John T Hale
email: jthale@uga.edu

Luca Campanelli

Jixing Li

Shohini Bhattasali

Christophe Pallier

Jonathan R Brennan

Neuro-computational models of language processing

Keywords: neurolinguistics, brain, computational model, deep learning, parsing, lexicon

HAL is

Linguistics as abstract neurobiology

"Verbing weirds language" says the boy Calvin to his friend Hobbes in a 1993 newspaper comic strip written by Bill Watterson. 1 The thing is, Hobbes (a stuffed tiger made animate by Calvin's imagination) has no problem understanding this sentence, and nor of course does the reader. Language users regularly deal with the abstractions of human language, whether turning adjectives into verbs, as in the comic strip, formulating a question from a declarative statement, or recognizing the dual meanings of the statement "Yesterday I saw an elephant in my pajamas." Linguists make sense of these patterns by recognizing that the human capacity for language rests on a complex set of abstractions: from phonological features to syntactic categories, Linguistics is abstract. As with other cognitive faculties, these linguistic abstractions are instantiated in the concrete neurobiology of language users.

The biological foundation of linguistic knowledge has been a pillar of generative linguistics since the 1960's [START_REF] Lenneberg | Topographic mapping of a hierarchy of temporal receptive windows using a narrated story[END_REF]). Yet, efforts to map between the abstract constructs of linguistics and concrete properties of the human brain face significant challenges. [START_REF] Embick | Towards a computational(ist) neurobiology of language: Correlational, integrated, and explanatory neurolinguistics[END_REF] discuss two specific barriers to grounding linguistics in neurobiology. The first concerns ontology, or the entities that are discussed in theories of linguistics versus theories of neuroscience: How does a noun relate to a neuron, or how do patterns of electrical discharge relate to syntactic dependencies? It is not plausible to simply draw a line from a primitive in one domain to a primitive in another. The second barrier concerns granularity: what is the "right" kind of primitive for connecting linguistic abstractions with neurobiological properties? Are there specific neuronal columns devoted to computing noun phrases, or do phrasal categories emerge from interactions between disparate cell assemblies? Is the noun phrase an appropriate unit to connect with neuronal function, or ought linkings be made to specific constructions, or perhaps to more general notions such as Merge? The field has not reached a consensus about how to tackle these questions, collectively called the Mapping Problem between linguistics and neurobiology [START_REF] Poeppel | The maps problem and the mapping problem: two challenges for a cognitive neuroscience of speech and language[END_REF].

The Mapping Problem: The challenge of linking linguistic and neural primitives in an explanatory way.

This paper reviews one approach to answering the Mapping Problem, one that relies on computational models of language processing. These models explicitly specify how properties of human language relate to real time processing, and also how such processes might affect observable neural signals. As such, this approach relates both to concrete properties of the brain and to live issues in the study of human language.

Box 1: Bluffer's Guide to Neurolinguistic Methods fMRI Functional Magnetic Resonance Imaging measures the flow of oxygenated blood inside the brain; more oxygenated blood flows in to serve the metabolic needs of neurons that are active, but with a delay of 4-8 seconds. Recordings from fMRI are thus blood-oxygen level dependent, and so commonly referred to as the BOLD signal. This tool offers high spatial but low temporal resolution. See [START_REF] Brennan | Hemodynamic methods[END_REF] or chapter 2 of [START_REF] Kemmerer | Cognitive neuroscience of language[END_REF]. EEG Electroencephalography measures electrical potentials on the scalp that are generated by tens of thousands of neurons from the cortex. These "brainwaves" are sensitive with high temporal resolution to the activity of synchronized neurons that share a common orientation, but the electrical signal is spatially imprecise (see [START_REF] Luck | An introduction to the event-related potential technique[END_REF] or page 56ff of Kemmerer 2014 chapter 2). ERP Event-related potentials are measured by averaging together EEG signals that have been aligned to an event such as stimulus presentation. Well-studied ERP components include the N400, a negative-going central-posterior voltage potential that increases in amplitude for semantically unexpected items, and the P600, a positive-going posterior potential that is related to violated syntactic expectations. See [START_REF] Swaab | Language-related ERP components[END_REF] for a review. MEG Magnetoencephalography measures the magnetic fluctuations associated with electrical activity. Like EEG, this signal tracks neural activity as it happens, millisecond by millisecond, but it also offers higher spatial resolution as magnetic fields can be used to triangulate the electrical "source" of activity in the brain. See [START_REF] Salmelin | Electromagnetic brain imaging[END_REF] for an overview to this technique, or [START_REF] Hansen | MEG: an introduction to methods[END_REF] for a textbook introduction. MVPA Multivariate pattern analysis refers to a diverse set of methods that analyze neural responses as patterns of activity that reflect the varying brain states. MVPA-based decoding analysis makes use of machine learning classifiers, whose inputs are vectors of observed neural signal levels, to guess whether a particular brain state falls into one of a finite number of discrete categories. These categories may label linguistic content in the stimuli that were presented to human participants. If the classifier achieves abovechance performance, then researchers infer that information corresponding to that category label was present within the brain, at the place and time the neural signals were taken (see [START_REF] Pereira | Machine learning classifiers and fMRI: A tutorial overview[END_REF]).

The scope of this review

As suggested above, the key element is computation. It is helpful to distinguish between the scientific use of computers in general and the computational theory of mind as a foundation for cognitive science. While computers may serve to deduce the consequences of linguistic theory or analyze neural signals, these sorts of computations are not the focus of this review. Rather, we focus on computations and representations that play an explanatory role in answering the question "how does an individual use language?" This garden-variety computationalism has been a meeting-point for linguists and brain scientists over the years (for a philosophical overview see [START_REF] Rescorla | The Computational Theory of Mind. The Stanford Encyclopedia of Philosophy Rohde DL[END_REF]. The present article specifically considers neuro-computational models that match timeseries brain data in order to shed light on questions of interest to linguists. It updates [START_REF] Stowe | Rethinking the neurological basis of language[END_REF] with computer models of language processing in particular regions of the brain. We consider neurolinguistic methods involving naturalistic text stimuli which have already been reviewed in [START_REF] Brennan | Naturalistic sentence comprehension in the brain: Naturalistic comprehension[END_REF], and we touch on classifier-based decoding of language in the brain, which is introduced in [START_REF] Murphy | Decoding language from the brain[END_REF].

The current review sets aside several closely related areas. One is the research tradition rooted aphasiology, which is often computational in exactly the sense evoked above (e.g. [START_REF] Caplan | Language: structure, processing, and disorders[END_REF]). Another closely-related area is computational psycholinguistics; for a general review of that field we refer the reader to [START_REF] Hale | Models of human sentence comprehension in computational psycholinguistics[END_REF]. We also set aside research using signals that reflect neural oscillations (e.g. [START_REF] Bastiaansen | Oscillatory neuronal dynamics during language comprehension[END_REF][START_REF] Meyer | Synchronous, but not entrained: Exogenous and endogenous cortical rhythms of speech and language processing[END_REF], while recognizing the promise of that direction of research when combined with explicit computational models of the kind considered in this review [START_REF] Martin | A mechanism for the cortical computation of hierarchical linguistic structure[END_REF]. 2 As Poeppel (2012, already cited above) emphasizes, scientific progress hinges upon our ability to explicitly link theorized linguistic concepts, such as phrase structure, and observable neurobiological phenomena, such as blood flow in the brain. These sorts of connections between theory and data inevitably reflect additional assumptions about cognitive processing. Using the methodologies in box 1, the next sections sketch several specific approaches to building such links. Current work using these methods is assessed with respect to its advantages and disadvantages for linguistics.

Guessing the next word or symbol

A prominent perspective on the brain construes it to be an "inference engine" that optimizes predictive accuracy [START_REF] Friston | The free-energy principle: A unified brain theory?[END_REF]. Neural signals, such as those described in box 1, certainly show exquisite sensitivity to deviations from an expected outcome. This sensitivity has been leveraged to probe the sorts of representations in use during language processing. In this type of work, computational models serve to define probability distributions over upcoming linguistic input.

The surprisal linking hypothesis A probability distribution over upcoming linguistic expressions quantifies how predictable those expressions are. This degree of predictability can be formalized as the self-information or surprisal of an event, such as a particular successor word following a given left-context string (see Hale 2016, for a review). One way to estimate such a probability distribution is by counting. For instance, the probability of the word "snow" in the phrase "look at the falling snow" can be estimated by counting in a large corpus of text how often the words "the falling" are followed by "snow." Counting word sequences by threes in this way is a trigram Markov Model (see box 2). This approach was taken by [START_REF] Willems | Prediction During Natural Language Comprehension[END_REF] to work out numerical surprisal values that quantify the unexpectedness of each successive word. Using fMRI they find increased activation for higher surprisal in the superior temporal lobe of the left and right hemisphere.

Surprisal: Deviation between an expected and actual linguistic input; equivalent to the bits of information conveyed.

Box 2: Language models

A language model assigns probabilities to sentences and their substrings.

ngram An ngram language model uses the previous n -1 symbols as a conditioning context for the probability of the n th symbol, a Markov assumption. The symbols may come from any discrete set including letters, phones, words, or parts of speech. The probabilities are estimated from corpora and are typically "smoothed" to handle unseen events (see Eisenstein 2019, §6.1, 6.2). The case n = 3 is called a trigram model. CFG Context-Free Grammar, also called Phrase-Structure Grammar, is a formal model of language based on symbolic rewriting rules. These symbols and rules induce equivalence classes over linguistic expressions, for instance noun phrases with or without an optional adjective. Such grammars served as the "base component" in transformational grammar (see Partee et al. 1993, chapter 16). Adding weights to CFG rules yields probabilistic context-free grammars (PCFG) whose derivations can be treated as a stochastic branching process (see Manning & Schütze 2000, chapter 11), from which a language model can be derived [START_REF] Jelinek | Computation of the probability of initial substring generation by stochastic context-free grammars[END_REF]. RNNs In recurrent networks, the same block of neural net architecture is "unrolled" so that it applies over and over again to each position in a string. This allows it to adaptively learn from examples using a variant of the backpropagation algorithm [START_REF] Werbos | Backpropagation through time: What it does and how to do it[END_REF]).

