Anjara Saloma

Maevatiana N Ratsimbazafindranahaka
email: maevatiana.ratsimbazafindranahaka@universite-paris-saclay.fr

Mathilde Martin

Aristide Andrianarimisa

Chloé Huetz

Olivier Adam

Isabelle Charrier

Alan Mcelligott

Social calls in humpback whale

Keywords: Animal Behavior, Marine Biology, Zoology Acoustic communication, Baleen whale, Vocal repertoire, Mother-offspring relationship

INTRODUCTION

Baleen whales, like other cetaceans (toothed whales), use acoustic communication in many social contexts such as predator alert, foraging cooperation, mating, individual identification, and parental care [START_REF] Edds-Walton | Acoustic communication signals of mysticete whales[END_REF][START_REF] Tyack | Communication and cognition[END_REF][START_REF] Dudzinski | Communication in marine mammals[END_REF]. Sounds used by baleen whales include vocalizations and sounds generated by active surface behaviors (breaching, tail or pectoral fins slapping, etc.). Some baleen whales' vocalizations, attributed to males, are organized in a repeated manner and in long bouts to constitute what is called a 'song' [START_REF] Edds-Walton | Acoustic communication signals of mysticete whales[END_REF][START_REF] Tyack | Communication and cognition[END_REF][START_REF] Dudzinski | Communication in marine mammals[END_REF]. Baleen whales also produce non-song vocalizations called 'social calls' [START_REF] Payne | Behavior and vocalizations of humpback whales (Megaptera sp.)[END_REF][START_REF] Tyack | Why do whales sing[END_REF][START_REF] Silber | The relationship of social vocalizations to surface behavior and aggression in the Hawaiian humpback whale (Megaptera novaeangliae)[END_REF]. Unlike songs, social calls have been described as variable through time, interrupted by silent periods, apparently unpredictable, and not showing the rhythmic, consistent and continuous temporal pattern of songs [START_REF] Tyack | Why do whales sing[END_REF][START_REF] Silber | The relationship of social vocalizations to surface behavior and aggression in the Hawaiian humpback whale (Megaptera novaeangliae)[END_REF]. While all social groups in all species of baleen whales potentially produces social calls, social calls produced by mother-calf pairs have been proposed to have the particularity of being produced at a low rate and/or at a low amplitude to minimize the risk of alerting predators [START_REF] Videsen | High suckling rates and acoustic crypsis of humpback whale neonates maximise potential for mother-calf energy transfer[END_REF][START_REF] Nielsen | Acoustic crypsis in southern right whale mother-calf pairs: infrequent, low-output calls to avoid predation[END_REF][START_REF] Parks | Acoustic crypsis in communication by North Atlantic right whale mother-calf pairs on the calving grounds[END_REF].

Humpback whales are baleen whales known for their complex, highly structured, and organized songs produced by males in breeding areas [START_REF] Payne | Songs of humpback whales[END_REF]) but they also generate social calls. Humpback whales' social calls have been the subject of several studies, both in feeding or breeding grounds, as well as on migration routes (D 'Vincent, Nilson & Hanna, 1985;[START_REF] Silber | The relationship of social vocalizations to surface behavior and aggression in the Hawaiian humpback whale (Megaptera novaeangliae)[END_REF][START_REF] Cerchio | Variation in feeding vocalizations of humpback whales Megaptera novaeangliae from southeast Alaska[END_REF][START_REF] Dunlop | The social vocalization repertoire of east Australian migrating humpback whales (Megaptera novaeangliae)[END_REF][START_REF] Dunlop | Non-song acoustic communication in migrating humpback whales (Megaptera novaeangliae)[END_REF][START_REF] Zoidis | Vocalizations produced by humpback whale (Megaptera novaeangliae) calves recorded in Hawaii[END_REF][START_REF] Stimpert | Non-song sound production and its behavioral context in humpback whales (Megaptera novaeangliae)[END_REF][START_REF] Stimpert | Common humpback whale (Megaptera novaeangliae) sound types for passive acoustic monitoring[END_REF][START_REF] Rekdahl | Temporal stability and change in the social call repertoire of migrating humpback whales[END_REF][START_REF] Fournet | Repertoire and classification of non-song calls in Southeast Alaskan humpback whales (Megaptera novaeangliae)[END_REF][START_REF] Rekdahl | Common nonsong social calls of humpback whales (Megaptera novaeangliae) recorded off northern Angola, southern Africa[END_REF]Epp et al., 2021;Epp, Fournet & Davoren, 2021;[START_REF] Indeck | Adult female-calf acoustic communication signals in migrating east Australian humpback whales[END_REF]. Distinct acoustic units of vocalization or "call types" were generally defined either using an automatic clustering method based of several acoustic parameters [START_REF] Stimpert | Non-song sound production and its behavioral context in humpback whales (Megaptera novaeangliae)[END_REF][START_REF] Stimpert | Common humpback whale (Megaptera novaeangliae) sound types for passive acoustic monitoring[END_REF][START_REF] Fournet | Repertoire and classification of non-song calls in Southeast Alaskan humpback whales (Megaptera novaeangliae)[END_REF] or an aural-visual classification [START_REF] Dunlop | The social vocalization repertoire of east Australian migrating humpback whales (Megaptera novaeangliae)[END_REF][START_REF] Dunlop | Non-song acoustic communication in migrating humpback whales (Megaptera novaeangliae)[END_REF][START_REF] Zoidis | Vocalizations produced by humpback whale (Megaptera novaeangliae) calves recorded in Hawaii[END_REF][START_REF] Rekdahl | Temporal stability and change in the social call repertoire of migrating humpback whales[END_REF][START_REF] Rekdahl | Common nonsong social calls of humpback whales (Megaptera novaeangliae) recorded off northern Angola, southern Africa[END_REF][START_REF] Fournet | Repertoire and classification of non-song calls in Southeast Alaskan humpback whales (Megaptera novaeangliae)[END_REF][START_REF] Fournet | Some things never change: multi-decadal stability in humpback whale calling repertoire on Southeast Alaskan foraging grounds[END_REF]Epp et al., 2021;Epp, Fournet & Davoren, 2021;[START_REF] Indeck | Adult female-calf acoustic communication signals in migrating east Australian humpback whales[END_REF]) validated using a supervised automatic classification (e.g., discriminant function analysis-DFA, Classification and Regression Tree-CART, or random forest-RF) or not. The recordings were obtained using moored or towed hydrophones [START_REF] Dunlop | The social vocalization repertoire of east Australian migrating humpback whales (Megaptera novaeangliae)[END_REF][START_REF] Dunlop | Non-song acoustic communication in migrating humpback whales (Megaptera novaeangliae)[END_REF][START_REF] Stimpert | Non-song sound production and its behavioral context in humpback whales (Megaptera novaeangliae)[END_REF][START_REF] Rekdahl | Temporal stability and change in the social call repertoire of migrating humpback whales[END_REF][START_REF] Rekdahl | Common nonsong social calls of humpback whales (Megaptera novaeangliae) recorded off northern Angola, southern Africa[END_REF][START_REF] Fournet | Repertoire and classification of non-song calls in Southeast Alaskan humpback whales (Megaptera novaeangliae)[END_REF][START_REF] Fournet | Some things never change: multi-decadal stability in humpback whale calling repertoire on Southeast Alaskan foraging grounds[END_REF]Epp et al., 2021;Epp, Fournet & Davoren, 2021), using hydrophone carried by divers [START_REF] Zoidis | Vocalizations produced by humpback whale (Megaptera novaeangliae) calves recorded in Hawaii[END_REF], or using animal borne tags [START_REF] Stimpert | Non-song sound production and its behavioral context in humpback whales (Megaptera novaeangliae)[END_REF][START_REF] Stimpert | Common humpback whale (Megaptera novaeangliae) sound types for passive acoustic monitoring[END_REF][START_REF] Indeck | Adult female-calf acoustic communication signals in migrating east Australian humpback whales[END_REF].

Humpback whales' social calls vary from low to high-frequency calls and differ in their general structure [START_REF] Silber | The relationship of social vocalizations to surface behavior and aggression in the Hawaiian humpback whale (Megaptera novaeangliae)[END_REF][START_REF] Dunlop | The social vocalization repertoire of east Australian migrating humpback whales (Megaptera novaeangliae)[END_REF][START_REF] Rekdahl | Temporal stability and change in the social call repertoire of migrating humpback whales[END_REF]. Social calls can be categorized into a broad class or category according to their general characteristics (e.g., [START_REF] Dunlop | The social vocalization repertoire of east Australian migrating humpback whales (Megaptera novaeangliae)[END_REF][START_REF] Zoidis | Vocalizations produced by humpback whale (Megaptera novaeangliae) calves recorded in Hawaii[END_REF][START_REF] Fournet | Some things never change: multi-decadal stability in humpback whale calling repertoire on Southeast Alaskan foraging grounds[END_REF]Epp, Fournet & Davoren, 2021). Humpback whale vocal repertoire has been described as a mix of discrete call types (call types acoustically distinct from one another and with little variation) and graded call types (call types with varying characteristics that form a continuum and represented by some common cases and by intermediate forms) (Epp, Fournet & Davoren, 2021). The variation of call types in humpback whale is likely related to factors such age, sex, or body mass [START_REF] Cerchio | Variation in feeding vocalizations of humpback whales Megaptera novaeangliae from southeast Alaska[END_REF]Epp, Fournet & Davoren, 2021).

Social calls are produced by the same sound generator as for songs, located inside the respiratory system, composed of the lungs (air source), the laryngeal sac (air source and acoustic resonator), and the nasal cavities (another resonator) [START_REF] Adam | New acoustic model for humpback whale sound production[END_REF]. Individuals in all group compositions produce social calls: lone individuals, multiple animals, pairs, and mother-calf pairs accompanied or not by escorts [START_REF] Dunlop | Non-song acoustic communication in migrating humpback whales (Megaptera novaeangliae)[END_REF].

