(j) as j → ∞ (ii) θ(0) = 0 (iii) for all ε > 0, there exists j ε ∈ N such that for all j ≥ j ≥ j ε , θ(j )-θ(j) ≤ ε(j -j).

A measure µ ∈ M([0, 1] d ) (or µ ∈ M(R d )) satisfies property (P) if there exist C, s 1 , s 2 > 0 such that: (P1) for all j ∈ N and λ ∈ D j , if µ(λ) = 0, then (5)
C -1 2 -js 2 ≤ µ(λ) ≤ C2 -js 1 .
(P2) There exists θ ∈ Θ such that for all j, j ∈ N with j ≥ j, and all λ, λ ∈ D j such that µ(λ) = 0, µ( λ) = 0, ∂λ ∩ ∂ λ = ∅, and λ ∈ D j such that λ ⊂ λ:

(6) C -1 2 -θ(j) 2 (j -j)s 1 µ(λ ) ≤ µ( λ) ≤ C2 θ(j) 2 (j -j)s 2 µ(λ ).
Heuristically, this last condition yields for every dyadic cube λ ∈ D j a control of the µ-mass of the cubes λ ∈ D j with j ≥ j and λ ⊂ 3λ. It is easily checked on self-similar measures satisfying an open-set condition for instance.

In [START_REF] Barral | Besov spaces in multifractal environement, and the Frisch-Parisi conjecture[END_REF], it is proved that there exist measures satisfying (P) for which the conclusion of Theorem 9 holds. 4.2. Prescription of multifractal formalism for functions. While the definition of the L q -spectrum for measures is quite standard and intuitive, finding a suitable formula for the L q -spectrum of functions is not straightforward. Indeed, one easily sees that equation ( 2) does not allow one to catch and describe the local regularity characteristics of smooth functions (with pointwise exponents greater than 1). Many alternative formulas have been proposed, and most of them are based on wavelets. It is thus useful at this point to set the notation concerning wavelets coefficients and wavelet leaders.

Let Φ : R d → R be a scaling function and consider an associated family of smooth wavelets Ψ = {ψ (i) } i=1,...,2 d -1 belonging to C r (R d ), with r ∈ N * (for a general construction, see [START_REF] Meyer | Ondelettes et opérateurs I[END_REF]Ch. 3]). For simplicity, we assume that Φ and the wavelets Ψ are compactly supported [START_REF] Daubechies | Ten Lectures on Wavelets[END_REF]. For every j ∈ Z, recall that D j is the set of dyadic cubes of generation j, i.e. if k = (k 1 , ..., k d ) ∈ Z d and

λ j,k := i=1,...,d [k i 2 -j , (k i + 1)2 -j ) ⊂ R d then D j = {λ j,k : k ∈ Z d } (
see the beginning of Section 4.1). Further we consider the set

Λ j = {λ = (i, j, k) : k ∈ Z d , i ∈ {1, ..., 2 d -1}},
and Λ = j∈Z Λ j . By abuse of notation, λ ∈ Λ j will still be called a dyadic cube of generation j and identified with λ = λ j,k ∈ D j .

For every cube λ = (i, j, k) ∈ Λ, we denote by ψ λ the function x → ψ (i) (2 j x -k). The set of functions 2 dj/2 ψ λ , j ∈ Z, λ ∈ Λ j , forms a Hilbert basis of L 2 (R d ), so that every f ∈ L 2 (R d ) can be expanded as

f = j∈Z λ∈Λ d λ ψ λ , with d λ = R d 2 dj ψ λ (x)f (x) dx,
where equality holds in L 2 (we will work with smooth functions, so equality will also hold pointwise). Observe that we choose an L ∞ normalization for the so-called wavelet coefficients

(d λ ) λ∈Λ of f ∈ L 2 (R d ) (more generally, of f ∈ L p (R d ) for some p ∈ [1, ∞]). For f ∈ L 2 (R d ), define also for k ∈ Z d (7) β(k) = R d f (x)Φ(x -k) dx.
Finally, for a function f ∈ L p (R d ) with p ∈ [1, ∞] whose wavelet coefficients are denoted by (d λ ) λ∈Λ , the wavelet leader associated with λ ∈ D j is

d L λ = sup λ ∈Λ, λ ⊂3λ |d λ |,
where for λ ∈ D j , 3λ stands for the cube with same center as λ and radius 3 2 2 -j (it is the cube that contains λ as well as its 2 d -1 neighbors in D j ). While wavelet coefficients are usually sparse (only a few coefficients carry the important information about f ), wavelet leaders possess a strong hierarchical structure since 0