Recurrent nets based on gated connections such as the Long Short-Term Memory of [START_REF] Hochreiter | Long Short-Term Memory[END_REF] fare better than their ungated counterparts when it comes to long-distance dependencies (see Goldberg 2017, chapter 15). Transformers Transformers are neural net architectures for sequence processing that use an "attention" mechanism based on multiplicative gating [START_REF] Vaswani | Attention is all you need[END_REF]. This allows distant symbols to influence the choice of fill-in-the-blank answers. They are typically trained using a masked language modelling objective which allows for influences from either left or right context. Transformers are built to maximize parallelism on modern computers, which entails an upper limit on the length of strings that they can handle (i.e. 512 tokens in BERT).

ral signals derived from "everyday" language processing (for discussion, see [START_REF] Brennan | Naturalistic sentence comprehension in the brain: Naturalistic comprehension[END_REF]. These sorts of studies elicit ecologically-natural language comprehension by presenting human participants with narrative language much as one might read a story or listen to an audiobook. At issue in subsequent statistical analysis is the degree to which the observed neural signals fit or fail to fit theoretical predictions about the processing difficulty of that text. A key requirement on predictors is broad coverage: the more stimulus words there are that receive a numerical difficulty prediction, the better the chance of being able to estimate a relationship between the predictor and the measured neural signal (e.g. BOLD see box 1). Language models such as the trigram mentioned above in connection with Willems et al.'s 2015 study are "broad coverage" in exactly this sense. But there are other language models that retain the trigram's breadth of coverage while at the same time sharpening the prob-ability distribution over successor words3 ; some key examples are summarized in box 2. Modern artificial neural network language models fall into this category (see [START_REF] Jurafsky | Speech and language processing[END_REF], chapter 7 or Goldberg 2017, chapter 9). Building on [START_REF] Elman | Finding structure in time[END_REF]), Frank et al. (2015) shows that neural network language models can account for a well-known expectancy effect in human electrophysiology, the N400 (see box 1).

Surprisal as a degree of change to internal representations [START_REF] Ettinger | Modeling N400 amplitude using vector space models of word representation[END_REF] and [START_REF] Rabovsky | Modelling the N400 brain potential as change in a probabilistic representation of meaning[END_REF] also model the N400 ERP component. In this line of work, the linking hypothesis is defined in terms of a componentwise changes to a neural network's internal representation. If these components were to receive an interpretation in terms of next-word distributions, then the proposal would reduce to (word) surprisal, as introduced above. But importantly for Rabovsky and colleagues, the representations inside their model are not just about word tokens. Rather they pair up a form of Case Grammar [START_REF] Fillmore | The case for case[END_REF]) with a decompositional semantics in the style of [START_REF] Katz | The structure of a semantic theory[END_REF]. Words and sentences are both represented using microfeatures that are conceptual in nature, following McClelland &Kawamoto (1986) andRohde (2002). This kind of modeling fleshes out, in computational detail, the established view of the N400 as having to do with dashed meaning expectations [START_REF] Kutas | Electrophysiology reveals semantic memory use in language comprehension[END_REF]. [START_REF] Brouwer | Neurobehavioral correlates of surprisal in language comprehension: A neurocomputational model[END_REF] extend this general approach in two ways. First, they model the P600 effect as well as the N400. Secondly, they do so using an internal representation that relates to models of propositional logic. By providing a 'microworld' interpretation for vector-valued internal states of a neural network, Brouwer et al. aim to capture the effect of real-world knowledge on human sentence processing.

The degree to which intermediate states of the human language comprehender should be viewed as "model-like" (as opposed to proof-like) remains open (see e.g. [START_REF] Hemforth | Language processing: Construction of mental models or more? In Mental Models and the Mind[END_REF]. So too is the question of whether or not neural sequence models should be thought of as possessing knowledge of grammar (see e.g. [START_REF] Steedman | Connectionist sentence processing in perspective[END_REF][START_REF] Linzen | Assessing the ability of LSTMs to learn syntax-sensitive dependencies[END_REF].

Structures & sequences Neural network sequence models of the type mentioned above share with ngram models a conceptualization of language as a time series of words, lacking any hierarchical structure. Chomsky's 1956 critique of this beads-on-a-string outlook is all too familiar to linguists. Cognizant of this critique, other researchers have used hierarchical syntactic analyses as an intermediate representation for modelling time series brain data.

For instance Brennan & Hale (2019) demonstrate that a phrase-structured model of part-of-speech tag sequences yields a better model of EEG data than does an ngram model or simple recurrent network in the style of Frank et al. (2015, already cited above). This project used the surprisal linking hypothesis. It uncovered a different kind of electrophysiological response, one that manifests earlier and on electrodes that are more anterior than the usual N400. Others have similarly confirmed the explanatory role of hierarchical syntax in other methodologies.

To take just one fMRI example, [START_REF] Shain | fMRI reveals language-specific predictive coding during naturalistic sentence comprehension[END_REF] use the surprisal linking hypothesis to work out processing difficulty predictions from a different sort of probabilistic phrasestructure grammar (see also [START_REF] Henderson | Language structure in the brain: A fixationrelated fMRI study of syntactic in reading[END_REF]. They also derive surprisals from a 5-gram Markov Model, one whose parameters can only be estimated with exceptionally large corpora. This point highlights a challenge in extending heavily data-reliant methods to broader cross-linguistic study, especially of under-resourced languages. Shain et al. find that these two sorts of surprisal, one based on hierarchical phrase-structure, and another based on word sequences, explain independent variability in the BOLD signal in a range of areas through the "language network" of the frontal and temporal lobes in the left hemisphere. That is, the data are consistent with processing mechanisms that are sensitive to hierarchical dependencies along-side local word-to-word transitions.

These studies reason backwards from the goodness of fit between fMRI data and hierarchically-structured language models to the scientific claim that hierarchical structure really is part of human language comprehension. This reasoning relies upon a comparison with a sequence-oriented, hierarchy-free language models. [START_REF] Hale | Finding syntax in human encephalography with beam search[END_REF] and [START_REF] Brennan | Localizing syntactic predictions using recurrent neural network grammars[END_REF] look at a subtler comparison, one that pits bona fide phrase structure composition against a "composition-free" kind of parsing. Bona fide phrase structure composition is formalized using the Recurrent Neural Network Grammars of Dyer et al. (2016, RNNG). The baseline is a language model whose outputs are not syntactic trees but merely bracketed strings. Neural networks using this "non-compositional" alternative representation (inspired by [START_REF] Vinyals | Grammar as a foreign language[END_REF][START_REF] Choe | Parsing as language modeling[END_REF][START_REF] Choe | Parsing as language modeling[END_REF], see the discussion of finite state transducers in [START_REF] Langendoen | Coordinate grammar[END_REF] are not compelled to form truly hierarchical internal representations. Statistical comparison of the surprisal values from the two models as regards human EEG highlights a time window about 200 to 300 milliseconds after a word is encountered when explicit hierarchical composition appears to be happening.

Multi-level prediction

Varying the symbols over which a language model is defined can also lead to insight. For instance [START_REF] Lopopolo | Using stochastic language models (SLM) to map lexical, syntactic, and phonological information processing in the brain[END_REF] use a quantity that is closely related to surprisal, the perplexity of the next symbol at three different levels of abstraction. The first level is phonological, where perplexity is defined in terms of the sequence of speech sounds. At the second level of abstraction, there is the familiar sequence of words. And at a third, the sequence contains part of speech tags. Using a trigram model at each of these levels, Lopopolo and colleagues identify three distinct collections of brain areas. Many of these are clustered in well-known perisylvian language regions but using set-theoretic operations these authors were able to identify, for instance, the Angular Gyrus as a region that simultaneously processes grammatical and phonological information.

In their EEG and MEG study, [START_REF] Heilbron | A hierarchy of linguistic predictions during natural language comprehension[END_REF] decompose word predictions from a transformer neural network (GPT-2) into different levels of linguistic analysis. First, like Lopopolo et al., they find dissociable patterns with morphosyntactic and phone-level predictions related to earlier responses (100-400 ms) in, mostly, temporal areas, while lexicalsemantic predictions drive later responses (> 400 ms) across a wider set of cortical areas. Second, neural responses at the level of speech sounds are modulated in a top-down way by predictions regarding word meaning. This supports theories in which linguistic predictions not only take place at multiple levels of abstraction, but also modulate each other.

The latter point is consistent with previous findings from the literature. To mention just one, [START_REF] Dikker | Predicting language: MEG evidence for lexical preactivation[END_REF] presented participants with picture-word sequence pairs in which the two elements of the pair could either match (high predictability; e.g., a picture of an apple followed by the word 'apple') or not match (low predictability; e.g., a picture of a banana followed by the word 'apple'). Participants were asked to indicate whether the word described the picture. Interestingly, Dikker & Pylkkänen find that noun predictability modulated MEG brain responses not only in fronto-temporal areas more directly associated with lexical access but also in lower level sensory areas like the visual cortex. All of these results accord with an "interactive" model of sentence processing, in which different linguistic levels modulate or co-constrain each other [START_REF] Marslen-Wilson | Sentence perception as an interactive parallel process[END_REF].

