Giuliana Banche

Vivian Tullio

Valeria Allizond

Narcisa Mandras

Janira Roana

Daniela Scalas

Fadwa El Fassi

Sergio D'antico

Anna Maria Cuffini

Nicola Carlone

Synergistic effect of erythromycin on polymorphonuclear cell antibacterial activity against erythromycin-resistant phenotypes of Streptococcus pyogenes 

Keywords:

streptococcal strains (cMLS B , M, and iMLS B A, B and C phenotypes) were assessed in the presence of the macrolide. The results showed that when erythromycin, PMNs and streptococci [both erythromycin-sensitive (ERY S ) and ERY R ] were simultaneously present in the culture medium, PMN phagocytic activity was similar to that of drug-free controls. In contrast, the results emphasised a significant high increase in intracellular killing by PMNs in the presence of erythromycin not only for ERY S streptococci but also for ERY R S. pyogenes with high (cMLS B , iMLS B A and iMLS B B phenotypes) and moderate (M and iMLS B C phenotypes) erythromycin resistance compared with controls without drug. From literature data it emerged that, even if intracellularly concentrated, erythromycin is relatively inactive because of its instability. The results indicate that the enhanced intra-PMN streptococcal killing detected is mainly attributable to PMN bactericidal systems that synergise with intracellular erythromycin in eradicating ERY R S. pyogenes strains (both with high and moderate resistance). These data confirm that the antibiotic resistance detected in vitro does not always imply a failure of antimicrobial treatment.

A c c e p t e d M a n u s c r i p t 4

Introduction

Streptococcus pyogenes, a common human pathogen that can cause a wide variety of infections, is globally susceptible to -lactams. Macrolides represent a good alternative in the treatment of S. pyogenes infections in patients who are allergic to penicillin or its derivates [START_REF] Bae | Phenotypes and genotypes of macrolide-resistant Streptococcus pyogenes isolated in Seoul, Korea[END_REF][START_REF] Gracia | Antimicrobial susceptibility of Streptococcus pyogenes in Central, Eastern, and Baltic European Countries, 2005 to 2006: the cefditoren surveillance program[END_REF]. Unfortunately, macrolide-resistant S. pyogenes have been isolated in many countries, including Europe and Asia [START_REF] Hart | Acute respiratory infections and antimicrobial resistance[END_REF][START_REF] Malbruny | Resistance to macrolides in clinical isolates of Streptococcus pyogenes due to ribosomal mutations[END_REF][START_REF] Shibl | Patterns of macrolide resistance determinants among S. pyogenes and S. pneumoniae isolates in Saudi Arabia[END_REF][START_REF] Varaldo | Genetic elements responsible for erythromycin resistance in streptococci[END_REF]; in Italy the rate of erythromycin resistance in group A streptococci (GAS), which notably increased in the 1990s, is now 25% [START_REF] Banche | In vitro compared activity of telithromycin and azithromycin against northwest Italian isolates of Streptococcus pyogenes and Streptococcus pneumoniae with different erythromycin susceptibility[END_REF][START_REF] Montagnani | Erythromycin resistance in Streptococcus pyogenes and macrolide consumption in a central Italian region[END_REF]. Two principal mechanisms are responsible for acquired resistance to macrolide-lincosamide-streptogramin B (MLS B ) antibiotics for GAS, namely target site modification and active efflux. Target site modification is due to a methylase, which can be constitutive (cMLS B phenotype) or inducible (iMLS B phenotype), that prevents the antibiotic binding to its ribosomal target. Active efflux (M phenotype) is related to a membrane protein responsible for efflux-mediated resistance [START_REF] Descheemaeker | Macrolide resistance and erythromycin resistance determinants among Belgian Streptococcus pyogenes and Streptococcus pneumoniae isolates[END_REF][START_REF] Leclercq | Mechanisms of resistance to macrolides and lincosamides: nature of the resistance elements and their clinical implications[END_REF].

However, this alarming in vitro erythromycin resistance does not always correlate with poor clinical efficacy in vivo as standard susceptibility testing methods do not take into account several host defence mechanisms that play a key role during infection in preventing the triggering and spread of a bacterial infection process [START_REF] Amsden | Pneumococcal macrolide resistance-myth or reality?[END_REF][START_REF] Cuffini | Clarithromycin mediated the expression of polymorphonuclear granulocyte response against Streptococcus pneumoniae strains with different patterns of susceptibility and resistance to penicillin and clarithromycin[END_REF][START_REF] Rondini | Bacteriological and clinical efficacy of various antibiotics used in the treatment of streptococcal pharyngitis in Italy. An epidemiological study[END_REF]. In fact, the ability of professional phagocytes such as polymorphonuclear cells (PMNs) to ingest and kill microorganisms is central to innate immunity and host defence [START_REF] Cuffini | The leading role of antimicrobial agents in modulating the binomial host-microorganism[END_REF][START_REF] Kobayashi | Regulation of the neutrophil-mediated inflammatory response to infection[END_REF][START_REF] Voyich | Genome-wide protective response used by group A Streptococcus to evade destruction by human polymorphonuclear leukocytes[END_REF]. Thus, the current trend in therapy requires the use of antibiotics that combine a high level of antibacterial activity and optimal pharmacodynamic and A c c e p t e d M a n u s c r i p t 5 pharmacokinetic properties with the capacity to act in concert with the immune system in a way that potentiates the host's defence mechanisms. The literature reports much evidence regarding the impact of erythromycin on the primary functions of phagocytes, namely human PMNs and macrophages [START_REF] Labro | Interference of antibacterial agents with phagocyte functions: immunomodulation or 'immuno-fairy tales'?[END_REF][START_REF] Martin | Uptake, accumulation, and egress of erythromycin by tissue culture cells of human origin[END_REF][START_REF] Miller | Erythromycin uptake and accumulation by human polymorphonuclear leukocytes and efficacy of erythromycin in killing ingested Legionella pneumophila[END_REF][START_REF] Prokesch | Antibiotic entry into human polymorphonuclear leukocytes[END_REF]. Since antibiotics that can interact positively with host defences might significantly contribute to improving the outcome of bacterial infection, this study focused on the potential synergy between human PMNs and erythromycin for antimicrobial activity, especially against erythromycin-resistant (ERY R ) S. pyogenes, in order to underline the differences in bacterial susceptibility to PMNs related to different antibiotic resistance phenotypes, e.g. cMLS B , M, and iMLS B subtypes A, B and C. The excellent penetration of erythromycin via an active (energy-requiring) process and its highly effective concentration within PMNs displaying a stimulating effect on phagocytic activities is already well known and largely documented in the literature [START_REF] Labro | Interference of antibacterial agents with phagocyte functions: immunomodulation or 'immuno-fairy tales'?[END_REF][START_REF] Martin | Uptake, accumulation, and egress of erythromycin by tissue culture cells of human origin[END_REF][START_REF] Miller | Erythromycin uptake and accumulation by human polymorphonuclear leukocytes and efficacy of erythromycin in killing ingested Legionella pneumophila[END_REF][START_REF] Prokesch | Antibiotic entry into human polymorphonuclear leukocytes[END_REF][START_REF] Cuffini | Comparative effects of roxithromycin and erythromycin on cellular immune functions in vitro. 3. Killing of intracellular Staphylococcus aureus by human macrophages[END_REF][START_REF] Cuffini | Comparative effects of roxithromycin and erythromycin on cellular immune functions in vitro. 1. Uptake of 3H-macrolides by human macrophages[END_REF].

