Belkaroui Rami
email: rami.belkaroui@gmail.com

Larodec

Rim Faiz
email: rim.faiz@ihec.rnu.tn

Aymen Elkhlifi
email: aymen.elkhlifi@paris4.sorbonne.fr

Conversation Analysis on Social Networking Sites

Keywords: Social Network, Twitter, Conversation retrieval, social media, user interactions

With the explosion of Web 2.0, people are becoming more communicative through expansion of services and multi-platform applications such as microblogs, forums and social networks which establishes social and collaborative backgrounds. These services can be seen as very large information repository containing millions of text messages usually organized into complex networks involving users interacting with each other at specific times. Several works focused only to retrieve separate tweets or those sharing same hashtags, but, it is not powerful enough if the goal of the search is to retrieve relevant tweets based on content. In addition, finding good results concerning the given subjects needs to consider the entire context. However, context can be derived from user interactions.

In this work, we propose a new method to retrieval conversation on microblogging sites. It's based on content analysis and content enrichment. The goal of our method is to present a more informative result compared to conventional search engine. To valid our method, we developed the TCOND system (Twitter Conversation Detector) which offers an alternative, results to keyword search on twitter and google. We have evaluated our method on collected social network corpus related to specific subjects, and we obtained good results.

I. INTRODUCTION

Recent years have revealed the accession of interactive media, which gave birth to a huge volume of data produced by users called User Generated Content (UGC) in blogs and microblogs more precisely. These Microblogging services like Twitter, attract more and more users due to the ease and the speed of information sharing especially in real time. In addition, microblogging services [START_REF] Jabeur | Uprising microblogs: A bayesian network retrieval model for tweet search[END_REF] gives people the ability to communicate, interact and collaborate with each other, reply to messages from others and create conversations. Furthermore, microblogs tend to become a solid media for simplified collaborative communication.

Twitter, the microblogging service addressed in our work, is a communication mean and a collaboration system that allow users to share short text messages, which doesn't exceed 140 characters with a defined group of users called followers. Users can reply to each other simply by adding @sign in front name user they are replying to. This set of socio-technical features has made possible for Twitter to host a wide range of social interactions from the broadcasting of personal thoughts to more structured conversations among groups of friends [START_REF] Boyd | Tweet, tweet, retweet: Conversational aspects of retweeting on twitter[END_REF]. While communicating people share different kind of information like common knowledge, opinions, emotions, information resources and their likes or dislikes. The analysis of those communications can be useful for commercial applications such as trends monitoring, reputation management and news broadcasting. In addition, one of main characteristic of Twitter is that users are not limited to produce contents, they can get involved indirectly in conversations with other users by liking and sharing user's posts.

This paper proposed a conversation retrieval method which can be used to extract conversation from twitter. Comparing with current methods, the new proposed not only extract directly reply tweets, but also relevant tweets which might be retweets or comments and other possible interactions. The method extract extensive posts beyond conventional conversation, which is much better called a discussion. In particular, the contributions of this paper are: first, the ability to provide an informative result for users' information needs based on user's content interactions analysis. Second, the definition of ranking function to order conversation results. Finally, the evaluation of the proposed method impact on keyword search results.

The rest of the document is organized as follows: we begin by presenting related work in related domains such as fourms discussion, Email threads. Then, we focus on more recent works addressing conversation retrieval on microblogging sites. In section 3, we propose our method allows to extract social user's content interactions. In section 4, we describe a set of ranking measures. The experimentation and evaluation results are detailed in section 5. Finally, we conclude and present same future works.

II. RELATED WORK

Conversation retrieval topic is relevant for three main domains: forum search, email/thread detection and Twitter, which is the main domain used in our work. We present following these domains.

A. Related Work in Forum/ Threads Search

An online forum is a Web application for holding discussions and posting User Generated Content in a particular domain, such as sports, recreation, techniques, travel, etc. In forums, conversations are represented as sequences of posts or threads, where the posts reply to one or more earlier posts. Several studies have looked at identifying the structure of a thread, question-answer pairs or responses that relate to a previous question in the thread.