Linguistic features within and across languages

The Decoding paradigm An alternative response to Poeppel's 2012 Mapping Problem views neural activity not as a indirect indication of work, but a direct characterization of a concept, linguistic or otherwise. This approach, pioneered by Tom Mitchell and Marcel Just is known as MVPA (see box 1). It treats BOLD levels as vector representations of concepts that are extensionally presented in the course of an fMRI experiment. [START_REF] Murphy | Decoding language from the brain[END_REF] review this tradition, which begins with [START_REF] Mitchell | Predicting human brain activity associated with the meanings of nouns[END_REF]. The central idea is to use a kind of learning program called a classifier to discover regularities within the brain data that align with distinctions between the experimental stimuli. These distinctions (say between the lexicalized concepts celery and dog or between Double Object vs Prepositional Dative phrase structures) simply become labels or features for classifiers of brain data. A key innovation in this approach is the use of held-out data; typically the observations are divided up into a training set and a testing set in order to assess the generalization performance of trained classifier. The results can be averaged across alternative divisions to make best use of the available observations while carefully avoiding circular reasoning ("cross-validation", a standard practice in machine learning). The spatial and/or temporal locality of the data can be leveraged to look for regions that are diagnostic for particular types of labels ("searchlight" see [START_REF] Kriegeskorte | Information-based functional brain mapping[END_REF].

Classifier: Statistical tool that learns to classify, or "decode", a stimulus feature based on brain signals.

However there are limitations, too. Decoding depends on a finite number of trials, and there are correspondingly a finite number of distinctions that can be drawn between all the labelled cases. It's always possible that a high-performing classifier has latched on to a pattern in the brain data that is accidentally, rather than lawfully, related to the intended meaning of the experimenter's label. This limitation is not specific to the decoding paradigm but is rather an instance of the classic problem of Construct Validity in psychology [START_REF] Cronbach | Construct validity in psychological tests[END_REF][START_REF] Smith | On construct validity: Issues of method and measurement[END_REF]. We take up this concern as it applies specifically to neuro-computational modeling in the conclusion.

Lexical semantics

In the seminal 2008 study mentioned above, those concepts were noun meanings, operationalized as "word embeddings", or vectors of co-occurrence counts with a handful of special verbs. This "distributional" approach generalizes the semantic markerese of [START_REF] Katz | The structure of a semantic theory[END_REF] already mentioned above (see [START_REF] Baroni | Frege in space: A program for composition distributional semantics[END_REF], on distributional models of meaning). A follow-up study argued that these representations are conceptual, and not lexical, by showing that they generalize across languages such as English and Portuguese [START_REF] Buchweitz | Identifying bilingual semantic neural representations across languages[END_REF].

Embedding: Vector representation of a word or other linguistic unit. [START_REF] Huth | Natural speech reveals the semantic maps that tile human cerebral cortex[END_REF] apply the same method to natural speech samples (in their case, excerpts from a popular podcast). Forming a distributional vector space based on word-cooccurrence, they observe that distinct lexical-semantic dimensions such as "social", "visual", or "numeric" map to distributed patterns of activity across diverse areas of the cortex. Opening the black box a little bit, they compare their word embeddings favorably against other embeddings that are derived from a trained neural network (word2vec; see Jurafsky & Martin 2021, chapter 6 or Goldberg 2017, §10.4 for an information-theoretic interpretation).

One approach to scaling these efforts to larger expressions is to simply average the single word vectors [START_REF] Pereira | Toward a universal decoder of linguistic meaning from brain activation[END_REF]; see also [START_REF] Li | Temporal lobes as combinatory engines for both form and meaning[END_REF]. However this "bag-ofwords" approach discards structural information, such as semantic roles. Tensor Product representations [START_REF] Smolensky | The Harmonic Mind[END_REF] and DisCoCat [START_REF] Coecke | Mathematical foundations for a compositional distributional model of meaning[END_REF]) are mathematical theories that offer ways of composing word embeddings into larger structures (see also [START_REF] Baroni | Frege in space: A program for composition distributional semantics[END_REF]).4 Another less-principled option is to use numerical vector representations that arise via training on tasks such as classifying entailment relations, or guessing a blanked-out word in a sentence. The assumption is that these vectors will encode whatever structural information is truly useful in these auxiliary tasks (e.g. [START_REF] Anderson | Deep artificial neural networks reveal a distributed cortical network encoding propositional sentence-level meaning[END_REF]). However, the details of particular training sets sometimes mean that there are superficial tricks a network can use to succeed at these sorts of auxiliary tasks without actually learning much in the way of sentence-structural information. 5 To avoid this uncertain inference from success on an auxiliary task to adequate linguistic representation with some set of vectors, structural features may also be encoded directly, as discussed in the following subsection.

Syntactic structure and thematic roles Sticking with the one-word, one-vector methodology described above, Wehbe et al. (2014a) extend the Mitchell/Just decoding paradigm to a broader range of linguistic features and structures that characterize naturalistic texts. Wehbe and colleagues assigned features to each word of chapter 9 of Harry Potter and the Sorcerer's Stone. These include referential features, i.e. about the identity of particular story characters as well as tags for the part of speech and choice of dependency-syntactic relation borne by a particular word. Human participants read the story one word at a time on a screen. This word at a time character of the presentation facilitates an analysis that considers each brain volume separately, relieving the analyst from any need to assume a particular hemodynamic response function (as in [START_REF] Brennan | Syntactic structure building in the anterior temporal lobe during natural story listening[END_REF][START_REF] Brennan | Abstract linguistic structure correlates with temporal activity during naturalistic comprehension[END_REF][START_REF] Li | Grammatical predictors for fMRI timecourses[END_REF] but cf. [START_REF] Shain | fMRI reveals language-specific predictive coding during naturalistic sentence comprehension[END_REF]. Classifiers are evaluated using a scheme which forces the system to differentiate between a passage's actual brain-data and a nonmatching 'foil' passage's brain data. As before this is done using held-out data in a cross-validated manner to assess whether any generalization beyond the training set would occur.

Wehbe et al.'s innovative study used simple textual features that are not controversial within linguistics. Its findings are consistent with the view that Broca's area subserves the recognition of grammatical dependencies such as Subject, Object, or Modifier (see e.g. Friederici 2017). [START_REF] Zhang | Decoding the silence: neural bases of zero pronoun resolution in Chinese[END_REF] push this MVPA approach in a referential direction, decoding the identity of story characters from naturalistic stimuli, even in "pro-drop" cases where there is no overt mention but instead they would be inferred from context by native speakers.

The promise of the approach lies in its synergy with artificial neural networks. By using regularization techniques such as ridge regression, it is possible to learn a linear relationship between (a) internal representations within neural networks and (b) observed brain data such as BOLD signals. This has been done with recurrent neural network sequence models in MEG (Wehbe et al. 2014b) The neural network models mentioned above form their own internal representations, in effect performing a corpus analysis on their training data via the backpropagation algorithm (Rumelhart et al. 1986a). As an alternative to this kind of discovery procedure one can instead start from linguistic features that are given in advance by an experimenter. [START_REF] Allen | Distinguishing grammatical constructions with fMRI pattern analysis[END_REF] did this by decoding neuroimages from human participants who had either read a verb in a prepositional dative structure ("gave a book to John") or a double-object structure ("gave John a book"). Another study by [START_REF] Frankland | An architecture for encoding sentence meaning in left mid-superior temporal cortex[END_REF] identified regions in the posterior temporal lobe that support above-chance decoding for the semantic roles Agent and Patient. This involved decoding which noun, out of four possible nouns, in fact filled a particular semantic role. [START_REF] Reddy | Can fMRI reveal the representation of syntactic structure in the brain? Under review[END_REF] encode phrase structure trees as numerical vectors in an approximate way, by sampling random walks that traverse those trees node by node. These encodings account for human fMRI observations in well-known language regions. Particularly when the trees include unexpanded top-down predictions about upcoming words, the resulting vectors capture aspects of the fMRI signal above and beyond univariate parser step counts (see section 5 below). These studies all share with Wehbe et al. (2014b) the use of linguistic features that are not controversial. Yet, the underlying methods can be further extended to test features that are more controversial, such as the thematic structure of experiencer predicates [START_REF] Pesetsky | Zero syntax: Experiencers and cascades[END_REF]. Bornkessel-Schlesewsky and Matthias Schlesewsky (2013;2016), typological differences between languages may correspond to different brain networks or different modes of operation of the same brain network, when it comes to language comprehension. Consistent with a universalist view, Honey et al. (2012) find that roughly the same brain regions show BOLD activity when native speakers of English or Russian perform a naturalistic task of comprehending stories in their native language. This converges with other work based on non-naturalistic methods such as [START_REF] Crinion | Language control in the bilingual brain[END_REF] and [START_REF] Correia | Brain-based translation: fMRI decoding of spoken words in bilinguals reveals language-independent semantic representations in anterior temporal lobe[END_REF]. Whereas Honey and colleagues do not identify the specific typological feature driving their result, [START_REF] Dunagan | Neural correlates of semantic number: A cross-linguistic investigation[END_REF] compares French and Chinese along a single feature: the way a language expresses singular versus plural number. In French, this marking is overt and obligatory whereas in Chinese it is optional. The results again support the postulation of one common set of language-relevant brain regions that subserve comprehension of number marking, regardless of how it is signaled within a language.

Typological comparison As persuasively argued by Ina

The universalist view supported by these data remains underspecified as to how shared processing resources are recruited by different language varieties. Neuro-computational models, especially those with interpretable components as discussed in the next section, can help unravel how brain mechanisms may be allocated differently across typologically diverse languages. How to specify these cross-linguistic models remains an urgent, open problem.

