Aurore Aubail
email: aaubail@yahoo.fr

Rune Dietz

Frank Rigét

Benoit Simon-Bouhet

Benoît Simon-Bouhet

Florence Caurant

An evaluation of teeth of ringed seals (Phoca hispida) from Greenland as a matrix to monitor spatial and temporal trends of mercury and stable isotopes

Keywords: mercury, stable isotopes, carbon, nitrogen, ringed seal, Phoca hispida, teeth, spatial and temporal trends

An evaluation of teeth of ringed seals (Phoca hispida)

from Greenland as a matrix to monitor spatial and temporal trends of mercury and stable isotopes.

Introduction

Mercury (Hg) is a global contaminant which is emitted to the atmosphere from both natural and anthropogenic sources all around the world [START_REF] Pacyna | Global emission of mercury from anthropogenic sources in 1995[END_REF]. The important volatility of the elementary form of Hg allows this metal to be long range transported [START_REF] Brooks | Transport pathways and processes leading to environmental exposure[END_REF], from its sources of emission to the most remote areas such as the Arctic regions [START_REF] Pacyna | Global emission of mercury from anthropogenic sources in 1995[END_REF]. The recently discovered phenomenon Atmospheric Mercury Depletion Event demonstrates that the cycle of Hg in the Arctic is complex, and involves processes at the polar sunrise that lead to the production of bio-available forms of Hg [START_REF] Brooks | Transport pathways and processes leading to environmental exposure[END_REF].

Mercury is known to accumulate in organisms and to biomagnify up the food chains [START_REF] Caurant | Mercury in pilot whales: Possible limits to the detoxification process[END_REF][START_REF] Dietz | Lead, cadmium, mercury and selenium in Greenland marine animals[END_REF][START_REF] Rigét | Temporal trends of mercury in marine biota of west and northwest Greenland[END_REF]. Due to its high position in the marine Arctic food chain and its long life span, the ringed seal (Phoca hispida) accumulates particularly great concentrations of Hg in its tissues [START_REF] Dietz | Lead, cadmium, mercury and selenium in Greenland marine animals[END_REF][START_REF] Dietz | Cadmium, mercury, zinc and selenium in ringed seals (Phoca hispida) from Greenland and Svalbard[END_REF], though with a large variability resulting from biological and ecological influences [START_REF] Aguilar | Biological factors affecting variability of persistent pollutant levels in cetaceans[END_REF]. This species has a circumpolar distribution and is thought not to be migratory [START_REF] Maxwell | Other seals of the northern hemisphere[END_REF], although some subadults have been reported to disperse over considerable distances (> 1000 km), as documented by [START_REF] Reeves | The Sierra Club Handbook of Seals and Sirenians[END_REF] and [START_REF] Kapel | Netting and conventional tagging used to study movements of ringed seals (Phoca hispida) in Greenland[END_REF]. Because of this supposed high degree of site fidelity and its ubiquitous occurrence, the ringed seal is a good candidate for monitoring contaminants for both spatial and temporal trends in Arctic marine food chains, and has therefore been selected by the international AMAP, Arctic Monitoring and Assessment Programme, as an essential monitoring species (AMAP, 2003).

Within the framework of the AMAP and other programmes, samples of ringed seals such as liver, kidney, muscle, and jaws have been collected from the local Inuit subsistence hunting in different regions of Greenland through the last twenty-five years. The liver is the main organ for Hg accumulation in marine mammals including pinnipeds [START_REF] Wagemann | Concentrations of heavy metals and organochlorines in marine mammals of northern waters: overview and evaluation[END_REF], and most studies dealing with Hg in seals have been carried out on liver among other soft tissues [START_REF] Dietz | Cadmium, mercury, zinc and selenium in ringed seals (Phoca hispida) from Greenland and Svalbard[END_REF][START_REF] Rigét | Circumpolar pattern of mercury and cadmium in ringed seals[END_REF][START_REF] Wagemann | Comparison of heavy metals in 2 groups of ringed seals (Phoca hispida) from the Canadian Arctic[END_REF].

The tooth is a hard tissue that presents the advantage, compared to soft tissues, to be long term preserved in ancient material, and to provide a good chemical record of life-history events of the individual. Indeed, Hg is incorporated with other trace elements into the crystalline apatite structure of teeth during their mineralisation [START_REF] Tvinnereim | Heavy metals in human primary teeth: some factors influencing the metal concentrations[END_REF]. As the teeth are relatively stable tissues, the incorporated elements are thought not to be remobilised, or very little, throughout the life of individuals (Eide et al., 1993;[START_REF] Haller | Dentin as a possible bioepidemiologic measure of exposure to mercury[END_REF][START_REF] Tvinnereim | Heavy metals in human primary teeth: some factors influencing the metal concentrations[END_REF]. Thus, because teeth appear to be excellent archives, they have been investigated for their content of trace elements in rats (Eide and Wesenberg, 1993), marine mammals [START_REF] Caurant | Lead contamination of small cetaceans in European waters -the use of stable isotopes for identifying the sources of lead exposure[END_REF][START_REF] Outridge | Teeth as biomonitors of soft tissue mercury concentrations in beluga Delphinapterus leucas[END_REF][START_REF] Outridge | A comparison of modern and preindustrial levels of mercury in the teeth of beluga in the Mackenzie Delta, Northwest territories, and walrus at Igloolik, Nunavut, Canada[END_REF][START_REF] Outridge | Long-term changes of mercury levels in ringed seal (Phoca hispida) from Amundsen Gulf, and beluga (Delphinapterus leucas) from the Beaufort Sea, western Canadian Arctic[END_REF], and humans (Eide et al., 1993;[START_REF] Tvinnereim | Heavy metals in human primary teeth: some factors influencing the metal concentrations[END_REF], and are considered to be reliable indicators of Hg exposure (Eide and Wesenberg, 1993;[START_REF] Outridge | Teeth as biomonitors of soft tissue mercury concentrations in beluga Delphinapterus leucas[END_REF][START_REF] Tvinnereim | Heavy metals in human primary teeth: some factors influencing the metal concentrations[END_REF].

Teeth have been used also to measure ratios of the naturally occurring stable isotopes of carbon ( 13 C/ 12 C) and nitrogen ( 15 N/ 14 N), which have demonstrated to provide insights into mammals feeding ecology [START_REF] Hobson | Investigating trophic relationships of pinnipeds in Alaska and Washington using stable isotope ratios of nitrogen and carbon[END_REF][START_REF] Hobson | Determination of trophic relationships within a high arctic marine food web using δ 13 C and δ 15 N analysis[END_REF]. The stable isotopic technique is based on the metabolic discrimination between the heavy and the light isotopes. Consumers are enriched in 15 N relative to their food and consequently δ 15 N measurements serve as indicators of a consumer trophic position [START_REF] Tieszen | Fractionation and turnover of stable carbon isotopes in animal tissues -implications for δ 13 C analysis of diet[END_REF]. By contrast, δ 13 C values vary little along the food chain and are mainly used to determine primary sources in a trophic network [START_REF] Deniro | Influence of diet on distribution of carbon isotopes in animals[END_REF]. In the marine environment, δ 13 C values can also indicate inshore versus offshore, or pelagic versus benthic, contribution to food intake (Hobson et al. 1994). The recent combined use of elemental and isotopic measurements appears to be a useful tool for the study of marine mammal ecology [START_REF] Das | Marine mammals from Northeast Atlantic: relationship between their trophic status as determined by δ 13 C and δ 15 N measurements and their trace metal concentrations[END_REF][START_REF] Hobson | Baleen as a biomonitor of mercury content and dietary history of North Atlantic Minke Whales (Balaenoptera acutorostrata): combining elemental and stable isotope approaches[END_REF] providing information on diet, geography, and trophic transfer of contaminants. Positive correlations between δ 15 N values and Hg concentrations have been described within Arctic marine food webs [START_REF] Atwell | Biomagnification and bioaccumulation of mercury in an arctic marine food web: insights from stable nitrogen isotope analysis[END_REF][START_REF] Rigét | Temporal trends of mercury in marine biota of west and northwest Greenland[END_REF], suggesting that part of the Hg variation may be linked to a biomagnification process, and several recent studies have been combining Hg and stable isotopes measurements to assess feeding variations for marine mammals [START_REF] Born | Population substructure of North Atlantic minke whales (Balaenoptera acutorostrata) inferred from regional variation of elemental and stable isotopic signatures in tissues[END_REF][START_REF] Dietz | Regional and inter annual patterns of heavy metals, organochlorines and stable isotopes in narwhals (Monodon monoceros) from West Greenland[END_REF][START_REF] Hobson | Baleen as a biomonitor of mercury content and dietary history of North Atlantic Minke Whales (Balaenoptera acutorostrata): combining elemental and stable isotope approaches[END_REF].

In this study, total Hg concentrations and δ 13 C and δ 15 N values have been determined in teeth of ringed seals from central West and central East Greenland. The aims of this work were: (i) to investigate the influence of biological factors such as gender or age, (ii) to assess potential temporal and spatial trends of Hg with support provided by the stable isotopes, and (iii) to evaluate tooth as a Hg monitoring matrix for this species.

Materials and methods

Sampling procedure and preparation

Tissue samples were obtained from 285 ringed seals collected from the traditional Inuit hunt in Qeqertarsuaq, Disko Bay, located at 69° 50′ N, 54° 00′ W (central West Greenland, CWG; n = 115), and in Ittoqqortoormiit, Scoresbysund, located at 70° 50′ N, 22° 00′ W (central East Greenland, CEG; n = 170). Sampling took place in May and June, in eight years within the last 25 years [START_REF] Zar | Biostatistical analysis: International edition[END_REF]and 1986for CWG and CEG respectively and 1994[START_REF] Aguilar | Biological factors affecting variability of persistent pollutant levels in cetaceans[END_REF], 2000, 2001[START_REF] Hohn | Age estimation[END_REF], 2004and 2006 in common to both regions) (Fig. 1, Table 1). Because of a greater hunting pressure in the Disko Bay area, the sampling was biased, and mainly juvenile seals were obtained from this area.

Prior to tissue sampling, the animals were measured, and other information such as the date of collection, location, and gender, were recorded. Soft tissues (liver, kidney, muscle) were sampled, and the results of Hg analyses carried out have been presented partly by [START_REF] Dietz | Cadmium, mercury, zinc and selenium in ringed seals (Phoca hispida) from Greenland and Svalbard[END_REF] and [START_REF] Rigét | Levels and spatial and temporal trends of contaminants in Greenland biota: an updated review[END_REF][START_REF] Rigét | Temporal trends of mercury in marine biota of west and northwest Greenland[END_REF]. In addition, the lower jaws were sampled, and later on went through a maceration process at the Zoological Museum in Copenhagen, Denmark, in order to be totally cleaned from flesh. Then, the bone tissues (mandibles) and the teeth were dried and stored in plastic bags or pap boxes until use.

The age of the animals was determinate at the National Environmental Research Institute (NERI, Roskilde, Denmark), by counting annual layers in the cementum of the canine teeth, after decalcification, thin sectioning (14 µm), and staining in toluidine blue as described by [START_REF] Dietz | Age determination in European Harbour seals Phoca vitulina[END_REF].

The second molar (from the front part) was taken from the lower right mandible for the Hg analytical purpose (n = 285), while the premolar of the same mandible was extracted for the stable isotopes analyses (n = 238).

Prior to analysis, each tooth was immersed in 10% nitric acid for 20s, and rinsed in several ultra-pure Milli-Q water baths. Teeth samples were dried for a minimum of 24h at room temperature, and subsequently stored in cleaned plastic flasks.

Analytical procedures and instrumentation for Hg analyses

Entire dry molar teeth were analysed for total Hg (Hg). The Hg measurements were performed on the whole molar teeth, at the laboratory of the NERI, using a solid sample atomic absorption spectrometer AMA-254 (Advanced Mercury Analyser-254 from LECO, Sweden).