≤ d L λ ≤ d L λ when λ ⊂ λ.
Remark 2. Although the notations for wavelet coefficients and wavelet leaders do not mention the function f , they highly depend on f and we should never forget about it! Wavelet coefficients and wavelet leaders characterize the pointwise Hölder exponents: indeed, if f ∈ C (R d ) for some > 0, then for every

x 0 ∈ [0, 1] d one has (8) h f (x 0 ) = lim inf j→∞ log d L λ j (x 0 ) log(2 -j ) ,
where λ j (x 0 ) is the unique cube λ ∈ D j that contains x 0 (see [START_REF] Jaffard | Wavelet techniques in multifractal analysis[END_REF]).

It was quite clear from the beginning that a formula based on increments like (2) was not stable neither mathematically nor numerically. To circumvent this difficulty, the idea of introducing wavelets (whose computation requires local means, bringing simultaneously a numerical stability crucial for applications and a natural connection with characterizations of standard functional spaces, see Section 5) was introduced by Alain Arnéodo and his collaborators. Two formulations are nowadays recognized to be the most relevant:

• Formula based on wavelets: [START_REF] Barral | From multifractal measures to multifractal wavelet series[END_REF] T f (q, j) =

λ∈Λ j :d λ =0 |d λ | q -→ η f (q) = lim inf j→+∞ log 2 T f (q, j) -j .
• Formula based on wavelet leaders:

(10) L f (q, j) = λ∈D j :d L λ =0 |d L λ | q -→ L f (q) = lim inf j→+∞ log 2 L f (q, j) -j .
Even if wavelets brought some stability in the computations, wavelet leaders are now recognized as the most efficient, relevant and numerically exploitable measurements of local and global regularity. In particular, the hierarchical structure of wavelet leaders (i.e. 0 ≤ d λ ≤ d λ as soon as λ ⊂ λ ) makes all computations easier and more stable [START_REF] Abry | Irregularities and scaling in signal and image processing: Multifractal analysis[END_REF]. Definition 6. The wavelet multifractal formalism WMF (resp. wavelet leader multifractal formalism WLMF) is satisfied for a function f on an interval

J ⊂ R + when D f (H) = (η f ) * (H) (resp. D f (H) = (L f ) * (H)) for every H ∈ J.
We also say that a function f satisfies the weak wavelet leader multifractal formalism (weak-WLMF) on an interval J ⊂ R + when there exists an increasing sequence (j n ) n≥1 of integers such that if L f (q) = lim inf n→+∞

log 2 L f (q,jn) -jn , then D f (H) = ( L f ) * (H) for every H ∈ J.
Remark 3. The above definition of formalisms depends a priori on the chosen wavelets Ψ. Actually it does not depend on Ψ in the increasing part of the multifractal spectrum [START_REF] Jaffard | Wavelet techniques in multifractal analysis[END_REF], but it does in the decreasing part. For simplicity, we do not mention this dependence in the notations.

Let S d be the set of admissible singularity spectra for functions satisfying a multifractal formalism, i.e. ( 11)

S d = σ : R + → [0, d] ∪ {-∞} :
σ is compactly supported in (0, +∞), concave, with maximum equal to d. .

We are now able to state the result on multifractal formalism prescription for functions.

Theorem 10. For every mapping σ ∈ S d , there exists a function f ∈ L 2 (R d ) satisfying the WLMF and whose singularity spectrum is equal to σ.

Proof. Observe that if a function f has its wavelet coefficients d λ given by µ(λ) for some probability measure µ ∈ M([0, 1] d ), then for every choice of α, β > 0, the function f α,β whose wavelet coefficients are dλ := d α λ 2 -jβ satisfies for every

H ≥ 0, D f α,β (H) = D f H -β α .
This simply follows from ( 8) and the fact that h f α,β (x 0 ) = αh f (x 0 ) + β for all x 0 .