Opening the black box to understand the mechanism

The linguistic features at issue in the previous section are outputs of syntactic and semantic computations that have been performed by candidate models of human language processing. This approach, as well as the prediction-based measures in section 3, are all extensional in nature. That is to say, we look at a model's outputs or a probability distribution over such outputs, and compare that theoretical prediction to human brain data in a way that is divorced from the internal process by which the model derived that prediction. This practice has become standard in what Marco Baroni calls "linguistically-oriented deep net analysis" (see [START_REF] Baroni | On the proper role of linguistically-oriented deep net analysis in linguistic theorizing[END_REF][START_REF] Li | Modeling pronoun resolution in the brain[END_REF]. Such extensional approaches make it possible to evaluate very large artificial neural nets, ones whose mode of operation defies succinct description. Of course, this cannot be the last word -we cannot simply stop having replaced one inscrutable black box (the brain) with another (deep neural nets). A better response to the mapping problem addresses not only outputs but also internal processing steps that make sense with respect to existing cognitive and linguistic theories.

Derivation steps dissociate from predictions One family of approaches ties directly to processing steps by tapping into parsing models discussed in section 3; these draw on properties of the parser itself, rather than the output representation. For example, the structure-based RNNG parser used by [START_REF] Hale | Finding syntax in human encephalography with beam search[END_REF] and [START_REF] Brennan | Localizing syntactic predictions using recurrent neural network grammars[END_REF], which was already discussed in terms of surprisal on page 7, may also be queried more directly in terms of the algorithm it implements. Working incrementally from one word to the next, that parsing algorithm includes a loop that iterates over possible syntactic analyses that are consistent with some partial linguistic input. Counting the number of iterations explored between words thus reflects processing effort in a direct way by summing the number of parser actions taken between words. This particular approach builds on a venerable tradition that ties cognitive load to properties of syntactic structure [START_REF] Kaplan | Augmented transition networks as psychological models of sentence comprehension[END_REF][START_REF] Frazier | Syntactic complexity[END_REF]. Indeed [START_REF] Brennan | Localizing syntactic predictions using recurrent neural network grammars[END_REF] find that such a measure fits with BOLD-signal recorded from the left temporal lobe and inferior frontal gyrus. That improvement in fit is seen only when the parser explicitly composes words into phrases (see page 7) and, moreover, the effect of parser actions is statistically independent of effects that reflect predictability. In other words, while an approach such as this demands a joint commitment to representational and algorithmic elements of a processing model, the relationship of these components to neural signals can be distinguished in a way that guides further, more focused, efforts.

Comparing process models [START_REF] Nelson | Neurophysiological dynamics of phrase-structure building during sentence processing[END_REF] takes on the challenge directly of comparing different process models while holding constant representational content (see also [START_REF] Brennan | MEG evidence for incremental sentence composition in the anterior temporal lobe[END_REF]. That work made use of exceptionally high-resolution data collected using inter-operative recordings from neurosurgery patients. These recordings come from electrodes placed directly onto the cortex to aid in the surgical treatment of epilepsy; this placement leads to data that can distinguish neural signals with a high degree of both spatial and temporal precision. Nelson et al. presented such patients with sentences that spanned a range of syntactic constructions. The sentences were generated by a context-free phrasestructure grammar (see box 2). A set of process models were constructed that differed in how eagerly, or predictively, they traversed these phrase-structures. These include the least eager "bottom-up" strategy which draws tree nodes only when all associated leaf nodes have been incrementally encountered, in contrast to the "left-corner" and "top-down"; strategies which each posit syntactic nodes more predictively (cf. [START_REF] Abney | Memory requirements and local ambiguities of parsing strategies[END_REF].

Correlations between the number of parser actions under each of these strategies with the electrophysiological signals reveal a cline such that the left-corner and bottom-up strategies show superior fits to left temporal data than the most eager top-down strategy. These differences are not equally distributed across language-related temporal and frontal regions, indicating that the brain bases of parsing must be understood in a more nuanced way as implementing a variable parsing strategy that is more eager in one region than another. Or there may be construction-specific variation such that certain linguistic expressions are understood in a more eager manner than others.

Process models for realistic grammars In mathematical linguistics, the degree of "expressivity" required for an adequate syntactic theory has been largely resolved: human grammar must be more than context-free, but not a lot more. [START_REF] Stabler | [END_REF] lays out this hidden consensus in an accessible way, building on seminal work by [START_REF] Joshi | How much context-sensitivity is required to provide reasonable structural descriptions: Tree adjoining grammars[END_REF] and [START_REF] Joshi | The convergence of mildly context-sensitive grammatical formalisms[END_REF].6 Kallmeyer (2010) offers a textbook introduction to parsing algorithms for these formalisms. Does a grammar whose level of expressivity is well-calibrated to human grammar in general yield processing-difficulty predictions that fit a particular person's brain? [START_REF] Brennan | Abstract linguistic structure correlates with temporal activity during naturalistic comprehension[END_REF] and [START_REF] Li | Grammatical predictors for fMRI timecourses[END_REF] offer an affirmative answer. They find that node-counts on X-bar structures derived by Minimalist Grammars predict unique variance in BOLD signal in the posterior temporal lobe, variance that is not captured by ngrams or naïve phrase structure in the style of the Penn Treebank (see [START_REF] Marcus | Building a large annotated corpus of English: The Penn Treebank[END_REF]). 7 Stanojević et al. (2021) go farther, operating directly on the derivations assigned by a near-context free formalism, Combinatory Categorial Grammar (CCG). They find that CCG improves BOLD modeling in six language-relevant brain regions. Adding a special parser action that facilitates late-attachment of modifier phrases improves the fit in both anterior temporal lobe and inferior frontal gyrus ("Broca's Area"). It would be premature to conclude on the basis of just these results that CCG or Minimalism is uniquely the right theory of human grammar. However, they do suggest that these competence theories capture distinctions that play some role in human language processing. Storage versus parsing A perennial question in linguistics concerns the distinction between atomic expressions that are simply remembered whole and structured expressions that are composed from smaller pieces (for a study with non-naturalistic stimuli see [START_REF] Lyu | Neural dynamics of semantic composition[END_REF] or Pylkkänen 2019 for a review of the composition operation). [START_REF] Bever | The cognitive basis for linguistic structures[END_REF] classic idea is that some expressions are understood as a matter of memory association; the link between a morpheme sequence and a functional label such as Actor-Action-Object is nothing less and nothing more than a habit. This idea has enduring appeal but there remains the issue of exactly where to draw the line between habits and rules [START_REF] Townsend | Sentence comprehension: The integration of habits and rules[END_REF][START_REF] Donnell | Productivity and reuse in language: A theory of linguistic computation and storage[END_REF][START_REF] Blache | Light-and-deep parsing: A cognitive model of sentence processing[END_REF]. [START_REF] Bhattasali | Localising memory retrieval and syntactic composition: An fMRI study of naturalistic language comprehension[END_REF] approach this question from standpoint of quantitative linguistics using naturalistic stimuli and employing statistical association metrics (see e.g. [START_REF] Evert | Corpora and collocations[END_REF]) to quantify the "stickiness" of word sequences that may correspond to memorized expressions. Entering these association metrics as predictors into a linear regression as in section 3 highlights the precuneus, a brain region that has been implicated in episodic memory. This result can be taken as supporting a variation on Bever's proposal, namely that human comprehension is interleaved such that sometimes memorized chunks are directly recalled in a way that depends on brain networks involving the precuneus and other times composed in a way that involves the temporal lobe (cf. [START_REF] Ullman | Contributions of memory circuits to language: The declarative/procedural model[END_REF]Hagoort 2016).

Reference and coreference Of course quite apart from their internal structure, expressions in a language have reference -they refer to people, places and things. [START_REF] Li | Modeling pronoun resolution in the brain[END_REF] studied this referential aspect of language using complexity metrics to rate the processing difficulty of particular coreference links, within a single literary narrative. As in section 3 this narrative served as an auditory stimulus for listeners who were undergoing fMRI. Across both English and Chinese translations of the same book, the results identified two brain regions in the temporal lobe. BOLD signal in these regions was explained best by a model that acknowledges distinct subject and nonsubject grammatical roles, and embraces the ACT-R memory theory [START_REF] Van Rij | How wm load influences linguistic processing in adults: A computational model of pronoun interpretation in discourse[END_REF]). This cognitive model outperformed neural network models at the task of accounting for the observed BOLD variation. Subsequent work on pro-drop sentences, where the pronoun is not even mentioned, implicates the same brain regions (see [START_REF] Zhang | Decoding the silence: neural bases of zero pronoun resolution in Chinese[END_REF], discussed above in section 4).

Lessons and next steps

The importance of interpretability The research highlighted in this review reaches for the goal of scientific explanation using computational models that are by-and-large interpretable. That is, the models connect with or implement theoretical constructs from linguistics. 8An alternative approach is suggested by the advent of high-performing deep neural networks that are trained end-to-end on specific natural language processing tasks such as language modeling (see RNN and Transformer entries in box 2). The success of such models relies upon their ability to extract fine-grained statistical regularities from exceptionally large training corpora. [START_REF] Schrimpf | The neural architecture of language: Integrative reverse-engineering converges on a model for predictive processing[END_REF] use the internal states that result from such training to decode fMRI and electrophysiological data (see section 4). They report highly reliable fits between the neural network representations and brain data, especially in the left temporal lobe. The best-fitting models in their comparison are also the largest neural networks that show the highest performance on non-brain related measures.