The use of this apparatus does not require a chemical pre-treatment which reduces considerably contamination risks or loss of Hg. The analytical process consists in a drying period at 120°C, prior to a combustion phase at 750°C which leads to the desorption of Hg from the samples. Subsequently, the Hg vapour produced is carried by an oxygen flow to a gold amalgamator, and trapped on its surface. Thereafter, the collected Hg is released from the amalgamator by a short heat-up to 900°C, and carried in a pulse through a spectrophotometer, where Hg is measured by UV absorption. The operating times used for this study, i.e. drying time, decomposition time, and waiting time, were respectively 50, 200, and 45s. The instrument is described in detail elsewhere [START_REF] Hall | Evaluation of a direct solid sampling atomic absorption spectrometer for the trace determination of mercury geological samples[END_REF].

As there is no commercial reference material with a matrix similar to teeth or bones and certified for Hg, a reference material was made from two commercial Standard Reference Materials (SRMs), the NIST 1400 Bone ash (National Institute of Standards and Technology, USA), and the DOLT-3 (Dogfish liver from the National Research Council of Canada). The Bone Ash SRM does not contain any Hg because it has been calcinated at great temperatures, but it brings the similar matrix to calcified tissues, while the DOLT-3 brings the organic matrix and the certified concentration of Hg. The adding of 0.96 g of DOLT-3 (Certified Hg concentration of 3.37 ± 0.14 in µg.g -1 dry weight) to 31.39 g of Bone Ash leads theoretically to a Hg concentration of 0.100 ± 0.004 µg.g -1 dry weight.

The calibration of the customised reference material has been carried out through two different techniques (AMA-254 and cold-vapour Atomic Absorption Spectrophotometry on a Perkin Elmer FIMS 100) in NERI. Moreover, an inter laboratory comparison was carried out for AMA-254 analysis between NERI and CCA (Centre Commun d'Analyses, University of La Rochelle, France). The measurement results of the customised reference material show a good accuracy (i.e. ratio between the recovery measured concentration and the theoretical concentration) of 102%, and a relative standard deviation of 6.3% (Table 2).

The analytical quality of the Hg measurements by the AMA-254 was ensured by including the customised reference material mentioned above at the beginning and at the end of each analytical cycle of 10 samples. The Hg measurements of this reference material were in good agreement with the assigned concentration (accuracy of 103.6% and relative standard deviation of 5.6% for n = 80).

The laboratory at NERI participates in the international inter-laboratory comparison exercises conducted by the EEC (QUASIMEME), and the latest 2007 results concerning analyses by AMA-254 were satisfactory (0 < z < 0.5).

All data are presented on a dry weight basis (dw), and the detection limit is 0.01 ng.g -1 dw.

Analytical procedures and instrumentation for stable isotope measurements

Each entire premolar was crushed into small pieces before being ground into homogenous powder using a ball mill (Retsch MM2000) for 2 minutes and the amplitude of 90. Then, powder was stored in small glass flasks. Carbonates of the teeth were removed by digesting the teeth with approximately 1 mL of a 4M-hydrochloric acid solution at 45°C for 48h. The digested contents were taken up in milli-Q ultrapur quality water, and homogenised before freezing to -20°C. Then, the samples were frozen at -80°C for a short time before freezedrying. Finally, an aliquot of each homogenised dried sample was taken, weighted, and loaded into tin capsules. Out of 285 individuals, only 238 samples were available for stable isotope analysis.

Relative abundance of stable isotopes of carbon ( 13 C and 12 C) and nitrogen ( 15 N and 14 N) were determined with an elemental analyser connected on-line to an isotope-ratio mass spectrometer (Micromass, Manchester, UK). Stable isotope results are expressed in delta notation (δ), defined as the part per thousand (‰) deviation from a standard material: δ 13 C or δ 15 N = [(Rsample / Rstandard) -1] × 10 3 , where R is 13 C/ 12 C or 15 N/ 14 N where Rsample and Rstandard are the fractions of heavy to light isotopes in the sample and standard, respectively. The international standards are the Pee Dee Belemnite (PDB) marine fossil limestone formation from South Carolina for δ 13 C, and atmospheric nitrogen for δ 15 N.

Data treatment

Prior to the statistical analyses, the Hg data (n = 285) were logarithmic-transformed (base e) to reduce skewness, and fit parametric requirements. Shapiro-Wilk and Bartlett test were applied to test the assumptions of parametric tests, such as analysis of variance (ANOVA) and linear regression analysis. In few cases, the assumptions were not fulfilled because of a couple of high Hg concentration values. However, ANOVA tests are robust to small deviations of the data from the normal distribution [START_REF] Zar | Biostatistical analysis: International edition[END_REF].

Standard parametric tests such as Pearson's correlation analysis, linear regression analysis, and ANOVA, were applied to test for the influence of factors (i.e. age, gender, location, and year) on the Log-transformed Hg concentrations or on the stable isotopic values. If the first runs of ANOVA test showed no significance of the interaction between two factors, the interaction factor was removed, and the test re-run.

The relationship between age and dental Hg concentrations or stable isotopic values were evaluated for each region with a scatterplot using LOWESS (Locally weighted polynomial regression) smoothing. In the same way, the analyses of temporal trends of Hg and stable isotopic values first were evaluated with scatterplots using LOWESS smoothing separately for each region, thus Log-linear regression analysis was applied to describe the linear component, and tested by means of an ANOVA.

The significance level was set to p = 0.05.

The statistical analyses were performed using the free software R, version 2.1.1 (R Development Core Team, 2008).

Results

Concentrations of Hg in molar teeth and influence of the biological factors, gender and age

Dental Hg concentrations ranged from 0.43 to 10.99 ng.g -1 dw for ringed seals from CWG and from 0.95 to 57.6 ng.g -1 dw for animals from CEG. Arithmetic mean (± standard deviation) was of 2.94 ± 1.99 ng.g -1 dw for CWG and 5.75 ± 6.20 ng.g -1 dw for CEG. Details for each location and sampling year are reported in Table 1. The coefficients of variation of dental Hg concentrations were very high for the CEG seals collected in 1999 (191%) and 2006 (197%) (due to two individuals displaying great Hg concentrations in those years, 49.5 ng.g -1 dw and 57.6 ng.g -1 dw, respectively), while they ranged from 21 to 74% for the other years for both regions.

No gender-related difference in dental Hg concentrations was observed (one-way ANOVA, p = 0.726), hence this parameter was discharged as a variable in the further statistical treatment.

The mean age was significantly lesser in seals from CWG (2.7 years old) than in seals from CEG (6.0 years old) (see Table 1) (One-way ANOVA, p < 0.001). Furthermore, no significant difference in mean age was observed for seals in CWG through the years (One-way ANOVA, p = 0.056) whereas the mean age differed significantly among sampling years of the ringed seals from CEG (One-way ANOVA, p = 0.028), with the greatest mean ages in 1986 and 2006, and the lowest in 2000 and 2004.

The plot smoother showed a clear non-linear relationship between age and Hg concentrations, with a decreasing trend in dental Hg concentrations for the first 5-7 years of age of the individuals from both regions (Fig. 2). The slope seemed thereafter to level off in older individuals, and a plateau is reached. Therefore the Linear Regression Model was run on two age groups of ringed seals, juveniles up to 5 years old and adults above 5 years old. The models showed that there was a significant regression between age and the Log-transformed Hg concentrations for the juveniles (≤ 5 years old) from both CWG (p < 0.001), and CEG (p = 0.024), with a steeper decrease observed for the West Greenland seals. However, no linear relation was found for adults from CEG (p = 0.961), while too few adults were available to test the age relationship for adult ringed seals from CWG.

Stable isotopes in teeth, influence of gender and age, and relation between elemental and isotopic values

Stable isotopic ratios of carbon and nitrogen determined in the ringed seal teeth exhibited respective mean values of -16.04 ± 0.52‰ and 16.35 ± 0.99‰ for CWG, and -17.23 ± 0.53‰ and 14.90 ± 1.14‰ for CEG (range of values, see Table 3). Both ratios were correlated across all individuals (r = 0.4, p < 0.001), but not when investigating CWG and CEG data separately.

Gender was shown to have no influence on the stable isotopic values, neither for δ 13 C (Oneway ANOVA, p = 0.74) nor for δ 15 N (p = 0.35). However, age was correlated significantly with both δ 13 C and δ 15 N values for seals from CWG (r = 0.25, p < 0.05 and r = -0.29, p < 0.05, respectively), and with δ 13 C values for seals from CEG (r = 0.42, p < 0.001) (Fig. 2). The δ 15 N age trends were observed on the graphs to be quite similar to the Hg ones while the δ 13 C relation with age showed an opposite trend to the Hg and δ 15 N ones.

Log-transformed Hg concentrations and δ 13 C and δ 15 N values were not correlated for the ringed seals from both CWG (p = 0.15 and p = 0.21, respectively) and CEG (p = 0.20 and p = 0.87, respectively), when the two regions were studied separately. However, when CWG and CEG data were pooled, relations between Log-transformed Hg concentrations and δ 13 C values (p < 0.001), and Log-transformed Hg concentrations and δ 15 N values (p < 0.01), were found to be significant.

These statistical results are summarised in Table 4.

Spatial and temporal trends

Although the plot smoother showed an overall decreasing trend in dental Hg concentrations for the first 5 to 7 years of age of the seals from both regions, the Hg concentrations measured in the first two years of life of the animals were observed to be quite stable. Therefore, in order to limit the age effect and to avoid the overlapping of the life periods of the seals, we decided to focus on the dental Hg concentrations of the youngest individuals, up to 2 years old, when assessing the spatial and temporal trends of Hg concentrations and stable isotopic ratios.

A spatial difference in Hg concentrations was found (Two-factor nested ANOVA (location nested sampling year), p < 0.001), with significantly greater concentrations in the seals from CEG compared to CWG (Table 1). However, this spatial trend was not consistent among years (p < 0.001).

Stable isotopic values differed significantly between regions, with both greater δ 13 C (p < 0.001) and δ 15 N values (p < 0.001) in CWG compared to CEG (Table 3). However, while the spatial trend in δ 13 C was consistent over time (p = 0.071), this was not found to be the case for the spatial trend in δ 15 N values (p = 0.039).

The temporal trends of dental Hg concentrations of ringed seals are shown on Fig. 3. The temporal trends were very similar for both regions, with an apparent decrease of Hg concentrations until the 1990's, followed by an increase until 2006, although the Hg dental concentrations of CEG seals were actually lesser in 2006 than in 2004. A Linear Regression Model was applied between 1994 and 2006 for seals from both CWG and CEG, and revealed a significant increase of dental Hg concentrations for this period in seals from both regions (p < 0.001 and p = 0.018, respectively).

Concerning the stable isotopes, the δ 13 C values of the ringed seals from CEG were showed to follow a significant decrease over time (4.7% per year, p < 0.001, Fig. 3), while δ 15 N values found in those animals were not (p = 0.65). Finally, neither δ 13 C nor δ 15 N values of the ringed seals from CWG did follow a temporal trend (p = 0.10 and p = 0.81, respectively).

Discussion

Influence of age on Hg concentrations in teeth

Mammalian teeth are composed of hydroxyapatite Ca 10 (PO 4 ) 6 (OH) 2 , and comprise three types of structures, i.e. cementum, dentine, and enamel. Cementum and dentine grow in well defined layers throughout life, and those layers are known to deposit annually in marine mammals [START_REF] Miyazaki | Encyclopedia of marine mammals[END_REF]. Mercury is believed to be deposited directly from the blood supply or the dentinal tubules into actively growing increments [START_REF] Haller | Dentin as a possible bioepidemiologic measure of exposure to mercury[END_REF], and not to be remobilised throughout life of individuals (Eide et al., 1993;[START_REF] Tvinnereim | Heavy metals in human primary teeth: some factors influencing the metal concentrations[END_REF].