Let σ : R + → [0, d]∪{-∞} ∈ S d be a mapping satisfying the conditions to be a singularity spectrum of a function satisfying a multifractal formalism.

Let α, β be two strictly positive real numbers such that the mapping σ α,β (H) = σ(αH +β) satisfies σ α,β (H) ≤ H and there exists H 0 > 0 such that σ α,β (H 0 ) = H 0 . The existence of (α, β) is an exercise (notice that (α, β) need not be unique).

Theorem 9 provides us with a measure µ satisfying the multifractal formalism for measures and D µ = σ α,β .

Then, Theorem 1 yields that the function F µ whose wavelet coefficients are given by d λ = µ(λ) has the same singularity spectrum as µ. In addition, comparing (3) with [START_REF] Barral | Besov spaces in multifractal environement, and the Frisch-Parisi conjecture[END_REF], and using the hierarchical structure of the measure (i.e. µ(λ ) ≤ µ(λ) whenever λ ⊂ λ), one sees that τ µ (q) = L Fµ (q) for every q ∈ R, hence F µ satisfies the WLMF.

Finally, using the first remark of this proof, the function F whose wavelet coefficients equal µ(λ) α 2 -jβ has its singularity spectrum equal to σ and satisfies the WLMF.

We thus have a complete answer for the prescription of multifractal formalism for functions. But at this point, one may have the feeling that the functions we built are mathematical toy examples. The purpose of the last sections is to explain that for any choice of concave admissible mapping σ, there are natural functional spaces in which typical functions have exactly σ as singularity spectrum. This confirms and strengthens the overall presence of multifractals in most of science fields, and reinforces the position of multifractal machinery as legitimate tool in signal processing and data analysis.

Typical multifractal behavior in classical functional spaces

As emphasized above, it is possible to find mathematical models that mimic large classes of multifractal behavior, in particular including all concave singularity spectra. This last part of the results is key, since for real-life data (multi-dimensional and/or multivariate signals, images, ...) only estimates for the L q -spectrum are numerically accessible (based on log-log plots on a well-chosen range of scales). Indeed, the standard paradigm is to assume that the discrete data f (say, a signal) is obtained from discrete samples of a mathematical model obeying a multifractal formalism, and to consider that the Legendre transform of the estimated L q -spectrum contains relevant information regarding the distribution of the singularities of f (somehow extrapolating on Frisch-Parisi heuristics). This Legendre transform is thus viewed as an "approximation" of the singularity spectrum of the data, although the meaning of the singularity spectrum of a discretized signal is not made precise. The obtained estimated singularity spectrum of the data f possesses various characteristics (values of the largest and the smallest exponents, locations of the maximum, curvature of the concave spectrum at its maximum,...) which are then used as classification tools between numerous samples of a physical, medical,... phenomenon. This has proven to be relevant in various fields going from medicine (heart-beat rate and X-ray analysis) and turbulence [START_REF] Lashermes | Comprehensive multifractal analysis of turbulent velocity using wavelet leaders[END_REF] to, recently, more surprising areas (paintings analysis [START_REF] Abry | When Van Gogh meets Mandelbrot: Multifractal classification of painting's texture[END_REF], text analysis [START_REF] Leonarduzzi | p-leader multifractal analysis for text type identification[END_REF]).

Inspired by these applications, it is thus key to investigate whether the mathematical objects we regularly meet satisfy a multifractal formalism (so that all these heuristics described above lie on solid mathematical grounds). In this survey, we focus on "typical" objects in the sense of Baire: in a Baire space E, a property P of elements x ∈ E is typical or generic when the set {x ∈ E : x satisfies P} is a residual set, i.e. its complement is included in a first Baire category set (a union of countably many nowhere dense sets in E).

Regularity properties of typical functions have been explored since the pioneer works of Banach [START_REF] Banach | Über die Baire'sche kategorie gewisser funktionenmengen[END_REF] or Mazurkiewicz [START_REF] Mazurkiewicz | Sur les fonctions non dérivables[END_REF] for instance. The seminal result concerning multifractal properties of typical functions is due to Buczolich and Nagy, who proved the following [START_REF] Buczolich | Hölder spectrum of typical monotone continuous functions[END_REF].