Social calls in adult humpback whales were first described as ranging from 50 Hz to over 10 kHz with fundamental frequencies below 3 kHz [START_REF] Silber | The relationship of social vocalizations to surface behavior and aggression in the Hawaiian humpback whale (Megaptera novaeangliae)[END_REF]. These social calls were reported to be produced when whales are predominantly in groups of three or more adults, in surface-active groups including both females and males [START_REF] Silber | The relationship of social vocalizations to surface behavior and aggression in the Hawaiian humpback whale (Megaptera novaeangliae)[END_REF]. Although up to now only few direct quantitative comparisons were conducted and most comparisons were based on aural and visual characteristics, several studies suggested that some calls types are shared between humpback whales from different regions [START_REF] Dunlop | The social vocalization repertoire of east Australian migrating humpback whales (Megaptera novaeangliae)[END_REF][START_REF] Stimpert | Common humpback whale (Megaptera novaeangliae) sound types for passive acoustic monitoring[END_REF][START_REF] Fournet | Repertoire and classification of non-song calls in Southeast Alaskan humpback whales (Megaptera novaeangliae)[END_REF][START_REF] Rekdahl | Common nonsong social calls of humpback whales (Megaptera novaeangliae) recorded off northern Angola, southern Africa[END_REF]Epp, Fournet & Davoren, 2021).

The first confirmed recording of humpback whale calves' social calls was performed by [START_REF] Zoidis | Vocalizations produced by humpback whale (Megaptera novaeangliae) calves recorded in Hawaii[END_REF] using a technique involving a two-element hydrophone array. It has been suggested that calves' vocal repertoire is limited and vocalizations are simple in structure, short, predominantly composed of low frequencies, and have a relatively narrow frequency bandwidth [START_REF] Zoidis | Vocalizations produced by humpback whale (Megaptera novaeangliae) calves recorded in Hawaii[END_REF]. Humpback whales are known to be vocal learners [START_REF] Tyack | Acoustic communication under the sea[END_REF][START_REF] Janik | Cetacean vocal learning and communication[END_REF]. Although mostly described for male songs, the vocal learning likely also apply to social calls. Calf might have to learn and master their calls, or aspects of them [START_REF] Tyack | Acoustic communication under the sea[END_REF]Epp, Fournet & Davoren, 2021).

Up to now, the biological functions of social calls remain unclear. Visual observations from surface activity suggested that social calls could serve either as a sign of aggression among males competing for the 'principal escort' status [START_REF] Tyack | Differential response of humpback whales, Megaptera novaeangliae, to playback of song or social sounds[END_REF][START_REF] Baker | Aggressive behavior between humpback whales (Megaptera novaeangliae) wintering in Hawaiian waters[END_REF][START_REF] Silber | The relationship of social vocalizations to surface behavior and aggression in the Hawaiian humpback whale (Megaptera novaeangliae)[END_REF] or, in some specific breeding competitive contexts, depending on sex and group composition, among adults as a deterrent to dissuade approach from other whales of same or opposite sex [START_REF] Tyack | Differential response of humpback whales, Megaptera novaeangliae, to playback of song or social sounds[END_REF].

Mother-calf pairs constitute the only stable social association on breeding grounds in humpback whales. Calves are born in a rich acoustic environment filled with songs and social calls. Recent acoustic studies on the social calls of humpback whales investigated vocal production from mother-calf pairs [START_REF] Dunlop | The social vocalization repertoire of east Australian migrating humpback whales (Megaptera novaeangliae)[END_REF][START_REF] Fournet | Repertoire and classification of non-song calls in Southeast Alaskan humpback whales (Megaptera novaeangliae)[END_REF][START_REF] Recalde-Salas | Non-song vocalizations of humpback whales in western Australia[END_REF][START_REF] Indeck | Adult female-calf acoustic communication signals in migrating east Australian humpback whales[END_REF] with an attempt to assign social calls to females or calves. Identifying the sound source remains a real challenge [START_REF] Saddler | Characterizing chilean blue whale vocalizations with DTAGs: a test of using tag accelerometers for caller identification[END_REF][START_REF] Indeck | Adult female-calf acoustic communication signals in migrating east Australian humpback whales[END_REF], and thus further investigations are needed to accurately assign social call types to mothers or calves.

The social calls of humpback whales in the South Western Indian Ocean remained poorly documented. This present study aimed to investigate the repertoire of social calls recorded by suction-cup acoustic tags (Acousonde 3B) attached to female humpback whales or their calf off Sainte Marie island, Madagascar, South Western Indian Ocean, during four breeding seasons. Our goal was to complement the knowledge of acoustic communication in poorly documented yet critical social groups such as mother-calf groups and to contribute to the global catalogue of humpback whale social calls. The social calls described in breeding and foraging grounds are likely to be different, as they are produced in different behavioral contexts. Such knowledge is thus crucial to better understand the biological function of social calls, but also to investigate dialects and thus vocal learning of social calls. In addition, the description of the repertoire of social calls in mother-calf groups is a crucial material for investigating the role of acoustic communication in humpback whale mother-calf interactions in the future.

MATERIALS AND METHODS

Study area

We collected acoustic data during winters 2013, 2014, 2016, and 2017 in the coastal waters off Sainte Marie island, Madagascar,South Western Indian Ocean (between latitudes 17 19′ and 16 42′ South,and longitudes 49 48′ and 50 01′ East), where mother-calf pairs come in these relatively calm and shallow waters [START_REF] Trudelle | Influence of environmental parameters on movements and habitat utilization of humpback whales (Megaptera novaeangliae) in the Madagascar breeding ground[END_REF].

Data collection and tagging procedure

We used Acousonde 3B (www.acousonde.com) attached to females or calves via four suction cups, a non-invasive attachment system, to record sounds. We deployed tags from a 6.40 m rigid motorboat using a 5 m handheld carbon-fiber pole. The boarding team consisted of staff experienced in successfully approaching mother-calf pairs: one operator (boat pilot), photographer, note-taker, and tagger. While the boarding team may change between outings, the team composition was always respected and each team member held one role within a whole outing. The tags were placed on the back, near the dorsal fin of the animal. Calves were tagged using one of the two approaches described in [START_REF] Stimpert | Tagging young humpback whale calves: methodology and diving behavior[END_REF] and in [START_REF] Huetz | Ontogeny and synchrony of diving behavior in Humpback whale mothers and calves on their breeding ground[END_REF] in order to minimize disturbance to the mother-calf pair. Tagging efforts were terminated if the pair displayed avoidance behavior or if the calf was not successfully tagged within 30 min. Immediate behavioral response of the animals to tagging was recorded as in [START_REF] Stimpert | Tagging young humpback whale calves: methodology and diving behavior[END_REF]. All mother-calf pairs were photoidentified to avoid double-sampling within the calving season. Tagged animals were not followed after tag deployment to avoid any further disturbance of their behavior. After tag deployment or an aborted attempt, the boat slowly moved away in the opposite direction of the mother-calf pair. The tag was retrieved after few hours or the following day, when it detached itself from the animal (usually as a consequence of rubbing against the mother, surface active behavior, etc.). The Fisheries Resources Ministry, Madagascar, fully approved all experimental protocols (Research and Collect permits #44/13-MPRH/SG/ DGPRH, #43/14-MRHP/SG/DGRHP, #28/16-MRHP/SG/DGRHP, and #26/17-MRHP/ SG/DGRHP). This present study complies with the European Union Directive on the protection of animals used for scientific purposes (EU Directive 2010/63/EU).

For each spotted group with mother-calf pair, approaches similar to touristic boats, complying with Madagascar's Code of Conduct for whale watching activities (Interministerial decree March 8th, 2000), were adopted: the boat's speed was reduced gradually at 800 m distance from the spotted group. The observation area for mother-calf groups was set at a 200 m radius around them. We used this distance to observe the groups before tag deployment. We noted the group composition: lone mother-calf pair (MC) or mother-calf pair accompanied by one or several escorts (MCE). Behavioral observations and photo-identification were obtained concurrently for each group, and the calf's relative age was estimated using the angle of furling of its dorsal fin [START_REF] Cartwright | Behavioral ontogeny in humpback whale (Megaptera novaeangliae) calves during their residence in Hawaiian waters[END_REF][START_REF] Huetz | Ontogeny and synchrony of diving behavior in Humpback whale mothers and calves on their breeding ground[END_REF]: C1 (neonate)-calf presenting some folds, scars, and skin color that tends to be light grey dorsally and white ventrally and with less than 44 dorsal fin furl (Faria et al., 2013), C2: very young but non-neonate calves having more than 45 but less than 72 dorsal fin furl, and C3: older calves that have unfurled dorsal fins (>72 ). Depending on the opportunity and on how the group behaved, we tagged the calf or the mother using either a passive or active approach, as described in [START_REF] Stimpert | Tagging young humpback whale calves: methodology and diving behavior[END_REF] and [START_REF] Huetz | Ontogeny and synchrony of diving behavior in Humpback whale mothers and calves on their breeding ground[END_REF].

We did not follow the animals after tag deployment to avoid further disturbance of their behavior. This means that change in group composition during the data recording is unknown. MC could be joined by other escort(s) or escort(s) in MCE might have left. Our tagging processes lasted 21 min on average and never exceeded 30 min (i.e., our maximum duration to tag a whale). Beside the mother-calf recordings, we also punctually recorded several male songs using a towed Aquarian H2a hydrophone deployed from the boat (engine off).

Vocal repertoire and acoustic analysis

We downloaded the audio files from the tags as à .MT files, converted them to à .WAV format using GoldWave software (GoldWave Inc., St. John's, Canada), and analyzed them using Avisoft SASLab Pro version 5.207 (Avisoft Bioacoustics, Nordbahn, Germany). We produced spectrograms of the acoustic recordings using 1,024-point Fast Fourier Transform, 75% overlap, and Hamming window.

Social calls that were clearly audible and distinguishable were aurally and visually identified, classified, and then compared with social call catalogues available in the literature [START_REF] Dunlop | The social vocalization repertoire of east Australian migrating humpback whales (Megaptera novaeangliae)[END_REF][START_REF] Dunlop | Non-song acoustic communication in migrating humpback whales (Megaptera novaeangliae)[END_REF][START_REF] Zoidis | Vocalizations produced by humpback whale (Megaptera novaeangliae) calves recorded in Hawaii[END_REF][START_REF] Stimpert | Common humpback whale (Megaptera novaeangliae) sound types for passive acoustic monitoring[END_REF][START_REF] Rekdahl | Temporal stability and change in the social call repertoire of migrating humpback whales[END_REF][START_REF] Rekdahl | Common nonsong social calls of humpback whales (Megaptera novaeangliae) recorded off northern Angola, southern Africa[END_REF][START_REF] Fournet | Repertoire and classification of non-song calls in Southeast Alaskan humpback whales (Megaptera novaeangliae)[END_REF]Epp, Fournet & Davoren, 2021;[START_REF] Indeck | Adult female-calf acoustic communication signals in migrating east Australian humpback whales[END_REF]. Call types qualitatively similar to call types described in these catalogues (reference catalogue hereafter) were given the same name. New names were assigned for the remaining call types onomatopoeically, as the behavioral context or biological function of these new call types is still unknown and the main acoustic structure can be common to several call types (e.g., several call types can be low-frequency upsweep calls). Distant song units could not be confused with social calls in our dataset as songs can be easily recognized by their well-organized temporal pattern. Additionally, we used 16 of the recorded male songs to determine if some social calls were similar to males' song units.