However, even if highly concentrated intracellularly, relative inactivity of erythromycin in intracellular killing has been detected compared with that of other macrolides owing to its instability at low intracellular pH values.

Materials and methods

Bacteria

Erythromycin-sensitive (ERY S ) and ERY R clinical isolates of S. pyogenes were cultured on Columbia agar supplemented with 5% sheep blood (Biolife Italiana Srl, Milan, Italy). Young colonies (18-24 h) were picked up to ca. Following inoculation, cryovials were kept at -80 C for extended storage [START_REF] Banche | Effect of dialysis membrane biocompatibility on polymorphonuclear granulocyte activity in dialysis patients[END_REF][START_REF] Cuffini | The erythromycin-resistance in S. pyogenes does not limit the human polymorphonuclear cell antimicrobial activity[END_REF].

Antimicrobial activity of erythromycin against Streptococcus pyogenes

Solutions of erythromycin (Sigma, St Louis, MO) were freshly prepared for each experiment and were shown to be free from endotoxin in a standard Limulus amoebocyte lysate assay (BioWhittaker, Walkersville, MD). Determination of the minimum inhibitory concentration (MIC) of erythromycin was carried by the microdilution broth method according to the latest Clinical and Laboratory Standards Institute (CLSI) guidelines [START_REF]Performance standards for antimicrobial susceptibility testing. Eighteenth informational supplement[END_REF]. Interpretation of the results was basically as outlined in the abovementioned CLSI guidelines [START_REF]Performance standards for antimicrobial susceptibility testing. Eighteenth informational supplement[END_REF].

Resistance phenotypes

ERY R phenotypes were determined by triple-disk diffusion testing [START_REF] Giovanetti | Phenotypes and genotypes of erythromycin-resistant Streptococcus pyogenes strains in Italy and heterogeneity of inducibly resistant strains[END_REF] on Mueller-Hinton agar supplemented with 5% sheep blood (Oxoid Ltd., Cambridge, UK).

Commercial disks (Oxoid Ltd.) of erythromycin (15 g), clindamycin (2 g) and josamycin (30 g) were used. The erythromycin disk was placed at the centre of the plate, with the clindamycin disk on the right and the josamycin disk on the left. A disk of penicillin G (10 units) (Oxoid Ltd.) was added on the bottom of the plate to confirm susceptibility of the GAS isolated strains. The antibiotic disks were placed 15-20 mm apart. After 18 h of incubation at 37 C in a 5% CO 2 atmosphere, absence of a significant zone of inhibition around the three disks was taken to indicate constitutive resistance (cMLS B phenotype). The presence of a zone of inhibition around the To induce erythromycin resistance in iMLS B S. pyogenes strains, bacteria were incubated in Todd-Hewitt broth (BD Becton Dickinson Italia S.p.A., Milan, Italy) containing an inducing subinhibitory concentration (0.25 MIC) of drug for 3.5 h at 37 C in a 5% CO 2 atmosphere [START_REF] Hyder | Inducible and constitutive resistance to macrolide antibiotics and lincomycin in clinically isolated strains of Streptococcus pyogenes[END_REF].

Polymorphonuclear cells

Peripheral venous blood was pooled from healthy donors negative for the presence of microbial and viral diseases (A.O.U. San Giovanni Battista, Turin, Italy). Blood was collected into sterile evacuated blood-collecting tubes containing lithium heparin (150 USPU/10 mL blood) and was settled at room temperature by gravity for 30 min in 2.5% dextran (500 000 molecular weight; Pharmacia S.p.A., Milan, Italy) in normal saline (1:1 ratio). The leukocyte-rich plasma supernatant was carefully layered on To obtain pure PMNs, residual erythrocytes were lysed by hypotonic shock for 30 s in sterile distilled water and then centrifuged further [START_REF] Cuffini | Entry of sanfetrinem into human polymorphonuclear granulocytes and its cell-associated activity against intracellular, penicillin-resistant Streptococcus pneumoniae[END_REF][30][START_REF] Tullio | Role of fosfomycin tromethamine in modulating non-specific defence mechanisms in chronic uremic patients towards ESBL-producing Escherichia coli[END_REF]. After counting in a Bürker cell counting chamber (Bürker, Marienfield, Germany), the density of PMNs was adjusted to 10 6 cells/mL in phosphate-buffered saline (PBS) supplied with 0.1% glucose and 0.1% human albumin (Sigma). PMNs were placed in sterile plasticcapped tubes treated with RPMI 1640 (Gibco Laboratories, Grand Island, NY) supplemented with 10% fetal calf serum (FCS) (Gibco Laboratories) and were incubated at 37 C in a shaking water-bath before addition of streptococci [10 7 colony-forming units (CFU)/mL]. Viability assayed by trypan blue exclusion before and after each experiment was >95%. The time between collection of blood and the beginning of the experiments did not exceed 3 h. The interval between PMN harvest and the start of experiments was <30 min [START_REF] Cuffini | Entry of sanfetrinem into human polymorphonuclear granulocytes and its cell-associated activity against intracellular, penicillin-resistant Streptococcus pneumoniae[END_REF][30][START_REF] Tullio | Role of fosfomycin tromethamine in modulating non-specific defence mechanisms in chronic uremic patients towards ESBL-producing Escherichia coli[END_REF].

Radioactive labelling protocol

A total of 200 L of the frozen culture was placed in fresh Todd-Hewitt broth containing 150 Ci of 3 H-uracil (specific activity 1.27 TBq/mmol) (NEN Products, Milan, Italy) at 37 C. Radiolabelled streptococci were centrifuged twice at 1200 rpm for 10 min with Todd-Hewitt broth and were re-suspended in fresh medium and adjusted to yield 10 7 CFU/mL as confirmed by colony counts in triplicate [START_REF] Cuffini | Entry of sanfetrinem into human polymorphonuclear granulocytes and its cell-associated activity against intracellular, penicillin-resistant Streptococcus pneumoniae[END_REF][30][START_REF] Tullio | Role of fosfomycin tromethamine in modulating non-specific defence mechanisms in chronic uremic patients towards ESBL-producing Escherichia coli[END_REF].

A c c e p t e d M a n u s c r i p t 9

Phagocytosis assay

In all experiments, the bacterium:PMN ratio was 10:1. Aliquots of 1 mL of S. pyogenes in RPMI 1640 with 10% FCS were added to PMNs in sterile plastic tubes (10 6 cells) and the tubes were then incubated at 37 C in a shaking water-bath. After incubation for a period of 30, 60 or 90 min, the tubes were centrifuged at 1200 rpm for 5 min. The pellet suspended in PBS was centrifuged at 1200 rpm for 5 min to remove free bacteria. Cells were then suspended in 1 mL of sterile distilled water for 5 min and 100 L samples of this suspension were placed in scintillation fluid (Atomlight; NEN Products) and counted by liquid scintillation spectrophotometry.

Radioactivity was expressed as the counts/min (cpm) per sample. The percentage of phagocytosis at a given sampling time was calculated as follows: phagocytosis (%) = (cpm in PMN pellet/cpm in total bacterial pellet)  100 [START_REF] Cuffini | Entry of sanfetrinem into human polymorphonuclear granulocytes and its cell-associated activity against intracellular, penicillin-resistant Streptococcus pneumoniae[END_REF][30][START_REF] Tullio | Role of fosfomycin tromethamine in modulating non-specific defence mechanisms in chronic uremic patients towards ESBL-producing Escherichia coli[END_REF].