Theorem 11. Let Mon([0, 1]) be the set of continuous monotone functions f : [0, 1] → R equipped with the supremum norm of functions. Typical functions in Mon([0, 1]) are multifractal with singularity spectrum equal to

D f (H) = H • 1 1 [0,1] (H) + (-∞) • 1 1 (1,+∞] (H).
Theorem 11 was the starting point of an abundant literature on the subject, examples of which are given in the following. The method consists first in finding an upper bound for the singularity spectrum of all functions in Mon([0, 1]) (here, the diagonal σ(H) = H)), then an explicit function F typ whose local behavior is the one suspected to be typical, and finally to construct a countable sequence (A n ) n≥1 of sets of functions, dense in Mon([0, 1]), which are for a given n, really close to F typ at a given scale (depending on n). If the parameters are correctly settled, the intersection of the (A n ) n≥1 will be the set of typical functions with multifractal behavior similar to that of F typ .

The proof is based on a careful analysis on local oscillations of functions, and simultaneous constructions of Cantor-like sets E f (H) carrying the sets of points with pointwise Hölder exponent equal to H, for every f ∈ n≥1 A n .

After Theorem 11, the first direction consisted in exploring the typical behavior in other standard functional spaces. The first, spectacular, results were obtained by Jaffard [START_REF] Jaffard | On the Frisch-Parisi conjecture[END_REF], who implemented the same strategy as [START_REF] Buczolich | Hölder spectrum of typical monotone continuous functions[END_REF] but added wavelet tools to deal with the important examples of Hölder and Besov spaces.

D f (H) = d • 1 1 {α} (H) + (-∞) • 1 1 [0,+∞]\{α} (H).
2) Let p ≥ 1 and s > d/p, and consider the Besov space B s,p q ([0, 1] d ). Typical functions in B s,p q ([0, 1] d ) are multifractal and satisfy

D f (H) = p(H -(s -d/p)) • 1 1 [s-d/p,s] (H) + (-∞) • 1 1 [0,+∞]\[s-d/p,s] (H).
In addition, typical functions satisfy the WLMF.

See figure 4 for an illustration. Theorems 11 and 12 are striking since they underline the preeminence of multifractal properties for "everyday" functions. Jaffard also described the multifractal behavior of typical functions belonging to countable intersections of Besov spaces, leading to a first answer to the Frisch-Parisi Conjecture. Although these results were a giant step in the domain, only increasing singularity spectra with restricted shapes can be obtained and the typical functions do not obey a satisfactory multifractal formalism. Let us also mention that Besov spaces with indices s < d/p were also considered in [START_REF] Jaffard | On the Frisch-Parisi conjecture[END_REF].

Other directions have been investigated. The most natural one concerns probability measures: typical multifractal properties were explored in [START_REF] Buczolich | Typical borel measures on [0, 1] d satisfy a multifractal formalism[END_REF] for measures supported on [0, 1] d and these results were extended by Bayart [START_REF] Bayart | Multifractal spectra of typical and prevalent measures[END_REF] for measures supported on general compact sets.

Theorem 13. Let K ⊂ R d be a compact set, and let M(K) be the set of probability measures on K.

A typical measure µ ∈ M(K) satisfies for any H ∈ [0, dim(K)), D µ (H) = H. In addition, when the dim(K)-Hausdorff measure of K is strictly positive, then typical measures satisfy D µ (dim(K)) = dim(K) and obey the multifractal formalism.

Another extension of typical monotone functions is provided by the set of monotone increasing in several variables: A function f : [0, 1] d → R is continuous monotone increasing in several variables (in short: MISV) if for all i ∈ {1, ..., d}, the coordinate functions f (i) (t) = f (x 1 , ..., x i-1 , t, x i+1 , ..., x d ) are continuous monotone increasing. The set of MISV functions is denoted by MISV d .

With Z. Buczolich, we also investigated the set CC d of continuous convex functions f :

[0, 1] d → R.
Equipped with the supremum norm • , CC d and MISV d are separable complete metric spaces. In [START_REF] Buczolich | Hölder spectrum of functions monotone in several variables[END_REF] and [START_REF] Buczolich | Multifractal properties of typical convex functions[END_REF] we obtained the following results. Theorem 14. 1) Typical functions in MISV d satisfy

D f (H) = (d -1 + H) • 1 1 [0,1] (H) + (-∞) • 1 1 [0,+∞]\[0,1] (H). 2) Typical functions f ∈ CC d satisfy D f (H) = (d -1) • 1 1 {0} (H) + (d -2 + H) • 1 1 [1,2] (H) + (-∞) • 1 1 [0,+∞]\[1,2]∪{0} (H).
See Figure 5 for a comparison between typical multifractal behavior in various functional spaces. This shall also be compared to Figure 3.