There are many works on searching forum threads that dealt with the reply-chains structure or reply-trees. [START_REF] Wang | Recovering implicit thread structure in newsgroup style conversations[END_REF] has concentrated on identifying the thread structure when explicit connections between messages are missing. Despite the fact that replies to posts in microblogging sites, are commonly explicit, it is proved that different autonomous conversations may be developed inside the same replies thread. Furthermore, distinct threads may belong related to macro-conversations. For example, being Twitter hashtags that connect separate threads by common topic. In [START_REF] Xi | Learning effective ranking functions for newsgroup search[END_REF] authors represent the principal differences between traditional IR tasks and searching in newsgroups. They use a mesures combination such as author metrics (posts number, number of replies, etc.) and features threads.

In question and answering, there are two streams of similar work: the first one is to find the best set of answers for a query, and the second is to identify question and answer pairs to build a querying system knowledge base.

In [START_REF] Feng | Learning to detect conversation focus of threaded discussions[END_REF], authors aimed to discover the most relevant answer in question threads. They implemented a discussionbot to automatically answer student queries in a threaded discussion by combining lexical similarity, speech acts and reputation of the author of posts into a similarity measure. To discover best potential answers, they used the HITS algorithm to find posts that are most likely to be answers to the initial question post. However, extract possible answers (the most informative message) using an algorithm with a rule-based traverse that is not optimal for selecting a best answer; consequently, the result may comprise redundant or incorrect information.

Similar to [START_REF] Feng | Learning to detect conversation focus of threaded discussions[END_REF], [START_REF] Hong | A classification-based approach to question answering in discussion boards[END_REF] detected threads in which first posts are questions and its corresponding answers belong to the thread. To detect answers, they used features including the positions of the candidate answer posts, authorship and likelihood models based on content. [START_REF] Seo | Online community search using thread structure[END_REF] tested different combinations of these features with an SVM classifier and found post position and authorship result in the highest accuracies. For answers, [START_REF] Ding | Using conditional random fields to extract contexts and answers of questions from online forums[END_REF] used language models to construct weighted similarity graphs between each question and the set of candidate answers. For each detected question, a page-rank like propagation algorithm was utilized to determine and rank the set of candidate answers.

B. Related Work in Email Threads Search

Previous research has been focused on using email structure especially emails threads [START_REF] Kerr | Thread arcs: an email thread visualization[END_REF], [START_REF] Lam | Exploiting E-mail Structure to Improve Summarization[END_REF]. Thread detection is an important task which has attracted significant attention [START_REF] Klimt | Introducing the enron corpus[END_REF].

Email is one of the most important tools for treating conversations between people. Generally, a typical user mailbox encloses hundreds of conversations. Few works indirectly address to the problem of thread reply reconstruction. Accorded to [START_REF] Kerr | Thread arcs: an email thread visualization[END_REF], the detection of these conversations has been identified as an important task. Clustering the messages into coherent conversations useful to applications, among them, it gives users the opportunity to see a messages greater context they are reading and collating related messages automatically. [START_REF] Lewis | Threading electronic mail -a preliminary study[END_REF] point out that by using text matching techniques to messages portions, it will be possible to detect threads effectively.

In [START_REF] Klimt | Introducing the enron corpus[END_REF] the authors suggested a method that allows to assemble messages having the same subject attributes and send them among the same group of people. However, conversations may span several threads with similar (but not exact) subject lines. Furthermore, a conversation not include all the participants in all the messages. In the same way, [START_REF] Cselle | Buzztrack: topic detection and tracking in email[END_REF] developed an email client extension that makes it possible to clusters messages by topic. However, their clustering approach is focused on topic detection, hence messages belonging to different conversations on the same topic will be clustered together. In addition, [START_REF] Yeh | Email thread reassembly using similarity matching[END_REF] recreated reply emails chains, called email threads. The authors suggested two approaches, one based on using header meta-information, the other based on timing, subject and emails content. But, this method is specific for emails and the features cannot be easily extended for microblogs conversation construction.