Despite a low affinity of Hg for the teeth, previous studies have demonstrated that this element can be measured reliably in teeth (Eide and Wesenberg, 1993;[START_REF] Tvinnereim | Heavy metals in human primary teeth: some factors influencing the metal concentrations[END_REF]. The dental Hg concentrations found in our study appear to be in the same order of magnitude than those reported in the current literature for marine mammals, i.e. ringed seal, with Hg mean values ± sd of 4.4 ± 1.7 to 8.6 ± 3.7 ng.g -1 dw in animals 5-25 years old [START_REF] Outridge | Long-term changes of mercury levels in ringed seal (Phoca hispida) from Amundsen Gulf, and beluga (Delphinapterus leucas) from the Beaufort Sea, western Canadian Arctic[END_REF], walrus (Odobenus rosmarus rosmarus), with Hg concentrations under 2 ng.g -1 dw [START_REF] Outridge | A comparison of modern and preindustrial levels of mercury in the teeth of beluga in the Mackenzie Delta, Northwest territories, and walrus at Igloolik, Nunavut, Canada[END_REF], and beluga (Delphinapterus leucas), with Hg mean values ± sd of 70.92 ± 45.35 ng.g -1 dw [START_REF] Kinghorn | Teeth as biomonitors of selenium concentrations in tissues of beluga whales (Delphinapterus leucas)[END_REF] and 98.4 ± 109 ng.g -1 dw [START_REF] Outridge | A comparison of modern and preindustrial levels of mercury in the teeth of beluga in the Mackenzie Delta, Northwest territories, and walrus at Igloolik, Nunavut, Canada[END_REF].

However, the trends of dental Hg concentrations with the age, described in the literature and in the present study, differ. Indeed, [START_REF] Outridge | Long-term changes of mercury levels in ringed seal (Phoca hispida) from Amundsen Gulf, and beluga (Delphinapterus leucas) from the Beaufort Sea, western Canadian Arctic[END_REF] found that Hg concentrations measured in the canine tooth (two upper thirds) of Canadian ringed seal was correlated positively with the age of the animal, whereas, in contrary, a significant decrease of dental Hg concentrations was observed the first years of life for the ringed seals in our study (p < 0.001 for CWG, and p = 0.024 for CEG; Fig. 2). In addition, [START_REF] Outridge | Teeth as biomonitors of soft tissue mercury concentrations in beluga Delphinapterus leucas[END_REF] showed that Hg concentrations in the tooth cementum increase with the age in beluga whales, while [START_REF] Kinghorn | Teeth as biomonitors of selenium concentrations in tissues of beluga whales (Delphinapterus leucas)[END_REF] did find no significant relation between age and Hg concentrations in the same tissue for the same species. Finally, [START_REF] Outridge | A comparison of modern and preindustrial levels of mercury in the teeth of beluga in the Mackenzie Delta, Northwest territories, and walrus at Igloolik, Nunavut, Canada[END_REF] reported an absence of correlation between age and cementum Hg concentrations in the walrus.

From this literature overview on the dental Hg in marine mammals, it seems that age accumulation of the Hg concentrations in the teeth differs within species (in relation to their dentition development), but also with the part of tooth (dentine, cementum or the entire tooth) or even the type of tooth (molar or canine, which may grow at distinct times during the dentition development) on which the Hg measurements are carried out.

Moreover, as marine mammal exposure to Hg occurs mainly via food [START_REF] Aguilar | Biological factors affecting variability of persistent pollutant levels in cetaceans[END_REF], the variation in dental Hg concentrations throughout life is expected to be related to changes in dietary regime with age. Ringed seals are diverse feeders on planktonic, nektonic, and benthic organisms, with a diet ranging from pelagic amphipods to different fishes, as polar cod (Boreagadus saida) or Arctic cod (Arctogadus glacialis) in the Greenland waters [START_REF] Siegstad | Diet of the ringed seal (Phoca hispida) in Greenland[END_REF]. Occasionally, immature ringed seals explore the water column down to the sea floor, and spend much of their time foraging at depths shallower than 50m [START_REF] Born | Habitat use of ringed seals (Phoca hispida) in the North Water Area (North Baffin Bay)[END_REF], showing a larger preference for crustaceans than adults [START_REF] Holst | Diet of ringed seals (Phoca hispida) on the east and west sides of the north water polynya, northern Baffin Bay[END_REF]. Furthermore, the diving capabilities of ringed seals increase with age, so that older and larger seals dive in general deeper than younger seals, consuming potentially more fish like polar cod while reducing the consumption of crustaceans [START_REF] Holst | Diet of ringed seals (Phoca hispida) on the east and west sides of the north water polynya, northern Baffin Bay[END_REF][START_REF] Reeves | The Sierra Club Handbook of Seals and Sirenians[END_REF].

However, this foraging behaviour does not explain why adults have lesser Hg concentrations compared to juvenile ringed seals. Crustaceans contain lesser Hg concentrations than fishes in Greenland waters [START_REF] Dietz | Lead, cadmium, mercury and selenium in Greenland marine animals[END_REF], and therefore, a greater crustacean intake of the younger individuals can not be responsible for greater Hg concentrations in teeth of juveniles compared to adult seals. In addition, the significant negative correlations found between age and δ 15 N values, for CWG individuals and for all ringed seals (Table 4, Fig. 2), do not match with the above mentioned crustacean-diet for juveniles and fish-diet for adults. Since the mineralisation of the permanent dentition occurs at the foetal stage for the ringed seal [START_REF] Reeves | The Sierra Club Handbook of Seals and Sirenians[END_REF][START_REF] Stewart | Mandibular dental ontogeny of ringed seals (Phoca hispida)[END_REF], a consistent explanation to both great Hg concentrations and δ 15 N values for the first years of life would be the dietary inputs from the mother during the period of gestation and lactation. Indeed, Hg is known to be transferred from the mother, to the foetus through the placenta, and to the newborn via milk [START_REF] Grandjean | Methylmercury: significance of intrauterine and postnatal exposures[END_REF]. Both transfers have been reported previously for seals, showing a more quantitatively significant Hg acquisition during the prenatal stage, compared to the postnatal one [START_REF] Wagemann | Trace-metals and methyl mercuryassociations and transfer in harp seal (Phoca groenlandica) mothers and their pups[END_REF].

Furthermore, the enriched δ 15 N values, measured in the first years of age, tend to support the hypothesis of these maternal transfers. Indeed, δ 15 N of a consumer are related to those of its prey [START_REF] Deniro | Influence of diet on distribution of carbon isotopes in animals[END_REF], and an enrichment of δ 15 N, of the order of 3‰ generally, occurs at each trophic transfer [START_REF] Hobson | Determination of trophic relationships within a high arctic marine food web using δ 13 C and δ 15 N analysis[END_REF]. Thus, the enriched δ 15 N values displayed by the youngest seals would reflect their consumer position relative to their mothers. Similarly to our results, enriched δ 15 N values in the early life have been previously reported in the teeth of other marine species, such as Steller sea lion (Eumetopias jubatus) [START_REF] Hobson | Stable isotope analyses of tooth annuli reveal temporal dietary records: An example using Steller sea lions[END_REF], longbeaked common dolphin (Delphinus capensis) [START_REF] Nino-Torres | Isotopic analysis of δ 13 C, δ 15 N, and δ 34 S "a feeding tale" in teeth of the longbeaked common dolphin, Delphinus capensis[END_REF], and narwhal (Monodon monoceros) [START_REF] Dietz | Regional and inter annual patterns of heavy metals, organochlorines and stable isotopes in narwhals (Monodon monoceros) from West Greenland[END_REF].

In addition, the depleted δ 13 C values observed in teeth of youngest individuals (Fig. 2) also corroborate the idea of a maternal transfer to foetus and young of the year. Indeed, body fat has been shown to be depleted in 13 C compared to other animal tissues due to selective fractionation [START_REF] Tieszen | Fractionation and turnover of stable carbon isotopes in animal tissues -implications for δ 13 C analysis of diet[END_REF], and the depleted δ 13 C values of the juvenile seals may reflect likely the mobilisation and the transfer of the carbon from maternal lipid stores, for the foetal development and milk production [START_REF] Dehn | Feeding ecology of phocid seals and some walrus in the Alaskan and Canadian Arctic as determined by stomach contents and stable isotope analysis[END_REF][START_REF] Polischuk | Use of stable-carbon and -nitrogen isotopes to assess weaning and fasting in female polar bears and their cubs[END_REF]. The same δ 13 C trend has been described previously by [START_REF] Hobson | Stable isotope analyses of tooth annuli reveal temporal dietary records: An example using Steller sea lions[END_REF] in the teeth of Steller sea lions, and explained by the transfer of maternal body lipids during the nursing period.

Following the great Hg concentrations observed in the teeth in early life, Hg concentrations are observed to drop for individuals from both regions (Fig. 2). This age Hg trend can be explained by a progressive "dilution" of the dentine deposited at the foetal and weaning stages by the addition of new post weaning dentine. In addition, the deposition of dentine, and its subsequent mineralisation, reduces the pulp area, and in some species, the pulp cavity becomes occluded at some point with subsequent suspension of dentinal accumulation [START_REF] Hohn | Age estimation[END_REF]. Thus, the plateau of Hg concentrations reached around 10 years old could actually reflect the time of the tooth occlusion. Indeed, up to 15-20 dentinal layers have been observed previously in the canines of ringed seals [START_REF] Stewart | Counts of growth layer groups in cementum and dentine in ringed seals (Phoca hispida)[END_REF], and the molars being smaller than canine teeth, it is likely that their occlusion would occur earlier in life.

Spatial variations of dental Hg concentrations in Greenland

As a general trend, Hg concentrations in teeth of ringed seals from Ittoqqortoormiit (CEG) were found to be significantly greater than those from Qeqertarsuaq (CWG). The same spatial difference in Hg concentrations between CEG and CWG has been reported previously for Hg concentrations in livers of ringed seals [START_REF] Dietz | Cadmium, mercury, zinc and selenium in ringed seals (Phoca hispida) from Greenland and Svalbard[END_REF][START_REF] Rigét | Circumpolar pattern of mercury and cadmium in ringed seals[END_REF]. Such a spatial variation can be caused by either differences in diet or differences in Hg concentrations in the environment among seals of both regions.

Feeding habits of the ringed seal have been documented to vary both spatially and temporally [START_REF] Reeves | The Sierra Club Handbook of Seals and Sirenians[END_REF][START_REF] Siegstad | Diet of the ringed seal (Phoca hispida) in Greenland[END_REF], which makes difficult to estimate the dietary Hg exposure of the ringed seals. However, differences in diet have been previously suggested by [START_REF] Dietz | Cadmium, mercury, zinc and selenium in ringed seals (Phoca hispida) from Greenland and Svalbard[END_REF], as a likely reason for such a spatial difference in Hg concentrations in ringed seals. In a study on stomach content of Greenland ringed seals, [START_REF] Siegstad | Diet of the ringed seal (Phoca hispida) in Greenland[END_REF] found both arctic and polar cods (65% by weight) to be important food items of this species in the area of Kong Oscars Fjord (CEG), while capelin (Mallotus villosus) and redfish (Sebastes spp.) (52.2 and 25.2% weight, respectively) have been reported to be among the main food items in the diet of ringed seals from Qeqertarsuaq area (CWG). Nevertheless, it is worth mentioning that these diets may only reflect seasonal preys since they have been determinate from seal stomachs collected in spring for the CWG individuals, and in summer for the CEG ones. [START_REF] Møller | Lipids and stable isotopes in marine food webs in West Greenland[END_REF] investigated stable isotopes in the CWG food webs and reported greater δ 15 N values for both Cottidae, arctic and polar cods, than for capelins or redfishes. Thus, although arctic cod exhibits a greater Hg content than other Greenland fish species [START_REF] Dietz | Lead, cadmium, mercury and selenium in Greenland marine animals[END_REF], which would have explained the greater Hg concentrations in teeth of ringed seals from the CEG, our δ 15 N isotopic data are paradoxically not in agreement with this difference in diet among seals from both areas, since δ 15 N values were lesser in CEG seals (Cottidae consumers) than in those from CWG (Table 3). On the contrary, δ 15 N values in teeth would be the diet's signature of the entire life of the individual. Thus, on a long time period, the seals would be more exposed to Hg in the CEG by eating at lower trophic levels, which implies that the food webs would be more contaminated in the CEG than in the CWG.