While remarkable, the conclusions that can be drawn from such a comparison are theoretically limited. Like the brain itself, the neural networks investigated are "black boxes" in the sense mentioned above in section 5; their internal states are not directly interpretable The approach faces the pitfall of trivial findings: systems that are highly sensitive to language statistics are in turn statistically similar to each other. Without interpretability, strong statistical relationships cannot form the foundation of explanatory theories. Examples of the latter can be found in related approaches, of course, as already seen with the attention payed to outputs of black-box models by [START_REF] Heilbron | A hierarchy of linguistic predictions during natural language comprehension[END_REF] and [START_REF] Caucheteux | Disentangling syntax and semantics in the brain with deep networks[END_REF], and the interpretable feature-spaces of Wehbe et al. (2014a). Another path forward is to unpack the internal representations of neural networks (e.g. [START_REF] Yu | Assessing Phrasal Representation and Composition in Transformers[END_REF][START_REF] Kuncoro | What do recurrent neural network grammars learn about syntax?[END_REF].

Model spaces In seeking explanatory models, this review has drawn on studies whose basic mode of operation is to identify correlations between properties of a computational model and brain activity signals. When such models are interpretable, reliable correlations are taken as evidence that particular features of the model underlie the neural system being measured. But, this inferential step requires careful scrutiny. As already mentioned in section 4 above, statistical classifiers and artificial neural networks may latch on to idiosyncratic data patterns rather than linguistic features of interest [START_REF] Heaven | Why deep-learning AIs are so easy to fool[END_REF] reviews this challenge as it faces modern AI more broadly.)

One strategy to strengthen inferencing relies on comparing alternative models of a particular linguistic phenomenon in terms of their statistical fit to neural data. This was done, for example, by [START_REF] Shain | fMRI reveals language-specific predictive coding during naturalistic sentence comprehension[END_REF]for word-prediction, Anderson et al. 2021 for decoding structure, and [START_REF] Hale | Finding syntax in human encephalography with beam search[END_REF] for process models. In all of these examples, claims for or against a particular theoretical model are based on explicit comparison to a set of alternatives which together define a space of possible models or hypotheses. The logic of such comparisons is familiar from experimental design, where baselines are established with control conditions, or from linguistic analyses guided by evaluating minimal pairs, and all such efforts face familiar challenges (e.g. Cronbach & Meehl's Construct Validity).

In principle, such a model space ought to span all possible hypotheses that are plausible given current understanding of some research question. In practice it is more likely that the models span only a small part of the relevant space and inferences based on such a comparison are accordingly limited. To give just one concrete example, while [START_REF] Brennan | Localizing syntactic predictions using recurrent neural network grammars[END_REF] argue that neural indices of prediction reflect hierarchy, not just word sequences, that inference is based on comparing just three models. An alternative word-sequence model, could, in principle, yield different results for such a comparison. As in other domains, more general conclusions follow when results from multiple approaches converge; in this case, [START_REF] Shain | fMRI reveals language-specific predictive coding during naturalistic sentence comprehension[END_REF] reach a theoretically similar conclusion from comparing a different set of sequence-based and hierarchical models. An opportunity granted by broad coverage computational models of naturalistic data is that researchers may compare alternative models against commonly shared datasets. Reusable data With a view toward encouraging further progress, we invite the participation of linguists, neurobiologists and AI researchers by sharing several datasets of our own. These include EEG and fMRI data evoked by the first chapter of Alice's Adventures in Wonderland [START_REF] Bhattasali | The Alice Datasets: fMRI & EEG observations of natural language comprehension[END_REF], and fMRI data evoked by presentation of The Little Prince to native speakers of French, English and Chinese [START_REF] Franzluebbers | The Little Prince fMRI dataset: English, French & Chinese[END_REF]. These latter data are described in more detail in supplementary section 7. These datasets join a growing set of neurolinguistic corpora, described in Table 1, which offer neural data along side rich, often naturalistic, linguistic stimuli and annotations.

Progress on the Mapping Problem In 2012 David Poeppel dramatically declared that "there is absolutely no mapping to date that we understand in even the most vague sense" (page 50). A decade later we can be more optimistic. Now, quantitative linking hypotheses based on expectancy (section 3), decoding (section 4) and step-counting (section 5) allow us to test more granular brain/language correlations. In our judgment Poeppel's Radical Decomposition strategy has been productive. When we decompose a well-specified computational model into smaller pieces, each of which would apply hundreds or thousands of times in the course of a naturalistic listening session, it becomes possible to quantitatively compare model variants that are based on different leading ideas from linguistics. A technical prerequisite is that the models being compared must all be sufficiently broad in coverage to actually apply to the experimental stimuli to which human participants are exposed in a study. And, as we have argued above, it is important that the models being compared are linguistically-interpretable.

In the future, we anticipate the field moving beyond single-language studies to larger and larger sets of languages (e.g. as envisaged by [START_REF] Stehwien | The little prince in 26 languages: Towards a multilingual neuro-cognitive corpus[END_REF][START_REF] Bornkessel-Schlesewsky | [END_REF]. Simultaneously considering multiple languages in a single neurocomputational model will be crucial for drawing general conclusions. With the availability of web-scale corpora, such as crowdsourced encyclopedias, we believe this is goal is now within reach.

  NarrativesfMRI datasets from English-speaking adults (N=345) who listened to a variety of spoken narratives.Nastase et al. (2020)Zurich Cognitive Language Processing CorpusEEG datasets from English-speaking adults (N=12) who read 4-6 hours of natural text.[START_REF] Hollenstein | ZuCo 2.0: A dataset of physiological recordings during natural reading and annotation[END_REF]

  Results to date seem to confirmStowe et al.'s 2005 suggestion that language processing involves both hemispheres and that it is organized in a processing cascade that proceeds from individual words to larger expressions (cf.[START_REF] Lenneberg | Topographic mapping of a hierarchy of temporal receptive windows using a narrated story[END_REF]. In virtue of their broad coverage Wehbe et al. demonstrate how that earlier work generalizes to more naturalistic language. They also open the door to explicit comparisons, analogous toHuth et al.'s 2016 comparison between their own word embeddings and embeddings from word2vec (see page 8).[START_REF] Anderson | Deep artificial neural networks reveal a distributed cortical network encoding propositional sentence-level meaning[END_REF] demonstrate how this work might procede for more abstract structures, by comparing fits between multiple lexical and sentence-based vector representations.

	and fMRI (Qian et al. 2016, with a somewhat simpler regression
	model). Later work extends this to the well-known Jabberwocky manipulation, which
	neuroscientists use to disentangle syntax from lexical semantics (see e.g. Stowe et al. 1998;
	Pallier et al. 2011; Matchin et al. 2016; Hashemzadeh et al. 2020) and to Transformers, a
	high-performing neural net building block for natural language processing (Toneva & Wehbe
	2019).

Table 1

 1 Openly available neurolinguistic datasets for use with neuro-computational models.

	Name		Description	Citation
	The	Little	Prince
	Datasets

Calvin and Hobbes -January

25, 1993. 2 Hale et al.

www.annualreviews.org • Neuro-computational models

This research follows[START_REF] Brennan | Syntactic structure building in the anterior temporal lobe during natural story listening[END_REF] and[START_REF] Yarkoni | Pictures of a thousand words: Investigating the neural mechanisms of reading with extremely rapid event-related fMRI[END_REF] in relying on neu-2 See also the commentaries collected in Language, Cognition & Neuroscience, 2020, Volume 35 Issue 9. 4 Hale et al.

These next-word probability distributions depend strongly on the genre-specific characteristics of their training data (see e.g.[START_REF] Hale | Text genre and training data size in human-like parsing[END_REF]).6Hale et al.

 Hale et al.

Tensor Product representations have been criticized for matching human similarity judgments less-well than other competing systems[START_REF] Mitchell | Composition in distributional models of semantics[END_REF][START_REF] Martin | Tensors and compositionality in neural systems[END_REF] and for being too expensive in terms of the required number of artificial neurons. SeeSmolensky & Legendre (2006, chapter 7) annd[START_REF] Smolensky | Four facts about Tensor Product Representations[END_REF] for replies.

[START_REF] Caucheteux | Disentangling syntax and semantics in the brain with deep networks[END_REF] introduce an intriguing alternative approach: average neural net activation vectors that arise when applied to ten different sentences that all share the same syntactic structure. In other words, cluster the states of deep neural nets by syntactic equivalence classes. These authors confirm via subsequent probing tasks that syntactic properties are indeed inferrable from these averaged representations.www.annualreviews.org • Neuro-computational models

 Hale et al.

This consensus spans Minimalism[START_REF] Stabler | Minimalist grammars and recognition[END_REF]), Combinatory Categorial Grammar ( §8.3 Steedman 2000) and Lexical-Functional Grammar[START_REF] Wedekind | Tractable Lexical-Functional Grammar[END_REF] to name just three frameworks.

This region where Minimalist Grammars make a distinguishable contribution is consistent with Wernicke's area.12Hale et al.

The conditions under which a language processing mechanism is interpretable with respect to linguistic theory have been debated.[START_REF] Stabler | How are grammars represented?[END_REF] rules out a naïve box-ology. However the hypothesis under which a generative grammar is "computed but not represented" (Stabler's "Hh") is still viable. For a brief summary of this debate seeHale (2017, §3.4).www.annualreviews.org • Neuro-computational models

 Hale et al.

 Hale et al.

 Hale et al.

 Hale et al.

 Hale et al.

Acknowledgments

This material is based upon work supported by the National Science Foundation under grant numbers 1903783 and 1607251, and the Jeffrey Sean Lehman Fund for Scholarly Exchange with China at Cornell University. The authors would like to thank Berta Franzluebbers for her assistance preparing the OpenNeuro archive.

Disclosure Statement

The authors are not aware of any affiliations, memberships, funding, or financial holdings that might be perceived as affecting the objectivity of this review.