[15] proposed an approach for conversation detection based on email attributes. They started by sharpening the distinction between email threads and conversations. The task was to assemble messages into consistent conversations using a function of similarity that takes all relevant email attributes, for example message subject, participants, date of submission, and message content. This method is similar to our method. We will use criteria for detecting tweets that will be in the same conversation.

C. Conversational Aspects on Microblogging Sites

Conversation retrieval is a new search paradigm for microblogging sites. It result from the intersection of Information Retrieval and Social Network Analysis (SNA). Most of Microblogs services provide a way to retrieve relevant information [START_REF] Cherichi | Relevant information discovery in microblogs -combining post's features and author's features to improve search results[END_REF], but lack the ability to provide all discussion tweets. In addition, existing conversation retrieval approaches for microblogging sites [START_REF] Cogan | Reconstruction and analysis of twitter conversation graphs[END_REF] have so far focused on the particular case of a conversation formed by directly replying tweets.

In [START_REF] Kumar | Dynamics of conversations[END_REF] the authors concentrated on different microblogging conversations aspects. They proposed a simple model that produces basic conversation structures taking into account the identities of each conversation member. Other related works focusing on different aspects of microblogging conversations are [START_REF] Huang | Conversational tagging in twitter[END_REF], [START_REF] Song | A spatio-temporal framework for related topic search in micro-blogging[END_REF] that deal respectively with conversations tagging and topics identification. These works presents limitations, most relevant messages don't even contain any hashtag.

Recently, various researches focused on the task of conversation retrieval in microblogs [START_REF] Magnani | Conversation retrieval from twitter[END_REF], [START_REF] Magnani | Information propagation analysis in a social network site[END_REF], [START_REF] Bruns | #Ausvotes: how twitter covered the 2010 australian federal election[END_REF], [START_REF] Magnani | Conversation retrieval for microblogging sites[END_REF]. [START_REF] Magnani | Conversation retrieval from twitter[END_REF] proposed a user-based tree model for retrieving conversations from microblogs. They considered only tweets that directly respond to other tweets by the use of @sign as a marker of addressivity. The advantage of this approach is to have a coherent conversation based on the direct links between users. The downside is that this method does not consider tweets that do not contain the @sign. Similarly [START_REF] Cogan | Reconstruction and analysis of twitter conversation graphs[END_REF] proposed a method to build conversation graphs, formed by users replying to tweets. In this case, a tweet can only directly reply to other tweet. However, users can get involved indirectly in conversations communities by commenting, liking, sharing user's posts.

III. A NEW METHOD FOR TWITTER CONVERSATION DETECTION: TCOND

A. Twitter Conversation Definition

[21] defined conversation as a tree where nodes represent short text messages posted by users at specific timestamps in reply to a parent nodes. Similarly [START_REF] Cogan | Reconstruction and analysis of twitter conversation graphs[END_REF] defined a conversation as a reply tree which is a graph where vertices are tweets and a directed edge represents one tweet that is a reply to another.

Contrary to [START_REF] Cogan | Reconstruction and analysis of twitter conversation graphs[END_REF], [START_REF] Magnani | Conversation retrieval from twitter[END_REF], we define a conversation as a set of short text messages posted by a user at specific timestamps on the same topic. This messages can be directly replied to other users by using "@username" or indirectly by liking, retweeting, commenting and other possible interactions. In Figure1, we have illustrated our system architecture that is made of four applications:

B. Our System Architecture

• One Web/Application Server (interface between users and system). • One Conversation Server (storage, indexing and search of conversations). • One Trend Server (gets trending topics and distribute them among tweet retrieval clients). • One Tweet Retrieval Clients (to retrieve tweets in parallel).

The Trend Server repeatedly gets from Twitter API the current trending topics, i.e., the most discussed topics. These topics are distributed to the Tweet Retrieval Clients that use Twitter Search API1 to get the corresponding tweets and also to retrieve the conversation chains. These are sent to the Conversation Server, that stores tweets and other information like the number of followers of every user participating to the conversations. The IR engine Lucene2 is used to index the conversations text and to associate it to their identifiers, from which they can be later efficiently retrieved. Users may then query the system through a Web application.