It appears clearly that in all the previous situations, the singularity spectra of typical functions have the same shape: it is an affine, increasing, mapping, with no decreasing part.

Other functional spaces, called S ν spaces were built in [START_REF] Aubry | The spaces sν : new spaces defined with wavelet coefficients and related to multifractal analysis[END_REF], in which typical functions all exhibit a singularity spectrum which is visibly increasing in the sense of [START_REF] Maman | Fixed points for the multifractal spectrum map[END_REF], enlarging the class of possible typical multifractal behavior in functional spaces. In addition, these typical functions do not satisfy a multifractal formalism in the sense of Definition 4.

In order to break this limitation (no decreasing part in the singularity spectrum), new (and natural) functional spaces have been introduced in [START_REF] Barral | Besov spaces in multifractal environement, and the Frisch-Parisi conjecture[END_REF]. Since standard functional spaces do not fulfill our requirements (i.e. typical functions in such spaces do not exhibit concave singularity spectra), it is natural to ask whether there are other functional spaces in which typical functions have any singularity spectrum given in advance, and satisfy a multifractal formalism. This solves the Frisch-Paris conjecture as stated in Conjecture 1.

Let B(R d ) be the Borel sets included in R d , and let us introduce the set of Hölder set functions

(12) C(R d ) := µ : B(R d ) → R + such that ∃ s 1 , s 2 ≥ 0, ∀ I ⊂ R d with |I| ≤ 1, |I| s 2 ≤ µ(I) ≤ |I| s 1 .
For µ ∈ C(R d ) and s ∈ R, we write

µ s (I) = µ(I) s , µ (s) (I) = µ(I)|I| s .
We will use the following notation: for x, y ∈ R d , B[x, y] is the smallest Euclidean ball that contains x and y. Definition 7. Let h ∈ R d , f : R d → R, and consider the finite difference operator ∆ h f :

x → f (x + h) -f (x). Define for n ≥ 2 by iteration ∆ n h f := ∆ h (∆ n-1 h f ). For every set function µ ∈ C(R d ), let us introduce for n ≥ 2 (13) ∆ µ,n h f (x) = ∆ n h f (x) µ(B[x, x + nh])
.

The µ-adapted n-th order modulus of continuity of f on R d is defined for t > 0 by

ω µ n (f, t) p = sup t/2≤|h|≤t ∆ µ,n h f L p (R d ) . (14)
It is trivial to check that that when µ(I) = 1 for every set I, then ω µ n (f, t) p coincides with the so-called homogeneous n-th order modulus of continuity of f

ω n (f, t) p = sup t/2≤|h|≤t ∆ n h f L p (R d ) .
Definition 8. Let µ ∈ C(R d ) associated with exponents 0 < s 1 ≤ s 2 in [START_REF] Bertoin | Solutions multifractales de l'équation de burgers[END_REF]. Let n ≥ s 2 . For 1 ≤ p, q ≤ +∞, the Besov space in µ-environment B µ,p q (R d ) is the space of those functions f : R d → R such that f L p (R d ) < +∞ and [START_REF] Buczolich | Typical borel measures on [0, 1] d satisfy a multifractal formalism[END_REF] |f | B µ,p q = 2 jd/p (ω µ n (f, 2 -j ) p ) j≥1 q (N) < +∞. Finally, let us introduce the spaces ( 16)

B µ,p q (R d ) = 0<ε<s 1 /2 B µ (-ε) ,p q (R d ).
The reader can check that B µ,p q (R d ), when endowed with the topology induced by the norm f

B µp q = (β(k)) k∈Z d p + |f | B µ,p
q , forms a Banach space (recall [START_REF]Inverse problems in multifractal analysis of measures[END_REF] for the definition of β(k)).

