Heike Jacobi
email: heike.jacobi@med.uni-heidelberg.de

Tamara Schaprian

Jan Beyersmann

Sophie Tezenas Du Montcel

Matthias Schmid

Thomas Klockgether

Risca Study Groups

Evolution of disability in spinocerebellar ataxias type 1, 2, 3, and 6

Objective: The aim was to study the evolution of disability in spinocerebellar ataxias (SCAs) type 1, 2, 3, and 6 (SCA1, 2, 3, 6). Methods: We analyzed data of two longitudinal cohorts (RISCA, EUROSCA) which recruited ataxic and non-ataxic SCA1, SCA2, SCA3, and SCA6 mutation carriers. To study disability, we used a five-stage system for ataxia defined by walking ability (stages 0-3) and death (stage 4). Transitions were analyzed using a multi-state model with proportional transition hazards. Based on the hazard estimates, transition probabilities and the expected lengths of stay in each stage were calculated. We further studied the effect of sex and CAG repeat length on progression. Results: Data of 3138 visits in 677 participants were analyzed. Median SARA scores for SCA1, SCA2, SCA3, and SCA6 ranged from 1.5 (interquartile range [IQR] = 0.0-3.5) to 3.5 (IQR = 1.4-6.1) in stage 0, 11.5 (IQR = 9.6-14.0) to 13.8 (IQR = 11.0-16.0) in stage 1, 19.0 (IQR = 17.0-21.0) to 23.8 (IQR = 19.5-27.0) in stage 2, and 28.5 (IQR = 26.0-32.5) to 34.0 (IQR = 32.6-37.1) in stage 3. Modeling allowed to calculate the subtype-specific probability to be in a certain stage at a given age and duration of each stage. CAG repeat length was associated with faster progression in SCA1 (HR, 95% CI: 1.1, 1.1-1.2), SCA2 (1.2, 1.1-1.3), and SCA3 (1.1, 1.0-1.2). In SCA6, female sex was associated with faster progression (1.7, 1.1-2.6). Interpretation: Our data are important for counselling of patients, assessment of the relevance of outcome markers, and design of clinical trials.

Introduction

The spinocerebellar ataxias (SCAs) are a genetically heterogeneous group of autosomal dominantly inherited progressive ataxia disorders. The most common, SCA1, SCA2, SCA3, and SCA6, are due to translated CAG repeat expansions that encode elongated polyglutamine tracts within the proteins associated with each subtype. SCA1, SCA2, and SCA3 mutation carriers typically develop ataxia in the fourth decade of life, whereas the onset of ataxia in SCA6 is about 20 years later. [START_REF] Klockgether | Spinocerebellar ataxia[END_REF] Survival after ataxia onset has been reported to be about 20 years. [START_REF] Monin | Survival and severity in dominant cerebellar ataxias[END_REF][START_REF] Kieling | Survival estimates for patients with Machado-Joseph disease (SCA3)[END_REF] A number of cohort studies assessed the progression rates of SCA1, SCA2, SCA3, and SCA6 patients by repeated administration of clinical scales. The majority of these studies used the Scale for the Assessment and Rating of Ataxia (SARA), [START_REF] Jacobi | Long-term disease progression in spinocerebellar ataxia types 1, 2, 3, and 6: a longitudinal cohort study[END_REF][START_REF] Ashizawa | Clinical characteristics of patients with spinocerebellar ataxias 1, 2, 3 and 6 in the US; a prospective observational study[END_REF][START_REF] Lee | Comparison of cerebellar ataxias: a three-year prospective longitudinal assessment[END_REF][START_REF] Yasui | A 3-year cohort study of the natural history of spinocerebellar ataxia type 6 in Japan[END_REF] one of the International Cooperative Ataxia Rating Scale, [START_REF] Franc ßa Mc | Progression of ataxia in patients with Machado-Joseph disease[END_REF] and another the neurological examination score for spinocerebellar ataxia (NESSCA). [START_REF] Jardim | Progression rate of neurological deficits in a 10-year cohort of SCA3 patients[END_REF] The maximal observation times of most of these studies ranged between 1 and 3 years. EUROSCA, a European longitudinal cohort study of 526 SCA1, SCA2, SCA3, and SCA6 patients had the longest follow-up with a median observation time of 49 months. [START_REF] Jacobi | Long-term disease progression in spinocerebellar ataxia types 1, 2, 3, and 6: a longitudinal cohort study[END_REF] The RISCA study followed 252 risk persons for SCA1, SCA2, SCA3, and SCA6 for a median period of 53 months and observed transition to manifest ataxia in 52% of the SCA1, 59% of the SCA2, 42% of the SCA3, and 13% of SCA6 mutation carriers. [START_REF] Jacobi | Biological and clinical characteristics of individuals at risk for spinocerebellar ataxia types 1, 2, 3, and 6 in the longitudinal RISCA study: analysis of baseline data[END_REF][START_REF] Jacobi | Conversion of individuals at risk for spinocerebellar ataxia types 1, 2, 3, and 6 to manifest ataxia (RISCA): a longitudinal cohort study[END_REF] These studies provided important quantitative information on the progression and on the transition of pre-ataxic mutation carriers to manifest ataxia. However, all studies covered only a small part of the entire disease span of SCA1, SCA2, SCA3, and SCA6. In all studies, progression rates were calculated as average values of the entire study population, although study participants entered the cohorts at different times of disease development. Another shortcoming of these studies is the reliance on clinical scales, although these scales may not always adequately reflect the consequences of ataxia on the daily life of affected individuals. [START_REF] Schmahmann | Development and validation of a patient-reported outcome measure of ataxia[END_REF] An alternative approach to assess disease progression in ataxia and its impact on patients´daily life is the use of staging systems defined by walking disability. To this end, a five-stage system for ataxia including four stages related to walking ability and a final stage defined by death was proposed. This staging system was applied in a retrospective survey of 466 ataxia patients including 286 patients with an autosomal dominantly inherited ataxia that revealed disease-specific differences of progression rates and identified CAG repeat length as risk factor for faster progression in SCA2 and SCA3. [START_REF] Klockgether | The natural history of degenerative ataxia: a retrospective study in 466 patients[END_REF] In the present study, we applied a multi-state model to analyze longitudinally acquired walking disability data of 677 SCA1, SCA2, SCA3, and SCA6 mutation carriers from the RISCA and EUROSCA cohorts. Our aim was to (1) better understand the relation between disability and clinical measures, (2) define the evolution of walking disability over the entire disease span, and (3) identify factors that influence transition into higher disability stages.

Methods

Participants

We analyzed longitudinal data of pre-ataxic and ataxic SCA 1, 2, 3, and 6 mutation carriers from the RISCA and EUROSCA cohorts. The RISCA study enrolled non-ataxic children or siblings of patients aged 18-50 for relatives of patients with SCA1, SCA2, or SCA3 and aged 35-70 years for relatives of patients with SCA6. [START_REF] Jacobi | Biological and clinical characteristics of individuals at risk for spinocerebellar ataxia types 1, 2, 3, and 6 in the longitudinal RISCA study: analysis of baseline data[END_REF][START_REF] Jacobi | Conversion of individuals at risk for spinocerebellar ataxia types 1, 2, 3, and 6 to manifest ataxia (RISCA): a longitudinal cohort study[END_REF] In the current analysis, only mutation carriers were included. EUROSCA enrolled ataxic SCA 1, 2, 3, and 6 mutation carriers aged 18 years or older. [START_REF] Jacobi | Long-term disease progression in spinocerebellar ataxia types 1, 2, 3, and 6: a longitudinal cohort study[END_REF][START_REF] Diallo | Prediction of survival with long-term disease progression in Most common spinocerebellar ataxia[END_REF] The data of all participants with at least one follow-up visit were considered.

To study the evolution of walking disability, we used the following staging system: stage 0 (no gait difficulties), stage 1 (gait difficulties), stage 2 (loss of independent gait, as defined by permanent use of a walking aid or reliance on a supporting arm), stage 3 (confinement to wheelchair), and stage 4 (death). [START_REF] Klockgether | The natural history of degenerative ataxia: a retrospective study in 466 patients[END_REF] At baseline, all RISCA participants were in stage 0 by definition, but in the course of the longitudinal study, a part of them converted to manifest ataxia. [START_REF] Jacobi | Conversion of individuals at risk for spinocerebellar ataxia types 1, 2, 3, and 6 to manifest ataxia (RISCA): a longitudinal cohort study[END_REF] As walking disability stages were not formally assessed in the RISCA study, we used the rating of the gait item (item 1) of SARA to infer walking disability, as follows: 0-1 points: stage 0; 2-4 points: stage 1; 5-7 points: stage 2; > 8 points: stage 3.

Clinical assessments

The EUROSCA and RISCA protocol included the following clinical assessments. SARA was used to assess the presence and severity of ataxia. [START_REF] Schmitz-Hubsch | Scale for the assessment and rating of ataxia: development of a new clinical scale[END_REF] The SARA sum score ranges from 0 to 40 with 0 indicating absence of ataxia and 40 the most severe degree of ataxia. In addition, a performance-based coordination test, the SCA Functional Index (SCAFI) was applied. [START_REF] Schmitz | SCA functional index: a useful compound performance measure for spinocerebellar ataxia[END_REF] SCAFI consists of three subtests: 8 m walk (8MW), 9-hole peg test (9HPT), and PATA rate (PATA). Each subtest is converted into a Zscore, the SCAFI score is generated as the arithmetic mean of all three Z-scores. The assessment of neurological signs other than ataxia was done using the Inventory of non-ataxia signs (INAS). [START_REF] Jacobi | Inventory of nonataxia signs (INAS): validation of a new clinical assessment instrument[END_REF] The INAS is a list of 30 items that are grouped into 16 non-ataxia signs, with the presence or absence of these signs resulting in a score of either 1 or 0. The INAS count denotes the number of non-ataxia signs with a range between 0 and 16. Moreover, patient-related outcome measures were assessed. As a measure of depressive symptoms, we applied the Patient's Health Questionnaire (PHQ-9). [START_REF] Spitzer | Validation and utility of a self-report version of PRIME-MD: the PHQ primary care study. Primary care evaluation of mental disorders. Patient health questionnaire[END_REF] The PHQ-9 sum score ranges from 0 (absence of depression) to 27 (severe depression). Health-related quality of life was assessed using the EQ-5D visual analogue scale (EQ-5D). [START_REF]EuroQol-a new facility for the measurement of healthrelated quality of life. The EuroQol Group[END_REF] For calculation, the patient is asked to mark the present day's health status on a scale with end points of 0 and 100, indicating the "the worst" (0) and "the best health you can imagine" (100).