Supplementary Information: Multilingual fMRI dataset

This section describes fMRI data elicited by the original French as well as English and Chinese translations of Saint-Exupéry's 1943 book The Little Prince. Upon acceptance of this article, the data will be publically-available at https://openneuro.org/datasets/ ds003643.

Participants

Participants for the English study were 51 young adults (32 females, mean age=21.3, SD=3.6) with no history of psychiatric, neurological or other medical illness that might compromise cognitive functions. They self-identified as native English speakers, and strictly qualified as right-handed on the Edinburgh handedness inventory [START_REF] Oldfield | The assessment and analysis of handedness: The Edinburgh inventory[END_REF]. All participants were paid, and gave written informed consent prior to participation, in accordance with the IRB guidelines of Cornell University.

Chinese participants were 35 healthy, right-handed young adults (15 females, mean age=19.3, SD=1.6). They self-identified as native Chinese speakers, and had no history of psychiatric, neurological, or other medical illness that could compromise cognitive functions. All participants were paid, and gave written informed consent prior to participation, in accordance with the IRB guidelines of Jiangsu Normal University.

French participants were 30 healthy, right-handed adults (16 Female, mean age=24.3; SD=4.9). They were all native French speakers and had no history of psychiatric, neurological, or other medical illness that could compromise cognitive functions. All participants gave written informed consent prior to participation, in accordance with the Regional Committee for the Protection 195 of Persons involved in Biomedical Research.

Experimental Procedure

After giving their informed consent, participants were familiarized with the MRI facility and assumed a supine position on the scanner. Auditory stimuli were delivered through MRI-safe, high-fidelity headphones inside the head coil. The headphones were secured


Ken ' Ichi Ohshika

Athanase Papadopoulos

Keywords: Mathematics Subject Classification. 53B40, 30F60, 32G15 Thurston metric, Teichmüller space, Teichmüller metric, Finsler manifold

TANGENT SPACES OF THE TEICHM ÜLLER SPACE OF THE TORUS WITH THURSTON'S WEAK METRIC

HIDEKI MIYACHI, KEN'ICHI OHSHIKA, AND ATHANASE PAPADOPOULOS Abstract. In this paper, we show that the analogue of Thurston's asymmetric metric on the Teichmüller space of flat structures on the torus is weak Finsler and we give a geometric description of its unit circle at each point in the tangent space to Teichmüller space. We then introduce a family of weak Finsler metrics which interpolate between Thurston's asymmetric metric and the Teichmüller metric of the torus (which coincides with the the hyperbolic metric). We describe the unit tangent circles of the metrics in this family.

The final version of this paper will appear in Annales Academiae Scientiarum Fennicae Mathematica.

Preliminaries

We shall use the following identification between the Teichmüller space of the torus and the upper half-plane model of the hyperbolic plane H:

Let T 2 be a two-dimensional torus and fix a pair of generators a, b of π 1 (T 2 ) represented by two simple closed curves on this surface intersecting at one point. The Teichmüller space of T 2 , denoted by T (T 2 ), is the set of equivalence classes of pairs (Σ, f ), where Σ is a Riemann surface and f : T 2 → Σ a homeomorphism, and where two pairs (Σ 1 , f 1 ), (Σ 2 , f 2 ) are defined to be equivalent when f 1 • f -1 2 is isotopic to a biholomorphism. From the uniformisation theorem, for every point x in T (T 2 ), there is a unique complex number ζ with Im(ζ) > 0 such that x is represented by the pair (C/(Z + ζZ), f ), where f is a homeomorphism taking the homotopy class of a, b to 1, ζ ∈ Z + ζZ = π 1 (C/(Z + ζZ)) respectively. In this way, T (T 2 ) is identified with H = {z ∈ C | Im(z) > 0}. This identification induces an isometry when the Teichmüller space T (T 2 ) is equipped with the Teichmüller metric and H is equipped with the metric of constant curvature -4. In the sequel, we shall refer to this metric on H as the hyperbolic metric d hyp . The isometry between the space T (T 2 ) equipped with the so-called Teichmüller metric and the space H equipped with the hyperbolic metric is a result of Teichmüller, see [START_REF] Teichmüller | Extremale quasikonforme Abbildungen und quadratische Differentiale[END_REF]Section 9] and [5, Section 9] for an English translation of Teichmüller's paper.

We also need the following notion: A weak metric δ on a set X is a map δ : X × X → R satisfying the following:

(1) δ(x, x) = 0 for every x in X;

(2) δ(x, y) ≥ 0 for every x and y in X;

(3) δ(x, y) + δ(y, z) ≥ δ(x, z) for every x, y and z in X.

In the paper [START_REF] Belkhirat | Thurston's weak metric on the Teichmüller space of the torus[END_REF], the following weak metric was introduced on H: First, for

ζ 1 , ζ 2 ∈ H, we let M (ζ 1 , ζ 2 ) = sup x∈R ζ 2 -x ζ 1 -x . (1.1)
The weak metric δ is then defined by setting

δ(ζ 1 , ζ 2 ) = log M (ζ 1 , ζ 2 ).
In the same paper, the following explicit expression of δ was obtained:

(1.2) δ(ζ 1 , ζ 2 ) = log |ζ 2 -ζ1 | + |ζ 2 -ζ 1 | |ζ 1 -ζ1 | .
We note that this implies that

(1.3) δ(ζ 2 , ζ 1 ) = δ(ζ 1 , ζ 2 ) + log Im(ζ 1 ) Im(ζ 2 ) .
It was also shown that this weak metric has the following two properties:

(1) The arithmetic symmetrisation of the weak metric δ, that is, the weak metric Sδ defined by

Sδ(ζ 1 , ζ 2 ) = 1 2 (δ(ζ 1 , ζ 2 ) + δ(ζ 2 , ζ 1 ))
is a genuine metric and coincides with the hyperbolic metric of the upper half-plane. (2) The weak metric δ is an analogue for the torus of Thurston's asymmetric metric on Teichmüller space.

The last statement needs some explanation, and we give it now. For any two points z 1 , z 2 in the Teichmüller space T (T 2 ), we take representatives

(Σ 1 = C/(Z + ζ 1 Z), f 1 ), (Σ 2 = C/(Z + ζ 2 Z), f 2 )
, and we regard them as tori equipped with the quotient flat metrics induced by the flat metric of the Euclidean plane. We set δ(z 1 , z 2 ) = δ(ζ 1 , ζ 2 ). In [START_REF] Belkhirat | Thurston's weak metric on the Teichmüller space of the torus[END_REF], a weak metric on T (T 2 ) was defined as follows. Let S(T 2 ) denote the set of homotopy classes of essential simple closed curves on the torus. We set,

(1.4) κ(z 1 , z 2 ) = log sup s∈S(T 2 ) length Σ 2 (f 2 (s)) length Σ 1 (f 1 (s)) ,
where length denotes the length of the closed geodesic in the corresponding homotopy class. The formula for κ(z 1 , z 2 ) is the analogue, in this Euclidean setting, of the formula for Thurston's metric in the hyperbolic setting given in [6, p. 8]. Theorem 3 of [START_REF] Belkhirat | Thurston's weak metric on the Teichmüller space of the torus[END_REF] says the following:

(1.5) κ(z 1 , z 2 ) = δ(ζ 1 , ζ 2 )
for any z 1 , z 2 ∈ T (T 2 ) and

z i = (Σ i = C/(Z + ζ i Z), f i ) for i = 1, 2.
The metric δ has another characterisation which is given in [START_REF] Belkhirat | Thurston's weak metric on the Teichmüller space of the torus[END_REF]. For two metrics g 1 , g 2 on T 0 = C/Z ⊕ iZ and a homeomorphism ϕ : T 0 → T 0 , we define

L(ϕ) = sup x =y d g 2 (ϕ(x), ϕ(y))/ length g 2 (s) d g 1 (x, y)/ length g 1 (s) ,
where s is a non-trivial simple closed curve, and set

λ(g 1 , g 2 ) = inf ϕ (log L(ϕ)).
The function λ is invariant under the action of homeomorphisms on T 0 homotopic to the identity. Hence, λ defines a weak metric on T (T 2 ). The metric is called the normalised weak Lipschitz distance. In [START_REF] Belkhirat | Thurston's weak metric on the Teichmüller space of the torus[END_REF], it was shown that κ

(ζ 1 , ζ 2 ) = λ(ζ 1 , ζ 2 ) for any ζ 1 , ζ 2 ∈ H.
In the rest of this paper, we investigate further properties of the weak metric κ = δ. We first show that the geodesics of the hyperbolic metric of H are geodesics with respect to this weak metric. We then show that this metric is weak Finsler (in a sense we shall make precise) and we give a geometric description of its unit circle at each point in the tangent space to Teichmüller space. We then introduce a family of weak Finsler metrics which interpolates between the weak metric δ and the hyperbolic metric (which coincides with the Teichmüller metric) which arises naturally from the construction given in this paper. We describe the unit tangent circle at each point for each weak metric in this family.

Geodesics for the weak metric δ

In this section, we give an explicit expression for the point where the supremum of (1.1) is attained for given ζ 1 , ζ 2 ∈ H and show its geometric meaning.