Measurement of antimicrobial activity of polymorphonuclear cells

In all the experiments, the bacterium:PMN ratio was 10:1. Aliquots of 1 mL of S. pyogenes (10 7 CFU) and PMNs in sterile plastic tubes (10 6 cells) were incubated in RPMI 1640 at 37 C in a shaking water-bath for 30 min to allow phagocytosis to proceed. The PMN/bacterium mixtures were centrifuged at 1200 rpm for 5 min and washed with phosphate saline to remove free extracellular bacteria. An aliquot of PMNs containing bacteria was taken, lysed by adding sterile water and then a viable count of intracellular GAS was performed (time zero). PMNs were then incubated further and at intervals (time x) viable counts of the surviving intracellular bacteria were measured in the same way. PMN killing values were expressed as the survival

Influence of erythromycin on phagocytosis and intracellular killing

The effects of erythromycin on either the phagocytosis of S. pyogenes strains or intracellular bacterial killing by PMNs were investigated by incubating bacteria and phagocytes (bacteria:PMN ratio 10:1) at 37 C in a shaking water-bath for periods of 30, 60 or 90 min in the presence of the drug (MIC and 0.25 MIC) [START_REF] Cuffini | Entry of sanfetrinem into human polymorphonuclear granulocytes and its cell-associated activity against intracellular, penicillin-resistant Streptococcus pneumoniae[END_REF][30][START_REF] Tullio | Role of fosfomycin tromethamine in modulating non-specific defence mechanisms in chronic uremic patients towards ESBL-producing Escherichia coli[END_REF].

Statistical analysis

Each test was performed in triplicate and the results were compared with those obtained with the control systems and expressed as the mean and standard error of the mean for 10 separate experiments. Statistical evaluation of the differences between test and control results was performed by analysis of variance (ANOVA) by Tukey's test. A P-value <0.01 was considered significant.

Results

In all experiments, PMN viability remained unchanged throughout the experiments.

The MIC of erythromycin for ERY S S. pyogenes was found to be 0.25 g/mL. Direct exposure of PMNs and ERY S bacteria to an inhibitory concentration of erythromycin A c c e p t e d M a n u s c r i p t 11 did not improve the phagocytosis until 90 min of incubation (Table 1). Conversely, in the same experimental conditions a marked increase in PMN intracellular killing was achieved with the addition of erythromycin (P < 0.01) (Table 2).

The erythromycin MICs of ERY R S. pyogenes were 128 g/mL for the cMLS B phenotype, 16 g/mL for the M phenotype, 128 g/mL both for iMLS B A and B phenotypes and 8 g/mL for the iMLS B C phenotype. Addition of erythromycin to human PMNs separately incubated with the different S. pyogenes resistant strains had no effect on the phagocytic capacity of PMNs throughout the observation period, as shown by phagocytosis percentages of ingested GAS comparable with that for controls without the drug (Table 1). In contrast, addition of the macrolide to PMNs after phagocytosis had occurred significantly enhanced phagocyte killing against ingested ERY R streptococci resulting in an increased number of killed bacteria for all three incubation times compared with the antibiotic-free systems, in which PMNs were totally unable to kill ingested streptococci at 60 min and 90 min of incubation (P < 0.01) (Table 2).

Discussion

Streptococcus pyogenes is a bacterium that has evolved sophisticated mechanisms to disrupt many critical aspects of PMN function; GAS successfully evades PMN phagocytosis and killing to cause human infections including pharyngitis, impetigo, cellulitis and necrotizing fasciitis [START_REF] Allen | Mechanisms of pathogenesis: evasion of killing by polymorphonuclear leukocytes[END_REF][START_REF] Voyich | Engagement of the pathogen survival response used by group A Streptococcus to avert destruction by innate host defense[END_REF]. It is well known that the rate of phagocytosis and killing of GAS by PMNs is maximal during initial host cell/pathogen interaction (within 30 min) and fails to increase thereafter [START_REF] Cuffini | The erythromycin-resistance in S. pyogenes does not limit the human polymorphonuclear cell antimicrobial activity[END_REF][START_REF] Voyich | Streptococcus pyogenes and human neutrophils: a paradigm for evasion of innate host defense by bacterial pathogens[END_REF].

These findings are consistent with the ability of S. pyogenes to regulate actively the number of differentially expressed genes that modulate PMN functions [START_REF] Kobayashi | Regulation of the neutrophil-mediated inflammatory response to infection[END_REF][START_REF] Voyich | Genome-wide protective response used by group A Streptococcus to evade destruction by human polymorphonuclear leukocytes[END_REF][START_REF] Voyich | Streptococcus pyogenes and human neutrophils: a paradigm for evasion of innate host defense by bacterial pathogens[END_REF][START_REF] Staali | Streptococcus pyogenes bacteria modulate membrane traffic in human neutrophils and selectively inhibit azurophilic granule fusion with phagosomes[END_REF][START_REF] Kobayashi | Bacterial pathogens modulate an apoptosis differentiation program in human neutrophils[END_REF]. Although S. pyogenes employs numerous mechanisms to block phagocytosis, the pathogen is only partially successful and is ingested by PMNs or in the presence of specific antibodies [START_REF] Voyich | Engagement of the pathogen survival response used by group A Streptococcus to avert destruction by innate host defense[END_REF][START_REF] Voyich | Streptococcus pyogenes and human neutrophils: a paradigm for evasion of innate host defense by bacterial pathogens[END_REF][START_REF] Kobayashi | Bacterial pathogens modulate an apoptosis differentiation program in human neutrophils[END_REF]. These data are consistent with the present results: in fact, in erythromycin-free systems, PMNs were able to engulf both ERY S and ERY R streptococci at values that remained constant during incubation and to kill them in only the first 30 min of observation (Tables 1 and2).

The results of this study showed that addition of erythromycin resulted in PMN phagocytosis which remained at levels similar that of drug-free controls, underlying that the macrolide did not adversely affect PMN functionality (Table 1). In contrast, the results emphasised a significant high increase of intracellular killing by PMNs in the presence of erythromycin for all S. pyogenes strains during the observation period (P < 0.01) (Table 2). The most interesting data pertain to the ability of phagocytes to kill the intracellular ERY R streptococci independently from their A c c e p t e d M a n u s c r i p t 13 different levels of resistance to MLS B antibiotics. Since erythromycin even highly concentrated within phagocytes [START_REF] Miller | Erythromycin uptake and accumulation by human polymorphonuclear leukocytes and efficacy of erythromycin in killing ingested Legionella pneumophila[END_REF][START_REF] Prokesch | Antibiotic entry into human polymorphonuclear leukocytes[END_REF] has a lower antimicrobial activity compared with that of other antibiotics owing to its intracellular instability in the acid medium [START_REF] Cuffini | Comparative effects of roxithromycin and erythromycin on cellular immune functions in vitro. 3. Killing of intracellular Staphylococcus aureus by human macrophages[END_REF][START_REF] Cuffini | Comparative effects of roxithromycin and erythromycin on cellular immune functions in vitro. 1. Uptake of 3H-macrolides by human macrophages[END_REF], these current results suggest that the enhanced intra-PMN streptococcal killing detected is mainly attributable to PMN bactericidal systems that tightly cooperate with intracellular erythromycin in eradicating the ERY R streptococci. In fact, PMNs were able to synergise with erythromycin for the intracellular bacterial killing of highly resistant streptococci of phenotypes cMLS B, iMLS B A and iMLS B B. During the entire time of exposure, the internalised constitutive and inducible resistant streptococci were more susceptible to the microbicidal intracellular mechanisms of human PMNs compared with drug-free controls, where the phagocytes were unable to counteract bacterial growth at 60 min and 90 min of incubation (P < 0.01) (Table 2). In the presence of erythromycin, also M phenotype and iMLS B C phenotype streptococci, characterised by a moderate resistance level, were killed at a rate higher than that observed for highly resistant phenotypes, highlighting a pattern quite similar to that of the ERY S strain (Table 2).