Data analysis and modeling

All analyzes were performed using the R Software for Statistical Computing, version 4.0.4. [START_REF] Core | R: A Language and Environment for Statistical Computing[END_REF] To describe the relation between disability stages and clinical assessments, we calculated the median and interquartile range (IQR) of each measure of all participants who were in a certain stage at any time. For those who had multiple visits in one disease stage, the median values of the clinical assessments were used. We used the Kruskal-Wallis Rank Sum test for comparison of clinical assessments (SARA, SCAFI, INAS, PHQ-9, and EQ-D5) across disability stages in a nonconfirmatory fashion, followed by post hoc procedure with Dunn's tests including Bonferroni correction to find differences in clinical assessments between the disability stages for each SCA. We did not make any further corrections to adjust p-values for multiple comparisons across score variables. The observational units in these tests refer to stage occupations. It is assumed that the distribution of the test statistic is not affected by possible dependencies occurring in persons occupying more than one stage.

Transitions between disability stages were analyzed using a multi-state modeling approach with proportional transition hazards. [START_REF] Putter | Tutorial in biostatistics: competing risks and multi-state models[END_REF][START_REF] De Wreede | The mstate package for estimation and prediction in non-and semiparametric multi-state and competing risks models[END_REF][START_REF] De Wreede | mstate: an R package for the analysis of competing risks and multi-state models[END_REF] Multi-state models describe how an individual moves between a series of states in continuous time. Each movement--referred to as transition-from one state to another state is characterized by a (possibly covariate-dependent) transition hazard describing the dynamics of the transition process between the two states. Here, we specified a separate multi-state model for each of the SCA subtypes. All models were defined by five states (stages 0-4). Age was used as time scale, and several sets of covariates were considered. Starting with a covariate-free model, we incorporated repeat length of expanded and normal allele and sex as covariates. As the exact age at conversion was not observed, we computed the time of conversion as the midpoint of the time interval defined by the visits before and after conversion. This adjustment was in line with earlier work on ataxia progression. [START_REF] Jacobi | Conversion of individuals at risk for spinocerebellar ataxia types 1, 2, 3, and 6 to manifest ataxia (RISCA): a longitudinal cohort study[END_REF] No adjustments were made if the last two (or even more) visits were observed within the same stage and no further follow-up data were available. Conversion times to stage 4 (death) were also left unadjusted, as the dates of death of deceased patients were known exactly.

Because SCA1, SCA2, SCA3, and SCA6 are progressive diseases, we restricted our analysis to conversions into higher disability stages. In a small number of visits, conversions to a lower stage were recorded (SCA1: 8, SCA2: 18, SCA3: 14, SCA6: 8). These were classified as stable. Conversions that skipped one stage were classified as conversion to the next higher stage (SCA1: 1, SCA2: 1, SCA3: 2, SCA6: 5). Due to small numbers, conversions from either stage 0 or 1 to stage 4 (death) were not included in the model (SCA1: 3, SCA2: 5, SCA3: 1, SCA6: 2). The multi-state model and the observed numbers of conversions are depicted in Figure 1.

Multi-state models were specified such that each covariate shared a common hazard ratio (HR) across all transitions, while baseline transition hazards were allowed to vary between states. To increase numerical stability, cumulative transition hazards were smoothed using shape-constrained additive models with monotone increasing P-splines. [START_REF] Pya | Shape constrained additive models[END_REF][START_REF] Pya | Scam: shape constrained additive models[END_REF] Based on the smoothed hazard estimates, state occupation probabilities, and expected lengths of stay (starting with the smallest time point and restricted by the last observed time point) in stages were calculated. [START_REF] Beyersmann | A note on computing average state occupation times[END_REF] Ninety-five percent confidence intervals (95% CIs) were calculated using the bias-corrected and accelerated bootstrap with 1000 samples.

Results are presented in terms of estimated transition HRs with 95% CIs. Likelihood-ratio tests were used to investigate the associations between covariates and transition hazards. The proportional hazards assumption was checked by a graphical analysis of Schoenfeld residuals (SSR).

Results

Data of 3138 visits in 677 participants, 525 from the EUROSCA and 152 from the RISCA study, were analyzed. The study population included 173 SCA1 mutation carriers with 759 visits, 207 SCA2 mutation carriers with 1011 visits, 172 SCA3 mutation carriers with 783 visits and 125 SCA6 mutation carriers with 585 visits. Participants had a median of 3.0 (IQR = 2.0-4.0) visits and a median observation time of 7.6 years (IQR = 3.3-10.1). Demographic and clinical characteristics of all participants are given in Table 1

Relation between disability stages and clinical assessments

In all SCA subtypes, median SARA scores steadily increased with increasing disability stages, and the IQR boundaries did not overlap (Fig. 2; Table S2). Similarly, median SCAFI scores steadily decreased, but the IQR boundaries of stage 1 and stage 2 in SCA6 overlapped. All SCAFI subscores steadily decreased. Among them, 8MW and 9-HPT had the closest relation to the disease stages. The median INAS count steadily increased in all subtypes with the exception of SCA6, in which it decreased from 2.3 in stage 2 to 2.0 in stage 3. IQR boundaries overlapped between stage 1, stage 2, and stage 3 in SCA1 and SCA2, and between all stages in SCA3 and SCA6 (Fig. 2; Table S2).

In all subtypes, median scores of patient-related outcome measures, PHQ-9, EQ-5D, worsened with increasing disability stages. Exceptions were median PHQ-9 score in SCA2 that decreased from 8.0 in stage 2 to 6.0 in stage 3, median PHQ-9 score in SCA6 that decreased from 4.5 in stage 2 to 2.0 in stage 3, and median EQ-5D score in SCA6 that increased from 51.5 in stage 2 to 54.0 in stage 3. In all subtypes, the PHQ-9 and EQ-5D IQR boundaries of all stages overlapped (Fig. 2; Table S2).

We further analyzed changes of each clinical measure of all transitions (Table S3). In all exceptional changes which seemed to show improvement of clinical measures, the changes of clinical measures between the stages were not significant.

Modeling of the evolution of disability

To determine the probability to be in a certain stage at a given age, we used multi-state modeling for each SCA subtype. The results of the covariate-free models are presented in Figure 3. In the first two decades of life, the estimated probability to be in stage 0 was 1. This result reflects the inclusion criteria of the RISCA and EUROSCA studies. In SCA1, the probability started to decrease around the age of 20 years reflecting onset of ataxia. It reached 0 at around the age of 50 years. In SCA2 and SCA3, the decrease started between 30 and 40 years, and the probability to be in stage 0 reached 0 between 50 and 60 years. The corresponding age values for SCA6 were 40 years and 70-80 years. The maximum probabilities of being in stages 1, 2, and 3 increased with time, with peaks ranging between 0.1 (stage 3 in SCA2 at age 48 years) and 0.6 (stage 3 in SCA6 at age 75 years). The estimated probability to be in stage 4 (death) started to rise from 0 around the age of 25 years in SCA1, 30 years in SCA2, 40 years in SCA3, and 55 years in SCA6. The probability to be in stage 4 at the age of 80 years was estimated to be 1 in SCA1 and SCA3, 0.8 in SCA2, and 0.5 in SCA6.

Multi-state modeling allowed to estimate the age of entering stages 1, 2, 3, and 4 and the expected duration of each stage, that is, duration for each SCA subtype. The expected age of entering stage 1-corresponding to ataxia onset-was 30.1 (95% CI: 25.9-34.9) years in SCA1, 37.1 (95% CI: 32.7-43.2) years in SCA2, 38.1 (95% CI: 26.5-43.0) years in SCA3, and 47.9 (95% CI: 41.0-67.7) years in SCA6. The expected duration of staying in stage 1 was estimated to be 7.0 (95% CI: 4.2-9.5) years in SCA1, 7.6 (95% CI: 5.7-9.7) years in SCA2, 6.5 (95% CI: 3.9-9.5) years in SCA3, and 6.0 (95% CI: 3.5-12.5) years in SCA6. The corresponding durations for stage 2 were 7. 2.

In multi-state modeling, we further studied the effect of sex, CAG repeat length of the expanded allele, and CAG repeat length of the normal allele on progression of walking disability in each SCA subtype. Analysis of SSR did not indicate major violations of proportional hazards assumption (Fig. S1).

In SCA6, female sex was associated with faster progression (HR: 1.63, 95% CI: 1.0-2.5), whereas sex was not found to have an effect on progression in the other subtypes (Table 3). CAG repeat length of the expanded allele was associated with faster progression in SCA1 (HR: 1.1, 95% CI: 1.1-1.2), SCA2 (1.2, 95% CI: 1.1-1.2), and SCA3 (1.1, 95% CI: 1.0-1.2). The CAG repeat length of the normal allele was not found to have an effect on progression in any SCA subtype (Table 3).

Discussion

This study provides a quantitative account of the disease evolution in SCA1, SCA2, SCA3, and SCA6. Our analysis is based on longitudinal data of the RISCA and EURO-SCA cohorts, which recruited SCA1, SCA2, SCA3, and SCA6 mutation carriers in different stages of the disease development. [START_REF] Jacobi | Long-term disease progression in spinocerebellar ataxia types 1, 2, 3, and 6: a longitudinal cohort study[END_REF][START_REF] Jacobi | Conversion of individuals at risk for spinocerebellar ataxia types 1, 2, 3, and 6 to manifest ataxia (RISCA): a longitudinal cohort study[END_REF][START_REF] Diallo | Prediction of survival with long-term disease progression in Most common spinocerebellar ataxia[END_REF] Whereas RISCA is a cohort of risk persons about half of whom were pre-ataxic mutation carriers at baseline, EUROSCA included patients over the entire spectrum of disease severity. Thus, two cohorts were merged that represent complementary aspects of the disease spectrum. Median observation times in these cohorts were longer than those of other SCA cohorts, but nevertheless cover only a small part of the entire disease span. To account for this, we modeled disease progression according to five disease stages defined by walking ability (stages 0-3) and death (stage 4). [START_REF] Klockgether | The natural history of degenerative ataxia: a retrospective study in 466 patients[END_REF] As gait problems in SCAs result in limited mobility, this staging system provides information that is directly meaningful for the real life of patients.