First we note the following, which can be shown easily from the definition of δ: Lemma 2.1. For λ > 0 and τ ∈ R, we have

δ(λζ 1 + τ, λζ 2 + τ ) = δ(ζ 1 , ζ 2 ), (2.1) δ(-ζ 1 , -ζ 2 ) = δ(ζ 1 , ζ 2 ). (2.2) For ζ 1 = a + ib and ζ 2 = α + iβ in H with ζ 1 = ζ 2 , we define (2.3) x ± = α 2 + β 2 -a 2 -b 2 2(α -a) ∓ (α -a) 2 + (β -b) 2 (α -a) 2 + (β + b) 2 2(α -a)
if a = α. When a = α, we define

x + = 0 (β > b) ∞ (β < b), x -= ∞ (β > b) 0 (β < b).
The following is an explicit expression for the supremum in (1.1):

Proposition 2.1. For ζ 1 , ζ 2 ∈ H with ζ 1 = ζ 2 , the supremum in (1.1) is attained at x + . Proof. Let ζ 1 = a + ib and ζ 2 = α + iβ.
The case where a = α can be easily dealt with. The case where α < a, from (2.2), by considering -ζ 1 and -ζ 2 instead of ζ 1 and ζ 2 respectively, is reduced to the case where a < α. Hence we only consider the case where a < α.

We first assume that a = 0. By assumption, we have α > 0. Set

f (x) = ζ 2 -x ζ 1 -x 2 = (x -α) 2 + β 2 x 2 + b 2 = 1 + A + Bx x 2 + b 2 ,
where

A = α 2 + β 2 -b 2 and B = -2α. Then, f (x) = - Bx 2 + 2Ax -Bb 2 (x 2 + b 2 ) 2 ,
and the critical points of f (x) are

x ± = - A B ± √ A 2 + b 2 B 2 B .
Since Re(ζ 2 ) = α > 0 and B = -2α < 0, we have x + < -A/B < x -. Therefore, f (x) attains its maximum at

x + = - A B + √ A 2 + b 2 B 2 B = α 2 + β 2 -b 2 2α - α 2 + (b -β) 2 α 2 + (b + β) 2 2α .
Suppose next that a = 0 and α > a. From the invariance (2.1) and the above calculation, by considering

ζ 1 -a = ib and ζ 2 -a = (α -a) + iβ instead of ζ 1 and ζ 2 , we see that the maximum is attained at a + (α -a) 2 + β 2 -b 2 2(α -a) - (α -a) 2 + (b -β) 2 (α -a) 2 + (b + β) 2 2(α -a)
which is equal to x + in (2.3). Proof. Set ζ 1 = a + ib and ζ 2 = α + iβ again. The case where a = α can be easily dealt with. Hence, as before, we may assume that a = 0 and α > 0, and set A = α 2 + β 2 -b 2 and B = -2α as in the proof of Proposition 2.1. Then

ζ 1 -- A B 2 = ib -- A B 2 = A 2 B 2 + b 2 = A 2 + b 2 B 2 B 2 , ζ 2 -- A B 2 = (α + iβ) -- A B 2 = A B + α 2 + β 2 = α 2 + β 2 -b 2 -2α + α 2 + β 2 = (α 2 -β 2 + b 2 ) 2 + 4α 2 β 2 4α 2 = (α 2 + (β -b) 2 )(α 2 + (β + b) 2 ) 4α 2 = A 2 + b 2 B 2 B 2 .
This means that -A/B is the centre of the Euclidean semicircle perpendicular to the real axis passing through the points ζ 1 and ζ 2 , and that x + , x 0 are the endpoints of this semicircle. Since

x + < -A/B < x -, x + , ζ 1 , ζ 2 and
x -lie on the semicircle in this order. Since such a semicircle is a hyperbolic geodesic, we have completed the proof.

M (ζ 1 , ζ 2 ) = |ζ 2 -x| |ζ 1 -x| , M (ζ 2 , ζ 3 ) = |ζ 3 -x| |ζ 2 -x| , and
M (ζ 1 , ζ 3 ) = |ζ 3 -x| |ζ 1 -x| . This implies that M (ζ 1 , ζ 3 ) = M (ζ 1 , ζ 2 )M (ζ 2 , ζ 3 ), hence δ(ζ 1 , ζ 3 ) = δ(ζ 1 , ζ 2 )+ δ(ζ 2 , ζ 3
). This means that γ is a geodesic with respect to δ. If x = ∞, then in the same setting, we have

M (ζ 1 , ζ 2 ) = M (ζ 2 , ζ 3 ) = M (ζ 1 , ζ 3 ) = 1,
and again γ is a geodesic with respect to δ.

Conversely, suppose that γ is a geodesic with respect to δ, and let ζ 1 , ζ 2 , ζ 3 be arbitrary three points lying on γ in this order. Then we have δ

(ζ 1 , ζ 3 ) = δ(ζ 1 , ζ 2 ) + δ(ζ 2 , ζ 3 ). By Eq. (1.3), this implies that δ(ζ 3 , ζ 1 ) = δ(ζ 3 , ζ 2 ) + δ(ζ 2 , ζ 1 ), hence Sδ(ζ 1 , ζ 3 ) = Sδ(ζ 1 , ζ 2 ) + Sδ(ζ 2 , ζ 3
) for the arithmetic symmetrisation Sδ. Since Sδ coincides with the hyperbolic metric, we see that γ is also a hyperbolic geodesic.

Before discussing the connection to Teichmüller theory, we shall give a brief comment on this heory. The ideal boundary ∂H = ∂T (T 2 ) is canonically identified with the Thurston compactification of T (T 2 ). Recall that the Thurston compactification of T (T 2 ) consists of the projective classes of measured foliations on the base surface (torus) T 0 = C/(Z + iZ). A measured foliation on T 0 is an equivalence class of a pair consisting of a foliation on T 0 together with a transverse measure. (Note in the general Thurston theory, the foliations may have singular points, where as in the case of the torus that we are discussing, the foliations are without singularities). Two such pairs are equivalent if either they are isotopic. (In the general case, one has to include Whitehead moves in the equivalence relation, but in the case of the torus, there are no such moves.) For α ∈ R, we define a measured foliation associated with α to be the pair consisting of the foliation obtained as integral curves of unit vectors satisfying (1 + |α|) -1 (dx + αdy) = 0 on T 0 and the transverse measure defined by (1 + |α|) -1 |dx + αdy|. α is called the slope of the foliation. Notice that when α = ∞, the associated measured foliation consists of the integral curves of the unit lines satisfying dy = 0 equipped with the transverse measure |dy|.

The point x + ∈ ∂H = ∂T (T 2 ) discussed at the beginning of this section corresponds to the slope of the horizontal foliation of the Teichmüller map from ζ 2 to ζ 1 . Geometrically, the leaves of the horizontal foliation are stretched under the deformation along the Teichmüller geodesic segment from ζ 1 to ζ 2 (see Figure 1). Combining Theorem 2.1 with Eq. (1.5), we have the following.

The weak Finsler structure of the weak metric δ

We recall now the notion of weak norm and weak Finsler metric on a manifold, adapted to the case we are dealing with. We start with a weak norm on a finite-dimensional vector space V . This is a map

V → [0, ∞), v → v , satisfying (1) 0 = 0; (2) v ≥ 0 for all v in V ; (3) v + v ≤ v + v for all v in V .
A metric on a smooth manifold M is said to be weak Finsler if M is equipped with a continuous field of weak norms defined on the tangent space at each point of M such that the distance between two points in M is equal to the infimum of the lengths of piecewise C 1 -paths joining them, the length of such a path being computed as the integral over this path of the weak norms of the tangent vectors. In this section, we show that the weak metric δ on T (T 2 ) is weak Finsler and we give a description of its induced weak norm on the tangent space of each point in this space. We start with the following proposition.

v δ = |v| + Im(v) 2Im(ζ) .
The meaning of the expression "induced weak norm" will be clear from the computation done in the proof, and it acquires its complete significance in Corollary 3.1 which follows.

Proof. Set ζ = ζ + tv (t > 0). Then, |ζ -ζ| = |ζ + tv -ζ| = t|v|; |ζ -ζ| = |ζ -ζ + tv| = |ζ -ζ| 1 + t v ζ -ζ = |ζ -ζ| 1 + tRe v ζ -ζ + o(t) = |ζ -ζ| 1 + tRe v 2iIm(ζ) + o(t) = 2Im(ζ) 1 + t Im(v) 2Im(ζ) + o(t) .
Hence, we have

δ(ζ, ζ + tv) = log 1 2Im(ζ) t|v| + 2Im(ζ) 1 + t Im(v) 2Im(ζ) + o(t) = log 1 + t |v| + Im(v) 2Im(ζ) + o(t) = t |v| + Im(v) 2Im(ζ) + o(t).
Thus, we obtain

lim t +0 δ(ζ, ζ + tv) t = |v| + Im(v) 2Im(ζ) .
Notice that as an invariant expression, the weak metric in Proposition 3.1 is presented as Consider now the hyperbolic space H equipped with the hyperbolic distance (of constant curvature -4) and let γ be an isometrically parametrised geodesic emanating from a point ζ 0 in H and converging to a point x 0 ∈ ∂H. We recall that the associated Busemann function associated with the geodesic ray γ is defined by [START_REF] Papadopoulos | Metric spaces, convexity and non-positive curvature[END_REF]Chapter 12]). Combining this with Theorem 2.1, we have the following corollary.

H ζ → lim t→∞ (d hyp (ζ, γ(t)) -t) (cf.
Corollary 3.1. The weak metric space (T (T 2 ), δ) is a weak Finsler metric space with the corresponding weak norm • δ given in Proposition 3.1.

Proof. We first show that (3.2)

θ 2 θ 1 γ(θ) δ dθ ≥ δ(ζ 1 , ζ 2 ) for ζ 1 , ζ 2 ∈ H and any piecewise C 1 -path γ : [θ 1 , θ 2 ] → H connecting ζ 1 to ζ 2 .
Indeed, from the invariant expression (3.1), the integration in the left-hand side of (3.2) is at least equal to the hyperbolic distance between ζ 1 and ζ 2 minus the difference of the Busemann functions at ζ 1 , ζ 2 (see (4.2)), which is the right-hand side of (3.1) (cf. §4). This observation also implies that the integration in the left-hand side of (3.2) is minimised only when it is done along the hyperbolic geodesic from ζ 1 to ζ 2 .