Interestingly, a similar picture was detected even at lower levels of erythromycin (0.25 MIC) where the synergism between PMNs and erythromycin resulted in SI values that overlapped with those observed in presence of an erythromycin level equal to the MIC (Table 2), without affecting phagocytosis (data not shown). The hypothesis of a tight bactericidal co-operation between PMNs and erythromycin is corroborated from our previous studies [START_REF] Cuffini | Comparative effects of roxithromycin and erythromycin on cellular immune functions in vitro. 3. Killing of intracellular Staphylococcus aureus by human macrophages[END_REF][START_REF] Cuffini | Comparative effects of roxithromycin and erythromycin on cellular immune functions in vitro. 1. Uptake of 3H-macrolides by human macrophages[END_REF], where it emerged that pre-exposure of human phagocytes to erythromycin before phagocytosis had no effect on In conclusion, according to the recent literature trend, the current methods of predicting whether or not an antibiotic will be effective against a bacterial pathogen in vivo and determining the optimal usage of an antimicrobial are still crude. Until we take account of inherent host defences and how antibiotics react with pathogens at the site of infection, the way we choose an antimicrobial agent over another will continue to be flawed. Our results, along with those from clinical trial studies [START_REF] Rondini | Bacteriological and clinical efficacy of various antibiotics used in the treatment of streptococcal pharyngitis in Italy. An epidemiological study[END_REF],

provide further evidence of the clinical success of erythromycin and indirectly confirm that the antibiotic resistance detected in vitro does not always imply in vivo treatment failure inasmuch as favourable pharmacokinetic and pharmacodynamic parameters and the complex interactions among antibiotic-resistant pathogens, administered drug and host defences play a key role as predictors of clinical outcome.

A c c e p t e d M a n u s c r i p t 2 *

 2 Corresponding author. Present address: Department of Public Health and Microbiology, Microbiology Section, University of Turin, Via Santena 9, 10126 Turin, Italy. Tel.: +39 011 670 5638; fax: +39 011 236 5638.E-mail address: annamaria.cuffini@unito.it (A.M. Cuffini).

A c c e p t e d M a n u s c r i p t 3 ABSTRACT

 3 To evaluate the synergistic activity of erythromycin and human polymorphonuclear cells (PMNs) on the binomial erythromycin-resistant (ERY R ) Streptococcus pyogenes/host, the phagocytic and bactericidal activities of PMNs against ERY R

3 - 4

 34 McFarlandstandard and were inoculated into cryovials containing both cryopreservative fluid

  to allow bacteria to adhere (Microbank; bioMérieux, Rome, Italy).

  disks and growth around the erythromycin disk were taken to indicate the M phenotype. A blunting zone of inhibition of clindamycin and josamycin proximal to the erythromycin disk was taken to indicate inducible resistance (iMLS B phenotype). The iMLS B A strain was characterised by absence of any zone of inhibition around both erythromycin and josamycin disks; the iMLS B B strain was characterised by blunting of the josamycin zone of inhibition proximal to the erythromycin disk and by no zone of inhibition around the erythromycin disk; and the iMLS B C strain was characterised by blunting of the josamycin zone of inhibition proximal to the erythromycin disk with a restricted zone of inhibition around the erythromycin disk[START_REF] Giovanetti | A novel efflux system in inducibly erythromycin-resistant strains of Streptococcus pyogenes[END_REF].

  A c c e p t e d M a n u s c r i p t 8 Ficoll-Paque (Pharmacia S.p.A.) and was centrifuged twice at 1200 rpm for 15 min.

  ), which was calculated by adding the number of surviving microorganisms at time zero to the number of survivors at time x and dividing by the number of survivors at time zero[START_REF] Cuffini | Entry of sanfetrinem into human polymorphonuclear granulocytes and its cell-associated activity against intracellular, penicillin-resistant Streptococcus pneumoniae[END_REF][30][START_REF] Tullio | Role of fosfomycin tromethamine in modulating non-specific defence mechanisms in chronic uremic patients towards ESBL-producing Escherichia coli[END_REF].

  Similar data were obtained by evaluating the effects of lower levels of erythromycin (0.25 MIC) on either PMN phagocytosis of radiolabelled streptococci (data not shown) or intracellular killing by incubating the bacteria and phagocytes for periods of 30, 60 or 90 min (Table2).

A

  c c e p t e d M a n u s c r i p t 14 subsequent intracellular bacterial killing. Once highly accumulated within acid lysosomes, erythromycin, being a weak base, showed a reduced limited antibacterial activity.

A c c e p t e d M a n u s c r i p t 2 a

 2 Erythromycin resistance phenotypes include: constitutive (cMLS B ) and inducible (iMLS B ) resistance due to a methylase target site modification; and resistance due to active efflux (M phenotype).

A c c e p t e d M a n u s c r i p t 1

 1

Table 1

 1 Effect of erythromycin (ERY) at the MIC on human polymorphonuclear cell phagocytosis of erythromycin-sensitive (ERY S ) and erythromycin-resistant (ERY R )

	Streptococcus pyogenes determined by phagocytosis assay
	S. pyogenes phenotype	Time (min) Phagocytosis (mean ± S.E.M.) (%)
			Drug-free control	1 MIC ERY
	ERY S (MIC = 0.25 g/mL)	30	51.2 ± 3.7	45.9 ± 4.16
		60	42.5 ± 1.64	46.6 ± 4.05
		90	44.9 ± 1.39	45.3 ± 4.1
	ERY R a
	cMLS B (MIC = 128 g/mL)	30	41.8 ± 3.8	40.0 ± 1.25
		60	46.8 ± 4.2	45.2 ± 4.2
		90	47.4 ± 4.5	46.9 ± 4.3
	M (MIC = 16 g/mL)	30	58.9 ± 1.9	55.1 ± 4.1
		60	48.3 ± 4.4	44.6 ± 4.4
		90	37.1 ± 3.7	35.1 ± 3.4
	iMLS B A (MIC = 128 g/mL) 30	31.8 ± 1.8	29.9 ± 2.7
		60	33.2 ± 2.4	33.0 ± 2.0
		90	33.9 ± 3.3	33.4 ± 2.8
	iMLS B B (MIC = 128 g/mL) 30	47.9 ± 0.6	41.1 ± 3.8
		60	38.2 ± 1.4	39.0 ± 1.9
		90	36.2 ± 3.6	37.7 ± 0.9
	iMLS B C (MIC = 8 g/mL)	30	76.0 ± 7.6	73.6 ± 5.6
		60	72.3 ± 4.01	70.8 ± 6.11
		90	69.9 ± 0.9	76.2 ± 3.7
	MIC, minimum inhibitory concentration; S.E.M., standard error of the mean; MLS B ,
	macrolide-lincosamide-streptogramin B.