C. Conversation Detection on Twitter Microblogs

We propose a method which combines a set of conversational features and the directly exchanged text messages in order to extract extensive posts beyond conventional conversation. In the following, we will present more details about our two approach steps.

1) Direct Conversation Detection: In this step, we aim to collect all tweets in reply directly to other tweets. Obviously, a reply to a user will always begin with "@username". Our goal in this step is to create reply tree. The reply tree construction process consists of two algorithms run in parallel Recursive Root Finder Algorithm and Iterative Search Algorithm.

Algorithm 1 Recursive Root Finder (A:twitter)

Let T be a tweet collected from Twitter (ID tweet) while (type (T i ) !=root) do Extract T i -1 by matching field "in reply to status id" end while A : twitter = A : twitter -1 Let T0 is the root (first tweet published) of the conversation C and T is a single tweet of the conversation retrieved. Let consider T i the type of tweet T. A tweet can have three types: root, reply or retweet. The goal of the Recursive Root Finder Algorithm is to identify the conversation root T0 given T. Note that when the algorithm starts,|T| is not known.

Algorithm 2 Iterative Search N = the set of all tweets in C for j = 1 to j ≤|N| do Search for tweets addressed to author. Extract replies to T i by matching field in reply to status id. end for Once, the conversation root T0 has been established, the Iterative Search Algorithm is used to seek the remainder of conversation C by searching all tweets (named N) which compose C addressed to T i using matching field "in reply to status id". It is run repeatedly until some conditions, indicating that the conversation has ended, are met.

2) Conversational Features:

To the best of our knowledge, there has not been previous work on the structure of reply-based on indirectly conversation. Therefore, we define a new features that may help to detect tweets related indirectly to a same conversation. The goal of this step is to extract tweets that may be relevant to conversation without the use of "@username". We use the following notations in the sequel:

• t i is a tweet present in direct conversation (tweets in reply to other tweets directly). • t j is a tweet that can be linked indirectly to conversation. The features we used are:

• Using the same URL: Twitter allows users to include URL as a supplement information to their tweets. By sharing an URL, an author would enrichment the information published in his tweet. This feature is applied to collect tweets that share the same URL. P1 is a binary function. P1(t i , t j ) = 1 if t contains the same URL. 0 otherwise.

• Hashtags Similarity: The # symbol, called hashtag, is used to mark a topic in a tweet or to follow conversation. Any user can categorize or follow topics with hashtags. We used this feature to collect tweets that share the same hashtags. P2 is a binary function.

P2(t i , t j ) =

1 if t contains the same hashtag. 0 otherwise.

(2)

• Tweets Time Difference:

The time difference is highly important feature for detecting tweets linked indirectly to conversation. We use the time attribute to efficiently remove tweets having a large distance in terms of time compared to conversation root. The difference in time, measured in seconds, between two tweets t i , t j .

• Tweets Publication dates: Date attribute are highly important for detecting conversations. Users tend to post tweets about conversational topic within a short time period. The Euclidean distance has been used to calculate how similar two posts publication dates are.

• Content:

The criterion Content refers to the thematic relevance traditionally calculated by IR systems standards. We compute the textual similarity between each element in t i , t j taking the maximum value as the similarity measure between two messages. The similarity between two elements is calculated using the well-known tf-idf cosine similarity, sim(t i , t j ).

• Similarity Function: Finally, the similarity between tweets indirectly linked to conversation and tweets which are present in the replytree is calculated by a linear combination between their attributes.

IV. CONVERSATION RANKING

In the last section, we defined a method to detect conversation. Now, we introduce a ranking function that can be used to rank results of conversations search task. This is an aggregation of other functions representing the relative importance of different conversation aspects. It's worth noting that most of the measures indicated in the following have been defined in other contexts, and their practical usefulness has been proved several times. Here we propose their joint application to ranking conversations microblogs search task.