To model the transitions between stages, we used a multi-state approach that was defined by a set of proportional transition hazards. This approach extends the wellestablished Cox regression model to a more general sequence of outcome events also referred to as states. In case of a covariate-free model, it has the advantage of resulting in non-parametric and thus fairly general estimates of the probabilities and expected duration of each stage. When including covariates such as sex and CAG repeat length, the model additionally requires the proportional hazards assumption to hold for all transitions. Analysis of SSR did not indicate major violations of this assumption. Furthermore, all models require the assumption of independent entry times and drop-outs, ensuring that the study population is consistently representative of the full, non-truncated, and uncensored population. [START_REF] Betensky | Recognizing the problem of delayed entry in time-to event studies: better late than never for clinical neuroscientists[END_REF] Regarding entry times, the independence assumption is likely to be met by the design of the RISCA and EURO-SCA studies. Regarding drop-outs, the multi-state model allows for stage-dependent censoring.

The expected duration of stay and corresponding age of entering each stage refer to the characteristics of the study population enrolled for RISCA and EUROSCA. This population, which has been recruited at ataxia referral centers, is likely to be skewed toward milder disease stages, as SCA patients in advanced stages are often not able to take the journey to a distant study center. This explains why the relative variability of stage duration is greater for stage 3 compared to the other stages. Although our study population may thus not be representative for the entire population of SCA1, SCA2, SCA3, and SCA6 patients, the study population is a representative target population for future clinical trials, which will preferentially be performed with patients in early disease stages. This study has a number of limitations: The staging system used is rather rough. Perhaps the use of the slightly finer grained Friedreich's Ataxia Rating Scale (FARS) Functional Staging of Ataxia would have given more detailed information, but this was not available when initiating the EUROSCA study. [START_REF] Subramony | Measuring Friedreich ataxia: interrater reliability of a neurologic rating scale[END_REF] Furthermore, despite the high number of participants, the number of transitions between stages, particularly between higher stages were limited.

Another limitation is that walking disability stages were not formally assessed in the RISCA study. Therefore, we used the rating of the gait item (item 1) of SARA to infer walking disability. We focused on the modelling of the evolution of gait disability and overall survival. Further analysis of other parameters including oculomotor abnormalities would have been desirable, [START_REF] Moscovich | Clinical evaluation of eye movements in spinocerebellar ataxias: a prospective multicenter study[END_REF] but were outside the scope of the present paper. In addition, the available RISCA and EUROSCA data are limited with respect to oculomotor dysfunction, as SARA, that does not include oculomotor items, was used as the primary outcome.

In general, all clinical and patient-related measures worsened with increasing disability. Exceptions occurred mainly in SCA6. In this subtype, INAS, PHQ-9, and EQ-5D, which are measures of general neurological involvement, depression, and health-related quality of life, did not further deteriorate with progression to stage 3, which may be due to the small number of participants in stage 3. This is best explained by the milder degree of ataxia and the later ataxia onset of SCA6 compared to SCA1, SCA2, and SCA3. [START_REF] Klockgether | Spinocerebellar ataxia[END_REF][START_REF] Geschwind | Spinocerebellar ataxia type 6. Frequency of the mutation and genotype-phenotype correlations[END_REF] As a result, walking disability due to ataxia is only one among several health problems in older SCA6 patients, which blurs the relationship between ataxia stages and these measures. Among all clinical measures, the clinical scale, SARA, [START_REF] Schmitz-Hubsch | Scale for the assessment and rating of ataxia: development of a new clinical scale[END_REF] had the closest relation to the disease stages underscoring the clinical relevance of SARA. SARA scores steadily worsened with increasing disease stage in all SCA subtypes. Further, SARA scores had no overlap between stages. This cannot be explained solely by the fact that gait is one of the eight SARA items and contributes up to eight of the maximally 40 SARA points. Although gait is one of the three timed tests of SCAFI, [START_REF] Schmitz | SCA functional index: a useful compound performance measure for spinocerebellar ataxia[END_REF] we observed overlap of SCAFI scores and its gait subscore, 8MW, between disease stages.

Patient-related outcome measures have been little studied so far in SCAs, and most studies were cross-sectional or covered only short periods of the entire disease span. [START_REF] Schmitz-H€ Ubsch | Self-rated health status in spinocerebellar ataxia-results from a European multicenter study[END_REF][START_REF] Schmitz-H€ Ubsch | Depression comorbidity in spinocerebellar ataxia[END_REF][START_REF] Lo | Depression and clinical progression in spinocerebellar ataxias[END_REF][START_REF] Jacobi | Long-term evolution of patient-reported outcome measures in spinocerebellar ataxias[END_REF] As ataxia-specific patient-related outcome instruments were not available during the conduct of the RISCA and EUROSCA studies, we have used generic measures for depression, PHQ-9, and health-related quality of life, EQ-5D. [START_REF] Spitzer | Validation and utility of a self-report version of PRIME-MD: the PHQ primary care study. Primary care evaluation of mental disorders. Patient health questionnaire[END_REF][START_REF]EuroQol-a new facility for the measurement of healthrelated quality of life. The EuroQol Group[END_REF] In SCA1, SCA2, and SCA3, these patient-related outcome measures deteriorated with increasing disease stages indicating that the degree of walking disability is a major determinant of limitations in the daily life of patients with these diseases. Recently, an ataxia-specific patient-reported outcome measure, PROM Ataxia has become available. [START_REF] Schmahmann | Development and validation of a patient-reported outcome measure of ataxia[END_REF] Properties of this new measure have to be determined in future longitudinal studies. In addition to clinical scales and patient-reported outcome measures instrumented assessments could help to quantify ataxia and disease progression. [START_REF] Kashyap | Objective assessment of cerebellar ataxia: a comprehensive and refined approach[END_REF] The multi-state modeling approach allowed to determine the likelihood to be in a certain disease stage at a given age and to calculate the length of each stage. The age of onset in SCA6 was higher than that of SCA1, SCA2, and SCA3, but the difference was smaller than that reported in the majority of previous cross-sectional studies that determined the age of onset based on patient report. [START_REF] Sch€ Ols | Spinocerebellar ataxia type 6: genotype and phenotype in German kindreds[END_REF][START_REF] Matsumura | Spinocerebellar ataxia type 6 -molecular and clinical features of 35 Japanese patients including one homozygous for the CAG repeat expansion[END_REF] Whereas the median survival times in SCA1, SCA2, and [START_REF] Klockgether | Spinocerebellar ataxia[END_REF] 1.0 0.9, 1.1 0.545 Expanded allele (repeat number) [START_REF] Klockgether | Spinocerebellar ataxia[END_REF] 1.1 1.1, 1.2 <0.001 SCA2 Sex (f) 0.9 0.6, 1.3 0.576 Normal allele (repeat number) [START_REF] Klockgether | Spinocerebellar ataxia[END_REF] 1.0 0.9, 1.1 0.819 Expanded allele (repeat number) [START_REF] Klockgether | Spinocerebellar ataxia[END_REF] 1.2 1.1, 1.2 <0.001 SCA3 Sex (f) 0.9 0.6, 1.3 0.634 Normal allele (repeat number) [START_REF] Klockgether | Spinocerebellar ataxia[END_REF] 1.0 1.0, 1.0 0.964 Expanded allele (repeat number) [START_REF] Klockgether | Spinocerebellar ataxia[END_REF] 1.1 1.0, 1.2 0.014 SCA6 Sex (f) 1.6 1.0, 2.5 0.032 Normal allele (repeat number) [START_REF] Klockgether | Spinocerebellar ataxia[END_REF] 0.1 0.9, 1.3 0.231 Expanded allele (repeat number) [START_REF] Klockgether | Spinocerebellar ataxia[END_REF] 1.1 0.8, 1.4 0.601 SCA, spinocerebellar ataxia; HR, transition hazard ratio; 95% CI, 95% confidence interval. SCA3 ranged between 18.1 and 22.4 years, which is in line with previous studies, we found a survival time of 34.8 years in SCA6 resulting an estimated age at death of 82.7 years. [START_REF] Monin | Survival and severity in dominant cerebellar ataxias[END_REF][START_REF] Kieling | Survival estimates for patients with Machado-Joseph disease (SCA3)[END_REF] Compared to the other SCAs, SCA6 patients stayed longer in disease stages 2 and 3, that is, in a state in which they required walking aids or a wheelchair. CAG repeat length had a modest effect on disease progression in SCA1, SCA2, and SCA3, but not SCA6. In SCA3, previous analyzes of the EUROSCA data and a Brazilian study did not show an accelerating effect of CAG repeat length on the progression of ataxia severity, whereas another study that use the NESSCA scale that assesses both ataxia and non-ataxia signs showed an effect. [START_REF] Franc ßa Mc | Progression of ataxia in patients with Machado-Joseph disease[END_REF][START_REF] Jardim | Progression rate of neurological deficits in a 10-year cohort of SCA3 patients[END_REF] Our study provides a comprehensive account of disease evolution in SCA1, 2, 3, and 6 based on prospective assessment of disability stages which are of immediate relevance for affected individuals. Our data are important for counselling of patients, for the assessment of the relevance of outcome markers, and for the design of clinical trials.

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  2022 The Authors. Annals of Clinical and Translational Neurology published by Wiley Periodicals LLC on behalf of American Neurological Association.

Figure 1 .

 1 Figure 1. The multi-state model, describing five states from stage 0 to absorbing stage 4. The numbers next to the arrows denote the numbers of transitions from the respective to the subsequent state for SCA1, SCA2, SCA3, and SCA6. SCA, spinocerebellar ataxia.