Stable isotopes of carbon provide information about the sources of primary productivity in ecosystems where distinct isotopically sources are present (for example phytoplankton vs. ice algae). Indeed, the 13 C enrichment of benthic compared to planktonic algae is substantial enough to allow distinguishing between benthic and planktonic food sources for coastal animals [START_REF] France | Differentiation between littoral and pelagic food webs in lakes using stable carbon isotopes[END_REF]; so that the more depleted δ 13 C values found in CEG seals could suggest a more pelagic diet for these animals compared to the CWG ones. But most likely, this difference in δ 13 C values can be attributed to distinct oceanography and characteristic spatial δ 13 C values of two different ecosystems. Indeed, the West Greenland area is influenced by a mixture of waters from the cold East Greenland Current and the warmer and more saline Irminger Current (a side branch of the Gulf stream, North Atlantic current), while the East Greenland is dominated by the East Greenland Current that brings cold low-saline polar water south along the eastern coast of Greenland [START_REF] Born | Population substructure of North Atlantic minke whales (Balaenoptera acutorostrata) inferred from regional variation of elemental and stable isotopic signatures in tissues[END_REF]. Previously, the temperature has been described as an important factor of variation of δ 13 C [START_REF] Rau | Plankton 13 C/ 12 C ratio changes with latitude -differences between Northern and Southern oceans[END_REF], resulting in a 13 C depletion with high latitudes. Thus, the warmer temperatures of the western Greenland waters may likely lead to more enriched δ 13 C values of the CWG food web, and explain the greater δ 13 C values found in the tissues of CWG ringed seals compared to the CEG ones.

Mercury is known to be transported on long distance in the atmosphere, and [START_REF] Pacyna | Sources of mercury in the Arctic[END_REF] described Europe, Asia and North America as the regions where are located most global and regional anthropogenic emissions of Hg with potential impact on the Arctic environment. The seasonal variability in long-range atmospheric transport to Greenland has been well demonstrated [START_REF] Kahl | Air mass trajectories to Summit, Greenland: a 44-year climatology and some episodic events[END_REF], and the influence of this transport may vary regionally, result in spatial differences in the deposition of atmospheric Hg, and subsequently, contribute to the spatial differences in Hg concentrations in the regional ecosystems of Greenland. However, it has also been reported that some natural geological formations may actually contribute substantial amounts of Hg to the surrounding environment [START_REF] Painter | Reconnaissance geochemistry and its environmental relevance[END_REF]. Thus, spatial differences in Hg concentrations found in seals and other marine mammals in the global Arctic have been thought to be related directly to the natural input from underlying geology [START_REF] Rigét | Circumpolar pattern of mercury and cadmium in ringed seals[END_REF][START_REF] Wagemann | Arctic marine mammals as integrators and indicators of mercury in the Arctic[END_REF]. In the same way, the distinct basement rock between CEG and CWG [START_REF] Watt | Early Neoproterozoic events in East Greenland[END_REF]) could lead to a different natural input of Hg.

Temporal trends of Hg concentrations in teeth

Significant increases in Hg concentrations were found in teeth of ringed seals from both regions between 1994 and 2006, although for CEG, the dental Hg concentrations of the animals were actually lesser in 2006 than in 2004 (Fig. 3). An increase in Hg concentrations in CWG region has been reported previously in liver of ringed seals for the period of 1999 to 2004 [START_REF] Rigét | Temporal trends of mercury in marine biota of west and northwest Greenland[END_REF]. The authors did attribute this increase to likely changes in the feeding behaviour of the animals that they investigated through stable isotopes analysis in muscle tissues. However, the increase was not significant anymore when the period was extended from 1994 to 2004. In addition, no temporal trend was detected in ringed seals from CEG, although some year-to-year differences were observed [START_REF] Rigét | Levels and spatial and temporal trends of contaminants in Greenland biota: an updated review[END_REF].

Mercury concentrations have been shown to increase with trophic levels (δ 15 N) in Arctic marine food web [START_REF] Atwell | Biomagnification and bioaccumulation of mercury in an arctic marine food web: insights from stable nitrogen isotope analysis[END_REF]. In our study, the δ 15 N analyses, which allow investigation of dietary shifts through time, did exhibit no temporal trends neither for the CWG nor for the CEG ringed seals (Fig. 3), so that a change in diet can not be a consistent explanation for the temporal variations of dental Hg concentrations. However, it can not be excluded that the shift in diet may not be reflected through the δ 15 N values, since changes in prey species do not imply necessarily changes in trophic level. Nevertheless, we did observe a significant decreasing temporal trend in δ 13 C values in the teeth of seals from CEG (Fig. 3).

Since pelagic consumers tend to be more depleted in 13 C compared with littoral consumers [START_REF] France | Differentiation between littoral and pelagic food webs in lakes using stable carbon isotopes[END_REF], this decreasing trend in δ 13 C values over time could likely result from a change toward a more pelagic feeding behaviour of the ringed seals. Alternatively, the decrease of δ 13 C values over time could reflect the dissolution of isotopically light carbon from anthropogenic CO 2 in the ocean, the so-known Seuss effect [START_REF] Quay | Oceanic uptake of fossil fuel CO 2 -13 C evidence[END_REF], which results in a reduction of the oceanic δ 13 C values, and subsequently, in animal tissues. This isotopically light carbon has been released largely to the atmosphere through the combustion of fossil fuels [START_REF] Quay | Oceanic uptake of fossil fuel CO 2 -13 C evidence[END_REF], and previous studies on teeth of fur seal and ringed seal have revealed a similar decrease in δ 13 C over a century [START_REF] Newsome | Historic decline in primary productivity in western Gulf of Alaska and eastern Bering Sea: isotopic analysis of northern fur seal teeth[END_REF][START_REF] Outridge | Long-term changes of mercury levels in ringed seal (Phoca hispida) from Amundsen Gulf, and beluga (Delphinapterus leucas) from the Beaufort Sea, western Canadian Arctic[END_REF].

The United States, Europe, Asia and Canada are considered to be the main sources of polluted air reaching Greenland [START_REF] Boutron | A forty year record of mercury in central Greenland snow[END_REF]. Although Hg anthropogenic emissions to the atmosphere from Europe and United-States have been reduced recently, the Asian Hg emissions have been reported to increase substantially these last decennials, especially from China and India, and mainly to be due to the burning of fossil fuels [START_REF] Pacyna | Global emission of mercury from anthropogenic sources in 1995[END_REF]. A work carried out by [START_REF] Kahl | Air mass trajectories to Summit, Greenland: a 44-year climatology and some episodic events[END_REF] on air mass trajectories to Greenland emphasised the importance of Asian sources. Thus, the recent Hg increase observed in teeth of ringed seals from Greenland may originate for example, from an increase of Hg emissions from coal combustion partly from China, and subsequent long-range transport of Hg to Greenlandic regions.

Conclusion

The present study outlines the important contribution of the prenatal period to the transfer of Hg to the teeth of the ringed seal. Age has been shown to influence Hg concentrations in juvenile seals with a significant decrease during the first 5 years. Although teeth exhibited much lower Hg concentrations than concentrations generally encountered in soft tissues, they appeared to be a good monitor of Hg environmental exposure, showing significant differences in Hg concentrations between both regions and pointing out the increasing temporal trends of Hg concentrations between 1994 and 2006. The ability to investigate temporal trends or to compare geographical areas is useful in the cases where soft tissues are not available.

Moreover, teeth constitute a preserved material in which pre-historic temporal trends of trace elements can be measured. However, the results and the comparison with literature also outline the uncertainties and difficulties for interpreting the concentrations and for using such material (e.g. unknown age or type of tooth) as a bio-monitoring media of Hg.

Finally, the combined use of the elemental and isotopic analyses provided important complementary information to conclude about the life exposure history of the animals.

Acknowledgement

This study was partly financed by The Dancea Programme and KVUG (Kommissionen for Videnskabelige Undersøgelser i Grønland). In addition, the Poitou-Charentes region and the Fund M.L. Furnestin-Faure have been contributed financially to this PhD work. We thank the numerous persons assisting with sample collection in the field, including the hunters who allowed us to sample their animals. We are also grateful to Peter Outridge, from the Canadian Geological survey, and Michael Goodsite, from the NERI, for discussing the techniques that we used. Special acknowledgement to Carine Churlaud from the Centre Commun d'Analyses (La Rochelle, France), Poul Bjerner Hansen from the University of Southern Denmark, and

*

  The mean measured concentration is the average of the Hg concentrations determined by the three sequences of analyses. ** The theoretical concentration is based on the DOLT-3 certified concentration in Hg.

Figure 1 .

 1 Figure 1. Map showing the two regions in Greenland from where ringed seals were sampled.

Figure 2 .

 2 Figure 2. Age (in years) vs. Hg concentrations (in ng.g -1 dw),  15 N and  13 C values (in ‰) in teeth of ringed seals from central West Greenland (CWG, left figures) and central East Greenland (CEG, right figures). The black lines are LOWESS smoother (robust, locally weighted scatterplot smoothing system based on the LOWESS algorithm). The y-axis for Hg concentrations is logarithmic-scaled.

Figure 3 .

 3 Figure 3. Temporal trend of Hg concentrations (in ng.g -1 dw),  13 C and  15 N values (‰) in teeth of ringed seals from central West Greenland (CWG, left figure) and from central East Greenland (CEG, right figure) ringed seals. The smoothing lines (robust, locally weighted scatterplot smoothing system based on the LOWESS algorithm) represent the fitted non-linear trends of the values. The y-axis for Hg concentrations is logarithmic-scaled.

Table 1 .

 1 Year of capture, number of ringed seals, their mean age (± standard deviation) in years, and their dental Hg concentrations range and mean (± standard deviation) in ng.g -1 dw, for the two regions, Qeqertarsuaq (Central West Greenland, CWG) and Ittoqqortoormiit (Central East Greenland, CEG).

		Qeqertarsuaq (CWG)		Ittoqqortoormiit, CEG
		Mean age	Mean [Hg] ±
	Year n	± sd	sd	Range [Hg]	Year n

Table 2 .

 2 Analytical quality data for the customised reference material measured concurrently with the tooth samples (units in µg.g -1 dw).

			Hg concentration	% Relative standard
		n	mean ± sd	deviation
	AMA CCA	8	0.0963 ± 0.0035	3.7
	AMA NERI	10	0.105 ± 0.006	5.7
	FIMS NERI	5	0.1063 ± 0.0036	3.4
	Mean measured concentration* 23	0.1023 ± 0.0064	6.3
	Theoretical concentration**	1	0.1 ± 0.

Hg analysis, and to Pierre Richard, Benoît Lebreton, and Gaël Guillou, from LIENSs (UMR 6250 CNRS-La Rochelle University, France), for their contribution and advice on the stable isotope analysis. Jeppe Møhl, Hans Baagøe, and Abdi Hedayat, from the Natural History Museum of Denmark in Copenhagen, are acknowledged for their help with collections and maceration process of mandibles. Finally, the two anonymous reviewers are thanked for their comments which added some more consistency and clarity to this paper; and Andrew Wright and Leslie Aroon Walsh are greatly acknowledged for their help and the corrections that they brought to this manuscript regarding the revisions and the use of the English language.


John Milnor's 80'th Birthday

THE QUADRATIC DYNATOMIC CURVES ARE SMOOTH AND IRREDUCIBLE

We reprove here the smoothness and the irreducibility of the quadratic dynatomic curves (c,

The smoothness is due to Douady-Hubbard. Our proof here is based on elementary calculations on the pushforwards of specific quadratic differentials, following Thurston and Epstein. This approach is a computational illustration of the power of the far more general transversality theory of Epstein.

The irreducibility is due to Bousch, Morton and Lau-Schleicher with different approaches. Our proof is inspired by the proof of Lau-Schleicher. We use elementary combinatorial properties of the kneading sequences instead of internal addresses.

Introduction

For c ∈ C, let f c : C → C be the quadratic polynomial

f c (z) := z 2 + c. A point z ∈ C is periodic for f c if f •n c (z) = z for some integer n ≥ 1; it is of period n if f •k c (z) = z for 0 < k < n. For n ≥ 1, set X n := (c, z) ∈ C 2 | z is periodic of period n for f c .
The objective of this note is to give new proofs of the following known results.