Table 2

 2 Effect of erythromycin (ERY) at the MIC and 0.25 MIC on human polymorphonuclear cell intracellular killing of erythromycin-sensitive (ERY S ) and erythromycin-resistant (ERY R ) Streptococcus pyogenes determined by intracellular

	killing assay
	S. pyogenes phenotype	Time	Survival index (mean ± S.E.M.)
		(min)	Drug-free	1MIC ERY 0.25 MIC
			control		ERY
	ERY S (MIC = 0.25	30	1.42 ± 0.09	1.18 ± 0.04	1.20 ± 0.02 *
	g/mL)			*
		60	>2	1.29 ± 0.06	1.26 ± 0.05 *
				*
		90	>2	1.30 ± 0.05	1.32 ± 0.04 *
				*
	ERY R a
	cMLS B (MIC = 128	30	1.76 ± 0.06	1.38 ± 0.11	1.37 ± 0.13 *
	g/mL)			*
		60	>2	1.75 ± 0.14	1.79 ± 0.10 *
				*
		90	>2	1.84 ± 0.17	1.81 ± 0.15 *
				*
	M (MIC = 16 g/mL)	30	1.69 ± 0.04	1.29 ± 0.10	1.32 ± 0.09 *
				*
		60	>2	1.32 ± 0.07	1.30 ± 0.05 *
				*
		90	>2	1.43 ± 0.12	1.42 ± 0.13 *
				*
	iMLS B A (MIC = 128	30	1.55 ± 0.11	1.45 ± 0.07 1.44 ± 0.04
	g/mL)	60	>2	1.67 ± 0.06	1.62 ± 0.08 *
				*

* Significantly different from the drug-free controls (P < 0.01).

a Erythromycin resistance phenotypes include: constitutive (cMLS B ) and inducible (iMLS B ) resistance due to a methylase target site modification; and resistance due to active efflux (M phenotype).

Acknowledgments

The authors are indebted to the staff of the Blood Center (A.O.U. San Giovanni Battista, Turin, Italy) for their technical assistance.

Funding

This work was supported by grants from Italian MUR (2005, 2007) and the Regione Piemonte Ricerca Sanitaria Finalizzata (2008 bis).

Competing interests

None declared.

Ethical approval

Not required.


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des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

13)

(1) Pôle pharmaceutique, University Hospital, Besançon, France An association between multiple sclerosis (MS) and IBD has been reproducibly reported and could occur more often than by chance alone [START_REF] Bernstein | The clustering of other chronic inflammatory diseases in inflammatory bowel disease: A population-based study[END_REF][START_REF] Gupta | Increased risk of demyelinating diseases in patients with inflammatory bowel disease[END_REF] . Geographical, case-control and prospective studies have suggested that sun exposure may protect from MS [START_REF] Van Der Mei | Regional variation in multiple sclerosis prevalence in Australia and its association with ambient ultraviolet radiation[END_REF][START_REF] Van Der Mei | Past exposure to sun, skin phenotype and risk of multiple sclerosis: case control study[END_REF][START_REF] Munger | Serum 25-hydroxyvitamin D levels and risk of multiple sclerosis[END_REF][START_REF] Munger | Vitamin D intake and incidence of multiple sclerosis[END_REF] . Therefore, we hypothesized that the association between MS and IBD is due to shared environmental factors, especially sun exposure, and thus we assessed the geographic distribution of CD and UC in relation to sunshine exposure.

MATERIALS AND METHODS

Setting

This study was conducted in metropolitan France (excluding Corsica and the overseas French territories) totalling 59,368,981 inhabitants on January 1, 2003. Its surface area is 535 280 km², with an average of 111 people per square kilometre. There are considerable variations, however, as half the population lives on just over 10% of the territory, and large areas remain sparsely populated.

Geographical units

This study was performed at the scale of "départements", i.e. the 94 administrative areas (subsequently called departments), as described previously [START_REF] Nerich | Geographical variations of inflammatory bowel disease in France: a study based on national health insurance data[END_REF] . In France, patients with IBD, whatever their place of residence, are fully reimbursed for their health care expenses by the national health insurance (NHI) system. The IBD data used for this study were obtained from the French statutory NHI system, which is based on professional status. Three main subsystems included in the NHI, cover 96% of the French population. The general fund insures salaried workers and their families in private secondary and tertiary industries and in the public sector (i.e. 84% of the population). The agricultural fund insures farmers and agricultural employees and their families (i.e. 7% of the population). The third fund provides medical coverage to selfemployed professionals and their families (i.e. 5% of the population). Unemployed and retired people remain affiliated to the last subsystem they were affiliated to when employed.

CD and UC are two of the 30 specified long-term illnesses which provide patients with a 100% reimbursement rate, thus covering the total cost of requested health care. Thus, all French IBD patients have equal access to free medical care. This full coverage by statutory health insurance is subject to prior NHI approval, after examination of an official form that certifies that the patient has CD or UC and that has been signed by the patient's physicians (general practitioner and/or gastroenterologist). Each health insurance fund records individual data on an annual basis for all patients who have obtained 100 percent coverage for each of the 30 specified long-term illnesses.

We contacted doctors in charge of national patient data files in the three insurance funds to request anonymous information concerning all new beneficiaries of 100 percent coverage for IBD granted between January 1, 2000 and December 31, 2002. We obtained centralized files covering the whole Ground radiometers are the most accurate method to measure sun exposure, but they are too few in number to offer a comprehensive geographical coverage. The surface UV radiation intensity can be obtained by combining modeling and satellite data from Meteosat ® , the European meteorological satellite. This method provides a comprehensive map of sunlight exposure with a precision of 5 km.

Correlation between measured and modelled erythemal daily doses was found to be highly significant [START_REF] Verdebout | A European satellite-derived UV climatology available for impact studies[END_REF] .

Statistical analysis

Incidence rates

The analysis was conducted separately for CD and UC. IBD patients were tabulated by sex, 5-year age classes and administrative area. The number of insured people in each area on January 1, 2003 was obtained from the health insurance funds and totalled to obtain the overall number of people affiliated with these funds within each area. For the population as a whole, incidence rates were calculated by sex for CD and UC, based on the number of all new beneficiaries of 100 percent coverage for CD or UC granted during the study period, with the person-years denominator being the sum of the insured population estimates for the 3 years in the three subsystems described above.

Influence of sun exposure on incidence rates

Ground UV dose from sunshine was expressed in kJ/m² and was provided by month and by administrative area for the years 1984 to 2002. We calculated a mean annual UV dose for the studied period, which was classified into quintiles, and linked with incidence data tabulated by area.

Age-and sex-adjusted incidence rate ratios (IRR), with their 95% confidence intervals (CI), were estimated using the Poisson regression, for CD and UC. We used the likelihood ratio statistics to calculate P-values for each variable.

Spatial analysis and disease mapping

A spatial analysis with a Bayesian approach was then performed, as described previously [START_REF] Nerich | Geographical variations of inflammatory bowel disease in France: a study based on national health insurance data[END_REF] . Sun exposure was introduced as a covariate into this model and tested for improvement of the model fit, We mapped the 94 administrative areas grouped into 3 classes according to the value and significance of smoothed RR: administrative areas with smoothed RRs significantly lower than 1, administrative areas with smoothed RRs not significantly different from 1, and administrative areas with smoothed RRs significantly higher than 1.

RESULTS

From

Crohn's disease

In the population as a whole, the incidence rate was 8.2 for 100,000 inhabitants (95% CI: 8.1-8.4) for the whole population, 7.1 (95% CI: 6.9-7.2) for men, and 9.4 (95% CI: 9.2-9.6) for women with no important variation between the three studied years. Table 1 shows the age and sex adjusted incidence rate ratios and 95% confidence intervals derived from the Poisson regression analysis for CD, according to quintiles of sun exposure, and taking as the reference the quintile of highest sun exposure. The two quintiles with the lowest sun exposure were associated with an increased IRR of

Ulcerative colitis

In the population as a whole, the incidence rate was 7.2 for 100,000 inhabitants (95% CI: 7.1-7.3), it was 7.7 (95% CI: 7.5-7.9) in men and 6.8 (95% CI: 6.6-7.0) in women. Table 2 shows the age and sex adjusted incidence rate ratios and 95% confidence intervals derived from the Poisson regression analysis for UC, according to quintiles of sun exposure. There was no statistically significant association between sun exposure and UC incidence (cf. Support Information).