For each identified social call, we measured nine temporal and spectral characteristics (Table 1). Similar to [START_REF] Dunlop | The social vocalization repertoire of east Australian migrating humpback whales (Megaptera novaeangliae)[END_REF], we categorized social calls as either low-frequency sounds (LF), mid-frequency harmonic sounds (MF), high-frequency harmonic sounds (HF), amplitude modulated sounds (AM), noisy and complex sound (NC), or pulsed sounds (PS). LF corresponds to calls with peak frequency below 160 Hz. MF corresponds to calls with a peak frequency ranging from 170 to 550 Hz. HF corresponds to calls with a peak frequency above 700 Hz. AM corresponds to sounds consisting of a combination of long harmonic and amplitude modulated components with peak frequency ranging from 20 to 300 Hz. NC corresponds to broadband calls or harmonic calls with additional noise-like features. PS corresponded to low-frequency sounds repeated rhythmically. Representative spectrograms (Hamming window, FFT window size: 1,024 pts, 90% overlap) for the identified social calls were generated using the Seewave package [START_REF] Sueur | Seewave, a free modular tool for sound analysis and synthesis[END_REF] in R software (R Core Team, 2021).

To obtain an overview of the intensity of sounds recorded in mother-calf groups, we measured the received level (in dB re 1mPa RMS) of the most common and aurally and visually easily identifiable call types using the Root Mean Square (RMS) function in Avisoft SASLab Pro. We considered only 10 good quality calls with a signal-to-noise ratio above 10 dB and without overlap for each selected call type. Using such criteria, calls selected for such measurements are likely produced by individuals constituting the focal mother-calf group (i.e., the mother, the calf, and possibly the escort if present). Mother-calf pair don't gather with other pairs, so the only calls detected from our recordings can only come from individuals of the focal groups. Songs are highly different, so no confusion could be made. A similar procedure was used previously on humpback whale mother-calf pairs in Western Australia [START_REF] Videsen | High suckling rates and acoustic crypsis of humpback whale neonates maximise potential for mother-calf energy transfer[END_REF].

Statistical analysis

To validate our aural-visual classification of calls by categories and by types, we performed random forest (RF) analyses [START_REF] Thiebault | At sea vocal repertoire of a foraging seabird[END_REF][START_REF] Indeck | Adult female-calf acoustic communication signals in migrating east Australian humpback whales[END_REF] using the randomForest [START_REF] Liaw | Classification and regression by randomForest[END_REF] package in R. We calculated the global accuracy of the RF classification of calls by categories and by types (defined as accuracy = 1 -OOB, where OOB stands for out-of-bag error, the misclassification error rate) and computed the Gini index, which gives the importance of the variables used for the classification. We only included acoustic variables that were primarily measurable for most of the calls: Dur, Fmax, Q25, Q50, Q75, and PR. We included only call types for which we had at least six exemplars. The number of variables to be randomly selected at each split was set at 2 (as we had only six variables), and the number of trees grown was set at 500. We used a balanced RF design to maintain equal sample sizes of each category or type in the classification and avoid the over-representation of the most represented classes [START_REF] Chen | Using random forest to learn imbalanced data[END_REF]. In this design, each tree of the RF is built with the same number of calls per category or per type (i.e., the smallest number of calls for a given category or type).

RESULTS

Tag deployments

We performed 62 successful tag deployments (35 on calves and 27 on mothers) during the four years of data collection. Acoustic data were usable for 21 deployments (21 different mother-calf groups): seven deployments on mothers and 14 deployments on calves (Table 2). The other deployments were not analyzed due to the high background noise level or their short durations (less than 30 min). Background noise was mainly present when the tag was placed in a higher position close to the dorsal fin and thus often out of the water, especially on calves for which surfacing activities occurred very often. Of the 21 studied groups, we identified 12 as MC and nine as MCE. Three groups had a C1 calf, three had a C2 calf, and 15 had a C3 calf. We detected social calls in all of the studied groups.

Social call classification

A total of 2,033 social calls were clearly distinguishable. Aural-visual characteristics allowed the classification of these calls into 30 call types representing five of the six main categories suggested by [START_REF] Dunlop | The social vocalization repertoire of east Australian migrating humpback whales (Megaptera novaeangliae)[END_REF] (Table S1): low-frequency sounds (LF), mid-frequency harmonic sounds (MF), high-frequency harmonic sounds (HF), amplitude modulated sounds (AM), and pulsed sounds (PS). We did not find any call corresponding to the noisy and complex sound (NC) category. Eleven call types were named after qualitatively similar calls in the reference catalogue we used and 17 were given new name as they appeared qualitatively different (Table S1). Audio examples of call types are provided (Audio S1).

Low-frequency sounds (LF)

LF was the most represented category (46%, N = 925/2,033). Eleven call types were within the LF category: "100 Hz sound", "bass", "boom", "gru", "snort", "burp", "guttural", "thowp", "wop", "bark", and "drum" (Fig. 1). "Bass" and "wop" (Figs. 1B and1I respectively) were the most common LF calls (heard in eight groups each), followed by "100 Hz" and "thowp" (Figs. 1A and1H) (seven groups), "gru" and "snort" (Figs. 1D and1E) (six groups), and by "boom" (Fig. 1C) (four groups). The remaining LF calls were rare (heard in two groups for "drum", Fig. 1K, and only one group for "guttural", "burp", and "bark", Figs. 1G, 1F and 1J respectively). "Burp", "bark", and "drum" were only heard in MC groups. The remaining LF calls were heard in both MC and MCE groups. "Bass" (Fig. 1B) was a harmonic sound with a fundamental frequency below 40 Hz on average, and can sometimes be masked by background noise. "Wop" (Fig. 1I) and "thowp" (Fig. 1H) were brief harmonic upsweep sounds similar in frequency but different in duration. The "100 Hz" call (Fig. 1A) was a long, relatively flat call. "Gru" (Fig. 1D) was a short harmonic sound like "snort" but with more spaced harmonics. "Snort" (Fig. 1E) can be produced in sequences. "Boom" (Fig. 1C) was a harmonic sound produced either in sequence or alone. "Boom" was frequently produced in series with "100 Hz" following a well-defined order (100 Hz-boom-boom-100 Hz). "Guttural" (Fig. 1G) appeared to be a "composite call" consisting of non-overlapping "gru" and "heek" (MF) not separated with a silence. "Drum" (Fig. 1K) was a very short call always produced in series, and "burp" (Fig. 1F) was a short harmonic sound with several close harmonics. "Bark" (Fig. 1J) was a short harmonic sound with ascending frequency modulation.

Mid-frequency harmonic sounds (MF)

MF was the second most represented category (35%, N = 718/2,033). Eight call types were within the MF category: "groan", "downsweep", "woohoo", "trumpet", "heek", "whoop", "wiper", and "creak" (Fig. 2). "Heek" was the most common MF call (heard in nine groups), followed by "whoop" (seven groups), "downsweep" and "woohoo" (five groups), and by "trumpet" (three groups). "Groan", "wiper", and "creak" were uncommon (heard in one group only) and were only heard in MC groups. The remaining MF calls were heard in both MC and MCE groups. "Heek" (Fig. 2E) was a short MF call with a variable frequency modulation pattern (ascending, descending, or modulated). In some instances, heek was produced with "gru" (LF), with a short silence separating the two vocalizations to constitute a "combined call". "Whoop" (Fig. 2F) was a long upsweep call starting with a flat Most of the LF sounds were harmonic-structure sounds, produced alone, except for boom and snort that can be produced in sequences. Drum sounds were always produced in series. Spectrogram parameters: Hamming window, FFT window size: 1,024 pts, 90% overlap. Generated using the Seewave package in R.

Full-size  DOI: 10.7717/peerj.13785/fig- 1 part and fast ascending frequency. "Downsweep" (Fig. 2B) was a long call showing a descending frequency slope with well-spaced harmonics. "Woohoo" (Fig. 2C) was a long-duration call (i.e., several seconds) showing a variable frequency-modulated pattern. "Trumpet" (Fig. 2D) was a call produced alone or associated with "gru" or "slight snort". "Downsweep", "woohoo", and "trumpet" were found in humpback whale songs recorded around the study site. "Groan" (Fig. 2A) was a long harmonic call. "Wiper" (Fig. 2G) was a short harmonic sound with a U-shape frequency modulation pattern, always produced in series (four to five repetitions) but with a random temporal pattern. "Creak" (Fig. 2H) was a composite call constituted by two different non-overlapping successive sounds without silence, and it showed the widest frequency bandwidth.

High-frequency harmonic sounds (HF)

HF was the third most represented category (10%, N = 202/2,033). Two social calls were within this category: "squeak" and "ascending shriek" (Fig. 3). Both were quite common: squeak (Fig. 3A) was heard in seven groups, and "ascending shriek" (Fig. 3B) was heard in four groups. "Squeak" and "ascending shriek" were heard in both MC and MCE groups.

"Squeak" was a very short call with frequencies above 1 kHz, and "ascending shriek" was one of the longest social calls with the highest frequencies amongst all. Both call types were also found in humpback whale songs recorded around the study site.