Figure 2 .

 2 Figure 2. Median SARA values for each stage in SCA1, SCA2, SCA3, and SCA6. SCA, spinocerebellar ataxia. (A) SARA; (B) INAS; (C) SCAFI; (D) PHQ-9; (E) EQ-5D. SARA, Scale for the Assessment and Rating of Ataxia; INAS, Inventory of Non-Ataxia Signs; SCAFI, SCA Functional Index; PHQ-9, Patient´s Health Questionnaire; EQ-5D, EQ-5D visual analogue scale.

Figure 3 .

 3 Figure 3. Estimated probabilities of being in stages 0-4, as obtained from the covariate-free multi-state model. (A) SCA1, (B) SCA2, (C) SCA3, (D) SCA6. SCA, spinocerebellar ataxia.

1 HR

 1 is given per additional repeat. ª 2022 The Authors. Annals of Clinical and Translational Neurology published by Wiley Periodicals LLC on behalf of American Neurological Association.

  and TableS1.

	At baseline, 210 subjects were in stage 0 (SCA1: 75;
	SCA2: 63; SCA3: 48; SCA6: 24), 290 in stage 1 (SCA1: 61;
	SCA2: 101; SCA3: 64; SCA6: 64), 137 in stage 2 (SCA1:
	25; SCA2: 36; SCA3: 45; SCA6: 31), and 40 in stage 3
	(SCA1: 12; SCA2: 7; SCA3: 15; SCA6: 6). Hundred and
	twenty-three participants died during the observation per-
	iod (SCA1: 37; SCA2: 38; SCA3: 34; SCA6: 14).

Table 1 .

 1 Demographic and clinical characteristics of participants at baseline.

		SCA1	SCA2	SCA3	SCA6
	Number of participants	173	207	172	125
	Men/women	89/84	104/103	87/85	63/62
	Age (years)

Table 2 .

 2 Expected duration of each stage and corresponding age of entering each stage in years.

	Stage 0	Stage 1		Stage 2		Stage 3		Stage 4
			Duration		Duration		Duration
	Duration [years] Age [years]	[years]	Age [years]	[years]	Age [years]	[years]	Age [years]

Table 3 .

 3 Risk factors for entering into a higher stage.

		Risk factor	HR	95% CI	p-value
	SCA1	Sex (f)	1.1	0.7, 1.6	0.794
		Normal allele (repeat number)

ª 2022 The Authors. Annals of Clinical and Translational Neurology published by Wiley Periodicals LLC on behalf of American Neurological Association. Disability Stages in SCA H. JACOBI ET AL.

ª 2022 The Authors. Annals of Clinical and Translational Neurology published by Wiley Periodicals LLC on behalf of American Neurological Association.

Acknowledgments

Several contributors of this publication are members of the European Reference Network for Rare Neurological Diseases -Project ID No 739510. The authors thank all individuals for their participation in the study. Open Access funding enabled and organized by Projekt DEAL.

Conflict of Interest

No conflict of interest related to the manuscript.

Author Contributions

Supporting Information

Additional supporting information may be found online in the Supporting Information section at the end of the article.

Figure S1

. Graphical checks of the proportional hazards assumption. Table S1. Demographic and clinical characteristics of RISCA and EUROSCA participants at baseline. Table S2. Outcome measures stratified by stage. Table S3. Changes of outcome measures between the stages.


Marc Chaperon

Michèle Audin

Abbas Bahri

Alain Chenciner

Nicole Desolneux-Moulis

Ivar Ekeland

Albert Fathi

Michel Herman

Misha Gromov

François Laudenbach

Claude Viterbo

Keywords: Mathematics Subject Classification (2010), Primary 99Z99; Secondary 00A00, Keywords, V, I, Arnold, Unfoldings, Implicit functions

I first met Claude at the seminar on contact and symplectic geometry organised from 1982 on by Daniel Bennequin at the École normale supérieure. It was much oriented towards the beautiful conjectures V.I. Arnold had stated in the mid-sixties, inspired by Poincaré's "last geometric theorem." What made the seminar seminal 1 is that its beginning coincided with the first breakthrough in that direction: at the end of 1982, Charles Conley and Eduard Zehnder proved [START_REF] Conley | The Birkhoff-Lewis fixed point theorem and a conjecture of V. I. Arnol[END_REF] the conjecture on fixed points of Hamiltonian transformations of the standard symplectic 2n-torus stated in [2, Appendix 9]. 2 It so happened that, the summer before, I had thought about this conjecture, seen how to deduce it from another statement about exact Lagrangian isotopies of the zero section in T * T n and proved a symplectic isotopy extension lemma [START_REF] Chaperon | Quelques questions de géométrie symplectique [d'après, entre autres[END_REF] implying that such an isotopy extends to a compactly supported Hamiltonian isotopy of the ambient space. Almost immediately after reading the preprint of [START_REF] Conley | The Birkhoff-Lewis fixed point theorem and a conjecture of V. I. Arnol[END_REF], I adapted the Conley-Zehnder proof to get [START_REF] Chaperon | Quelques questions de géométrie symplectique [d'après, entre autres[END_REF] the more general statement, of which I had just learned that a slightly less precise form had also been conjectured by Arnold [START_REF] Arnold | Sur une propriété topologique des applications globalement canoniques de la mécanique classique[END_REF][START_REF] Avez | Problèmes ergodiques de la mécanique classique[END_REF]Appendice 33].

The two weeks spent on the proof of this Arnold conjecture brought me more recognition than the two years of very hard work on my 1980 thèse d' État [START_REF]Géométrie différentielle et singularités de systèmes dynamiques[END_REF][START_REF]C k -conjugacy of holomorphic flows near a singularity[END_REF][START_REF]A forgotten theorem on Z k ×R m -action germs and related questions[END_REF]: soon after the Bourbaki seminar [START_REF] Chaperon | Quelques questions de géométrie symplectique [d'après, entre autres[END_REF], I lectured on the Arnold conjectures in Marseille-Luminy and published with Edi Zehnder an expanded version [17] of these lectures. Having felt ill at ease when teaching the fact (established by Amann and Zehnder) that the non C 2 infinite dimensional action functional, once reduced to finite dimensions, is as smooth as the Hamiltonian of the isotopy, I found it urgent to design a purely finite dimensional proof of "my" Arnold conjecture; this again took me two weeks [START_REF]Une idée du type "géodésiques brisées" pour les systèmes hamiltoniens[END_REF][START_REF]An elementary proof of the Conley-Zehnder theorem in symplectic geometry[END_REF] and brought me much more recognition.

François Laudenbach liked this new proof; he had an extremely bright (and nice) student of his, Jean-Claude Sikorav, work on its generalisations and consequences. Jean-Claude first proved with François [START_REF] Laudenbach | Persistance d'intersection avec la section nulle au cours d'une isotopie hamiltonienne dans un fibré cotangent[END_REF] what had been the true aim of [START_REF]Une idée du type "géodésiques brisées" pour les systèmes hamiltoniens[END_REF][START_REF]An elementary proof of the Conley-Zehnder theorem in symplectic geometry[END_REF], namely, the extension of my result from the cotangent bundle of the torus to that of an arbitrary closed manifold. 3 He then noticed that formula [START_REF]Une idée du type "géodésiques brisées" pour les systèmes hamiltoniens[END_REF] in [START_REF]Une idée du type "géodésiques brisées" pour les systèmes hamiltoniens[END_REF] means that my discretised action is a generating phase for the deformed Lagrangian submanifold, and extended this to arbitrary closed manifolds [START_REF]Sur les immersions lagrangiennes dans un fibré cotangent[END_REF]. In [START_REF]Problèmes d'intersections et de points fixes en géométrie hamiltonienne[END_REF], he generalised the result to Lagrangian immersions and gave an easy proof of the Arnold conjecture on fixed points in situations including surfaces, first obtained more painfully in [START_REF] Sikorav | Points fixes d'une application symplectique homologue l'identité[END_REF].

At about the same time, the 25 years old Claude, who had been the student of Laudenbach and Ekeland, solved [START_REF] Viterbo | A proof of Weinstein's conjecture in R 2n[END_REF] a big problem: the Weinstein conjecture in R 2n . Oddly enough, he did not use Jean Claude's generating phases, with which he would soon do wonders [START_REF]Symplectic topology as the geometry of generating functions[END_REF].

The last of my favourite Arnold conjectures had been proved [START_REF] Gromov | Pseudo holomorphic curves in symplectic manifolds[END_REF] via holomorphic disks, and Floer theory 4 had taken off [START_REF] Floer | Morse theory for Lagrangian intersections[END_REF], leaving me with my fear of flying. My belief that some room was left for earthlier methods 5 now rested mostly on Claude's shoulders. He did not disappoint me.

Subharmonic bifurcations in real or complex dimension one

We first recall the simplest case of the most basic fact.

The period doubling bifurcation

Let h : (u, x) → h u (x) be a C k local map (R 2 , 0) → (R, 0), k ≥ 2, such that the derivative h 0 (0) = ∂ x h(0, 0) equals -1.
The fixed point 0 of h 0 is robust, meaning that every h u with u small enough has a fixed point ϕ(u) nearby, depending C k on the parameter u: Proposition 1.1. The fixed points of the unfolding h : (u, x) → u, h u (x) form near 0 the graph of a C k function ϕ : (R, 0) → (R, 0).

Proof. This follows from the implicit function theorem applied to the C k equation F (u, x) := x-h(u, x) = 0, as F (0, 0) = 0 and ∂ x F (0, 0) = 2 = 0.

Of course 0 is a 2-periodic point of h 0 , i.e., a fixed point of h 2 0 := h 0 • h 0 .

3 A little sooner, Helmut Hofer had done this [START_REF] Hofer | Lagrangian embeddings and critical point theory[END_REF] in an infinite dimensional framework. Whatever its proof, I thought the result would be more central than it turned out to be. 4 Whose idea owes as much to Charlie Conley as to Edi Zehnder, see the foreword of [START_REF]On generating families[END_REF]. 5 A similar belief is at the origin of the present article.