DISCUSSION

The present study assessed the geographic distribution of IBD in relation to sun exposure. It appears that CD incidence is higher in geographical areas with a low sunshine exposure than in areas with high or medium sunshine exposure, where CD incidence is either lower than or not significantly different from the mean national incidence rate. Thus, there seems to be a threshold of sun exposure under which CD incidence increases. By contrast, no association between UC incidence and sun exposure was found.

Our study is based on data derived from administrative health care records and relies on ecological data. These limitations have already been discussed in our previous papers [START_REF] Nerich | Geographical variations of inflammatory bowel disease in France: a study based on national health insurance data[END_REF][START_REF] Nerich | Fine-scale geographic variations of inflammatory bowel disease in France: correlation with socioeconomic and house equipment variables[END_REF] . Moreover, in a recent paper concerning geographic mapping of IBD, Green and Bernstein suggest to use small geographic areas in order to minimize the misclassification of populations by risk categories [START_REF] Green | Lessons from geographic mapping in France[END_REF] .

However, the sun exposure data are available at the scale of "departments" and not at a smaller scale. Additionally, the association between sun exposure and CD is not necessarily causal. It can be due to an underlying genetic or environmental factor which has a geographical distribution similar to that of sun exposure, such as chemicals, water or soil microbial ecology and diet. In a recent prospective study, high total protein intake, specifically animal protein, was associated with a significantly increased risk of IBD in middle-aged French women 13. It could be questioned whether this could confound the geographical variation of CD. However, the findings for protein intake were similar for CD and UC, unlike the present association between CD and sun exposure. If a role of vitamin D is suspected, the role of dietary vitamin D could also be questioned. It must be emphasized that in France, very few dairy products are fortified with vitamin D and about 90% of vitamin D is related to sun exposure. Thus a potential opposite effect of specific foods such as dairies through proteins on one hand and vitamin D on the other is most likely very limited.

On the other hand, the association between CD and sun exposure could also be causal, as suggested by several epidemiological and biological data. Firstly, there is a reproducible North-South gradient of CD incidence which has been found in the USA, Scotland and France [START_REF] Nerich | Geographical variations of inflammatory bowel disease in France: a study based on national health insurance data[END_REF][START_REF] Sonnenberg | Geographic variation of inflammatory bowel disease within the United States[END_REF][START_REF] Armitage | Incidence of juvenile-onset Crohn's disease in Scotland: association with northern latitude and affluence[END_REF] . Sun exposure might well explain the North-South gradient found in these countries. Secondly, the geographic distribution of multiple sclerosis (MS) appears to be similar to that of CD in France and several lines of evidence suggest that sun exposure protects from MS [START_REF] Vukusic | Regional variations in the prevalence of multiple sclerosis in French farmers[END_REF] . A strong inverse correlation between UV radiation and MS prevalence has been found in a geographical study performed in Australia [START_REF] Van Der Mei | Regional variation in multiple sclerosis prevalence in Australia and its association with ambient ultraviolet radiation[END_REF] . A case-control study has shown that exposure to the sun in infancy may protect against multiple sclerosis [START_REF] Van Der Mei | Past exposure to sun, skin phenotype and risk of multiple sclerosis: case control study[END_REF] . It has also been shown that people with the highest blood levels of vitamin D had the lowest risks of MS [START_REF] Munger | Serum 25-hydroxyvitamin D levels and risk of multiple sclerosis[END_REF] . Furthermore, results from a cohort study of 187,500 nurses showed that women who consumed at least the current daily recommended dose of vitamin D have a lower risk of developing MS [START_REF] Munger | Vitamin D intake and incidence of multiple sclerosis[END_REF] . Exposure to the sun may explain the French North-South gradient of MS and CD. Thirdly, there is a biological plausibility for our findings [START_REF] Peyrin-Biroulet | Crohn's disease: the hot hypothesis[END_REF] . Sun and vitamin D have been found to act as immune regulators. UV radiation and vitamin D have been shown to induce regulatory T cell function [START_REF] Maeda | Phenotypic and functional characterization of ultraviolet radiation-induced regulatory T cells[END_REF] . UV radiation induces immune suppression via induction of IL-4 and IL-10 and suppression of IL-12 production [START_REF] Ullrich | Sunlight and skin cancer: lessons from the immune system[END_REF] . 1,25-dihydroxyvitamin D inhibits dendritic cell differentiation and maturation, which induces a shift in T cell differentiation from Th1 and Th17 to a Th2 phenotype [START_REF] Baeke | Vitamin D signaling in immunemediated disorders: evolving insights and therapeutic opportunities[END_REF] . Recent studies have found that vitamin D is also a key regulatory factor of innate immunity. Vitamin D, produced in response to UV radiation, stimulates the production of [START_REF] Liu | Toll-Like Receptor Triggering of a Vitamin D-Mediated Human Antimicrobial Response[END_REF] . More recently, a study performed in primary cultures and lines of human monocytic and epithelial intestinal cells has found that 1,25-dihydroxyvitamin D stimulates the expression of NOD2 in the presence of its agonist muramyl dipeptide [START_REF] Wang | Direct and indirect induction by 1,25dihydroxyvitamin D3 of the NOD2/CARD15-defensin beta2 innate immune pathway defective in Crohn disease[END_REF] . Taken together, these results suggest that low sunshine exposure and vitamin D deficiency impair innate immunity and induce a shift in T cell differentiation towards TH1/TH17 phenotype, which are both characteristic of CD.

Our study suggests that low sunlight exposure is associated with higher CD risk. This potential association should be further investigated within case-control and cohort studies. If a protective role of sun exposure on CD risk is confirmed, prevention of CD through vitamin D supplementation should be tested in high-risk subjects. Vitamin D supplementation may also reduce the risk of relapse in patients with CD, as shown by a recent randomized trial [START_REF] Jørgensen | Clinical trial: vitamin D3 treatment in Crohn's disease-a randomized double-blind placebo-controlled study[END_REF] .

F

( 2 )

 2 INSERM U645 EA-2284 IFR-133, School of Medicine and Pharmacy, Besançon, France (3) Service de Pédiatrie, University Hospital, Besançon, France (4) INSERM, UMRS 1018, Team 9, Centre for Research in Epidemiology and Population Health, Université Paris Sud, Institut Gustave Roussy, 94805. Villejuif, France (5) Département de Santé Publique, University Hospital, Besançon, France (6) UPRES EA 4266 Agents Pathogènes et Inflammation, Université de Franche Comté, Besançon, France (7) Caisse Nationale d'Assurance Maladie des Travailleurs salariés (CNAMTS), Paris, France (8) Caisse Centrale de la Mutualité Sociale Agricole (CCMSA), Bagnolet, France (9) Caisse Nationale d'Assurance Maladie des Professions indépendantes (CANAM), Saint Denis (10) Caisse Nationale Militaire de Sécurité Sociale (CNMSS), Paris, France (11) Etablissement National des Invalides de la Marine (ENIM), Paris, France (12) Laboratoire Espace Santé et Territoire, Université Paris X, Paris, France (13) Service d'Hépatogastroentérologie, University Hospital of Bicêtre, Assistance Publique Hôpitaux de Paris, Université Paris Sud, Le Kremlin Bicêtre, France disease (CD) and ulcerative colitis (UC) are chronic inflammatory disorders of the gastrointestinal tract. Genetic predisposition of inflammatory bowel disease (IBD) is established, particularly in CD, in which several susceptibility genes that affect bacterial clearance (including autophagy), the IL23 pathway, and T cell homeostasis have been identified 1 . Environmental factors also contribute to IBD pathogenesis 2 . Smoking, appendectomy and intestinal infections have been reproducibly linked with susceptibility to IBD 2 . It is likely that other environmental factors increase IBD risk 2 . These environmental factors could possibly be found through the association between IBD and other diseases for which environmental factors are already known.