Amplitude modulated sounds (AM)

AM was the fourth most represented category (8%, N = 167/2,033). Five social calls were within the AM category: "door", "whine", "trill", "bug sound", and "AM grunt" (Fig. 4). "AM grunt" (Fig. 4E) and trill (Fig. 4C) were the most common MF calls (heard in six and five groups, respectively). The remaining calls were relatively uncommon since they were heard only in one group each. "Door" (Fig. 4A) was heard in an MC group. "Whine" (Fig. 4B) and bug (Fig. 4D) were found in MCE groups only. "Trill" and "AM grunt" were heard in both MC and MCE groups. "Trill", "door", "whine", and "bug" are long calls ranging from 1 to 5 s duration with a peak frequency ranging from 100 to 400 Hz. They were produced in bouts of random durations. "AM grunt" was short and, while commonly produced alone, it was sometimes associated with LF calls such as "gru" or "snort". "Whine", "trill", and "bug" were also found in humpback whale songs recorded around the study site. Pulsed sounds (PS)

PS was the least represented category (1%, N = 21/2,033). Four social calls were classified in this category: "fry", "bubble sound", "moped", and "gloop" (Fig. 5). These calls consisted of a repetition of very short, low-frequency sounds, and they were pretty uncommon. "Fry" (Fig. 5A) was heard in two groups and the remaining PS calls were heard in only one group each. "Fry" was heard in both MC and MCE groups. "Bubble sound" (Fig. 5B) was heard in a MC group. "Moped" (Fig. 5C) and "gloop" (Fig. 5D) were only found in MCE groups.

Aural-visual classification validity

For the validation of our classification RFs, 757 social calls representing 17 call types were used for the analyses (social calls for which Dur, Fmax, Q25, Q50, Q75, and PR were all measured). These analyzed social calls exclude call types for which we had less than six exemplars. The analyzed call types were: 100 Hz, bass, boom, gru, snort, burp, thowp, wop, downsweep, woohoo, trumpet, heek, whoop, squeak, ascending shriek, trill, and fry. The RF showed a global accuracy of prediction of 93% for classifying the calls into the five Except for AM grunts, AM sounds are long calls ranging from 1 to 5 s with a peak frequency between 100 and 400 Hz, produced in bouts of random durations. Whine, trill, and bug sounds were also found in humpback whale songs recorded around the study site. Spectrogram parameters: Hamming window, FFT window size: 1,024 pts, 90% overlap. Generated using the Seewave package in R.

Full-size  DOI: 10.7717/peerj.13785/fig- 4 defined main categories (OOB error rate = 7%, Table 3). The acoustic variables showing the highest importance for the classification were Q25, Fmax, and PR (Gini index: 10.24, 10.18, and 6.93 respectively, Table 3). Most call categories showed low individual classification error rates. For classifying the calls by types, the RF showed a global accuracy of prediction of 77% (OOB error rate = 23%, Table 4). The acoustic variables showing the highest importance for classification were Fmax, Q25, and Dur (Gini index: 32.10, 31.78, and 26.94 respectively, Table 4). Of the 17 calls included in the RF analysis, only four call types showed exceptionally high error rates (≥50%): gru, snort, thowp, and wop. These call

Note:

The overall error rate (out-of-bag error rate, OOB) was 7%. The last column indicates the classification error for each main call category. The bottom lines show the used acoustic variables along with the Gini index reflecting their relative importance in the classification.

DISCUSSION

Humpback whales' vocal activity is well known for its diversity and complexity at individual, group, and population levels. Social calls occur in all group compositions, in both breeding and foraging grounds as well as on migratory routes [START_REF] Dunlop | The social vocalization repertoire of east Australian migrating humpback whales (Megaptera novaeangliae)[END_REF][START_REF] Dunlop | Non-song acoustic communication in migrating humpback whales (Megaptera novaeangliae)[END_REF][START_REF] Zoidis | Vocalizations produced by humpback whale (Megaptera novaeangliae) calves recorded in Hawaii[END_REF][START_REF] Stimpert | Non-song sound production and its behavioral context in humpback whales (Megaptera novaeangliae)[END_REF][START_REF] Rekdahl | Temporal stability and change in the social call repertoire of migrating humpback whales[END_REF][START_REF] Fournet | Repertoire and classification of non-song calls in Southeast Alaskan humpback whales (Megaptera novaeangliae)[END_REF][START_REF] Recalde-Salas | Non-song vocalizations of humpback whales in western Australia[END_REF]Epp et al., 2021;Epp, Fournet & Davoren, 2021;[START_REF] Indeck | Adult female-calf acoustic communication signals in migrating east Australian humpback whales[END_REF]. The aural-visual analysis allowed us to identify 30 social calls distributed into five main call categories (LF, MF, HF, AM, PS) for mother-calf groups off Sainte Marie island. Our RF analyses showed a globally high agreement rate that demonstrates our aural-visual classification's robustness. However, some call types had low agreement rate in the RF compared to others (e.g., "gru", "snort", "thowp", and "wop"). This may be due to the fact that the repertoire is composed of discrete call types and graded call types (Epp, Fournet & Davoren, 2021). Call types with low agreement in the RF may correspond to graded calls. The existence of graded calls in mother-calf groups is not surprising since a high variation is expected. These groups are composed of individuals with different attributes (young and small individual in the process of maturing its calls, adult female, and potentially an adult male) that are likely related to call characteristics [START_REF] Cerchio | Variation in feeding vocalizations of humpback whales Megaptera novaeangliae from southeast Alaska[END_REF]Epp, Fournet & Davoren, 2021).

We could not establish if sounds were produced either by the mother or the calf or by any nearby conspecifics (i.e., escort) in our acoustic recordings. The accelerometer data of the tag [START_REF] Goldbogen | Using accelerometers to determine the calling behavior of tagged baleen whales[END_REF] could not be used to assign caller identity as the sampling rate used was 10 Hz, and even if our sampling rate was higher than 10 Hz, the close spatial proximity between a mother and her calf makes such methodology unreliable [START_REF] Saddler | Characterizing chilean blue whale vocalizations with DTAGs: a test of using tag accelerometers for caller identification[END_REF]. On the other hand, received levels alone are insufficient for assigning caller identity as most calls may show low amplitude levels, and several animals may be present around the tagged animal [START_REF] Stimpert | Variations in received levels on a sound and movement tag on a singing humpback whale: implications for caller identification[END_REF]. The calls described in the present study are thus considered as the acoustic output of mother-calf groups (including possible escort). Further investigations are still needed to assign each recorded social call to an individual. We are currently planning to explore the possibility of using simultaneous deployment of Acousonde tags on the mother and the calf to determine the caller's identity. Combining the received level of the same call on two different tags and the vertical distance between the mother and her calf (obtained from the diving profile) may allow the attribution social call to the corresponding individual.

Nine call of types out of 30 we aurally and visually identified were similar to song units recorded off the Sainte Marie island between 2013 and 2017 and were detected even in groups identified as MC, except for whine and bug sound. Assuming that mother-calf group composition did not change through the recordings' duration, the detection of sounds similar to song units in groups composed only of a female and a calf (MC groups) suggests that female humpback whales (or even calves) are able produce sounds with similar acoustic features as males' song units. Some social calls recorded in mother-calf groups off Sainte Marie island presented qualitative similarities to those described in other geographic areas during the breeding, feeding seasons, or migrating routes and were assumed to be the same call type. Those calls included snort, thowp, wop (also known as whup), bark, groan, trumpet, squeak, ascending shriek, trill, and AM grunt [START_REF] Dunlop | The social vocalization repertoire of east Australian migrating humpback whales (Megaptera novaeangliae)[END_REF][START_REF] Dunlop | Non-song acoustic communication in migrating humpback whales (Megaptera novaeangliae)[END_REF][START_REF] Zoidis | Vocalizations produced by humpback whale (Megaptera novaeangliae) calves recorded in Hawaii[END_REF][START_REF] Stimpert | Common humpback whale (Megaptera novaeangliae) sound types for passive acoustic monitoring[END_REF][START_REF] Rekdahl | Temporal stability and change in the social call repertoire of migrating humpback whales[END_REF][START_REF] Rekdahl | Common nonsong social calls of humpback whales (Megaptera novaeangliae) recorded off northern Angola, southern Africa[END_REF][START_REF] Fournet | Repertoire and classification of non-song calls in Southeast Alaskan humpback whales (Megaptera novaeangliae)[END_REF]Epp, Fournet & Davoren, 2021;[START_REF] Indeck | Adult female-calf acoustic communication signals in migrating east Australian humpback whales[END_REF]. Given the wide range of contexts within which these calls were detected (different group types, from breeding areas to feeding areas), these social calls are probably among the most common in humpback whales, and they may have important social roles. These social calls may constitute a global repertoire shared by humpback whales around the world. Further studies are needed to determine their behavioral context and roles, especially for mother-calf pairs. Social calls were detected even in mother-calf groups with neonate calf (C1 class), suggesting that vocal exchanges between mother and calf occur very soon right after birth. Such vocal interactions may be a way to reinforce the calf's social bond with the mother and to imprint the calf's voice on the mother. Calves have been reported to vocalize, and they can produce series of grunts, predominantly low-frequency sounds with a relatively narrow bandwidth [START_REF] Zoidis | Vocalizations produced by humpback whale (Megaptera novaeangliae) calves recorded in Hawaii[END_REF]. In our acoustic recordings, one call type, heek, is very similar to the amplitude modulated frequency sounds described by [START_REF] Zoidis | Vocalizations produced by humpback whale (Megaptera novaeangliae) calves recorded in Hawaii[END_REF], and thus, heek may be potentially assigned to calves.

We identified calls that can be combined or mixed with a given call type. We can assume that the calls were not successive calls from different individuals vocalizing one after the other due to the quasi absence of silence (and absence of overlap) in all instances. Call combination has not been previously described in humpback whales. Composite or concatenated calls have only been documented for few species [START_REF] Koren | Complex call in male rock hyrax (Procavia capensis): a multi-information distributing channel[END_REF][START_REF] Ouattara | Campbell's monkeys concatenate vocalizations into context-specific call sequences[END_REF][START_REF] Jansen | Segmental concatenation of individual signatures and context cues in banded mongoose (Mungos mungo) close calls[END_REF], 2013;[START_REF] Déaux | The bark, the howl and the bark-howl: identity cues in dingoes' multicomponent calls[END_REF]. Concatenated calls often have different biological functions than calls produced separately, as shown for the bark-howl vocalizations in dingo (Canis familiaris dingo) [START_REF] Déaux | The bark, the howl and the bark-howl: identity cues in dingoes' multicomponent calls[END_REF].