Theorem 1.1 (Douady-Hubbard). For every n ≥ 1, the closure of X n in C 2 is a smooth affine curve.

Theorem 1.2 (Bousch, Morton and Lau-Schleicher). For every n ≥ 1 the closure of X n in C 2 is irreducible.

Milnor [START_REF] Milnor | Tsujii's monotonicity proof for real quadratic maps[END_REF] reformulated in the language of quadratic differentials a proof of Tsujii showing that the topological entropy of the real quadratic polynomial x → x 2 + c varies monotonically with respect to the parameter c. Our approach here to prove Theorem 1.1 is similar. We use elementary calculations on quadratic differentials and Thurston's contraction principle (instead of parabolic implosion techniques used in the original proof of Douady-Hubbard). Our calculation is a computational illustration of the far deeper and more conceptual transversality theory of Epstein [E].

Theorem 1.2 has been proved by Bousch [B] and by Morton [Mo] using a combination of algebraic and dynamical arguments, and by Lau and Schleicher [LS, Sc] using dynamical Date: July 23, 2011. The first author is partially supported by the Institut Universitaire de France.

arguments only. Our approach here follows essentially [LS, Sc], except we replace their argument on internal addresses [START_REF] Schleicher | Internal addresses of the Mandelbrot set and Galois groups of polynomials[END_REF]Lemma 4.5] by a purely combinatorial argument on kneading sequences (Lemma 4.2 below). Also, we make use of a result of Petersen-Ryd [PR] instead of Douady-Hubbard's parabolic implosion theory.

The somewhat similar curve consisting of cubic polynomial maps with periodic critical orbit is smooth of known Euler characteristic, due to the works of [START_REF] Bonifant | Cubic polynomial maps with periodic critical orbit. II. Escape regions[END_REF]). But the irreducibility question remains open.

In Section 2 we prove that X n is an affine curve by introducing dynatomic polynomials defining the curve, in Section 3 we prove the smoothness while in Section 4 we prove the irreducibility. Sections 3 and 4 can be read independently.

Acknowlegements. We thank Adam Epstein and John Milnor for helpful discussions, and William Thurston for his encouragement.

Dynatomic polynomials

In this section, we define the dynatomic polynomials Q n ∈ Z[c, z] (see [START_REF] Milnor | Geometry and dynamics of quadratic rational maps[END_REF] and [Si]) and show that

X n = (c, z) ∈ C 2 | Q n (c, z) = 0 .
For n ≥ 1, let P n ∈ Z[c, z] be the polynomial defined by P n (c, z) := f •n c (z) -z. The dynatomic polynomials Q n will be defined so that

P n = k|n Q k .
Example 1. For n = 1 and n = 2, we have

P 1 (c, z) = z 2 -z + c, P 2 (c, z) = z 4 + 2cz 2 -z + c 2 + c, Q 1 (c, z) = z 2 -z + c, Q 2 (c, z) = z 2 + z + c + 1 P 1 (c, z) = Q 1 (c, z), P 2 (c, z) = Q 1 (c, z) • Q 2 (c, z).
Further examples may be found in [START_REF] Silverman | The arithmetic of dynamical systems[END_REF]Table 4.1].

With an abuse of notation, we will identify polynomials in Z[c, z] and polynomials in Z [z] with Z = Z[c]. In particular, we shall write R(c, z) when R ∈ Z [z]. Note that P n ∈ Z[z] is a monic polynomial (its leading coefficient is 1) of degree 2 n . Proposition 2.1. There exists a unique sequence of monic polynomials Q n ∈ Z[z] n≥1 , such that for all n ≥ 1, we have

P n = k|n Q k .
Proof. The proof goes by induction on n. For n = 1, it is necessary and sufficient to define

Q 1 (c, z) := P 1 (c, z) = z 2 -z + c.
Note that Q 1 ∈ Z[z] is indeed monic. Assume now that n > 1 and that the polynomials

Q k are defined for 1 ≤ k < n. Set A := k|n,k<n Q k .
Since the polynomials Q k ∈ Z[z] are monic, the polynomial A ∈ Z[z] is also monic. So, we may perform a Euclidean division to find a monic quotient Q ∈ Z[z] and a remainder R ∈ Z[z] with degree(R) < degree(A), such that P n = QA + R. We need to show that R = 0, which enables us to set

Q n := Q. Let ∆ ∈ Z[c] be the discriminant of A. We claim that ∆(0) = 0. Indeed, for each k < n, the polynomial Q k (0, z) ∈ Z[z] divides P k (0, z) = z 2 k -z whose roots are simple. So, the roots of Q k (0, z) are simple. In addition, a root z 0 of Q k (0, z) is a periodic point of f 0 whose period m divides k. We have m = k since otherwise, Q k (0, z) • P m (0, z) = Q k (0, z) • j|m Q j (0, z)
would have a double root at z 0 and would at the same time divide

P k (0, z) = j|k Q j (0, z) whose roots are simple. So, if 1 ≤ k 1 < k 2 < n, then Q k 1 (0, z) and Q k 2 (0, z)
do not have common roots. This shows that the roots of A(0, z) are simple, whence ∆(0) = 0. Fix c 0 ∈ C such that ∆(c 0 ) = 0 (since ∆ does not identically vanish, this holds for every c 0 outside a finite set). Then, the roots of A(c 0 , z) ∈ Z[z] are simple. Such a root z 0 is a periodic point of f c 0 , with period dividing n, whence a root of P n (c 0 , z). As a consequence, A(c 0 , z) divides P n (c 0 , z) in C [z]. It follows that R(c 0 , z) = 0 for all z ∈ C. Since this is true for every c 0 outside a finite set, we have that R = 0 as required.

Remark 1. The proof we gave shows that the dynatomic polynomials Q n have no repeated factors (otherwise Q n (0, z) ∈ Z[z] would have a double root) and moreover, if

k 1 = k 2 , then Q k 1 and Q k 2 do not have common factors (otherwise Q k 1 (0, z) ∈ Z[z] and Q k 2 (0, z) ∈ Z[z]
would have a common root). Those facts will be used later.

Remark 2. The degree of Q k is at most that of P k , that is 2 k . It follows that the degree of A := k|n,k<n Q k is at most 2 n -2, and so, the degree of Q n = P n /A is at least 2. In particular, for n ≥ 1, the set X n is non-empty.

We extensively used the properties of roots of Q n (0, z) ∈ Z[z]. We will now study the properties of the roots of Q n (c 0 , z) ∈ C[z] for an arbitrary parameter c 0 ∈ C. Proposition 2.2. Let n ≥ 1 be a positive integer and c 0 ∈ C be an arbitrary parameter. Then,

z 0 ∈ C is a root of Q n (c 0 , z) ∈ C[z]
if and only if one of the following three exclusive conditions is satisfied:

(1) z 0 is periodic for f c 0 , the period is n and the multiplier is not 1; in that case Q n (c 0 , z) has a simple root at z 0 , or

(2) z 0 is periodic for f c 0 , the period is n and the multiplier is equal to 1; in that case Q n (c 0 , z) has a double root at z 0 , or (3) z 0 is periodic for f c 0 , the period m < n is a proper divisor of n and the multiplier of z 0 as a fixed point of f •m c 0 is a primitive n m -th root of unity; in that case Q n (c 0 , z) has a root of order n m at z 0 .

Proof. If Q n (c 0 , z 0 ) = 0, then P n (c 0 , z 0 ) = 0 and so, z 0 is periodic for f c 0 and the period m divides n. Conversely, if z 0 is periodic of period m for f c 0 , then

P k (c 0 , z 0 ) = 0 if and only if k is a multiple of m. In particular, if k is not a multiple of m, then Q k (c 0 , z 0 ) = 0. Since 0 = P m (c 0 , z 0 ) = k|m Q k (c 0 , z 0 ), we deduce that Q m (c 0 , z 0 ) = 0.
Case 1. If the multiplier ρ of z 0 as a fixed point of f •m c 0 is not a root of unity, then P n (c 0 , z) has a simple root at z 0 whenever n is a multiple of m. In that case, Q m (c 0 , z) is a factor of P n (c 0 , z) and so, no other factor of P n can vanish at z 0 . As a consequence, Q n (c 0 , z 0 ) vanishes if and only if n = m. In addition, Q m (c 0 , z) ∈ C[z] has a simple root at z 0 .

Next, if the multiplier ρ of z 0 as a fixed point of f •m c 0 is a primitive s-th root of unity, then the multiplier of z 0 as a fixed point of f •mk c 0 is ρ k . It is equal to 1 if and only if k is a multiple of s. In that case, z 0 is a multiple root of P mk (c 0 , z) of order s + 1. Indeed, f c 0 has only one cycle of attracting petals since this cycle must attract the unique critical point of f c 0 .

Case 2. If s = 1, then P n (c 0 , z) has a double root at z 0 whenever n is a multiple of m. As above, Q n (c 0 , z 0 ) vanishes if and only if n = m, but this time, Q m (c 0 , z) ∈ C[z] has a double root at z 0 .

Case 3. If s ≥ 2, then P n (c 0 , z) has a simple root at z 0 whenever n is a multiple of m but not a multiple of ms, and a multiple root at z 0 of order s + 1 whenever n is a multiple of ms. So, Q n (c 0 , z 0 ) vanishes if and only if n = m or n = ms; the polynomial Q m (c 0 , z) ∈ C[z] has a simple root at z 0 and the polynomial Q ms (c 0 , z) ∈ C[z] has a root of order s at z 0 .

Proposition 2.3. For all n ≥ 1, we have

X n = (c, z) ∈ C 2 | Q n (c, z) = 0 . If (c, z) ∈ X n -X n ,
then z is periodic for f c , its period m is a proper divisor of n, and the multiplier of z as a fixed point of f •m c is a primitive n m -th root of unity.

Proof. Let Y n ∈ C 2 be the affine curve defined by Q n :

Y n := (c, z) ∈ C 2 | Q n (c, z) = 0 .
According to the previous Proposition,

• if (c, z) belongs to X n , then Q n (c, z) = 0 and so, X n ⊆ Y n .

• if (c, z) ∈ Y n -X n , then z is periodic for f c , its period m is a proper divisor of n, and the multiplier of z as a fixed point of f •m c is a primitive n m -th root of unity. In particular, Q m (c, z) = 0. Since Q n and Q m do not have common factors, this only occurs for a finite set of points (c, z) ∈ Y n . Thus, Y n ⊆ X n .

Remark 3. We saw that X n -X n is finite. More generally, the set of points (c 0 , z 0 ) ∈ X n such that f •n c 0 (z 0 ) = z 0 and (f •n c 0 ) (z 0 ) = 1 is finite. Indeed, for such a point, c 0 is a root of the discriminant of P n ∈ Z[z]. Since the roots of P n (0, z) are simple, this discriminant does not vanish at c = 0, and so, its roots form a finite set.

Smoothness of the dynatomic curves

Our objective is now to give a proof of Theorem 1.1. We will prove the following more precise version. We shall denote by π c : C 2 → C the projection (c, z) → c and by π z : C 2 → C the projection (c, z) → z.

Theorem 3.1. For every n ≥ 1, the affine curve X n is smooth. More precisely, for (c 0 , z 0 ) ∈ X n , we have:

(1) if z 0 ∈ X n has multiplier different from 1, then π c : X n → C is a local isomorphism; (2) if z 0 ∈ X n has multiplier 1, then π z : X n → C is a local isomorphism; in addition, π c : X n → C has local degree 2. (3) if z 0 ∈ X n -X n has multiplier a primitive s-th root of unity, then π z : X n → C is a local isomorphism; in addition, π c : X n → C has local degree s.
The idea is to apply the Implicit Function Theorem. In particular, we will prove that ∂Q n ∂z (c 0 , z 0 ) = 0 in Case 1 (this is almost immediate), and that ∂Q n ∂c (c 0 , z 0 ) = 0 in Cases 2 and 3. This is where we have to work: following Epstein, we will first relate this partial derivative to the coefficient of a quadratic differential of the form (f c 0 ) * qq; we will then show that (f c 0 ) * q = q by using (a generalization of) Thurston's Contraction Principle. This approach is fundamentally different from Douady-Hubbard's original proof, where Fatou-Leau's flower theorem on parabolic periodic points as well as Douady-Hubbard's parabolic implosion theory play an essential role.