  French territory. They contained the following individual information: age, sex, postcode of the place of residence, diagnosis that conferred the 100 percent coverage, coded either CD (ICD code K50) or UC (ICD code K51) with the tenth revision of the International Classification of Diseases (ICD). This study was approved by the "Commission Nationale de l'Informatique et des Libertés" (CNIL, Paris, France), a national commission which provides legal protection of computerized personal data.

  for age and sex. The model provides smoothed Relative Ratios (RR) which are more accurate and steady than the traditional standardized incidence ratios.

  January 1, 2000 to December 31, 2002, 14,113 new patients with CD (including 8,138 women and 5,975 men) and 12,339 new patients with UC (including 5,959 women and 6,380 men) were registered by the three health insurance funds, among the overall French insured population of 57,098,525. The map of France showing the average annual erythemal daily dose (kJ/m²) by administrative area is displayed in figure 1.

  CD. The intermediate quintile had an IRR not significantly different from 1. The second quintile with the highest sun exposure was associated with a decreased IRR of CD.Local UV dose improved the Bayesian model of geographic distribution of CD, and the effect of local UV dose was stronger than the effect of local spatial autocorrelation. Variations of smoothed RR of CD in both sexes according to sunlight exposure are displayed in Figure2. Areas with a higher than 1 corresponded to areas with low sunlight exposure (North of France), whereas those with high or medium sunlight exposure had smoothed RRs either lowers than 1 (centre of France) or not significantly different from 1 (figure2).

  can kill various viruses and bacteria, including the mycobacteria

15 FIGURESFigure 1 : 16 Figure 2 :

 151162 FIGURES

Table 1 :

 1 Age-and sex-adjusted incidence rate ratios (IRR) and 95% confidence intervals (CI) derived from Poisson regression analysis for Crohn's Disease according to the average annually

	o r
	P
	e
	e r
	R
	e
	v i e
	w

Table 2 :

 2 Age-and sex-adjusted incidence rate ratios (IRR) and 95% confidence intervals (CI) derived from Poisson regression analysis for Ulcerative Colitis according to the average annually

	erythemal daily dose
			Ulcerative colitis
		IRR	95% CI p-value
	Sex		0.01
	Female	Reference
	Male	1.13 1.03 -1.25
	Age (years)		< 10 -4
	25 -49	Reference
	0 -24	0.26	0.22 -0.30
	50 -74	0.69	0.62 -0.77
	≥ 75	0.46	0.37 -0.57
	Average erythemal daily dose (kJ/m²)		0.25
	1.729 -1.928	Reference
	1.606 -1.725	0.86	0.73 -1.01
	1.488 -1.585	0.85	0.71 -1.01
	1.397 -1.484	0.87	0.75 -1.01
	1.272 -1.396	0.93	0.81 -1.07


Ai-Hua Fan
email: ai-hua.fan@u-picardie.fr

J Örg Schmeling

Meng Wu
email: meng.wu@u-picardie.fr

MULTIFRACTAL ANALYSIS OF MULTIPLE ERGODIC AVERAGES

Keywords: 2010 Mathematics Subject Classification. Primary 37C45, 42A55; Secondary 37A25, 37D35 Multiple ergodic averages, Hausdorff dimension, non-linear transfert operator

In this paper we present a complete solution to the problem of multifractal analysis of multiple ergodic averages in the case of symbolic dynamics for functions of two variables depending on the first coordinate.

Introduction and results

Let T : X → X be a continuous map on a compact metric space X. Let f 1 , • • • , f ( ≥ 2) be real valued continuous functions defined on X. We consider the following possible limits (for different x ∈ X):

(1)

M f1,••• ,f (x) = lim n→∞ 1 n n k=1 f 1 (T k x)f 2 (T 2k x) • • • f (T k x).
Such limits are widely studied in ergodic theory. It was proposed in [START_REF] Fan | Level sets of multiple ergodic averages[END_REF] to give a multifractal analysis of the multiple ergodic average M f1,••• ,f (x). The authors of [START_REF] Fan | Level sets of multiple ergodic averages[END_REF] succeeded in a very special case where X = {-1, 1} N , f k (x) = x 1 for all k and T is the shift, by using Riesz products. In this note, we shall study the shift map T on the symbolic space X = Σ m = S N with S = {0, 1, • • • , m -1} (m ≥ 2). We assume that = 2 (the case ≥ 3 seems more difficult) and f 1 and f 2 are Hölder continuous. We endow Σ m with the standard metric: d(x, y) = m -n where n is the largest k ≥ 0 such that

x 1 = y 1 , • • • , x k = y k .
The Hausdorff dimension of a set A in Σ m will be denoted by dim A.

For any α ∈ R, define

L(α) = {x ∈ Σ m : M f1,f2 (x) = α}.
Let α min = min x,y∈Σm f 1 (x)f 2 (y) and α max = max x,y∈Σm f 1 (x)f 2 (y). Our question is to determine the Hausdorff dimension of L(α). We further assume that α min < α max (otherwise both f 1 and f 2 are constant and the problem is trivial). From classical dynamical system point of view, the set L(α) is not standard and its dimension can not be described by invariant measures supported on it. Let us first examine the largest dimension of ergodic measures supported on the set L(α) by introducing the so-called invariant spectrum:

F inv (α) = sup {dim µ : µ ergodic, µ(L(α)) = 1 } .
Recall that (see [START_REF] Wu | Fan Sur les dimension de measure[END_REF])

dim µ = inf{dim B : B Borel set, µ(B c ) = 0}.
The dimension F inv (α) is in general smaller than dim L(α) (compare the next two theorems). It is even possible that no ergodic measure is supported on L(α).

Theorem 1.1. Let f 1 and f 2 be two Hölder continuous functions on Σ m . If L(α) supports an ergodic measure, then

F inv (α) = sup dim µ : µ ergodic, f 1 dµ f 2 dµ = α .
It is known [START_REF] Fan | Peyrière Generic points in systems of specification and Banach valued Birkhff ergodic average[END_REF] that the above supremum is the dimension of the set of points x such that

lim n→∞ 1 n n k=1 f 1 (T k x) • lim n→∞ 1 n n k=1 f 2 (T k x) = α.
Assume that f 1 and f 2 are the same function f . As a corollary of Theorem 1.1, µ(L(α)) = 1 for some ergodic measure µ implies α ≥ 0. So, if f takes a negative value α < 0, then Theorem 1 shows that there is no ergodic measure supported on L(α). However, Theorem 2 shows that dim L(α) > 0.

In the following we assume that both f 1 and f 2 depend only on the first coordinate. For any s ∈ R, consider the non-linear transfer equation

(2) t s (x) 2 =
T y=x e sf1(x)f2(y) t s (y).

It can be proved that the equation admits a unique solution t s : Σ m → R + , which depends only on the first coordinate. Let dx denote the measure of maximal entropy for the shift on Σ m and let

P (s) = log Σm t s (x)dx + log m.
Also it can be proved that P is an analytic convex function and even strictly convex when α min < α max (Lemma 3.1).