Compared to the East-Australian catalogue [START_REF] Dunlop | The social vocalization repertoire of east Australian migrating humpback whales (Megaptera novaeangliae)[END_REF] for which recordings were performed on migrating humpback whales of different group compositions (i.e., with or without calves), our repertoire contained fewer main categories (five vs. six), a similar number of call types (30 vs. 34, with eight shared call types), and a lower proportion of social calls also used in songs within the studied area (nine out of 32 in Madagascar vs. 22 out of 34 in Eastern Australia; [START_REF] Dunlop | The social vocalization repertoire of east Australian migrating humpback whales (Megaptera novaeangliae)[END_REF]. A repertoire with 16 call types has been described for the Southeast Alaskan humpback whales, with three calls likely shared with our repertoire [START_REF] Fournet | Repertoire and classification of non-song calls in Southeast Alaskan humpback whales (Megaptera novaeangliae)[END_REF]. The repertoire described in Newfoundland (Canada) consisted of 13 calls types and shared only one call type with our repertoire. A standardized comparative study using the same recording methods among these different areas is needed to accurately determine if a given call type is really unique to a population/area, as well as to confirm if the described call types are indeed new ones or a variation of one (previously described) call type. Our results, however, along with these previous descriptions of the repertoire of the humpback whale, support the existence of a highly diversified repertoire of social calls in a humpback whale. Our results also suggest that there is likely as much call diversity in mother-calf groups as in other social groups. Other acoustic studies focusing on mother-calf pairs also support the occurrence of such diversity [START_REF] Indeck | Adult female-calf acoustic communication signals in migrating east Australian humpback whales[END_REF].

Regarding our analysis on received levels, we found that calls recorded in mother-calf groups (mother-calf pairs accompanied or not by an escort) were produced at a low amplitude level, as found previously [START_REF] Tyack | Differential response of humpback whales, Megaptera novaeangliae, to playback of song or social sounds[END_REF]. The received levels ranged from 132 to 145 dB re 1mPa RMS, which is lower compared to the estimated received level of songs produced by singers of 149 to 169 dB re 1mPa [START_REF] Au | Acoustic properties of humpback whale songs[END_REF] and from our own recordings of singers (>165 dB re 1mPa, hydrophone clipped). Our results are consistent with a recent study on mother-calf pairs in Australia (136 to 141 dB re 1mPa RMS; [START_REF] Videsen | High suckling rates and acoustic crypsis of humpback whale neonates maximise potential for mother-calf energy transfer[END_REF]. Such low-amplitude vocal production in a mother-offspring pair is quite common in mammals, such as in pinnipeds (walrus ;[START_REF] Miller | Airborne acoustic communication in the walrus odobenus[END_REF]Charrier, Aubin & Mathevon, 2010), sheep (low-pitch bleats;[START_REF] Sèbe | Early vocal recognition of mother by lambs: contribution of low-and high-frequency vocalizations[END_REF], and cats (purring sounds; [START_REF] Peters | Purring and similar vocalizations in mammals[END_REF]. Low amplitude level in the context of mother-offspring interactions is not surprising as the communication between the mother and her calf is short-to mediumrange, and the purpose is likely to maintain social contact, to reinforce the social bond and maternal attachment with the calf, and to coordinate behaviors (e.g., side by side swimming, nursing sessions, etc.). Social purpose includes specific role such as individual identification. In a short-range communication context, why yelling when talking is sufficient? Antipredator strategy and male escort avoidance may also be hypothesized to explain such low-amplitude calls [START_REF] Videsen | High suckling rates and acoustic crypsis of humpback whale neonates maximise potential for mother-calf energy transfer[END_REF].

CONCLUSIONS

Our study provides a first assessment of the vocal repertoire of humpback whale mother-calf groups off Sainte Marie island, Madagascar, South Western Indian Ocean. We found that social calls recorded in these mother-calf groups are highly diversified and may be as diverse as those previously described in social groups not including calves for instance. A low acoustic intensity level characterized these social calls. The results suggest important vocal interactions between mother-calf pair, and mother-calf pairs with escorts. Our study contributes to the global catalogue of humpback whale calls and is a starting point in investigating the role of acoustic communication in humpback whale mother-calf interactions.

Figure 1

 1 Figure1Spectrograms of low-frequency sounds (LF). (A) 100 Hz, (B) bass, (C) boom, (D) gru, (E) snort, (F) burp, (G) guttural sound, (H) thwop, (I) wop, (J) bark, (K) drum. Most of the LF sounds were harmonic-structure sounds, produced alone, except for boom and snort that can be produced in sequences. Drum sounds were always produced in series. Spectrogram parameters: Hamming window, FFT window size: 1,024 pts, 90% overlap. Generated using the Seewave package in R.Full-size  DOI: 10.7717/peerj.13785/fig-1

Figure 2

 2 Figure 2 Spectrograms of mid-frequency harmonic sounds (MF). (A) Groan, (B) downsweep, (C) woohoo, (D) trumpet, (E) heek, (F) whoop, (G) wiper, (H) creak. Downsweep, woohoo and trumpet calls were also found in humpback whale songs recorded around the study site. Heeks were produced in association with Grus (LF) in some instances. Spectrogram parameters: Hamming window, FFT window size: 1,024 pts, 90% overlap. Generated using the Seewave package in R. Full-size  DOI: 10.7717/peerj.13785/fig-2

Figure 3

 3 Figure3Spectrograms of high-frequency sounds (HF). (A) Squeak, (B) ascending shriek. Squeaks were very short calls with frequencies above 1 kHz and ascending shrieks were among the longest social call types with the highest frequencies among all call types. Spectrogram parameters: Hamming window, FFT window size: 1,024 pts, 90% overlap. Generated using the Seewave package in R.Full-size  DOI: 10.7717/peerj.13785/fig-3

Figure 4

 4 Figure4Spectrograms of amplitude-modulated sounds (AM). (A) Door, (B) whine, (C) trill, (D) bug sound, (E) AM grunt. Except for AM grunts, AM sounds are long calls ranging from 1 to 5 s with a peak frequency between 100 and 400 Hz, produced in bouts of random durations. Whine, trill, and bug sounds were also found in humpback whale songs recorded around the study site. Spectrogram parameters: Hamming window, FFT window size: 1,024 pts, 90% overlap. Generated using the Seewave package in R.Full-size  DOI: 10.7717/peerj.13785/fig-4

Figure 5

 5 Figure 5 Spectrograms of pulsed sounds (PS). (A) Fry, (B) bubble sound, (C) moped, (D) gloop. PS are repetitive short and low frequency sounds. Spectrogram parameters: Hamming window, FFT window size: 1,024 pts, 90% overlap. Generated using the Seewave package in R. Full-size  DOI: 10.7717/peerj.13785/fig-5

Table 1

 1 Measured temporal and spectral characteristics for each identified social call.

	Measurements	Abbreviation Description
	Manual measurements
	Duration (s)	Dur	Total duration of the social call (a)
	Fundamental frequency (Hz)	F0	Fundamental frequency (b)
	Frequency excursion (Hz)	Fexc	Difference between the higher and lower frequencies (c)
			Measured only when applicable
			For harmonic-structured calls, Fexc was measured on the first visible frequency band
	Automatic measurements
	Peak frequency (Hz)	Fmax	Frequency at which the maximum amplitude level occurs (b)
	1 st energy quartile frequency (Hz) Q25	Frequency below which 25% of the total energy occurs (b)
	2 nd energy quartile frequency (Hz) Q50	Frequency below which 50% of the total energy occurs (b)
	3 rd energy quartile frequency (Hz) Q75	Frequency below which 75% of the total energy occurs (b)
	Frequency bandwidth (Hz)	Bdw	Frequency bandwidth within which the total energy fell within 12 dB of Fmax (b)
	Pulse rate (Hz)	PR	Number of pulse per second as measured using the pulse train analysis function in Avisoft
			SASLab Pro
			PR was considered to be zero for calls without a pulsed structure

Note:

(a) Measured on the oscillogram. (b) Measured on the averaged spectrum. (c) Measured on the spectrogram. Saloma et al. (2022), PeerJ, DOI 10.7717/peerj.13785 6/22

Table 2

 2 Details of the tagging sampling effort. MC: lone mother-calf pair. MCE: mother-calf pair accompanied by one or several escorts.

	Date	Tagged individual	Group type	Calf's relative age	Analyzed recording
					duration (hh:mm)
	07/08/2013	Mother	MCE	C3	04:05
	16/08/2013	Mother	MCE	C1	01:22
	05/09/2013	Mother	MC	C3	01:26
	09/09/2013	Mother	MCE	C3	00:39
	12/09/2013	Calf	MC	C3	03:14
	15/09/2013	Calf	MCE	C3	01:06
	05/08/2013	Mother	MC	C3	05:15
	24/08/2014	Mother	MC	C3	02:00
	26/08/2014	Mother	MCE	C2	14:23
	29/08/2014	Calf	MC	C3	00:27
	08/09/2014	Calf	MCE	C3	03:27
	09/09/2014	Calf	MC	C3	05:00
	10/09/2014	Calf	MC	C1	02:15
	11/09/2014	Calf	MCE	C2	00:35
	17/09/2014	Calf	MC	C3	00:28
	11/08/2016	Calf	MCE	C2	05:38
	17/08/2016	Calf	MCE	C3	07:14
	18/08/2016	Calf	MC	C1	10:14
	05/09/2016	Calf	MC	C3	07:46
	28/08/2017	Calf	MC	C3	05:32
	01/09/2017	Calf	MC	C3	02:30

Note:

C1 (neonate): calf presenting some folds, scars, and skin color that tends to be light grey dorsally and white ventrally and with less than 44 dorsal fin furl. C2: very young but non-neonate calves having more than 45 but less than 72 dorsal fin furl. C3: older calves that have unfurled dorsal fins (>72 ).

Table 3

 3 Random forest classification matrix for mother-calf groups' social call categories.

			Predicted class
			AM	HF	LF	MF	PS	Error
	True class	AM	21	0	0	2	0	0.09
		HF	0	94	0	0	0	0
		LF	0	0	438	16	0	0.04
		MF	17	7	8	143	0	0.18
		PS	0	0	0	0	11	0
	Variables		Q25	Fmax	PR	Dur	Q50	Q75
	Gini index		10.24	10.18	6.93	6.85	5.1	4.68

Table 5

 5 Received level measured for the most common aurally and visually easily identifiable call types.

	Calls types	Mean ± SD amplitude received level	N	Tagged individual
		(in dB re 1mPa RMS)
	100 Hz	141 ± 4	10	Mother
	Bass	154 ± 6	10	Mother
	Boom	135 ± 3	10	Mother
	Trill	132 ± 2	10	Mother
	Heek	145 ± 6	10	Calf

Saloma et al. (2022), PeerJ, DOI 10.7717/peerj.13785

Saloma et al. (2022), PeerJ, DOI 10.7717/peerj.13785 20/22

ACKNOWLEDGEMENTS

We thank everyone involved in humpback whale research in Madagascar: Cétamada team (François-Xavier Mayer, Henry Bellon, and Didier Cabocel) and all the eco-volunteers who dedicated their time and energy to this project.