The intuition behind Definition 8 consists in introducing some space-dependent constraints that will create heterogeneity at all scales. Indeed, when a function f belongs to B µ,p q (R d ), its oscillations ∆ n h f (x) must be very small in certain regions (around points x where µ(B(x, r)) ∼ r α with α large), while in other regions (where µ(B(x, r)) ∼ r α with α small) the control of the oscillations can be relaxed.

In [START_REF] Barral | Besov spaces in multifractal environement, and the Frisch-Parisi conjecture[END_REF], a wavelet characterization of B µ,p q (R d ) and B µ,p q (R d ) is proved when µ is an almost-doubling set function satisfying condition (P) (recall equation ( 4) and Remark 1). Observe indeed that Definition 5 of the condition (P) for measures can easily be extended for set functions µ ∈ C(R d ).

For this, let us introduce a second semi-norm for f ∈ L p (R d ) : we set |f | p,q,µ = (A j ) l≥1 q (N) , where

A j =   λ∈Λ j d λ µ(λ) p   1/p .
The following inequalities are proved in [START_REF] Barral | Besov spaces in multifractal environement, and the Frisch-Parisi conjecture[END_REF].

Theorem 15. Let µ ∈ C(R d ) be an almost doubling set function satisfying condition (P), and let Φ be a scaling function associated with wavelets Ψ (see Section 4.2). Let p ≥ 1, and q ∈ [1, +∞].

Assume that the wavelets Ψ are compactly supported, belong to the standard Besov space B s,p q (R d ) for some s > d/p + s 2 , and possess at least s 2 + 1 vanishing moments (s 1 and s 2 are the exponents associated with µ in (12)).

For every 0 < ε < s 1 , there exists a constant C > 1 (not depending on f ) such that

f L p + |f | B µ,p q ≤ C( f L p + |f | µ (+ε) ,p,q ) (17) f L p + |f | µ,p,q ≤ C( f L p + |f | B µ (+ε) ,p q ). ( 18
)
Moreover, when µ is doubling, (17) and (18) hold for ε = 0, and the norms f L p +|f | p,q,µ and f L p + |f | B µ,p q are equivalent.

Last theorem supports the idea that B µ,p q (R d ) is the right space to work with, since it is characterized by wavelet coefficients, while the spaces B µ,p q (R d ) are not (unless µ is doubling).

The main theorem in [START_REF] Barral | Besov spaces in multifractal environement, and the Frisch-Parisi conjecture[END_REF] is the following. Theorem 16. Let σ ∈ S d be an admissible singularity spectrum (recall [START_REF] Bayart | Multifractal spectra of typical and prevalent measures[END_REF]). Call H s the smallest value at which σ(H) = d.

There exists an almost doubling set function µ ∈ C(R d ) satisfying condition (P) and p ∈ [1, +∞] such that for every q ∈ [1, +∞], typical functions f ∈ B µ,p q (R d ) possess the following properties:

• D f = σ • f satisfies the WLMF for every H ≤ H s .

• f satisfies the weak-WLMF for every H > H s . In addition:

• when p = +∞, typical functions in B µ,p q (R d ) satisfy D f = D µ . • when µ is doubling, the same holds for B µ,p q (R d ) instead of B µ,p q (R d ).

Also, given σ ∈ S d , from the proof in [START_REF] Barral | Besov spaces in multifractal environement, and the Frisch-Parisi conjecture[END_REF] it can be checked that the couple (µ, p) in Theorem 16 is not unique.

Theorem 16 brings a solution to the Frisch-Parisi conjecture (Conjecture 1). The fact that the (strong) multifractal formalism holds only for the increasing part of the singularity spectrum (when H ≤ H s ) seems to be unavoidable. A heuristic explanation of the weak validity of the multifractal formalism in the decreasing part of the spectrum (and not the full validity) is that functions have usually sparse wavelet representations, generating very large values for negative values of q for L f (q, j) on some values of j.

Let us conclude this section by mentioning that a deeper study of the B µ,p q and B µ,p q spaces is performed in [START_REF] Barral | Besov spaces in multifractal environement, and the Frisch-Parisi conjecture[END_REF], leading to results that have their own interest. More precisely, a uniform upper bound for the singularity spectrum of all functions in B µ,p q and B µ,p q is found, as well as the singularity spectrum of typical functions in these spaces for large classes of almost-doubling measures µ. Without giving details on the results, it appears that the singularity spectra D f of typical functions f may have very different shapes depending on the initial measure µ, and the proofs involve many arguments coming from geometric measure theory, ergodic theory and harmonic analysis.