The first aspect regards the exchanged text message. To rank text messages we can compute their relevance with regard to some information requirements. However, text relevance of single tweets can be evaluated using any IR model, and to evaluate the relevance of an entire conversation we can calculate the average relevance of its interactions. Many standard models such as the boolean, vector-space or more complex models can be used, but this is a traditional topic in IR for which we do not present details here. In our implementation, we use the Appache Lucene library with its built-in ranking functions. In addition, the messages popularity can be defined in several different ways to evaluate the ranking conversations. This can be usually computed easily in Social Network, e.g., counting the number of likes, sharings or retweets received by the message. In the same way, we can use conversation frequency (number of interactions) that may tell us something more than a single message can. Finally, the same people may exchange messages, but at different times this may be more or less important and the rate at which messages are exchanged can be indicative of the level of interest/emotion attached to conversation. Therefore, we will also use time-related measures. In our case, computed the difference between an input timestamp and an internal timestamp of conversation (starting, medium or ending).

V. EXPERIMENTS AND RESULTS

The following experiment has been designed to gather some knowledge on the impact of our results on endusers. For this experiment we have selected two events and queried our dataset using Google 3 , Twitter search engine 4 and our method. Then we have asked a set of assessors to rate the top-10 results of every search task, to compare these approaches. In order to measure the quality of the results, we use the Normalized Discounted Cumulative Gain (NDCG) at 10 for all the judged event. In addition, we used a second metric which is the Precision at top 10. In the following, we first describe the experimental setting, then we present the results and finally we provide an interpretation of the data.

A. Experimental Settings

The analysis presented in this section is based on a social database collected over a period of the first two weeks of July 2013 by monitoring microbloging system Twitter posts (tweets). In particular, we used a sample of about 63 000 posts containing trending topic keywords. Trending topics have been determined directly by Twitter and we have selected the most frequent ones during the monitoring period.

To evaluate the results of our search tasks we have used a set of 60 assessors with three relevance levels, namely highly relevant (value equal to 2), relevant (value equal to 1) or irrelevant (value equal to 0). The assessors selected among students and colleagues of the authors (with backgrounds in computing and social sciences), on a voluntary base, and no user was aware of the underlying systems details. Every user was informed of two events happened during the sampling period: the first event is "the 100st edition of the Tour France" and the second is "the death of computer mouse inventor Douglas Engelbart". For each event we performed three searches: 1. One using Google. 2. One using Twitter Search. 3. One using our method (TCOND).

The evaluators were not aware of which systems had been used. Every user for each search task was presented with two conversations selections, one for each of the previous options with the corresponding top-10 results.

B. Experimental Outcomes and Interpretation Results

P@10 (Aver-age%) We compare our conversation retrieval method with the results returned by Google and by Twitter search engine using two metrics namely the P@10 and the NDCG@10. From this comparison, we obtained the values summarized in Table 1, where we notice that our method overcomes the results given by both of Google and Twitter. The reason of these promising values is the fact that we combine a set of conversational features and direct replies method to retrieve conversation may have a significant impact on the users' evaluation.

Concentration on the first messages selection (related to the Tour de France), conversations obtained with our method receive higher scores with compared to Google and Twitter's selection. By switching to the second event selection (related to the death of computer mouse inventor), we can see a similar scenario that our method's selection is the one with the higher scores. According to the free comments of some users and following the qualitative analysis of the posts in the two selections we can see that Google and twitter received lower scores not because they contained posts judged as less interesting, but because some posts were considered not relevant with regard to the searched topic.

Focusing on the two messages selection, we observe that both conversations selections obtained with twitter search has higher scores with respect to Google's selection. These results lead us toward a more general interpretation of the collected data. It appears that the usage of social metrics have a significant impact on the users' degree interest in the retrieved posts. In addition, the process of retrieving conversations from Social Network differs from traditional Web information retrieval, it involves human communication aspects, like the degree interest in the conversation explicitly or implicitly expressed by the interacting people.

C. Properties of Conversations

In this part, we state the main observations about the top-10 conversations results detected using our conversation retrieval method. We study the conversations distribution duration (number of hours since the original tweet until the last tweet) and conversations frequency (the number of messages that compose conversation).