Proposition 1.2. Assume that α(u) := h u ϕ(u) satisfies α (0) = 0.Then, near 0, the 2-periodic points of h (solutions of h 2 u (x) = x) form the union of two curves intersecting transversally at 0: of course graph ϕ, and a C k-1 curve W of which h| W is an involution, implying that T 0 W is the x-axis.

Proof. Conjugating h by the local diffeomorphism (u, x) → u, x -ϕ(u) , we may assume ϕ = 0-the new h u (0) is the old h u ϕ(u) . By Taylor's formula, h u (x) = xg u (x) near 0, where g u (x) = 1 0 h u (tx) dt, hence g u (0) = α(u) and therefore g 0 (0) = -1; the map g :

(u, x) → g u (x) is C k-1 and the equation h 2 u (x) = x writes xg u (x)g u h u (x) = x, which means x = 0 (the fixed points) or G(u, x) := g u (x)g u h u (x) -1 = 0. As G(0, 0) = 0 and ∂ u G(0, 0) = d du u=0 g u (0) 2 = -2α (0) = 0, there exist open neighbourhoods U, V of 0 in R such that the zeros of G| U ×V form the "graph" W = {u = ψ(x)} of a C k-1 implicit function ψ : V → U .
The map h is by definition an involution of its set of 2-periodic points, of which

W \{0} is an open subset, which becomes h-invariant if W is replaced by W ∩ h(W ) (this means restricting conveniently the open subset V ). Invariance writes ψ(x) = ψ h ψ(x) (x) , hence ψ (0) = lim x→0 ψ h ψ(x) (x) -ψ(x) h ψ(x) (x) -x = 0 since the only fixed point of h lying in W is 0; thus, T 0 W is the x-axis.
Examples. The curve W can have various positions with respect to T 0 W :

• If h u (x) = α(u)
x, where α : (R, 0) → (R, -1) is a C k function with ±α (0) > 0, then W is the x-axis; the fixed point 0 of h u is attracting for ±u < 0, repulsing for ±u > 0, and this cannot be called a bifurcation.

• If h u (x) = -(1 + u -x 2 )
x, then the fixed point 0 of h u is attracting for u < 0, repulsing for u > 0, and W is the parabola u = x 2 ; for u > 0, the attracting 2-periodic orbit {-√ u, √ u}, born for u = 0, takes the place of 0 as an attractor of h u , a genuine bifurcation.

• If h u (x) = -(1 + u + x 2 )
x then, for u < 0, the repulsing 2-periodic orbit {-√ -u, √ -u} gradually "throttles" the attracting fixed point 0, so that for u ≥ 0 no attractor of h u persists near 0, a true catastrophe.

The generic situations look like the last two examples (Figure 1). 1.2. Subharmonic bifurcations, holomorphic case Proposition 1.3. Let h : (u, z) → h u (z) be a local map (C 2 , 0) → (C, 0), holomorphic and such that h 0 (0) = ∂ z h(0, 0) is a q th root of unity ρ = e 2πi p q , 0 < p < q. Then:

i) The fixed points of the unfolding h : (u, z) → u, h u (z) form near 0 the graph of a holomorphic function ϕ : (C, 0) → (C, 0). ii) Assume that α(u) := h u ϕ(u) satisfies α (0) = 0. Then, near 0, the qperiodic points of h (solutions (u, z) of the equation h q u (z) = z) form the union of two holomorphic curves intersecting transversally at 0: the fixed point set graph ϕ and a curve W on which h induces a Z/qZ-action. iii) When ρ is a primitive q th root of unity, the curve W and the z-plane are tangent to order q -1 at 0.

Proof. i) As h 0 (0) = ρ = 1, just apply the holomorphic implicit function theorem to the equation z -h(u, z) = 0. ii) As in the proof of Proposition 1.2, one can assume ϕ = 0, hence h u (z) = z g(u, z) with g holomorphic this time and g(u, 0) = α(u). The equation

h q u (z) = z writes zg u (z)g u h u (z) • • • g u h q-1 u (z) = z, which means either z = 0 (the fixed points) or G(u, z) := g u (z)g u h u (z) • • • g u h q-1 u (z) -1 = 0.
As G(u, 0) = α(u) q -1 and α(0) = ρ, one has ∂ u G(0) = qρ q-1 α (0) = 0 and G(0) = 0, hence there exist open neighbourhoods U, V of 0 in C such that the zeros of G| U ×V form the "graph" W = {u = ψ(z)} of a holomorphic implicit function ψ : V → U . The map h induces by definition an action of Z/qZ on its set of qperiodic points, of which

W \ {0} is an open subset, that becomes h-invariant if W is replaced by W ∩ h-1 (W ) ∩ • • • ∩ h1-q (W
) (as before, this means restricting conveniently the open subset V ). Invariance writes ψ(z) = ψ h ψ(z), z .

(

iii) Still assuming ϕ = 0, if we derivate (1.1), we get

ψ (z) = ψ h ψ(z), z ∂ 1 h ψ(z), z ψ (z) + h ψ(z) (z) .
For z = 0, as the identity h(u, 0) = 0 implies that ∂ 1 h(u, 0) = 0, this reads ψ (0) = ψ (0) h 0 (0), that is, (ρ -1)ψ (0) = 0, hence ψ (0) = 0, which proves our result if q = 2. Otherwise assuming inductively that ψ vanishes to order k -1 at 0 for some k ∈ {2, . . . , q -1} and derivating k times (1.1) at 0, the Faà di Bruno formula and the identity ∂ 1 h(u, 0) = 0 yield

ψ (k) (0) = ψ (k) (0) h 0 (0) k , that is, (ρ k -1) ψ (k) (0) = 0, hence ψ (k) (0) = 0 as ρ is a primitive q th root of unity. Examples. If h u (z) = α(u)z, where α : (C, 0) → (C, ρ) is a holomorphic function such that α (0) = 0, then W is the z-plane. If h u (z) = (ρ + u -z q )z, then W is the curve u = z q . 1.3. Opening Pandora's box Under the hypotheses of Proposition 1.3 ii)-iii), α is a holomorphic local diffeomorphism (C, 0) → (C, ρ).
Viewing it as a local parameter change and performing the variable changes in the proof of Proposition 1.3, the following hypotheses are verified with u 0 = ρ:

Hypotheses. Given u 0 ∈ S 1 , set ũ0 := (u 0 , 0) ∈ C 2 and let h : (u, z) → h u (z) be a holomorphic local map (C 2 , ũ0 ) → (C, 0) such that h u (0) = 0 and h u (0) = u. Proposition 1.

now reads as follows:

Proposition 1.4. If u 0 = e 2πi p q , 0 < p < q, gcd(p, q) = 1, then the q-periodic points of h near ũ0 form the union of {z = 0} and the h-invariant "graph" W p/q = {u = ψ p/q (z)} of a holomorphic ψ p/q : (C, 0) → (C, u 0 ) such that ψ p/q (j) (0) = 0 for 1 ≤ j < q. The function ψ := ψ p/q verifies (1.1), and h generates a Z/qZ-action on

W p/q , namely m, ψ(z), z -→ hm ψ(z), z = ψ(z), h ψ(z) m (z) , induced by the Z/qZ-action (m, z) → h ψ p/q (z) m (z) on Dom ψ p/q .
When u 0 is not a root of unity, the following result can apply to f = h u0 :

Theorem 1.5 (Brjuno [5], Yoccoz [START_REF] Yoccoz | Petits diviseurs en dimension 1[END_REF]). If u 0 = e 2πiω with ω ∈ [0, 1] \ Q, the following two conditions are equivalent: i) ω is a Brjuno number, meaning that the convergents pn qn of its continued fraction expansion verify

log qn+1 qn < ∞. ii) Every holomorphic germ f : (C, 0) → (C, 0) such that f (0) = u 0 is holomorphically linearisable.
Notes. The implication i)⇒ii) is Brjuno's. In 1942, Siegel [START_REF] Siegel | Iteration of analytic functions[END_REF] had proved ii) under the stronger condition sup log qn+1 log qn < ∞. This already defines a full measure set of numbers u 0 ∈ S 1 , but Theorem 1.5 provides the optimal set. Back to families, in the trivial case h u (z) = uz, every h u is linear(isable). However, in general, h u0 is linearisable if u 0 = e 2πiω with ω Brjuno.

In that case, linearisability means that there exists a holomorphic local coordinate (conjugacy)

Z ω : (C, 0) → (C, 0) such that Z ω • h u0 = u 0 Z ω ; as the rotation z → u 0 z preserves each circle S r = {|z| = r}, every closed curve C r = Z -1
ω (S r ) with r > 0 small enough is h u0 -invariant and, of course, Z ω | Cr conjugates h u0 | Cr to the rotation z → e 2πiω z restricted to S r . Question 1.6. Is this the limit of what happens near u = e 2πipn/qn ? Do the holomorphic functions ψ pn/qn tend to the constant ψ ω = u 0 = e 2πiω in some uniform neighbourhood of 0 and, for z ∈ C close to 0, do the periodic orbits

ψ pn/qn (z), h ψ pn /qn(z) k (z) : 0 ≤ k < q n of h tend to the closed h-invariant curve {u 0 } × C r such that z ∈ C r ?
More precisely, does the (holomorphic) standard linearisation 6

Z pn/qn (z) = 1 q n qn-1 k=0 e -2πikpn/qn h ψ pn /qn (z) k (z) of the Z/q n Z-action (m, z) → h ψ pn /qn (z) m (z) tend to Z ω when n → ∞?
Notes. If h u (z) = uz, the answer is trivially positive even when ω is not Brjuno. The question is whether this holds for arbitrary families h. My hope would be to deduce the Siegel-Brjuno theorem from the uniform convergence of ψ pn/qn and maybe Z pn/qn in a uniform neighbourhood of 0, at least for some well-chosen family h. One should get invariant fractals at the limit when ω is not Brjuno, as in [START_REF]Bifurcations de points fixes elliptiques, II. Orbites périodiques et ensembles de Cantor invariants[END_REF] -the Pérez-Marco hedgehogs [START_REF] Pérez-Marco | Fixed points and circle maps[END_REF], independent of any arithmetic conditions apart from ω being irrational, might be obtained by this method.