Perspectives

First of all, we are far from being exclusive on generic dimensional results in analysis (see for instance [START_REF] Fraysse | Quelques propriétés génériques en analyse[END_REF][START_REF] Gruslys | Dimensions of prevalent continuous functions[END_REF]), and many other regularity properties shall definitely be studied from the Baire genericity standpoint.

In this survey we focused on the notion of Baire genericity -the same issues can (and must) be addressed in the prevalence sense. Many results regarding prevalent multifractal properties have been obtained, see [START_REF] Aubry | Local behavior of traces of besov functions: Prevalent results[END_REF][START_REF] Fraysse | How smooth is almost every function in a Sobolev space?[END_REF][START_REF] Fraysse | Regularity criteria of almost every function in a Sobolev space[END_REF][START_REF] Olsen | Fractal and multifractal dimensions of prevalent measures[END_REF][START_REF] Olsen | Prevalent L q -dimensions of measures[END_REF] amongst many references, and asking whether prevalent properties coincide with generic ones can sometimes bring some surprises (when they do not coincide).

Finally, one challenging research direction consists in establishing multifractal properties for (classes of) solutions to ordinary or partial differential equations, as well as for the stochastic counterparts. Indeed, multifractal ideas originate from the study of turbulence and other physical phenomena that are ruled by ODEs, SDEs or (S)PDEs, and it would be a fair return to demonstrate the multifractality of (some of) those functions that are solutions to such equations. A few examples already exist (i.e., Burgers equation with a Brownian motion as initial condition [START_REF] Bertoin | Solutions multifractales de l'équation de burgers[END_REF] and large classes of stochastic jump diffusions [START_REF] Barral | A pure jump Markov process with a random singularity spectrum[END_REF][START_REF] Yang | Multifractality of jump diffusion processes[END_REF]), but they are only a first step toward a systematic multifractal analysis of solutions to (partial) differential equations, which will certainly require the development of new techniques and approaches.

Date:

  December 20, 2019. Research partly supported by the grant ANR MULTIFRACS. .

Figure 1 .

 1 Figure 1. Image and estimated multifractal spectrum of different color levels of a satellite image. Courtesy H. Wendt.

Figure 2 .

 2 Figure 2. Two FMRI signals of a resting (in black) and acting (in red) patient. Comparison between their estimated multifractal spectrum. Courtesy H. Wendt.

  (i) What are the mappings σ : R + → [0, d] ∪ {-∞} that are admissible to be a multifractal spectrum, i.e. there exists a function f : R d → R such that D f = σ? (ii) What are the mappings σ : R + → [0, d] ∪ {-∞} that are admissible to be a homogeneous multifractal spectrum, i.e. there exists a function f : R d → R such for every cube I ⊂ R d with non-empty interior, D f I = σ where f I stands for the restriction of f on I? (iii) Given an admissible (homogeneous or not) singularity spectrum σ : R + → [0, d] ∪

Theorem 9 .

 9 Let τ : R → R be concave, non-decreasing, with -d = τ (0 + ) ≤ τ (1) = 0. Then there exists an HM almost doubling measure µ ∈ M([0, 1] d ) with full support in [0, 1] d such that τ µ = τ and µ satisfies the multifractal formalism, i.e. D µ = τ * .

Theorem 12 . 1 )

 121 Let α > 0 and consider the space C α ([0, 1] d ) of α-Hölder functions on [0, 1] d . Typical functions in C α ([0, 1] d ) are monofractal and satisfy

Figure 4 .

 4 Figure 4. Typical singularity spectra of measures supported on [0, 1] d (Left) and of functions in B s,p q (R d ) (Right).

2 Figure 5 . 6 .

 256 Figure 5. Typical singularity spectra for measures, MISV and convex functions

Stéphane Seuret, Université Paris-Est, LAMA (UMR 8050), UPEMLV, UPEC, CNRS, F-94010, Créteil, France E-mail address: seuret@u-pec.fr

Acknowlegments

The author wish to thank the organizers of the fantastic conference "Fractal Geometry and Stochastics 6", and the referee for all her/his relevant and useful comments on this text.