Funding

This work was supported by the CNRS, the Total Foundation, the WWF Education Fund for nature program, and the CeSigma Company. The funders had no role in

Data Availability

The following information was supplied regarding data availability:

The raw measurements for all humpback whale social calls reported in the study are available in the Supplemental File.

types may share features with other call types (short, harmonic, and low-to-medium frequency calls).

Received level

Five call types were selected for the calculation of the received level: three LF ("100 Hz", "bass", and "boom"), one AM ("trill"), and one MF sound ("heek"). The received level ranged from 132 to 154 dB re 1mPa RMS with an average of 141 dB re 1mPa RMS (N = 50, 10 per call type; Table 5).

Note:

The overall error rate (out-of-bag error rate, OOB) was 23%. The last column indicates the classification error for each main call type. The bottom lines show the used acoustic variables along with the Gini index reflecting their relative importance in the classification.

ADDITIONAL INFORMATION AND DECLARATIONS

study design, data collection and analysis, decision to publish, or preparation of the manuscript.

Grant Disclosures

The following grant information was disclosed by the authors: CNRS. Total Foundation. WWF Education Fund for Nature Program. CeSigma Company.

Competing Interests

The authors declare that they have no competing interests.

Author Contributions

Animal Ethics

The following information was supplied relating to ethical approvals (i.e., approving body and any reference numbers): All methods and approaches were carried out in accordance with relevant national guidelines and regulations in force in Madagascar and were fully approved by the Ministry of Fisheries Resources, Madagascar (the approving body for all marine-related research in Madagascar) (National research and collect permits #44/13-MPRH/SG/DGPRH, #43/14-MRHP/SG/DGRHP, #28/16-MRHP/SG/DGRHP, and #26/17-MRHP/SG/DGRHP). The study complies with the European Union Directive on the Protection of Animals Used for Scientific Purposes (EU Directive 2010/63/EU).

Field Study Permissions

The following information was supplied relating to field study approvals (i.e., approving body and any reference numbers):

Field studies were fully approved by the Ministry of Fisheries Resources, Madagascar.

Supplemental Information

Supplemental information for this article can be found online at http://dx.doi.org/10.7717/ peerj.13785#supplemental-information.


Andr És

IRMAR Bert Wiest
email: bertold.wiest@univ-rennes1.fr

Andrés Navas
email: andres.navas@usach.cl

NIELSEN-THURSTON ORDERINGS AND THE SPACE OF BRAID ORDERINGS

Keywords: 2000 Mathematics Subject Classification, 20F36, 20F60; 06F15, 1

We study the topological space of left-orderings of the braid group, and its subspace of Nielsen-Thurston orderings. Our main result is that no Nielsen-Thurston ordering is isolated in the space of braid orderings. In the course of the proof, we classify the convex subgroups and calculate the Conradian soul for any Nielsen-Thurston ordering of Bn. We also prove that for a large class of Nielsen-Thurston orderings, including all those of infinite type, a stronger result holds: they are approximated by their own conjugates. On the other hand, we suggest an example of a Nielsen-Thurston ordering which may not be approximated by its conjugates.

Introduction

In recent years, some progress in the theory of orderable groups has been achieved by studying the so-called space of orderings (throughout, the word ordering stands for a left-invariant total order relation on a group). This corresponds to the set of all orderings that may be defined on a given group Γ, and carries a natural topology: given finitely many group elements, a neighborhood of a prescribed ordering ≺ is the set of all orderings which coincide with ≺ on this finite set. It is easy to see that this space is totally disconnected and compact; moreover, it is metrizable when the underlying group is countable. It is a nontrivial fact that this space cannot be countably infinite [START_REF] Linnell | The topology on the space of left orderings of a group[END_REF][START_REF] Navas | On the dynamics of left-orderable groups[END_REF][START_REF] Navas | with an appendix by A. Clay. A new characterization of Conrad's property for group orderings, with applications[END_REF]. Notice that an isolated point in this space corresponds to an ordering which is completely determined by finitely many inequalities.

Remarkable examples of orderable groups are the braid groups B n . Historically, the first ordering on B n (for n 3) was defined by Dehornoy. It was shown in [START_REF] Navas | On the dynamics of left-orderable groups[END_REF] that the Dehornoy ordering ≺ D is non isolated in the space of braid orderings. This is to be contrasted with the work [START_REF] Dubrovina | On braid groups[END_REF], where Dubrovina and Dubrovin show, by means of a slight modification of the Dehornoy construction, that B n admits isolated orderings. For example, on B 3 there is a unique ordering ≺ DD verifying the inequalities σ 1 σ 2 ≻ DD 1 and σ -1 2 ≻ DD 1. Dehornoy's construction involves some deep ideas which are purely algebraic. However, there is an alternative geometric approach to braid orderability. Indeed, ≺ D can be seen as a particular member of a family of uncountably many orderings, each of which is associated to a completely separating geodesic of the punctured disk (endowed with a hyperbolic metric). These so-called Nielsen-Thurston orderings on braid groups where largely studied and classified by H. Short and the second-named author in [START_REF] Short | Orderings of mapping class groups after Thurston[END_REF]. In the present paper we pursue this study from the point of view of the space of orderings. Our main result may be stated as follows.

Theorem. No Nielsen-Thurston ordering is isolated in the space of braid-orderings.

We recall that the positive cone of an ordering is the semigroup consisting of elements greater than the identity. As a direct consequence of this theorem and [8, Proposition 1.8] we have:

Corollary. There is no Nielsen-Thurston ordering whose positive cone is finitely generated as a semigroup.

To show the theorem, we first determine the convex subgroups of Nielsen-Thurston orderings (where a subset S is said to be convex if g ∈ S whenever f 1 ≺ g ≺ f 2 and both f 1 , f 2 are in S). We prove that the only convex subgroups are the obvious ones, coming from the way in which the geodesic cuts up the surface. For example, for the Nielsen-Thurston orderings of full infinite type (to be defined below), we prove that they have no convex subgroups other than {1} and B n . This solves by the affirmative the second half of Question 2.21 in [START_REF] Dehornoy | Ordering Braids[END_REF]Chapter XVI].

With the whole list of convex subgroups at hand, for each Nielsen-Thurston ordering we determine the Conradian soul, that is the maximal convex subgroup restricted to which the ordering satisfies the so-called Conrad property. This notion was introduced by the first-named author as a tool for studying the possibility of approximating a given ordering by its conjugates (where the conjugate of an ordering ≺ by a group element h is the ordering ≺ h for which f ≺ h g if and only if h -1 f h ≺ h -1 gh). For example, by [START_REF] Navas | On the dynamics of left-orderable groups[END_REF] every ordering on a countable group having trivial Conradian soul may be approximated by its conjugates. As a consequence, Nielsen-Thurston orderings of full infinite type can be approximated by their conjugates.

For Nielsen-Thurston orderings whose Conradian soul is isomorphic to Z (as is the case of the Dehornoy ordering), we show that the property of accumulation by its conjugates still holds. In the rest of the cases, we show that the Conradian soul is isomorphic to Z k for some k 2. Although we ignore whether these orderings may be approximated by their conjugates (actually, we present evidence that some of them might be isolated in the "space of Nielsen-Thurston orderings"), we show that they are not isolated in the space of braid orderings. This is achieved by means of a classical and very simple convex extension type argument.

A brief reminder on Nielsen-Thurston orderings

Nielsen-Thurston orderings of B n , which were introduced in [START_REF] Short | Orderings of mapping class groups after Thurston[END_REF], are the total orderings arising from the natural action of B n on the open interval ]0, π[. We briefly recall the definition. Let D n denote the unit disk, with n punctures lined up on the horizontal diameter. We can equip D n with a hyperbolic structure (in which the punctures correspond to cusps). Indeed, many such hyperbolic structures exist, but we fix one choice, for the rest of the paper.

A Nielsen-Thurston ordering is, by definition, induced by a finite or infinite geodesic ray γ : [0, T ] → D n (with T ∈ R + ) or γ : [0, ∞[→ D n , starting at a basepoint on the boundary of D n , and (in the case of a finite ray) terminating at some other point of ∂D n .

The ordering induced by γ is defined as follows. Any geodesic starting on the basepoint in ∂D n has in its starting point a certain angle, belonging to the interval ]0, π[, with the boundary ∂D n . Conversely, the geodesic is uniquely determined by the value of this angle. More generally, even a non-geodesic ray starting on the basepoint is homotopic, relative to its end points, to a unique geodesic, so homotopy classes of such rays are still in one-to-one correspondence with angles in ]0, π[. Now the braid group, viewed as the mapping class group of D n , acts on the geodesic γ; this yields a partial ordering ≺ of B n , by defining that β 1 ≺ β 2 if and only if the angle of β 1 .γ is smaller than the angle β 2 .γ.

If γ is not invariant under the action of any braid, then this recipe determines a total ordering of B n , and this ordering is called the Nielsen-Thurston ordering induced by γ. Notice that a necessary (but not sufficient) condition for γ not being invariant under any braid action is that γ cuts up the surface, in the sense that no two punctures are in the same path components of D n -γ([0, T ])).

An important property of Nielsen-Thurston orderings is the so-called subwordproperty, or property S. This means that for any two braid words w and w ′ , where w ′ is obtained from w by inserting positive generators σ i , we have : w ≺ w ′ . Geometrically, this is nothing but the fact that the left half-twist along any arc connecting two punctures pushes any curve intersecting the arc further to the left. In particular this half-twist is a positive element for the ordering.

The properties of the Nielsen-Thurston orderings are closely related to the geometric nature of the corresponding geodesic. Let us consider, for a maximal m ∈ {0, 1, . . . , n -1}, the m points in time such that some pair of punctures is in the same path component of D n -γ([0, t i -ǫ]), but in separate path components of D n -γ([0, t i ]). We shall call this the sequence of separating moments.

If m = n -1, i.e. if a finite initial segment of γ is sufficient to cut up the surface, then the ordering is said to be of finite type. If m < n -1 then the ordering is of infinite type. If m = 0, i.e. if γ has no self-intersections and no finite initial segment of γ separates any pair of punctures, then we say the ordering is of full infinite type.