We now show that the distance between any two points ζ 1 and ζ 2 ∈ H is given by integrating the weak norm • δ along a parametrised geodesic joining these two points.

To this end, we first assume that Re(ζ 1 ) = Re(ζ 2 ). As in the proof of Proposition 2.1, we may assume that

ζ 1 = ib and ζ 2 = α+iβ with α > 0. Let A = α 2 +β 2 -b 2 , B = -2α and R = √ A 2 + b 2 B 2 /|B|. Define θ 1 , θ 2 ∈ (0, 2π) by e iθ 1 = (ζ 1 -(-A/B))/R and e iθ 2 = (ζ 2 -(-A/B))/R. Note that θ 2 < θ 1 . The geodesic from ζ 1 to ζ 2 is parametrised as γ(θ) = (-A/B) + Re i(θ 1 +θ 2 -θ) (θ 2 ≤ θ ≤ θ 1 )
. Hence by setting φ to be θ 1 + θ 2 -θ, we have

θ 1 θ 2 γ(θ) δ dθ = θ 1 θ 2 | γ(θ)| + Im( γ(θ)) 2Im(γ(θ)) dθ = θ 1 θ 2 R -R cos(θ 1 + θ 2 -θ) 2R sin(θ 1 + θ 2 -θ) dθ = θ 1 θ 2 1 -cos(φ) 2 sin(φ) dφ = 1 2 log(1 + cos θ 2 ) - 1 2 log(1 + cos θ 1 ) = 1 2 log 1 - 2α 2 -A √ A 2 + b 2 B 2 -log 1 + A √ A 2 + b 2 B 2 .
An easy calculation shows that this expression is equal to

log α 2 + (β -b) 2 + α 2 + (β + b) 2 2b that is, to δ(ζ 1 , ζ 2 ).
This gives what we wanted.

We now suppose that Re Next we describe the unit circle in the tangent space with respect to the weak norm • δ . Proposition 3.2. The unit circle of the tangent space at ζ ∈ H with respect to • δ is expressed as a parabola with focus at the origin and vertex at iIm(ζ).

Proof. Let ζ = α + iβ. When v = v 1 + iv 2 ∈ C ∼ = T ζ
H (as real vector spaces) lies on the unit circle of the tangent space at ζ, we have

1 = |v| + Im(v) 2Im(ζ) = v 2 1 + v 2 2 + v 2 2β
, which is equivalent to

v 2 1 + v 2 2 = (2β -v 2 ) 2 = 4β 2 -4βv 2 + v 2 2
. This means that the unit tangent circle at ζ = α + iβ ∈ H is the parabola

v 2 = - v 2 1 4β + β,
which implies the desired result.

Note that the fact that the unit tangent circle of the weak Finsler norm has an infinite direction expresses the fact that the distance function is degenerate in this direction (that is, we have, in this direction, δ(x, y) = 0 for x = y).

Deforming δ to the Teichmüller metric

In this section, we consider a family of weak Finsler metrics which interpolate between δ and the hyperbolic distance (which, as is well known, coincides with the Teichmüller distance). We then describe the unit tangent circle of each of these metrics.

Consider the family of weak metrics δ p (0 ≤ p ≤ 1) defined by

δ p (ζ 1 , ζ 2 ) = log sup x∈R (Im(ζ 1 )) p/2 (Im(ζ 2 )) p/2 ζ 2 -x ζ 1 -x = p 2 log Im(ζ 1 ) Im(ζ 2 ) + log sup x∈R ζ 2 -x ζ 1 -x . (4.1) Note that the function (4.2) H ζ → 1 2 log Im(ζ 0 ) Im(ζ) = - 1 2 ζ ζ 0 d log y
is the Busemann function associated with the geodesic ray emanating some fixed point ζ 0 ∈ H converging to x = ∞ ∈ ∂H of the hyperbolic metric of curvature -4, which is the Teichmüller distance. Hence, the function

1 2 log Im(ζ 1 ) Im(ζ 2 ) = 1 2 log Im(ζ 0 ) Im(ζ 2 ) - 1 2 log Im(ζ 0 ) Im(ζ 1 )
appears in (4.1) is the difference of the Busemann functions. Using the same proof as in Theorem 2.1, the hyperbolic geodesic from ζ 1 to ζ 2 is the geodesic of the metric δ p . The arithmetic symmetrisation of δ p is the hyperbolic metric of curvature -4, like for δ = δ 0 (cf. (1) in §1).

As we did in Proposition 3.1, we can calculate the infinitesimal form of the metric δ p :

ζ ∈ H. For v ∈ T ζ H ∼ = C, log Im(ζ) Im(ζ + tv) = -log 1 + t Im(v) Im(ζ) = -t Im(v) Im(ζ) + o(t)
as t +0. We obtain Notice that v δp > 0 when v = 0 and p > 0. The unit tangent circle with respect to the weak norm • δp in the tangent space T ζ H is the ellipse with foci 0 and -4Im(ζ)(1 -p)/(p(2 -p)) (see Figure 2). As an invariant expression, the weak metric • δp is presented as A discussion similar to that of the proof of Corollary 3.1 and (4.3) shows that the weak metric space (T (T 2 ), δ p ) is a weak Finsler metric with associated weak norm • δp and that the hyperbolic geodesic from ζ 1 to ζ 2 ∈ H is a unique geodesic for δ p . Notice that • δ 1 is the norm from the hyperbolic metric. From (4.3), v δ 1 ≤ v 2 δp ≤ (2 -p) v δ 1 for 0 < p ≤ 1. Hence δ p and δ q are bi-Lipschitz-equivalent for 0 < p, q ≤ 1. In particular, for 0 < p ≤ 1, δ p is complete and separates two points in H.

It follows that {δ p } 0≤p≤1 is a continuous family of weak Finsler metrics giving a deformation from δ = δ 0 to the hyperbolic metric δ 1 (which is Teichmüller metric). Notice that |ζ -x| 2 Im(ζ) coincides with the extremal length of the measured foliation corresponding to x ∈ R up to a constant factor (depending only on x). (We recall that the extremal length of a simple closed curve c on a Riemann surface is defined to be the infimum of the reciprocals of the moduli of the annuli whose core curves are homotopic to c, and the extremal length function can be extended continuously to the space of measured foliations.) Hence, δ 1 coincides with Kerckhoff's formula for the Teichmüller distance [START_REF] Kerckhoff | The asymptotic geometry of Teichmüller space[END_REF] adapted to the case of the torus.

Proposition 2 . 2 .

 22 The points x + and x -in Proposition 2.1 are the endpoints at infinity of the hyperbolic geodesic line in H passing through ζ 2 and ζ 1 . The point x + lies on the side of ζ 1 , and x -lies on the side of ζ 2 .

Theorem 2 . 1 .

 21 Hyperbolic geodesics in H are geodesic with respect to the weak metric δ. Conversely, every geodesic with respect to δ is a hyperbolic geodesic. Proof. Suppose that ζ 1 , ζ 2 and ζ 3 lie on a hyperbolic geodesic γ in this order. By Proposition 2.2, the endpoint at infinity x of γ which lies on the side of ζ 1 not containing ζ 2 , ζ 3 attains the supremum of (1.1) for M (ζ 1 , ζ 2 ), M (ζ 2 , ζ 3 ) and M (ζ 1 , ζ 3 ), provided that x = ∞. Then by Section 1, we have

Corollary 2 . 1 .

 21 Suppose that z, z ∈ T (T 2 ) correspond to ζ 1 , ζ 2 ∈ H respectively. Then the distance κ(z, z ) = δ(ζ 1 , ζ 2 ) is attained by the slope of the horizontal foliation for the Teichmüller map from z to z .

Figure 1 .

 1 Figure 1. The Teichmüller ray (the hyperbolic geodesic ray) from ζ 1 = i to ζ 2 = λi with λ > 1 is the vertical ray emanating from from ζ 1 . In this case, x + = 0, and the leaves of the horizontal foliation are defined by dx = 0, which is stretched along the deformation from ζ 1 to ζ 2 .

Proposition 3 . 1 .

 31 Let ζ be a point in H, and v a tangent vector at ζ. The weak metric δ induces on v a weak norm v δ expressed by

(3. 1 ) • δ = dx 2 +

 12 dy 2 + dy 2y = ds hyp + 1 2 d log y on T ζ H and ζ = x + iy ∈ H, where ds hyp is the length element of the hyperbolic metric on H of constant curvature -4.

  (ζ 1 ) = Re(ζ 2 ). Then δ(ζ 1 , ζ 2 ) is equal to zero when Im(ζ 2 ) < Im(ζ 1 ) and to δ(ζ 1 , ζ 2 ) = d hyp (ζ 1 , ζ 2 ) otherwise. From (3.1),v δ is equal to zero when Im(v) < 0 and to v hyp := ds 2 hyp (v, v) otherwise. Hence, the integral of the δ-norm along the hyperbolic geodesic connecting from ζ 1 to ζ 1 coincides with the δ-distance from ζ 1 to ζ 2 .

Figure 2 .

 2 Figure 2. The infinitesimal unit circle of the weak norm • δp at ζ = 0 for p = 1, 0.3, and 0. Each infinitesimal unit circle has the origin as a focus, and the lower dot is another focus for the infinitesimal unit circle for p = 0.3.

(4. 3 )

 3 • δp = dx 2 + dy 2 + (1 -p)dy 2y = ds hyp + 1 -p 2 d log y.

Acknowledgements The authors would like to thank the referee for his careful reading and his insightful remarks and suggestions which improved the paper.

16K05202, partially, 16H03933, 17H02843.