Once we know that in Cases 2 and 3, the projection π z :

X n → C is a local isomorphism, the local degree of the projection π c : X n → C follows from Proposition 2.2. Indeed, if Q n (c 0 , z) ∈ C[z] has a root of order ν at z 0 and if ∂Q n ∂c (c 0 , z 0 ) = 0, then Q n (c 0 + η, z 0 + ε) = aη + bε ν + o(η) + o(ε ν ) with a = 0 and b = 0
and the solutions of Q n (c, z) = 0 are locally of the form c(z), z with

c(z 0 + ε) = c 0 - b a ε ν + o(ε ν ). 3.1. Case 1 of Theorem 3.1. Let (c 0 , z 0 ) ∈ X n be such that the multiplier of z 0 as a fixed point of f •n c 0 is not 1. Then, Q n (c 0 , z 0 ) = 0, for all k < n, Q k (c 0 , z 0 ) = 0 and (f •n c 0 ) (z 0 ) = 1. Since k|n Q k (c, z) = P n (c, z), we have ∂Q n ∂z (c 0 , z 0 ) • k|n,k<n Q k (c 0 , z 0 ) = ∂P n ∂z (c 0 , z 0 ) = (f •n c 0 ) (z 0 ) -1 = 1.
As a consequence, ∂Q k ∂z (c 0 , z 0 ) = 0. By the Implicit Function Theorem, X n is smooth near (c 0 , z 0 ) and the projection π c :

X n → C is a local isomorphism. 3.2. Case 2 of Theorem 3.1. Let (c 0 , z 0 ) ∈ X n be such that the multiplier of z 0 as a fixed point of f •n c 0 is 1. As previously, since Q n (c 0 , z 0 ) = 0 and k|n Q k (c, z) = P n (c, z), we have ∂Q n ∂c (c 0 , z 0 ) • k|n,k<n Q k (c 0 , z 0 ) = ∂P n ∂c (c 0 , z 0 ). Since for all k < n, Q k (c 0 , z 0 ) = 0 it is enough to prove that ∂P n ∂c (c 0 , z 0 ) = 0.
We shall use the following notations: for n ≥ 0, we let ζ n : C → C be defined by

ζ n (c) := f •n c (z 0 ), and we set z n := ζ n (c 0 ) = f •n c 0 (z 0 ) and δ n := f c 0 (z n ) = 2z n . Since P n (c, z 0 ) = f •n c (z 0 ) -z 0 = ζ n (c) -z 0 , we have ∂P n ∂c (c 0 , z 0 ) = ζ n (c 0 ).
Lemma 3.2 (Compare with [START_REF] Milnor | Tsujii's monotonicity proof for real quadratic maps[END_REF]). We have

ζ n (c 0 ) = 1 + δ n-1 + δ n-1 δ n-2 + . . . + δ n-1 δ n-2 • • • δ 1 . Proof. The function ζ 0 is constant (equal to z 0 ). From ζ n (c) = ζ n-1 (c) 2 + c, we obtain ζ n (c 0 ) = 1 + δ n-1 ζ n-1 (c 0 ) with ζ 0 (c 0 ) = 0.
The result follows by induction.

In order to prove that

1 + δ n-1 + δ n-1 δ n-2 + . . . + δ n-1 δ n-2 • • • δ 1 = 0
we shall now work with meromorphic quadratic differentials.

3.2.1. Quadratic differentials. A meromorphic quadratic differential q on C takes the form q = q dz 2 with q a meromorphic function on C. We use Q(C) to denote the set of meromorphic quadratic differentials on C whose poles (if any) are all simple. If q = q dz 2 ∈ Q(C) and U is a bounded open subset of C, the norm

q U := U q(x + iy) dxdy
is well defined and finite.

Example 2.

dz 2 z D(0,R) = 2π 0 R 0 1 r r drdθ = 2πR .
3.2.2. Pushforward. For f : C → C a non-constant polynomial and q = q dz 2 a meromorphic quadratic differential on C, the pushforward f * q is defined by

f * q := T q dz 2 with T q(z) := f (w)=z q(w) f (w) 2 . If q ∈ Q(C), then f * q ∈ Q(C) also.
Lemma 3.3 (Compare with [START_REF] Milnor | Tsujii's monotonicity proof for real quadratic maps[END_REF] or [L]). For f = f c , we have

(3.1)          f * dz 2 z = 0 f * dz 2 z -a = 1 f (a) dz 2 z -f (a) - dz 2 z -c if a = 0. Proof. If f (w) = z, then w = ± √ z -c and dw 2 = dz 2 4(z -c) .
We then have

f * dz 2 z -a = dz 2 4(z -c) 1 √ z -c -a + 1 - √ z -c -a = a dz 2 2(z -c)(z -f (a))
.

If a = 0, we get the first equality in (3.1). Otherwise, we get the second equality using

1 f (a) 1 z -f (a) - 1 z -c = f (a) -c 2a(z -c)(z -f (a)) = a 2(z -c)(z -f (a))
.

3.2.3.

A particular quadratic differential. Consider the quadratic differential q ∈ Q(C) defined by

q := n-1 k=0 ρ k z -z k dz 2 with ρ k = δ n-1 δ n-2 • • • δ k .
The multiplier of z 0 as a periodic point of

f c 0 is ρ 0 = 1. So, ρ n-1 δ n-1 = 1 = ρ 0 .
In addition, for 0 ≤ k ≤ n -2, we have

ρ k δ k = ρ k+1 .
Applying Lemma 3.3, and writing f for f c 0 , we obtain

f * q = n-1 k=0 ρ k δ k dz 2 z -z k+1 - dz 2 z -c 0 = q - n-1 k=0 ρ k • dz 2 z -c 0 .
As mentioned earlier,

∂P n ∂c (c 0 , z 0 ) = ζ n (c 0 ) = n-1 k=0 ρ k .
It is therefore enough to prove that f * q = q. This is done in the next paragraph using a Contraction Principle.

3.2.4. Contraction Principle. The following lemma is a weak version of Thurston's contraction principle (which applies to the setting of rational maps on P 1 ).

Lemma 3.4 (Contraction Principle). For a non-constant polynomial f and a round disk V of radius large enough so that U := f -1 (V ) is relatively compact in V , we have

f * q V ≤ q U < q V , ∀ q ∈ Q(C).
Proof. The strict inequality on the right is a consequence of the fact that U is relatively compact in V . The inequality on the left comes from

f * q V = x+iy∈V f (w)=x+iy q(w) f (w) 2 dxdy ≤ x+iy∈V f (w)=x+iy q(w) f (w) 2 dxdy = u+iv∈U q(u + iv) dudv = q U . Corollary 3.5. If f : C → C is a polynomial and if q ∈ Q(C), then f * q = q.
3.3. Case 3 of Theorem 3.1. Let (c 0 , z 0 ) ∈ X n -X n be such that z 0 is periodic for f c 0 with period m < n dividing n and multiplier ρ, a primitive s-th root of unity with s := n m . According to Proposition 2.2 point 3, the polynomial

Q n (c 0 , z) ∈ C[z] has a root of order s ≥ 2 at z 0 , so that ∂Q n ∂z (c 0 , z 0 ) = 0.
We want to show that

∂Q n ∂c (c 0 , z 0 ) = 0. Let us write (3.2) P n (c, z) = P m (c, z) • R(c, z) with R(c, z) = k|n,k m Q k (c, z).
On the first hand, since Q n (c 0 , z 0 ) = 0, we have Lemma 3.6. We have ∂R ∂c

∂R ∂c (c 0 , z 0 ) = ∂Q n ∂c (c 0 , z 0 ) • k|n,k m,k<n Q k (c 0 , z 0 ). Since for all k < n with k = m, Q k (c 0 , z 0 ) = 0, it is enough to prove that ∂R ∂c (c 0 , z 0 ) = 0. 3.3.1. Variation along X m . Note that (c 0 , z 0 ) ∈ X m
(c 0 , z 0 ) = s ρ ρ(ρ -1)
.

Proof. Differentiating Equation (3.2) with respect to z, and evaluating at c, ζ(c) , we get:

ρ s c -1 = (ρ c -1) • R c, ζ(c) + P m c, ζ(c) =0 • ∂R ∂z c, ζ(c) = (ρ c -1) • R c, ζ(c) .
Differentiating with respect to c and evaluating at c 0 , we get:

sρ s-1 ρ = ρR(c 0 , z 0 ) =0 + (ρ -1) ∂R ∂c (c 0 , z 0 ) + (ρ -1) ∂R ∂z (c 0 , z 0 ) =0 ζ (c 0 ) = (ρ -1) ∂R ∂c (c 0 , z 0 ).
The result follows since ρ s = 1 and so, ρ s-1 = 1/ρ.

Thus, we are left with proving that ρ = 0. This will be done by using a particular meromorphic quadratic differential having double poles along the cycle of z 0 .

Quadratic differentials with double poles.

Lemma 3.7 (Compare with [L]). For f = f c , we have

f * dz 2 (z -a) 2 = dz 2 (z -f (a)) 2 - 1 2a 2 dz 2 z -f (a) - dz 2 z -c if a = 0. Proof. If f (w) = z, then w = ± √ z -c and dw 2 = dz 2 4(z -c) .
Then

f * dz 2 (z -a) 2 = dz 2 4(z -c) 1 ( √ z -c -a) 2 + 1 (- √ z -c -a) 2 = (z -c + a 2 ) dz 2 2(z -c)(z -c -a 2 ) 2 = (z -c + a 2 ) dz 2 2(z -c)(z -f (a)) 2 .

Decomposing the last expression into partial fractions gives

z -c + a 2 2(z -c)(z -f (a)) 2 = A (z -f (a)) 2 + B z -f (a) + C z -c with A = f (a) -c + a 2 2(f (a) -c) = 2a 2 2a 2 = 1, C = c -c + a 2 2(c -f (a)) 2 = a 2 2a 4 = 1 2a 2 and B = -C = - 1 2a 2 . Set f := f c 0 , z k := f •k (z 0 ), δ k := f (z k ) = 2z k , ζ k (c) := f •k c ζ(c) and ζk := ζ k (c 0 ). Then ζ k+1 (c) = f c ζ k (c) and ζ n = ζ 0 . Since δ 0 δ 1 • • • δ m-1 = ρ = 1
, there is a unique m-tuple (µ 0 , . . . , µ m-1 ) such that

µ k+1 = µ k 2z k - 1 2z 2 k ,
where the indices are considered to be modulo m.

Now consider the quadratic differential q (with double poles) defined by

q := m-1 k=0 1 (z -z k ) 2 + µ k z -z k dz 2 .
Lemma 3.8 (Levin). We have

f * q = q - ρ ρ • dz 2 z -c 0 .
Proof. By construction of q and the calculation of f * q in Lemma 3.3, the polar parts of q and f * q along the cycle of z 0 are identical. But f * q has an extra simple pole at the critical value c 0 with coefficient

m-1 k=0 - µ k 2z k + 1 2z 2 k = - m-1 k=0 µ k+1 .
We need to show that this coefficient is equal to -ρ ρ .

Using

ζ k+1 (c) = ζ k (c) 2 + c, we get ζk+1 = 2z k ζk + 1. It follows that ζk+1 µ k+1 -µ k+1 = 2z k ζk µ k+1 = ζk µ k - ζk z k . Therefore m-1 k=0 µ k+1 = m-1 k=0 ζk+1 µ k+1 -ζk µ k + ζk z k = m-1 k=0 ζk z k = ρ ρ ,
where last equality is obtained by evaluating at c 0 of the logarithmic derivative of

ρ c := m-1 k=0 2ζ k (c).
To complete the proof that ρ = 0, we will use a generalization of the Contraction Principle due to Epstein.

Lemma 3.9 (Epstein). We have f * q = q.