Convex subgroups of Nielsen-Thurston orderings of finite type

Theorem 3.1. Suppose ≺ is a Nielsen-Thurston ordering of finite type, associated to a geodesic γ. Let 0 < t 1 < t 2 < . . . < t n-1 be the sequence of separating moments of γ. Then the chain of convex subgroups is

{1} ⊂ G n-1 ⊂ . . . ⊂ G 1 ⊂ B n
where G i denotes the subgroup of B n consisting of all braids which preserve the geodesic segment γ([0, t i ]).

Proof. It is immediate that the subgroups thus described are convex, we only have to prove that they are the only ones.

The proof is by downwards induction. The start of the induction is with the cases where all components of D n -γ([0, t n-i ]) contain at most two punctures. In these cases, the subgroup consisting of elements of B n leaving γ n-i invariant is isomorphic to Z i , and is lexicographically ordered. Such an ordered group has no convex subgroups except the ones obtained by successively removing the largest generator -and these subgroups do indeed appear in the chain of convex subgroups

G n-1 ⊂ . . . ⊂ G n-i .
Let us now suppose inductively that for some index i there are no convex subgroups in

G n-i except G n-j with j < i. Let G ′ be a convex subgroup of G n-i-1 such that G n-i G ′ ⊂ G n-i-1 . Our aim is to prove that G ′ = G n-i-1 .
Let us recall the structure of the group G n-i-1 : the geodesic γ([0, t n-i-1 ]) cuts D n into a number of components, some of which contain punctures. If we denote n 1 , . . . , n k the number of punctures contained in each of these path components, then the subgroup

G n-i-1 is isomorphic to a product G n-i-1 ∼ = B n1 × . . . × B n k . Moreover, there is exactly one connected component of D n -γ([0, t n-i-1 ]) which is further cut by γ([t n-i-1 , t n-i ]
) into pieces one of which contains λ punctures and another one µ punctures (with λ, µ 1, and at least one of the two being 2). Without loss of generality let us say the component being cut is the first one, so that

n 1 = λ + µ. λ punctures γ([tn-i-1, tn-i]) µ punctures Figure 1. Fat lines: γ([0, t n-i-1 ]). Dashed line: γ([t n-i-1 , t n-i ]).
Now there are three disks with geodesic boundary contained in the region under discussion: one containing λ punctures, one containing µ punctures, and one containing both sets of punctures. Possibly after a conjugation we can assume that the three disks are round (i.e. their boundaries intersect the horizontal diameter only twice, see Figure 1).

We shall study the action of the hypothetical subgroup G ′ on the geodesic γ([0, t n-i ]), keeping in mind that it fixes the initial segment γ([0, t n-i-1 ]). Let

γ max = sup g ′ ∈G ′ g ′ .γ
(where the supremum is in the sense that every geodesic corresponds to a point of the real line, and we're taking the supremum of real numbers). What does the first intersection of γ max with our critical component of

D n -γ([0, t n-i-1 ]) look like? Notice that γ max must be G ′ -invariant. γmax γmax i+1 σi i i+1 σi.γmax i (a) (b) Figure 2.
The geodesic segment γ max cannot pass between two punctures among the λ punctures of the left disk, nor can it pass between two of the µ punctures of the right disk, for if it did, then the action of an appropriate element in the smaller convex subgroup B λ or B µ would displace the geodesic even further to the left, which is excluded by construction (see Figure 2(a)). Suppose, on the other hand, that γ max avoids entirely the two discs with geodesic boundary containing the set of λ or µ punctures, but intersects the horizontal segment between the two disks. Then we have g ′ .γ max > γ max or (g ′ ) -1 .γ max > γ max , where g ′ is any element of G ′ which moves the segment γ([t n-i-1 , t n-i ]) (such an element must exist, since G n-i G ′ ). This is also impossible.

In summary, the geodesic γ max remains disjoint from the disk with geodesic boundary containing the λ + µ punctures; thus it spirals onto its boundary, as indicated in Figure 2(b). But this means that g ′ .γ can get as big (far to the left) as any geodesic g.γ with g ∈ G n-i-1 . This implies that G ′ = G n-i-1 . 2

Convex subgroups of Nielsen-Thurston orderings of full infinite type

We recall that a geodesic ray γ in D n gives rise to an ordering of full infinite type if it starts at the basepoint in ∂D n , has no self-intersections, separates the punctures, and is not stabilized by any nontrivial element of B n .

The next result should be compared with [START_REF] Clay | An exotic left ordering of the free group from the Dehornoy ordering[END_REF].

Theorem 4.1. Suppose γ is an infinite geodesic ray in D n giving rise to a Nielsen-Thurston ordering ≺ of B n of full infinite type. Then ≺ has no convex subgroups except {1} and B n .

Proof. The strategy is to prove that B n can be generated by elements which are "arbitrarily close to the identity", which implies that any nontrivial convex subgroup contains all of B n .

Here are the details. The geodesic γ can be specified by an infinite word, where the kth letter (k ∈ N) specifies between which pair of punctures of D n the kth intersection of γ with the horizontal diameter occurs, and in which direction (up or down).

We shall prove that the braid group can be generated by n-1 elements τ 1 , . . . , τ n-1 which all leave an arbitrarily long initial segment of γ invariant: the words describing τ i .γ all coincide with the word describing γ on an arbitrarily long initial segment.

Here is the construction of the elements τ i : we trace out the geodesic γ, for at least the whole required initial segment, and continue until we pass close to a puncture (so close that the line we have drawn so far does not intersect the short segment from our current position to the puncture). At that moment, we deviate from the geodesic γ, and drop into the puncture. We replace the curve we have just drawn by a geodesic which is homotopic to it relative to its endpoints, and call this geodesic γ ′ . The puncture at its end will be called the central puncture. Let k be the number of intersections of γ ′ with the horizontal diameter (not counting the final drop into the central puncture).

There are geodesic arcs a 1 , . . . , a n-1 disjoint from γ ′ and from each other, connecting the central puncture to the n -1 other punctures. For later use, we shall equip these arcs with an orientation pointing towards the central puncture. The braid τ i will be a positive or negative half Dehn-twist around the arc a i -the sign remains to be specified, depending on the relative position of terminal segments of γ ′ and a i .

We shall consider three cases. Firstly, if the last intersections of these two arcs with the horizontal diameter do not occur between the same pairs of punctures, then the sign of the half Dehn twist can be chosen arbitrarily. Indeed, in this case the first k intersections of τ i .γ with the horizontal diameter coincide with those of γ, independently of whether τ i is taken to be the positive or negative half twist along a i -see Figure 3(a).

The second case to consider is where the last intersections of the the arcs γ ′ and a i with the horizontal diameter lie between the same pairs of punctures, and where, moreover, the terminal segment of a i lies to the right of the terminal segment of γ ′ (as seen while approaching the central puncture along γ ′ ) -see Figure 3(b) for examples. In this case, we take τ i to be the positive half twist along a i . Again, we observe that with this choice the word describing the first k intersections of τ i .γ with the horizontal diameter coincides with the corresponding word for γ.

The third and final case is where the last intersections of the the arcs γ ′ and a i with the horizontal diameter lie between the same pairs of punctures, and where the terminal segment of a i lies to the left of the terminal segment of γ ′ . In this case, we take τ i to be the negative half twist along a i -see Figure 3(c). 2

Convex subgroups of general Nielsen-Thurston orderings

By combining the techniques of proof of Theorems 3.1 and 4.1, one can easily obtain the following unified result:

Theorem 5.1. Suppose ≺ is a Nielsen-Thurston ordering, associated to a geodesic γ. Let 0 < t 1 < t 2 < . . . < t m , with m ∈ {0, 1, . . . , n -1}, be the sequence of separating moments of γ. Then the chain of convex subgroups is

{1} ⊂ G m ⊂ . . . ⊂ G 1 ⊂ B n
where G i denotes the subgroup of B n consisting of all braids which preserve the geodesic segment γ([0, t i ]). 2

The Conradian soul of Nielsen-Thurston orderings

Recall that a group ordering ≺ is said to be Conradian if for every positive f, g there exists k ∈ N so that f g k ≻ g (see [START_REF] Botto-Mura | Orderable groups[END_REF][START_REF] Kopitov | Right ordered groups. Siberian School of Algebra and Logic[END_REF]). Using structure theorems for Conrad orderable groups and the description of the convex subgroups from the previous sections, one can easily show that no Nielsen-Thurston ordering is Conradian (c.f. Corollary 6.4). However, we prefer giving a longer proof which avoids these structure theorems. In this proof, the cases of finite type orderings on B 3 and B 4 need a special argument (compare Remark 6.5).

Example 6.1. The Dehornoy ordering ≺ D on B 3 is not Conradian. This is shown in [START_REF] Navas | On the dynamics of left-orderable groups[END_REF]Example 3.21], through a very indirect argument. A shorter proof works as follows: using the identity σ 2 σ 1 σ -1

2 = σ -1 1 σ 2 σ 1 we get, for all k ∈ N, that 1 ≺ D σ 2 σ 1 σ -(k+1) 2 = σ -(k+1) 1 σ 2 σ 1 . Therefore, (σ -1 2 σ 1 )(σ k 1 ) ≺ D σ 1 for all k ∈ N, yet both σ -1
2 σ 1 and σ 1 are ≺ D -positive. Notice that, according to [START_REF] Short | Orderings of mapping class groups after Thurston[END_REF], every Nielsen-Thurston ordering of finite type on B 3 is conjugate to the Dehornoy ordering. Since the Conrad property is stable under conjugacy, no Nielsen-Thurston ordering of finite type on B 3 is Conradian. Example 6.2. Up to conjugacy, there are only three Nielsen-Thurston orderings of finite type on B 4 , namely those given by the geodesics of Figure 4 below (see again [START_REF] Short | Orderings of mapping class groups after Thurston[END_REF]). None of them gives rise to a Conradian ordering. Indeed, (a) corresponds to the Dehornoy ordering, and the above argument applies. For the geodesics in (b) and (c), one can directly check the inequality 1

≺ σ 3 σ 2 σ -(k+1) 3
, which allows to conclude as in the previous example. Following [START_REF] Navas | On the dynamics of left-orderable groups[END_REF][START_REF] Navas | with an appendix by A. Clay. A new characterization of Conrad's property for group orderings, with applications[END_REF], the Conradian soul of an ordering is defined as the maximal convex subgroup restricted to which the order satisfies the Conrad property. It was already shown in [START_REF] Navas | On the dynamics of left-orderable groups[END_REF] that the Conradian soul of the Dehornoy ordering on B n corresponds to σ n-1 . Next, we describe the Conradian soul of any Nielsen-Thurston ordering, thus solving a problem raised in [START_REF] Navas | On the dynamics of left-orderable groups[END_REF]Remark 3.40].