Subharmonic bifurcations, Arnold tongues and KAM circles

Here, smooth means real analytic or C ∞ .

Subharmonic bifurcations in real dimension two

Let h : (u, z) → h u (z) be a smooth local map (R 2 × R 2 , 0) → (R 2 , 0
) such that the eigenvalues of the derivative Dh 0 (0) = ∂ z h(0, 0) are primitive q th roots of unity ρ = e 2πi p q , ρ = e -2πi p q , 1 ≤ p < q, q ≥ 3.

Proposition 2.1.

i) The fixed points of the unfolding h : (u, z) → u, h u (z) form near 0 the graph of a smooth function ϕ : (R 2 , 0) → (R 2 , 0). ii) There is a smooth local function α : (R 2 , 0) → (C, ρ) such that the eigenvalues of Dh u ϕ(u) are α(u), α(u). iii) If Dα(0) : R 2 → C is bijective then, near 0, the q-periodic points of h form the union of two surfaces intersecting transversally at 0: of course graph ϕ, plus a C q-3 surface W on which h| W induces a Z/qZ-action.

Proof. i) follows from the implicit mapping theorem applied to the smooth equation

F (u, z) := z -h(u, z) = 0, as ∂ z F (0, 0) : R 2 → R 2 is invertible.
ii) follows from the formula for the eigenvalues of a real 2 × 2 matrix with no real eigenvalue.

iii) We may assume ϕ = 0, and the new Dh u (0) is the old Dh u ϕ(u) .

Lemma 2.2. a) An R-linear change of variables J(u) : R 2 → C, depending smoothly on u, yields h : (R 2 × C, 0) → (C, 0) and Dh u (0)z = α(u)z. b) Modulo a change of variables, polynomial of degree q -1 with respect to z, z, whose coefficients are smooth functions of u, the Taylor polynomial Q u of h u to order q -1 at 0 for small u is of the form

Q u (z) = z α(u) + [ q-1 2 ] k=1 b k (u)|z| 2k + β(u)z q-1 .
6 The other holomorphic local linearisations Z satisfy Z • Z -1 pn/qn (z) = za(z qn ), a(0) = 0.

Proof of the lemma. a) The isomorphism J(u) ∈ L(R 2 , C) is an eigenvector of Dh u (0) T : λ → λ • Dh u (0) associated to the eigenvalue α(u). Under the condition, e.g., J(u)(1, 0) = 1, it is unique and depends smoothly on u. b) By normal form theory or direct computation, one can assume that Q u (z) -α(u)z is a C-linear combination (depending smoothly on u) of monomials z j zk with 1 < j + k ≤ q -1 and u j 0 ūk 0 = u 0 , that is, e 2πi(j-k-1)p/q = 1, which writes (j -k -1)p = q with ∈ Z. As gcd(p, q) = 1, one has = mp, m ∈ Z, hence j -k -1 = mq and either m = 0, hence z j zk = z|z| 2k , or m = -1 and j = 0, yielding z j zk = zq-1 . By Taylor's formula,

h u (z) = Q u (z) + q j=0 z j zq-j 1 0 (1 -t) q-1 (q -1)! q j ∂ j z ∂ q-j z h u (tz)dt = z a(u, z) + b(u, z)
zq-1 z where a, b are smooth, a(u, 0) = α(u) and b(u, 0) = β(u). It follows that h u (z) = zg u (z) with g : (u, z) → g u (z) only C q-3 in general and g u (0) = α(u).

• For q > 3, the same arguments as in the proof of Proposition 1.3 yield a C q-3 implicit function ψ : (C, 0) → (R 2 , 0) whose graph W has the required properties near the origin-in particular, (1.1) holds.

• If q = 3, then h u (z) = A u (z)z near 0, where A u (z) = 1 0 Dh u (tz) dt (hence A u (0)z = α(u)z)
, and one can similarly apply the implicit map theorem along r = 0 after dividing by r the equation h 3 u (re iθ ) = re iθ . 7The details are left to the reader.

Example. If h u (z) = (ρ + u)z -zq-1 , u, z ∈ C, then W is the surface u = zq-1 /z, which is C q-3 but not C q-2 . Thus, our bound for the differentiability of W is sharp. No such problem arised in the holomorphic case.

Arnold tongues

Under the hypotheses of Proposition 2.1 iii), α is a smooth local diffeomorphism (R 2 , 0) → (C, ρ). Viewing it as a local parameter change, the following hypotheses are verified with u 0 = ρ, modulo the variable changes in the proof of Proposition 2.1:

Hypotheses. For u 0 ∈ S 1 , setting ũ0 := (u 0 , 0) ∈ C 2 , let h : (u, z) → h u (z) be a smooth local map (C 2 , ũ0 ) → (C, 0) such that h u (0) = 0 and Dh u (0)z = uz. Proposition 2.

now reads as follows:

Proposition 2.3. If u 0 = e 2πi p q , 0 < p < q, gcd(p, q) = 1, then, near ũ0 , the q-periodic points of h form the union of {z = 0} and the h-invariant "graph" W p/q = {u = ψ p/q (z)} of a C q-3 function ψ p/q : (C, 0) → (C, u 0 ). The function ψ := ψ p/q verifies (1.1), and h| W p/q generates a Z/qZ-action on W p/q as in Proposition 1.4.

The functions ψ = ψ p/q of Proposition 1.4, being holomorphic, are either constant or open. Thus the invariant manifold W p/q , projected into parameter space, is either {e 2πip/q } or (in general) an open neighbourhood of e 2πip/q . The non-holomorphic case is altogether different: Proposition 2.4. If u 0 is a primitive q th root of unity e 2πip/q , 0 < p < q, q > 4, then: i) Up to a smooth local change of variables C 2 , ũ0 → C 2 , ũ0 , of the form (u, z) → (u, Z u (z)) with Z u (real) polynomial of degree q -2, one has the following: near ũ0 , the unfolding h is tangent to order q -2 along C × {0} to a smooth unfolding P (u, z) = (u, P u (z)) of the form

P u (z) = z u - [ q-3 2 ] k=1 b k (u)|z| 2k .
ii) For b 1 (u 0 ) = 0, the principal part of ψ p/q (z) is u 0 + b 1 (u 0 )|z| 2 . Thus, for ū0 b 1 (u 0 ) = 0, the set Im ψ p/q of those u near u 0 for which h u has a q-periodic orbit lies on one side of S 1 .

iii) The function ψ p/q is tangent to order q -3 at ũ0 to a normal form

ψp/q (z) = u 0 + [ q-3 2 ] k=1 a k |z| 2k =: χ p/q (|z|), a k ∈ C, a 1 = b 1 (u 0 ).
Thus, when the first Birkhoff invariant b 1 (u 0 ) is non-zero, restricting Dom ψ p/q if necessary, the set Im ψ p/q is contained near u 0 in an "Arnold tongue" 0≤t≤ε u ∈ C : |u -χ p/q (t)| ≤ δ ε t q-3 along the curve χ p/q ([0, ε]), with ε > 0 small and lim ε→0 δ ε = 0.

Proof. i) follows from Lemma 2.2 b).

ii)-iii) As ψ p/q (z) = ψ p/q h ψ p/q (z), z by (1.1), the Taylor polynomial ψp/q (z) = u 0 + 1≤j+ ≤q-3 c j z j z =: u 0 + ĉ(z) satisfies ψp/q (z) = ψp/q P ψp/q (z) (z) up to terms of degree greater than q -3.

Denoting the Taylor expansion of b

k (u 0 + v) at v = 0 by bk (v) = m≥0 b kmn v m vn ,
this means that, up to terms of degree greater than q -3,

ĉ(z) = ĉ   z u 0 + ĉ(z) - [ q-3 2 ] k=1 bk ĉ(z) |z| 2k    .
ii) It follows that c 10 = u 0 c 10 = 0, c 01 = ū0 c 01 = 0, c 20 = u 2 0 c 20 = 0, c 02 = ū2 0 c 02 = 0; thus, the first c j that can be nonzero is c 11 =: a 1 , and it is equal to b 100 = b1 (0) = b 1 (u 0 ).;

iii) Inductively, one can see that ĉ(z) = ĉ(u 0 z), hence ĉ(z) =

[ q-3 2 ] k=1 a k |z| 2k .
The reader can fill in the details as an exercise.

Example.

If h u (z) = z u - [ q-3 2 ]
k=1 a k |z| 2k then ψ p/q = ψp/q ; thus, near u 0 , Im ψ p/q is the curve χ p/q ([0, ε]).

Opening Pandora's box wider

Question 2.5. For diophantine ω with convergents p n /q n , if b 1 (u 0 ) = 0, one can wonder as in the holomorphic case whether one has the following:

• The smooth functions ψ pn/qn tend to some ψ ω : (C, 0) → (C, u 0 ) in a uniform neighbourhood of 0; thus, the h-invariant surfaces W pn/qn tend to the h-invariant surface W ω = {u = ψ ω (z)}.