Proof. The proof rests again on the contraction principle, but we can not apply directly Lemma 3.4 since q is not integrable near the cycle z 0 , . . . , z m-1 . Consider a sufficiently large round disk V so that U := f -1 (V ) is relatively compact in V . Given ε > 0, we set

V ε := m k=1 f •k D(z 0 , ε) and U ε := f -1 (V ε ).
When ε tends to 0, we have

f * q V -Vε ≤ q U -Uε = q V -Vε -q V -U + q Uε-Vε -q Vε-Uε .
If we had f * q = q, we would have 0 < q V -U ≤ q Uε-Vε .

However, q Uε-Vε tends to 0 as ε tends to 0, which is a contradiction. Indeed, q = q dz 2 , the meromorphic function q being equivalent to 1 (z-z 0 ) 2 as z tends to z 0 . In addition, since the multiplier of z 0 has modulus 1,

D(z 0 , ε) ⊂ U ε -V ε ⊂ D(z 0 , ε ) with ε ε -→ ε→0 1.
Therefore,

q Uε-Vε ≤ 2π 0 ε ε 1 + o(1) r 2 rdrdθ = 2π(1 + o(1) log ε ε -→ ε→0 0.

Irreducibility of the dynatomic curves

Our objective is now to give a proof of Theorem 1.2. Note that since the affine curve X n is defined by a polynomial Q n which has no repeated factors, this proves that Q n is irreducible.

Since X n is smooth, we may equivalently prove the following result.

Theorem 4.1. For every n ≥ 1, the set X n is connected.

4.1. Kneading sequences. Set T = R/Z and let τ : T → T be the angle doubling map

τ : T θ → 2θ ∈ T.
By abuse of notation we shall often identify an angle θ ∈ T with its representative in [0, 1[. In particular, the angle θ/2 ∈ T is the element of τ -1 (θ) with representative in [0, 1/2[ and the angle (θ + 1)/2 is the element of τ -1 (θ) with representative in [1/2, 1[. Every angle θ ∈ T has an associated kneading sequence ν(θ) = ν 1 ν 2 ν 3 . . . defined by

ν k =                1 if τ •(k-1) (θ) ∈ θ 2 , θ + 1 2 , 0 if τ •(k-1) (θ) ∈ T - θ 2 , θ + 1 2 , if τ •(k-1) (θ) ∈ θ 2 , θ + 1 2 .
For example, ν(1/7) = 11 and ν(7/31) = 1100 . We shall say that an angle θ ∈ T, periodic under τ , is maximal in its orbit if its representative in [0, 1) is maximal among the representatives of τ •j (θ) in [0, 1) for all j ≥ 1. If the period is n and the binary expansion of θ is .ε 1 . . . ε n , then θ is maximal in its orbit if and only if the periodic sequence ε 1 . . . ε n is maximal (in the lexigographic order) among its iterated shifts, where the shift of a sequence

ε 1 ε 2 ε 3 indexed by N is σ(ε 1 ε 2 ε 3 . . .) = ε 2 ε 3 ε 4 . . . .
Example 3. The angle 7/31 = .00111 is not maximal in its orbit but 28/31 = .11100 is maximal in the same orbit.

The following lemma indicates cases where the binary expansion and the kneading sequence coincide. Lemma 4.2. Let θ ∈ T be a periodic angle which is maximal in its orbit and let .ε 1 . . . ε n be its binary expansion. Then, ε n = 0 and the kneading sequence ν(θ) is equal to ε 1 . . . ε n-1 .

For example, 28 31 = .11100 and ν(θ) = 1110 .

Proof. Since θ is maximal in its orbit under τ , the orbit of

θ is disjoint from ]θ/2, 1/2]∪]θ, 1].
It follows that the orbit τ •j (θ), j = 0, 1, . . . , n -2 have the same itinerary relative to the

two partitions T -0, 1 2 and T - θ 2 , θ + 1 2 .
The first one gives the binary expansion whereas the second gives the kneading sequence. Therefore, the kneading sequence of θ

is ε 1 . . . ε n-1 . Since τ •(n-1) (θ) ∈ τ -1 (θ) = θ 2 , θ + 1 2 and since θ + 1 2 ∈ ]θ, 1], we must have τ •(n-1) (θ) = θ 2 < 1 2
. So ε n , as the first digit of τ •(n-1) (θ), must be equal to 0. 4.2. Filled-in Julia sets and the Mandelbrot set. We will use results proved by Douady and Hubbard in the Orsay Notes [DH] (see also [PR] for simpler proofs of some results) that we now recall.

For c ∈ C, we denote by K c the filled-in Julia set of f c , that is the set of points z ∈ C whose orbit under f c is bounded. We denote by M the Mandelbrot set, that is the set of parameters c ∈ C for which the critical point 0 belongs to K c .

If c ∈ M , then K c is connected. There is a conformal isomorphism φ c : C -K c → C -D which satisfies φ c • f c = f 0 • φ c . The dynamical ray of angle θ ∈ T is R c (θ) := z ∈ C -K c | arg φ c (z) = 2πθ .
If θ is rational, then as r tends to 1 from above, φ -1 c (re 2πiθ ) converges to a point γ c (θ) ∈ K c . We say that R c (θ) lands at γ c (θ). We have f c • γ c = γ c • τ on Q/Z. In particular, if θ is periodic under τ , then γ c (θ) is periodic under f c . In addition, γ c (θ) is either repelling (its multiplier has modulus > 1) or parabolic (its multiplier is a root of unity).

If c /

∈ M , then K c is a Cantor set. There is a conformal isomorphism φ c : U c → V c between neighborhoods of ∞ in C, which satisfies φ c • f c = f 0 • φ c on U c . We may choose U c so that U c contains the critical value c and V c is the complement of a closed disk. For each θ ∈ T, there is an infimum r c (θ) ≥ 1 such that φ -1 c extends analytically along R 0 (θ)∩ z ∈ C | r c (θ) < |z| . We denote by ψ c this extension and by R c (θ) the dynamical ray

R c (θ) := ψ c R 0 (θ) ∩ z ∈ C | r c (θ) < |z| .
As r tends to r c (θ) from above, ψ c (re 2πiθ ) converges to a point x ∈ C. If r c (θ) > 1, then x ∈ C -K c is an iterated preimage of 0 and we say that R c (θ) bifucates at x. If r c (θ) = 1, then γ c (θ) := x belongs to K c and we say that R c (θ) lands at γ c (θ). Again, f c • γ c = γ c • τ on the set of θ such that R c (θ) does not bifurcate. In particular, if θ is periodic under τ and R c (θ) does not bifurcate, then γ c (θ) is periodic under f c .

The Mandelbrot set is connected. The map

φ M : C -M c → φ c (c) ∈ C -D is a conformal isomorphism. For θ ∈ T, the parameter ray R M (θ) is R M (θ) := c ∈ C -M | arg φ M (c) = 2πθ .
It is known that if θ is rational, then as r tends to 1 from above, φ -1 M (re 2πiθ ) converges to a point γ M (θ) ∈ M . We say that R M (θ) lands at γ M (θ). The dynamical rays R c (θ) and R c (η) land at a common point x 1 := γ c (θ) = γ c (η). This point x 1 is the point of the parabolic cycle whose immediate basin contains the critical value c. The dynamical rays R c (θ) and R c (η) are adjacent to the Fatou component containing c. The curve R c (θ) ∪ R c (η) ∪ {x 1 } is a Jordan arc that cuts the plane in two connected components. One component, denoted V 0 , contains the dynamical ray R c (0) and all the points of the parabolic cycle, except x 1 . The other component, denoted V 1 , contains the critical value c.

Since V 1 contains the critical value, its preimage U := f -1 c (V 1
) is connected and contains the critical point 0. It is bounded by the dynamical rays R c (θ/2), R c (η/2), R c (θ + 1)/2 and R c (η + 1)/2 . Two of those dynamical rays land at the point x 0 of the parabolic cycle whose immediate basin contains the critical point 0. The two other dynamical rays land at -x 0 . Since V 0 does not contain the critical value, its preimage has two connected components. One component, denoted U 0 , contains the dynamical ray R c (0). The other component is denoted U 1 .

Lemma 4.3. Let θ ∈ T be a periodic angle which is maximal in its orbit. Then,

• either θ = .11 . . . 10,

• or γ M (θ) is the root of a primitive hyperbolic component.

Proof. If θ = 0, then γ M (0) = 1/4 is the root of a hyperbolic component. So, without loss of generality, we may assume that θ = 0. Let n ≥ 2 be the period of θ under τ , let η be the companion angle of θ and let U 0 and U 1 and U be defined as above.

Since θ is maximal in its orbit, τ •(n-1) (θ) = θ/2 (see Lemma 4.2). So, R c (θ/2) lands on x 0 . One of the two rays R c (η/2) and R c (η + 1)/2 lands on x 0 . Since U is connected

U 1 Rc(η) Rc(θ)
Rc( η+12 )

x 1

Rc( θ+1 2 )

x 0 -x 0 Rc(0) U 0 U Rc( θ 2 ) Rc( η 2 )
and contains dynamical rays with angles in between η/2 and θ/2 and dynamical rays with angles in between (η + 1)/2 and (θ + 1)/2, the ray landing on x 0 has to be R c (η + 1)/2 . It follows that (η + 1)/2 is in the orbit of η under τ .

Since θ/2 < θ < (θ + 1)/2 and since R c (θ) avoids U , we have θ ≤ (η + 1)/2. On the one hand, if θ < (η + 1)/2, then the orbit of θ under τ does not contain (η + 1)/2 since otherwise θ would not be maximal in its orbit. In that case, the orbits of θ and η are disjoint and γ M (θ) is the root of a primitive hyperbolic component. On the other hand, if θ = (η + 1)/2, then the rays R c (θ) and R c (η + 1)/2 are equal. In that case, their landing point is the same, so x 0 = x 1 = f c (x 0 ) is a fixed point of f c . The rays landing at this fixed point are permuted cyclically. The dynamical rays R c (θ) and R c (η) are consecutive among the rays landing at x 0 , η < (η + 1)/2 = θ and R c (θ) is mapped to R c (2θ) = R c (η). It follows that each dynamical ray landing at x 0 is mapped to the one which is once further clockwise. Consequently, the kneading sequence of θ is 1 . . . 1 and according to Lemma 4.2, the binary expansion of θ is .1 . . . 10.

4.3.

Outside the Mandelbrot set. The projection π c : X n → C is a ramified covering. According to Proposition 3.1, the critical points are the points (c, z) Note that each connected component of X n is unbounded (because X n is an affine curve), and so, intersects W n . Thus, in order to prove that X n is connected, it is enough to prove that the closure W n of W n in X n is connected. We shall say that two components of W n are adjacent if they have a common boundary point in X n . 4.4. Labeling components of W n . Here, we explain how the components of W n may be labelled dynamically.

∈ X n such that f •n c (z) = z and (f •n c ) (z) = 1. So,
A parameter c ∈ W belongs to a parameter ray R M (θ) with θ = 0 not necessarily periodic. The dynamical rays R c (θ/2) and R c (θ + 1)/2 bifurcate on the critical point. The Jordan curve R c (θ/2) ∪ R c (θ + 1)/2 ∪ {0} separates the complex plane in two connected components. We denote by U 0 (c) the component containing the dynamical ray R c (0) and by U 1 (c) the other component.