Theorem 6.3. The Conradian soul of a Nielsen-Thurston ordering on B n corresponds to the maximal Abelian convex subgroup, that is, to the largest copy of

B 2 × • • • × B 2 which is convex.
Proof. For a geodesic of infinite type, the smallest non-trivial convex subgroup is a copy of B m for some m with 3 m n. Such a group has no convex subgroup. If it was Conradian, then it would be Archimedean, and hence by Hölder's theorem, the underlying group would be Abelian (see [START_REF] Botto-Mura | Orderable groups[END_REF][START_REF] Kopitov | Right ordered groups. Siberian School of Algebra and Logic[END_REF]). This is absurd as B m , for m 3, is not Abelian.

If the geodesic is of finite type, then it is clear that the maximal convex copy of a product of (say, k) B 2 factors is contained in the Conradian soul (indeed, any bi-invariant ordering -and thus any ordering defined on an Abelian groupis Conradian). The next larger convex subgroup of B n corresponds either to a product

P = B n1 × • • • × B n k-1 ,
where there is only one index different from 2, which is equal to 4, or to a product

P = B n1 × • • • × B n k ,
where only one index is different from 2, and it is equal to 3. If the restriction of the ordering to P was Conradian, then the restriction to B 4 or B 3 , respectively, would be also Conradian. However, we have seen in Examples 6.1 and 6.2 that this cannot be the case.

An alternative argument for the end of the proof. To see that the restriction of the ordering to P cannot be Conradian, first note that P has only finitely many convex subgroups. By [9, Proposition 1.7] and its proof, if P was Conradian then the underlying group would be solvable. However, neither B 3 nor B 4 are solvable. 2 Corollary 6.4. No Nielsen-Thurston ordering is Conradian.

Proof. If a Nielsen-Thurston ordering was Conradian, then it would coincide with its Conradian soul. However, according to Theorem 6.3, this is never the case. 2 Remark 6.5. Although stated for Nielsen-Thurston orderings on general braid groups B n (with n 3), the preceding corollary gives new information only in the cases of B 3 and B 4 . Indeed, these groups do admit Conradian orderings, whereas for n 5 there is no Conradian ordering on B n at all (see [2, pp. 287-289]).

7. Nielsen-Thurston orderings of infinite type are approximated by their conjugates

The relevance of the Conradian soul stems from the following result which we cite from [8, Proposition 4.7] (see also [START_REF] Clay | Isolated points in the space of left-orderings of a group[END_REF]Theorem 1.2] and [9, Theorem 2.7]).

Theorem 7.1. If the Conradian soul of an ordering on an infinite countable group is trivial, then this ordering is an accumulation point of its conjugates.

Using this result we may easily prove the following Theorem 7.2. Any Nielsen-Thurston ordering of infinite type on B n may be approximated by its conjugates.

Proof. By Theorem 6.3, the Conradian soul of the ordering is trivial. The result then follows from Theorem 7.1. 2

Nielsen-Thurston orderings of finite type are not isolated

We now show our main result.

Theorem 8.1. Nielsen-Thurston orderings are not isolated in the topological space of left-invariant orderings of B n .

Proof. In the case of orderings of infinite type, the ordering may be approximated by its conjugates (see Theorem 7.2).

In the finite type case, for a maximal value of k there exists a convex subgroup of the form

B 2 × • • • × B 2 ∼ Z k . If k 2,
then the restriction of ≺ to Z k is not isolated in the space of orderings of Z k (see for instance [START_REF] Sikora | Topology on the spaces of orderings of groups[END_REF]). A convex extension argument (see [START_REF] Navas | On the dynamics of left-orderable groups[END_REF]Section 3.3.5]) then shows that ≺ is not isolated in the space of braid orderings. If k = 1, then the smallest convex subgroup strictly containing the maximal Abelian convex subgroup B 2 is a copy of B 3 . By [START_REF] Short | Orderings of mapping class groups after Thurston[END_REF], the restriction of ≺ to this copy of B 3 is a conjugate of the Dehornoy ordering, which is not isolated in the space of orderings of B 3 : see Example 8.2 below. Once again, a convex extension argument shows that ≺ is not isolated in the space of braid orderings. 2 Example 8.2. Dehornoy's ordering is approximated by its conjugates. This was first shown in [START_REF] Navas | On the dynamics of left-orderable groups[END_REF] throughout a very indirect argument. A simpler proof appears in [START_REF] Dehornoy | Ordering Braids[END_REF]Chapter XIV]. Here we propose an even simpler argument. Let ≺ j be the Dehornoy ordering, conjugated by σ -j 2 σ 1 . Thus, a word w is positive in the ordering ≺ j if and only if σ -1 1 σ j 2 wσ -j 2 σ 1 ≻ D 1. We claim that the sequence ≺ j tends to ≺ D in the space of orderings. Indeed, if w = σ k 2 for some k > 0, then σ -1 1 σ j 2 wσ -j 2 σ 1 ≻ D 1 since the Dehornoy order has Property S. If, on the other hand, w is a σ 1 -positive word:

w = σ k1 2 σ 1 σ k2 2 σ 1 . . . σ k ℓ-1 2 σ 1 σ k ℓ 2 , then we calculate σ -1 1 σ j 2 wσ -j 2 σ 1 = σ -1 1 σ j 2 σ k1 2 σ 1 σ k2 2 σ 1 . . . σ k ℓ-1 2 σ 1 σ k ℓ 2 σ -i 2 σ 1 = σ 2 σ j+k1 1 σ -1 2 σ k2 2 σ 1 . . . σ k ℓ-1 2 σ 1 σ k ℓ 2 σ -n 2 σ 1 . Thus σ 1 σ -j
2 wσ j 2 σ 1 is σ 1 -positive for sufficiently large j (namely for j > -k 1 ), which proves the desired convergence.

Finally, ≺ j is different from ≺ D for all positive integers j, since its smallest positive element is the conjugate of σ 2 by σ -j 2 σ 1 , and this is different from σ 2 .

Remark 8.3. The choice of the sequence σ -j 2 σ 1 is quite natural. Indeed, the sequence σ -j 2 σ 1 approaches the Conradian soul from above: for any element c of the Conradian soul σ 2 and for any σ 1 -positive word w we have c ≺ D σ -j 2 σ 1 ≺ D w, where the second inequality is only true for sufficiently large j. In terms of geodesics, if γ is the standard geodesic inducing the Dehornoy-order on B 3 (c.f. Figure 4(a)), then the sequence of geodesics σ -j 2 σ 1 .γ is eventually further to the right than any prescribed geodesic β.γ, for β a σ 1 -positive braid -see Figure 5.

σ -7 2 σ1
Figure 5. The geodesic σ -j 2 σ 1 .γ deviates from γ to the left, but "by as little as possible": it goes "over" the leftmost puncture (thus deviating to the left), but then it goes as much to the right as possible (it turns j times counterclockwise around the two rightmost punctures, with j arbitrarily large) Example 8.2 can be generalized as follows.

Theorem 8.4. Every Nielsen-Thurston ordering ≺ of finite type whose Conradian soul is a copy of B 2 can be approximated by its conjugates.

Proof. If σ is a positive generator of the Conradian soul, then σ corresponds to a left half-twist. According to [8, Proposition 4.9] and its proof (see also [START_REF] Navas | with an appendix by A. Clay. A new characterization of Conrad's property for group orderings, with applications[END_REF]Theorem 2.9] and its proof), there exists a sequence ≺ j = g j (≺) of conjugates of ≺ which converges either to ≺ or to ≺ ′ , where ≺ ′ is the ordering whose set of positive elements is the union of {σ k : k < 0} and {g : g ≻ 1, g / ∈ σ }. If the limit point was ≺ ′ , then the conjugates g -1 j σg j would be negative for j large enough. This contradicts the fact that the Nielsen-Thurston ordering ≺ has the subword property S.

Remark 8.5. The argument of the proof above actually shows the stronger result that for any given Nielsen-Thurston ordering of finite type ≺, there exists another finite-type Nielsen-Thurston ordering ≺ 1 , obtained from the given one simply by permuting the lexicographic order of the Conradian soul, which is approximated by conjugates of itself (compare [8, Proposition 4.9] and [9, Corollary 2.10]).

The previous results show that the only Nielsen-Thurston orderings which may fail to be approximated by their conjugates are those of finite type whose Conradian soul is higher-rank Abelian. this order can be approximated by other orders, but we do not know whether it can be approximated by its conjugates. Imitating Example 8.2 and the proof of Theorem 8.4, we could try to conjugate the order by elements which are as small as possible while being larger than all elements of the Conradian soul. This, however, yields an approximation not of the desired ordering ≺, but of one where the lexicographic order of Z 3 has been permuted. Question. Can the ordering induced by the geodesic of Figure 6(a) be approximated by its conjugates? If not, can it be approximated by a sequence of Nielsen-Thurston orderings? The calculation above suggests that the answer to both questions may be "no".

Figure 3 .

 3 Figure 3.

Figure 4 .

 4 Figure 4. The three conjugacy classes of finite type Nielsen-Thurston orderings of B 4 .

Example 8 . 6 .

 86 Consider the ordering ≺ induced by the geodesic shown in Figure 6(a). The Conradian soul of this ordering is σ 1 , σ 3 , σ 5 = Z 3 . By Theorem 8.1,

Figure 6 .

 6 Figure 6. (a) A geodesic whose associated ordering of B 6 seems not to be a limit point of Nielsen-Thurston orderings. Note the convex subgroups σ 5 ⊆ σ 3 , σ 5 ⊆ σ 1 , σ 3 , σ 5 . (b) Conjugating by σ -N3 σ 4 (a sequence which becomes eventually smaller than any positive element outside the Conradian soul) yields an approximation of a different ordering with convex jumps σ 1 ⊆ σ 1 , σ 5 ⊆ σ 1 , σ 3 , σ 5 .

Acknowledgments. The first-named author was funded by PBCT-CONICYT Research Network on Low Dimensional Dynamics.