• For small z, the periodic orbits ψ pn/qn (z), h ψ pn/qn (z) k (z) : 0 ≤ k < q n of h tend to a closed h-invariant curve {ψ ω (z)} × C ωz such that z ∈ C ωz and that the rotation number of h ψ(z) | Cωz is ω. • The standard linearisation of the Z/q n Z-action (m, z) → h ψ pn /qn (z) m (z) tends to a local transformation Z ω linearising the local diffeomorphism z → h ψω(z) (z). Hence, the T-action (θ, z) → Z -1 ω e 2πiθ Z ω (z) leaves ψ ω invariant, 8 implying that Im ψ ω is a curve (with boundary), limit of the narrower and narrower Arnold tongues Im ψ pn/qn . Example (KAM invariant curves). Assume that h possesses the following properties near some u 0 = e 2πiω0 with ω 0 ∈ R \ Q: i) If |u| = 1, the transformation h u preserves the area. ii) h u = |u| h u/|u| , hence h u multiplies the area by |u| 2 .

iii) One has b 1 (u 0 ) = 0. By ii), no h u with |u| = 1 can have a closed invariant curve near 0. Thus, if the answer to Question 2.5 is positive, then every ψ ω has modulus one, hence b 1 (u 0 ) = iλu 0 , λ ∈ R-which already follows from i). Figure 2 shows what happens for (u, z) = (e 2πiω , z) ∈ S 1 × C close to ũ0 , in local coordinates (ω, z). The ω-axis is in red and the "paraboloids" are the surfaces W ω with ω Diophantine, which do lie in S 1 × C as |ψ ω (z)| = 1. These surfaces intersect the slice u = u 0 at the h u0 -invariant closed curves ("KAM circles") C ωz with ψ ω (z) = u 0 , which occupy most of the room near z = 0, with maybe complicated dynamics in between. Note. The limit surfaces W ω and the linearisations Z ω in Question 2.5 might be obtained as in [START_REF] Chenciner | Bifurcations de points fixes elliptiques, I. Courbes invariantes[END_REF] (where, however, the typical situation is b 1 (e 2πiω ) / ∈ iR, yielding normally hyperbolic invariant circles). Figure 2, which I like a lot, most probably follows from standard KAM theory [START_REF] Féjoz | Introduction to KAM theory with a view to celestial mechanics[END_REF]. The case where h is holomorphic will be referred to as the holomorphic case.

Higher dimensions

Note. A general situation reduces to these hypotheses. Let h : (u, x) → h u (x) be a smooth local map (R 2d × R 2d , 0) → (R 2d , 0) such that the eigenvalues of Dh 0 (0) are simple and not real. Near 0, the fixed points of h form the graph x = ϕ(u) of a smooth implicit function, which we may assume to be 0.

There is [START_REF] López De Medrano | Regularities and singularities appearing in the study of polynomials and linear operators[END_REF][START_REF]Singularities in dynamics: a catastrophic viewpoint[END_REF] a smooth local map J of R 2d into the space of R-linear isomorphisms R 2d → C d , defined near 0, such that each

J(u)Dh u (0)J(u) -1 is a diagonal automorphism diag α(u) : z → α 1 (u)z 1 , . . . , α d (u)z d of C d (thus, the eigenvalues α j (u), α j (u) of Dh u (0), 1 ≤ j ≤ d, depend smoothly on u). Via the identification (u, x) → u, J(u)x , we can view h as a local map (R 2d × C d , 0) → (C d , 0) such that Dh u (0) = diag α(u).
Setting α(u) := α 1 (u), . . . , α d (u) and assuming Dα(0) : R 2d → C d invertible, the smooth local map α : (R 2d , 0) → C d is a local diffeomorphism. If we view it as an identification, then u 0 := α(0) satisfies our hypotheses.

Periodic orbits

Proposition 3.1. Assume that u 0 = ρ = (ρ 1 , . . . , ρ d ), where ρ j = e 2πipj /q , 0 < p j < q. Let π : R + ×S 2d-1 , {0}×S 2d-1 → (C d , 0) be the oriented blowup π(r, y) := ry ("polar coordinates"). Then, setting ȗ0 := (u 0 , 0) ∈ C d × R + and denoting by S2d-1 the complement of the coordinate hyperplanes in S 2d-1 : i) Near ũ0 , the map h lifts to a smooth local map h : (u, r, y) → hu (r, y) of

C d × R + × S 2d-1 , {ȗ 0 } × S 2d-1 into R + × S 2d-1 , {0} × S 2d-1 such that π • hu = h u • π and h(ȗ 0 , y) = 0, (diag u 0 )y for all y ∈ S 2d-1 .
ii) The q-periodic points of the unfolding h : (u, r, y) → u, hu (r, y) contain {ȗ 0 } × S 2d-1 and the h-invariant "graph" W = {u = ψ(r, y)} of a smooth local map ψ = ψp/q : R + × S2d-1 , {0} × S2d-1 → (C n , u 0 ). iii) Hence, the non-fixed q-periodic points of h contain the h-invariant "graph"

W = {u = ψ(z)} of a smooth ψ = ψ p/q : π(Dom ψ) \ {0} → C n \ {u 0 }. iv) In the holomorphic case, W is holomorphic. Proof. i) The relation π• hu (r, y) = h u •π(r, y) writes hu (r, y) = (R u , Y u )(r, y) with R u (r, y) = |h u (ry)| and Y u (r, y) = h u (ry)/|h u (ry)| for r > 0; now, by Taylor's formula, h u (ry) = rA u (ry)y, where A u (ry) := 1 0 Dh u (try) dt, hence Y u (r, y) = A u (ry)y/|A u (ry)y| wherever A u (ry)y = 0, including r = 0 near u = u 0 since A u (0) = diag u.
ii) One has that h q u (ry) = ry if and only if rG u (r, y) = 0, where G u (r, y) = G(u, r, y) = A u h q-1 u (ry) • • • A u (ry)y -y, hence in particular G(u, 0, y) = (diag u) q y -y. Forgetting the fixed points r = 0, the equation h q u (ry) = ry reads G(u, r, y) = 0. Now, all y ∈ S 2d-1 verify G(u 0 , 0, y) = 0 and ∂ u G(u 0 , 0, y) = q diag(ρ 1 y 1 , . . . , ρd y d ), invertible if and only if y 1 • • • y d = 0, i.e., y ∈ S2d-1 . Hence, there exist open neighbourhoods U of u 0 in C d and V of {0} × S2d-1 in R + × S2d-1 such that the zeros of G| U × V form the "graph" of a smooth implicit map ψ : V → U ; as before, this graph W becomes

h-invariant if it is replaced by W ∩ h-1 ( W ) ∩ • • • ∩ h1-q ( W ).
iii) Recall that π is a diffeomorphism off the "exceptional divisor" π -1 (0). iv) We can therefore "read" the equation G u (r, y) = 0 via this diffeomorphism, that is, write it g u (z) := h q u (z) -z = 0 for z = 0; as the unfolding (u, r, y) → u, G u (r, y) is a local diffeomorphism at every point of W , so is (u, r, y) → u, rG u (r, y) , hence the unfolding g : (u, z) → u, g u (z) is a diffeomorphism at every point of W ; the map g being holomorphic, its local inverses are, implying that W is holomorphic.

Note. A nicer way to prove iv) is to use the complex blowup π C : (D, z) → z, z ∈ D, D ⊂ C d complex line through 0;9 the implicit function theorem yields a holomorphic ψC "upstairs", defined on an open subset of the complement of the closure of {(D, z)

: z = 0, z 1 • • • z d = 0} and equal to u 0 on π -1 C (0). Example. If h u (z) = diag u + χ(z q1 1 , . . . , z q d d ) z
, where q j is the denominator of p j /q in irreducible form and χ :

(C d , 0) → (C d , 0) is holomorphic, then ψ(z) = ρ -χ(z q1 1 , . . . , z q d d )
, which has contact of order at least min q k with the constant ρ at 0. Proposition 3.2. The automorphism diag ρ lifts via π to the diffeomorphism diag ρ : (r, y) → r, (diag ρ)y .

i) Restricting W if required, there is a smooth diffeomorphism Z = Zp/q of W onto an open diag ρ-invariant subset Ω ⊃ {0} × S2d-1 of R + × S2d-1 ,
conjugating h| W to ( diag ρ)| Ω, with Z(ȗ 0 , y) = (0, y) for all y ∈ S2d-1 . ii) The map Z induces a smooth diffeomorphism Z = Z p/q of W onto the open diag ρ-invariant "trefoil" Ω := π( Ω) \ {0}, conjugating h| W to diag ρ| Ω and tending to 0 when the variable in W tends to ũ0 . iii) If h is holomorphic, so is Z.

Proof. i) The conjugacy Zp/q is as in Question 1.6, but in polar coordinates: Zp/q ψ(r, y), (r, y) = r |C(r, y)|, C(r, y)

|C(r, y)| , where C(r, y) = 1 q q-1 k=0 (diag ρ) -k A ψ(r,y) h k-1 ψ(r,y) (ry) • • • A ψ(r,y) (ry)y.
For all y ∈ S2d-1 , one has that C(0, y) = y, hence Z(ȗ 0 , y) = (0, y) and

D Zp/q (0, y) = 1 0 * id y ⊥ : R × y ⊥ → R × y ⊥
is invertible. It follows that Zp/q is a smooth local diffeomorphism, whose domain can be made h-invariant as usual. It is not difficult to check that it is a conjugacy, see equation (3.1) hereafter. ii) is obvious; by definition, the conjugacy Z p/q is as in Question 1.6:

Z p/q ψ p/q (z), z = 1 q q-1 k=0 (diag ρ) -k h ψ p/q (z) k (z). (3.1)
iii) follows at once. Hence, denoting again by W the inverse image of this new Ω by Z, the map h| W is invariant under the T d -action Z * ȇ : t, Z-1 (r, y) → Z-1 ȇ(t, r, y); in particular, it preserves each orbit, which orbits constitute a foliation of W by d-tori

Z-1 {r} × (x 1 S 1 × • • • × x d S 1 ) with x j > 0 and x 2 1 + • • • + x 2 d = 1.
In general, these tori of course do not lie each in a slice u = constant like the orbits of h| W . The foliation, like the new W , depends on the choice of Z, which is far from unique since the set of diag ρ-invariant smooth diffeomorphism germs R + × S 2d-1 , {0} × S 2d-1 → R + × S 2d-1 , {0} × S 2d-1 is infinite dimensional. 10 However, when p/q tends to some diophantine ω ∈ [0, 1] d , the orbits of h| Wp/q should "become denser and denser in such invariant tori":

where k ∈ N d , |k| = k 1 + • • • + k d and kω = k 1 ω 1 + • • • + k d ω d . Then, every holomorphic germ f : (C d , 0) → (C d , 0) such that Df (0) = diag u 0 is holomorphically linearisable: there exists a holomorphic local diffeomorphism Z ω : (C d , 0) → (C d , 0) such that Z ω • h u0 = (diag u 0 )Z ω . 11 As the rotation z → (diag u 0 )z = (e 2πiωj z j ) 1≤j≤d preserves each d-torus T r = {|z 1 | = r 1 , . . . , |z d | = r d }, every embedded torus T ωr = Z -1
ω (T r ) with r j > 0 small enough is h u0 -invariant and, of course, Z ω | Tωr conjugates h u0 | Tωr to the rotation z → (diag u 0 )z restricted to T r . Question 3.4. Applied to f = h u0 , is this the limit of what happens near u = (e 2πipj /q ) 1≤j≤d when p/q ∈ Q d tends to ω? 12 Do the maps ψ p/q tend to ψ ω = u 0 in some uniform neighbourhood of 0? For z ∈ C d close to 0, does the periodic orbit ψ p/q (z), h ψ p/q(z) k (z) : 0 ≤ k < q of h tend to the closed h-invariant torus {u 0 } × T ωr such that z ∈ T ωr ? More precisely, does the holomorphic linearisation (3.1) of h| W p/q tend to Z ω when n → ∞?