Theorem 1.2. Let f 1 and f 2 be two functions on Σ m depending only on the first coordinate. For α ∈ [P (-∞), P (+∞)], we have

L(α) = ∅. For α ∈ [P (-∞), P (+∞)], we have dim L(α) = 1 2 log m (P (s α ) -s α P (s α ))
where s α is the unique solution of P (s) = α.

We can prove that α min ≤ P (-∞) ≤ P (+∞) ≤ α max and that α min = P (-∞) if and only if there exist i

0 , i 1 , • • • , i ∈ S ( ≥ 1) with i 0 = i such that f 1 (i k )f 2 (i k+1 ) = α min (similar criterion for α max = P (+∞)).
Let us look at two examples on Σ 2 . For 1 for the graphs of dim L(α) and F inv (α). Remark that F inv (α) = 0 but dim L(α) > 0 for -1 ≤ α < 0. Also remark that dim L(α) was computed in [START_REF] Fan | Level sets of multiple ergodic averages[END_REF] by using Riesz products. See Figure 2 for the graphs of dim L(α) and F inv (α) when f 1 (x) = f 2 (x) = x 1 . In the second case F inv (α) = H( √ α) and dim L(α) can be numerically computed through P (s) = 2 log t 0 (s) where x = t 0 (s) is the real solution of the third order algebraic equation

f 1 (x) = f 2 (x) = 2x 1 -1, we have dim L(α) = 1 2 + 1 2 H 1 + α 2 , F inv (α) = H 1 + √ α 2 where H(x) = -x log 2 x -(1 -x) log 2 (1 -x). See Figure
x 3 -2x 2 -(e s -1)
x + (e s -1) = 0.

These two examples show that F inv (α) < dim L(α) except for some special α's.

The proof of Theorem 1.2 is based on the following observation. If f 1 and f 2 depend only on the first coordinate

x 1 , k f 1 (T k x)f 2 (T 2k
) can be decomposed into the sum of j f 1 (T i2 j x)f 2 (T i2 j+1 x) with odd i, which have independent coordinates. This observation was used in [START_REF] Fan | Level sets of multiple ergodic averages[END_REF] to compute the box dimension of X 0 = {x : ∀n, x n x 2n = 0} which is a subset of L(0) (here f 1 (x) = f 2 (x) = x 1 is considered). The Hausdorff dimension of X 0 was later computed in [START_REF] Kenyon | Hausdorff dimension for fractals invariant under the multiplicative integers[END_REF] where a non-linear transfer operator characterizes the measure of maximal Hausdorff dimension for X 0 .

We have stated the results for functions of the form f 1 (x 1 )f 2 (y 1 ) (product of two functions depending on the first coordinate). But the results with obvious modifications hold for functions of the form f (x 1 , y 1 ).

Proof of Theorem 1.1

Let µ be an ergodic measure such that µ(L(α)) = 1. Then

α = lim n→∞ 1 n n k=1 E µ [f 1 (T k x)f 2 (T 2k x)] = lim n→∞ 1 n n k=1 E µ [f 1 (x)f 2 (T k x)] = E µ [f 1 (x)M f2 (x)].
(The first and third equalities are due to Lebesgue convergence theorem and the second one is due to the invariance of µ). Since µ is ergodic,

M f2 (x) = E µ f 2 for µ-a.e. x. So, α = E µ f 1 E µ f 2 . It follows that F inv (α) ≤ sup {dim µ : µ ergodic, E µ f 1 E µ f 2 = α } .
To obtain the inverse inequality, it suffices to observe that the above supremum is attained by a Gibbs measure ν which is mixing and that the mixing property implies M f1,f2 (x) = E ν f 1 E ν f 2 ν-a.e..

Proof of Theorem 1.2

We will prove a result which is a bit more general than Theorem 1.2. Our proof is sketchy and a full proof is contained in [START_REF] Fan | Multiple ergodic averages and nonlinear transfer operaters[END_REF] where other generalizations are also considered.

Here is the setting. Let ϕ : S × S → R be a non constant function with minimal value α min and maximal value α max . For α ∈ R, define

E(α) = x ∈ Σ m : lim n→∞ n -1 n k=1 ϕ(x k , x 2k ) = α .
Lemma 3.1. For any s ∈ R, the system

t 2 i = m-1 j=0 e sϕ(i,j) t j (i = 0, 1, • • • , m -1)
admits a unique solution (t 0 (s), t 1 (s), • • • , t m-1 (s)) with strictly positive components, which is an analytic function of s. The function

P (s) = log m-1 j=0 t j (s)
is strictly convex.

The proof of the lemma is lengthy. The existence and uniqueness of the solution are based on the fact that the square roots of right members of the system define an increasing operator on a suitable compact hypercube. The analyticity of the solution is a consequence of the implicit function theorem. Theorem 3.1. For any α ∈ [P (-∞), P (+∞)], we have

E(α) = 1 2 log m (P (s α ) -s α P (s α ))
where s α is the unique solution of P (s) = α.

The solution (t 0 (s), t 1 (s), • • • , t m-1 (s)) of the above system allows us to define a Markov measure µ s with initial probability (π(i)) i∈S and probability transition matrix (p i,j ) S×S defined by

π(i) = t i (s) t 0 (s) + t i (s) + • • • + t m-1 (s) , p i,j = e sϕ(i,j) t j (s) t i (s) 2 .
Now decompose the set of positive integers N * into Λ i (i being odd) with Λ i = {i2 k } k≥0 so that Σ m = i:2 |i S Λi . Take a copy µ s on each S Λi and then define the product measure of these copies. This gives a probability measure P s on Σ m . Let D(P s , x) be the lower local dimension of P s at x. where s α is the solution of P (s) = α. From the lemma, we can deduce that L(α) = ∅ if α ∈ [P (-∞), P (+∞)]. In order to get the inverse inequality, we only have to show that P sα is supported on L(α). We first prove the following law of large numbers by showing the exponential correlation decay of (F n ) under P s .

Lemma 3.3. Let (F n ) be a sequence of functions defined on S × S such that sup n sup x,y |F n (x, y)| < ∞. For P s -a.e. x ∈ Σ m , we have

lim n→∞ 1 n n k=1 (F k (x k , x 2k ) -E Ps F k (x k , x 2k )) = 0.
Applying the above lemma to F n (x n , x 2n ) = ϕ(x n , x 2n ) for all n and computing E Ps ϕ(x n , x 2n ), we get Thus we finished the proof for α ∈ (P (-∞), P (+∞)). If α = P (-∞) (resp. P (+∞)), as in the standard multifractal analysis, we use the probabilities P s et let s tend to -∞ (resp. +∞).

Figure 1 .

 1 Figure 1. When f1(x) = f2(x) = 2x1 -1.

Figure 2 .

 2 Figure 2. When f1(x) = f2(x) = x1.

Lemma 3 . 2 . 1 2

 321 For any x ∈ E(α), we have D(P s , x) ≤ 1 2 log m [P (s) -αs]. It follows that dim E(α) ≤ log m [P (s) -αs]. Minimizing the right hand side gives rise to dim E(α) ≤ 1 2 log m [P (s α ) -αs α ]

Lemma 3 . 4 .

 34 For P s -a.e. x ∈ Σ m , we havelim n→∞ 1 n n k=1 ϕ(x k , x 2k ) = P (s).

Acknowledgement The author would like to thank B. Solomyak for his careful reading of the note and for the information that the Hausdorff dimension of L(α) is computed in a different way for the special case m = 2, f 1 (x) = f 2 (x) = x 1 in [6].