The orbit of a point z ∈ K c has an itinerary with respect to this partition. In other words, to each z ∈ K c , we can associate a sequence ι c (z) ∈ {0, 1} N whose j-th term is equal to 0 if f

•(j-1) c (z) ∈ U 0 and is equal to 1 if f •(j-1) c (z) ∈ U 1 . A point z ∈ K c is periodic for f c if
and only if the itinerary ι c (z) is periodic for the shift with the same period. The map ι c :

K c → {0, 1} N is a bijection. Let us define ι n : W n → {0, 1} N by ι n (c, z) := ι c (z).
As c varies in W , the periodic points of f c , the dynamical ray R c (0) and the Jordan curve R c (θ/2) ∪ R c (θ + 1)/2 ∪ {0} move continuously. As a consequence, the map ι n : W n → {0, 1} N is locally constant, whence constant on each connected component of W n . So, each connected component V of W n may be labelled by the itinerary ι n (V ). Since ι c : K c → {0, 1} N is injective, distinct components have distinct labels. Since ι c : K c → {0, 1} N is surjective, each periodic itinerary of period n is the label of a Proof. Set c 0 := γ M (θ). Since c 0 is the root of a primitive hyperbolic component, the periodic point x 1 := γ c 0 (θ) has period n and multiplier 1. According to Theorem 3.1 Case 2 (see also [START_REF] Douady | Etude dynamique des polynômes complexes[END_REF]Exposé XIV,Proposition 3]), X n is smooth at (c 0 , x 1 ), the projection to the first coordinate has degree 2 and the projection to the second coordinate has degree 1. So, in a neighborhood of (c 0 , x 1 ) in C 2 , X n can be written as

(c 0 + δ 2 , x(δ)), (c 0 + δ 2 , x(-δ))
where x : (C, 0) → (C, x 1 ) is a holomorphic germ with x (0) = 0. In particular, as c moves away from c 0 , the periodic point x 1 of f c 0 splits into a pair of nearby periodic points x(± √ c -c 0 ) for f c , that get exchanged when c makes a small turn around c 0 . So, it is enough to show that for c ∈ C -M close to c 0 , those two periodic points have itineraries ε 1 . . . ε n-1 0 and ε 1 . . . ε n-1 1.

Let us denote by V 0 , V 1 , U 0 , U 1 and U the open sets defined in Section 4.2. For j ≥ 0, set x j := f •j c 0 (x 0 ) and observe that for j ∈ [1, n -1], we have x j ∈ U ε j .

For c ∈ R M (θ), consider the following compact subsets of the Riemann sphere C ∪{∞}: R(c) := R c (θ) ∪ {c, ∞} and S(c) := R c (θ/2) ∪ R c (θ + 1)/2 ∪ {0, ∞}.

Denote by U 0 (c) the component of C -S(c) containing R c (0) and by U 1 (c) the other component. From any sequence c j ∈ R M (θ) converging to c 0 , we can extract a subsequence so that R(c j ) and S(c j ) converge respectively, for the Hausdorff topology on compact subsets of C ∪ {∞}, to connected compact sets R and S. Since S(c

) = f -1 c R(c) , we have S = f -1 c 0 (R).
According to [START_REF] Petersen | Convergence of rational rays in parameter spaces[END_REF]Sections 2 and 3], R ∩ (C -K c 0 ) = R c 0 (θ), the intersection of R with the boundary of K c 0 is reduced to {x 1 }, and the intersection of R with the interior of K c 0 is contained in the immediate basin of x 1 , whence in V 1 . It follows that as c ∈ R M (θ) tends to c 0 , any Hausdorff accumulation value of the family of compact sets R(c) is contained in V 1 and so, any accumulation value of the family of compact sets S(c) is contained in U . In other words, any compact subset of

C -U is contained in C -S(c) for c ∈ R M (θ) close enough to c 0 . More precisely, every compact subset of U 0 is contained in a connected compact set L ⊂ U 0 whose interior intersects R c 0 (0); for c ∈ R M (θ) close enough to c 0 , L intersects R c (0) and is contained in C -S(c), whence in U 0 (c). As a consequence, every compact subset of U 0 is contained in U 0 (c) for c ∈ R M (θ) close enough to c 0 . Similarly, every compact subset of U 1 is contained in U 1 (c) for c ∈ R M (θ) close enough to c 0 . Fix j ∈ [1, n -1]
and let D j be a sufficiently small disk around x j so that

D j ⊂ U ε j ⊂ C -U . According to the previous discussion, if c ∈ R M (θ) is close enough to c 0 , we have f •(j-1) c x(± √ c -c 0 ) ⊂ D j ⊂ U ε j (c).
So, the itineraries of x(± √ c -c 0 ) are of the form ε 1 . . . ε n-1 ε ± with ε ± ∈ {0, 1} and ε + = ε -(each itinerary corresponds to a unique point in K c ). The result follows.

Corollary 4.5. Let ε 1 . . . ε n-1 be the kneading sequence of an angle θ ∈ T -{0} which is periodic of period n. If γ M (θ) is the root of a primitive hyperbolic component, then, the components of W n with labels ε 1 . . . ε n-1 0 and ε 1 . . . ε n-1 1 are adjacent.

Proof. According to the previous proposition, the closures of those components both contain the point (c 0 , x 1 ) with c 0 := γ M (θ) and x 1 := γ c 0 (θ).

Proposition 4.6. Let θ = 1/(2 n -1) = .1 . . . 10 be periodic of period n ≥ 2. If one follows continuously the periodic points of period n of f c as c makes a small turn around γ M (θ), then the periodic points in the cycle of ι -1 c (1 . . . 10) get permuted cyclically.

Proof. Set c 0 := γ M (θ). As mentioned earlier, all the dynamical rays R c 0 τ •j (θ) land on a common fixed point x 0 . This fixed point is parabolic and each ray landing at x 0 is mapped to the one which is once further clockwise. It follows that the multiplier of f c 0 at x 0 is ω := e -2πi/n .

According to Theorem 3.1 Case 3, X n is smooth at (c 0 , x 0 ), the projection to the first coordinate has local degree n and the projection to the second coordinate has local degree 1. It follows that in a neighborhood of (c 0 , x 0 ) in C 2 , X n can be written as (c 0 + δ n , x(δ)), (c 0 + δ n , x(ωδ)), . . . , (c 0 + δ n , x(ω n-1 δ)) Equivalently, we must show that there is a sequence c j ∈ C -M converging to c 0 , such that the sequence of periodic point y j = ι -1 c j (1 . . . 10) converges to x 0 . Let c j ∈ R M (θ) converge to c 0 . As in the previous proof, consider the following compact subsets of the Riemann sphere C ∪ {∞}: R(c j ) := R c j (θ) ∪ {c j , ∞} and S(c j ) := R c j (θ/2) ∪ R c j (θ + 1)/2 ∪ {0, ∞}.

Rc(4θ) U Rc(θ) Rc(2θ) Rc(2 n-1 θ) U 2 1 U 3 1 Rc(2 3 θ) U 0 U n-1 1 -x 0 x 0 U 1 1 =V 1
Denote by U 0 (c j ) the component of C -S(c j ) containing R c j (0) and by U 1 (c j ) the other component. Without loss of generality, extracting a subsequence if necessary, we may assume that the sequence y j converges to a point y, and that the sequence R(c j ) and S(c j ) have Hausdorff limits R and S. Passing to the limit on f •n c j (y j ) = y j , we see that f •n c 0 (y) = y, and so, y is periodic for f c 0 with period dividing n. In particular, it is contained in the boundary of K c 0 . We must show that y = {x 0 }.

It follows from [START_REF] Petersen | Convergence of rational rays in parameter spaces[END_REF]Sections 2 and 3] that R ∩ (C -K c 0 ) = R c 0 (θ), the intersection of R with the boundary of K c 0 is reduced to {x 0 } and the intersection of R with the interior of K c 0 is contained in the immediate basin of x 0 . We cannot quite conclude that L is contained in U , but rather that it is contained in U ∪ • K c 0 . As in the previous proof, it follows that the Hausdorff limit of U 1 (c j ) is contained in U ∪ U 1 ∪ • K c 0 . Since ι c j (y j ) = 1 . . . 10, we know that y j , f c j (y j ), . . . , f •(n-2) c j (y j ) belong to U 1 (c j ). So the points y, f c 0 (y), . . . , f •(n-2) c 0 (y) belong to U ∪ U 1 ∪ • K c 0 . Since y is in the boundary of K c 0 , we deduce that y, f c 0 (y), . . . , f

•(n-2) c 0 (y) belong to U ∪ U 1 .
The dynamical rays landing at x 0 divide U 1 in n -1 connected components U j 1 labelled clockwise so that

V 1 = U 1 1 fc 0 -→ U 2 1 fc 0 -→ • • • fc 0 -→ U n-1 1 fc 0 -→ C -U 1 .
The component U maps with degree 2 to U 1 1 = V 1 (see Figure 7). Now, we claim that the orbit of y intersects U . Indeed, either y itself is in U , or y is in U j 1 for some j ≥ 1. Then, f

•(n-j) c 0 (y) ∈ C -U 1 . Since it cannot be in U 0 , it belongs to U .

The map f •n c 0 : U → C -U 1 is a proper map of degree 2 and U ⊂ C -U 1 . Note that x 0 ∈ U is a multiple fixed point of f •n c 0 and that there is an attracting petal contained in U . It follows from a version of the Lefschetz fixed point formula (see [START_REF] Goldberg | Fixed points of polynomial maps. Part II. Fixed point portraits[END_REF]Lemma 3.7]) that x 0 is the only fixed point of f •n c 0 contained in U . As a consequence, the orbit of the periodic point y contains x 0 , and since x 0 is a fixed point, we have y = x 0 as required.

Corollary 4.7. The components of W n whose labels contain a single 0 are adjacent.

Proof. Let θ := .1 . . . 10 and (c 0 , x 0 ) := (γ M (θ), γ c 0 (θ)). By the above proposition every component of W n whose label is a shift of 1 . . . 10 contains (c 0 , x 0 ) in its boundary. But every periodic label of period n containing a single 0 is indeed a shift of 1 . . . 10. 4.6. Proof of Theorem 4.1. We will finally deduce that W n is connected. According to Corollary 4.7, components of W n whose label contain a single 0 have a common boundary point. So, it is enough to show that component of W n whose label has at least two 0 has a common boundary point with a component of W n whose label has one less 0.

The map F : C 2 → C 2 defined by F (c, z) := c, f c (z) restricts to an isomorphism F : X n → X n . It permutes the components of W n as follows: the label of F (C) is the shift of the label of C. In addition, two components C 1 and C 2 of W n are adjacent if and only if F (C 1 ) and F (C 2 ) are adjacent.

Let C be a connected component of W n whose label ι contains at least two 0. Let ε 1 . . . ε n = σ •k (ι) be maximal (in the lexigographic order) among the iterated shifts of ι. Then, the angle θ := .ε 1 . . . ε n is maximal in its orbit. According to Lemma 4.2, ε n = 0 and the kneading sequence ν(θ) is ε 1 . . . ε n-1 . According to Lemma 4.3, γ M (θ) is the root of a primitive hyperbolic component. According to Corollary 4.5, the component F •k (C) which is labeled ε 1 . . . ε n-1 0 is adjacent to the component C of W n which is labeled ε 1 . . . ε n-1 1. Then, F •(n-k) (C ) is a component of W n adjacent to F •(n-k) F •k (C) = C, and its label contains one less 0 than the label of C. This completes the proof of Theorem 4.1.

  and the multiplier ρ of z 0 as a fixed point of f •m c 0 is ρ = 1. Thus, according to Case 1, X m is locally the graph of a function ζ(c) defined and holomorphic near c 0 with ζ(c 0 ) = z 0 . The point ζ(c) is periodic of period m for f c . We denote by ρ c its multiplier and set ρ := dρ c dc c 0 .

Figure 1 .

 1 Figure 1. The kneading sequence of θ = 1/7 is ν(1/7) = 11 .

Figure 2 .

 2 Figure 2. The kneading sequence of θ := 28/31 = .11100 is ν(28/31) = 1110 .

Figure 3 .

 3 Figure 3. The parameter rays R M (27/31) and R M (28/31) land at a common root of a primitive hyperbolic component.

Figure 4 .

 4 Figure 4. The filled-in Julia set K c and a cycle of dynamical rays for c := γ M (28/31). The dynamical rays R c (28/31) and R c (27/31) both land at the same point.

Figure 5 .

 5 Figure 5. We have 30/31 = .11110 and the parameter ray R M (30/31) lands on the boundary of the main cardioid.

Figure 6 .

 6 Figure 6. The regions U 0 and U 2 for a parameter c belonging to R M (28/31).

Figure 7 .

 7 Figure 7. For c 0 := γ M (.11110), the dynamical rays R c 0 τ •j (θ) land on a common fixed point x 0 .