Note. This is not as simple as Question 1.6: indeed, unless I am mistaken, the maps ψ p/q are not a priori defined in a neighbourhood of 0, so that part of the question is whether Dom ψ p/q tends to such a neighbourhood. On the other hand, it follows from normal form theory that, as in the case d = 1, the map ψ = ψ p/q has more and more contact with u 0 at 0 when p/q → ω. 13 10 Indeed, the set of diag ρ-invariant smooth diffeomorphism germs 11 One can assume DZω(0) = Id. Pöschel [START_REF]On the Siegel-Sternberg linearization theorem[END_REF] attributes Theorem 3.3 to Siegel, who certainly proved its analogue for vector fields [START_REF]Über die Normalform analytischer Differentialgleichungen in der Nähe einer Gleichgewichtslösung[END_REF]. The same applies to its improvement by Brjuno. This "Siegel-Brjuno" theorem for maps and much more is proved in [START_REF] Rüssmann | On the convergence of power series transformations of analytic mappings near a fixed moint into a normal form[END_REF][START_REF] Pöschel | On invariant manifolds of complex analytic mappings near fixed points[END_REF][START_REF]On the Siegel-Sternberg linearization theorem[END_REF][START_REF] Stolovitch | Family of intersecting totally real manifolds of (C n , 0) and germs of holomorphic diffeomorphisms[END_REF]. 12 For example, p j /q = p jn /qn can be the n th convergent of ω j . 13 If one prefers, ψC has more and more contact with u 0 at points of π -1 C (0).

(C d , 0) → (C d , 0) is infinite dimensional, as any smooth germ η : (C d , 0) → (C d , 0) yields the diag ρ-invariant germ 1 q q 1 (diag ρ) -k • η • (diag ρ) k .
3.4. Passing to the limit in the smooth case? If u 0 = (e 2πiω1 , . . . , e 2πiω d ), where ω = (ω 1 , . . . , ω d ) ∈ T d is non-resonant, meaning that ω 1 , . . . , ω d ∈ T are independent over Z, then, by normal form theory, one has the following: for each positive integer N , up to smooth local conjugacy (u, z) → u, Z u (z) , every h u with u -u 0 small enough is tangent to order 2N + 1 at 0 to a polynomial map

P u (z) = diag u + N =1 b (u) |z 1 | 2 , . . . , |z d | 2 z
with b (u) : R d → R d homogeneous of degree , depending smoothly on u. As for d = 1, it follows that when p/q tends to ω the map ψ p/q of Proposition 3.1 is tangent to higher and higher order at 0 to a polynomial normal form

14 ψp/q (z) = χ p/q |z 1 | 2 , . . . , |z d | 2 , χ p/q (0) = ρ, Dχ p/q (0) = b 1 (ρ).
Thus, if b 1 (u 0 ) (and therefore b 1 (ρ) for small p q -ω) is invertible then, restricting ψ p/q , the set Im ψ p/q lies near ρ in a thinner and thinner "Arnold tongue" along the smooth d-fold with corner χ p/q ([0, ε) d ) for small ε > 0. ) is invertible, one can wonder as in the holomorphic and one-dimensional cases whether one has the following when p/q tends to ω:

• The ψp/q 's tend to a map ψω of R + × S2d-1 , {0} × S2d-1 into (C n , u 0 ) in a uniform neighbourhood of {0}× S2d-1 ; thus, the h-invariant surfaces Wp/q tend to the h-invariant 2d-fold Wω = {u = ψω (r, y)}.

• For each (r, y), the periodic orbits ψp/q (r, y), hk ψp/q (r,y) (r, y) 0≤k<q of h tend to a h-invariant embedded d-torus { ψω (r, y)} × T ωry such that (r, y) ∈ T ωry . • The "linearisation" Zp/q of Proposition 3.2 i) tends to a local transformation Zω "linearising" the local diffeomorphism (r, y) → h ψω(r,y) (r, y).

Hence, the T d -action θ, Z-1 ω (r, y) → Z-1 ω r, e 2πi diag θ y leaves ψω invariant, implying that Im ψω is a d-fold (with corner), limit of the narrower and narrower subsets Im ψp/q .

Example (KAM invariant tori). Assume that h possesses the following properties near some u 0 = e 2πiω0 with ω 0 non-resonant: iii) The linear map b 1 (u 0 ) is an isomorphism. Then, if the answer to question 3.5 is positive, it follows from ii) that every ψ ω takes its values in {|u 1 | = • • • = |u d | = 1}, yielding a 3d-dimensional analogue of Figure 2, see [START_REF] Féjoz | Introduction to KAM theory with a view to celestial mechanics[END_REF][START_REF] Massetti | A normal form à la Moser for diffeomorphisms and a generalization of Rüssmann's translated curve theorem to higher dimensions[END_REF]. 15

Comments and references

My interest in this part of the program sketched in [START_REF]Singularities in dynamics: a catastrophic viewpoint[END_REF] awoke when I heard Abed Bounemoura talk about [START_REF] Bounemoura | A Diophantine duality applied to the KAM and Nekhoroshev theorems[END_REF].

The dimension of both parameter and phase space, minimal here, can be much higher 16 . Proposition 1.3 has been known (at least) to me for thirty years, as well as the "blown-up" version of Proposition 2.1. 17 I have no reference for the higher dimensional results in subsection 3.4. The excision of the coordinate hyperplanes in Propositions 3.1-3.2 corresponds to the closure of manifolds of periodic orbits of lower period, which might tend to (manifolds of) lower dimensional KAM tori à la Eliasson [START_REF] Eliasson | Perturbations of stable invariant tori for Hamiltonian systems[END_REF][START_REF] Féjoz | Introduction to KAM theory with a view to celestial mechanics[END_REF].

It is known that "good" periodic orbits accumulate on KAM tori. My naive hope is to do it the other way round and get the mysterious objects as limits of obvious ones, which would clarify a very intricate situation.

One of the sources of this article is an awfully biased reading of the two papers [START_REF] Chenciner | Bifurcations de points fixes elliptiques, I. Courbes invariantes[END_REF][START_REF]Bifurcations de points fixes elliptiques, II. Orbites périodiques et ensembles de Cantor invariants[END_REF] by Alain Chenciner, to whom my debt cannot be overestimated, though he certainly does not share my viewpoint that conservative systems are essentially meant to deny the existence of death (and birth. . . ).

Last but not least, I thank Jacques Féjoz and Laurent Stolovitch for very useful discussions and comments.

Figure 1 .

 1 Figure 1. Bifurcation and catastrophe.

8

  Conjugating everything by Zω, one can assume h ψω (z) (z) = e 2πiω z, hence (1.1) reads ψω(z) = ψω(e 2πiω z), which yields ψω(z) = ψω(e 2πikω z) for every integer k and therefore ψω(z) = ψω(e 2πiθ z) for all θ ∈ T by density.

Figure 2 .

 2 Figure 2.

3. 1 .

 1 Statement of the hypotheses Hypotheses. Given u 0 ∈ C d , d > 1, whose components are nonzero and all different, set ũ0 := (u 0 , 0) ∈ C d × C d and let h : (C d × C d , ũ0 ) → (C d , 0) be a smooth local map (u, x) → h u (x) such that h u (0) = 0 and Dh u (0) = diag u : z → u 1 z 1 , . . . , u d z d .

Note.

  The diagonal action e : (t, z) → e 2πi diag t z of T d on C d preserves diag ρ and lifts to the action ȇ : (t, r, y) → (r, e 2πi diag t y) =: ȇt (r, y) of T d on R + × S 2d-1 , which preserves diag ρ. The open subset Ω becomes ȇ-invariant (and still diag ρ-invariant) if it is replaced by t∈T d ȇt ( Ω), which contains {0} × S2d-1 and is open because T d is compact.

3. 3 .

 3 Passing to the limit in the holomorphic case? In the holomorphic case, if u 0 = (e 2πiω1 , . . . , e 2πiω d ), ω = (ω 1 , . . . , ω d ) ∈ T d , the following result may apply to h u0 : Theorem 3.3. Assume ω diophantine in the sense that, for some large τ , inf 1≤j≤d inf |k|≥2 |k| τ e 2πikω -e 2πiωj > 0,

Question 3 . 5 .

 35 Assume ω diophantine in the sense that, for some large τ ,inf m =0 |m| τ e 2πimω -1 > 0, where m ∈ Z d , |m| = m 1 + • • • + m d and mω = m 1 ω 1 + • • • + m d ω d . If b 1 (u 0

  i) If |u 1 | = • • • = |u d | = 1, the transformation h u preserves the standard symplectic form σ = 1 2i (dz 1 ∧ dz 1 + • • • + dz d ∧ dz d ).ii) One has thath u = h (u1/|u1|,...,u d /|u d |) • diag(|u 1 |, . . . , |u d |) and therefore h * u σ = 1 2i (|u 1 | 2 dz 1 ∧ dz 1 + • • • + |u d | 2 dz d ∧ dz d ).

The "blown-up" surface W = graph ψ in polar coordinates is smooth, see section 3.

In the standard j th local chart of CP d-1 , this blowup reads z j , (w k ) k =j → z with z k = z j w k for k = j; the "forbidden" closed subset is the union of the hyperplanes w k = 0.

If one prefers, ψ has more and more contact with ψp/q • π at points of π -1 (0).