• Conversations Frequency We examined the conversations'frequency which is the length of the maximum path to a leaf from the root (Figure2). Most conversations that occur in Twitter appear to be dyadic exchanges of three to five messages sent over a period of 15 to 30 minutes. Of all tweets that generated a reply, 84.81% have only one reply. Another 10.7% attracted a reply to the original reply the conversation was two levels deep. Only 1.53% of Twitter conversations are three levels deep after the original tweet, there is a reply, reply to the reply, and reply to the reply of reply.

• Conversations Duration

The analysis we made has demonstrated that the majority of conversations are not continued if the oldest tweet in conversation is more than 5 hours old. We found that 97.87% of @replies take place within the first hour of the original tweet being published, while an additional 0.98% of replies happen in the second hour. Subsequently, reply activity dramatically declines as it is shown in Figure3. There is no way to know for certain that a conversation will not be replied to at some indeterminate time in the future.

VI. CONCLUSION

In this paper, we explored a new method for detecting conversations on microblogging sites: an information retrieval activity exploiting a set of conversational features in addition to the directly exchanged text messages to retrieve conversation. In particular, we have defined a set of metrics as relevance, popularity, timeliness and frequency to be used in the computation of the ranking of a conversation.

The previous observations indicate that conversations are typically short and do not provide all the context of users' interactions. In addition, Our experimental results show the importance of using conversational features and considering all the possibilities of interactions between the participants in order to provide the entire conversation that has been published as well as its context.

We believe that the type of conversations described in this work can benefit applications that rely on microblogging posts from the end user's perspective. Future work will further research the conversational aspects by including human communication aspects, like the degree of interest in the conversation explicitly or implicitly expressed by the interacting people and their influence/popularity by gathering data from multiple sources from Social Networks in real time.

Figure 1 .

 1 Figure 1. Our System "TCOND" Architecture

Figure 2 .

 2 Figure 2. Conversation Levels Deep

Figure 3 .

 3 Figure 3. Conversations Duration

Table I TABLE OF VALUES

 IOF FOR COMPUTING OUR WORKED EXAMPLE

			NDCG (Aver-
			age%)
	Task1
	Google	59.62	56.86
	Twitter	65.73	59.71
	TCOND	73.28	64.52
	Task2
	Google	57.31	56.02
	Twitter	62.78	58.45
	TCOND	67.27	62.73

http://dev.twitter.com/doc/get/search

www.lucene.apache.org

www.google.com

Search.twitter.com.


Barr Sylvain
email: sylvain.barre@univ-ubs.fr@www.univ-ubs.fr

Abdelghani Zeghib
email: zeghib@umpa.ens-lyon.fr@www.umpa.ens-lyon.fr

REAL AND DISCRETE HOLOMORPHY: INTRODUCTION TO AN ALGEBRAIC APPROACH

We consider spaces for which there is a notion of harmonicity for complex valued functions defined on them. For instance, this is the case of Riemannian manifolds on one hand, and (metric) graphs on the other hand. We observe that it is then possible to define an "amusing" notion of holomorphic functions on them, and show how rigid it is in some cases.

Resumé. On peut parler d'harmonicité des fonctions à valeurs complexes, définies sur des variétés riemanniennes, ou sur des graphes métriques, ainsi que sur d'autres espaces plus généraux. Nous observons ici, que dans tous ces cas, on peut aussi parler d'holomorphie de ces fonctions. Nous décortiquons le cas classique des variétés riemanniennes, et la cas discret de certains graphes. Nous montrons, en particulier, une rigidité dans le cas discret, qui le distingue du cas classique.

1. Introduction 1.1. Harmonic and holomorphic. It is a usual practice to compare harmonicity with holomorphy in any context where they are simultaneously defined. In the classical case of real and complex valued mappings defined on open subsets of C, the dictionary is realized, locally, by means of the notion of conjugate part.

In the general case of hermitian complex manifolds, since one notion is metric, and the other is differentiable, the comparison is available, only if some compatibility between the metric and the complex structure is fulfilled. Roughly speaking this corresponds to the Kaehler property.

1.1.1. General framework, Support of harmonicity. Classically, harmonic functions are defined by means of a metric on a Riemannian manifold. However, there is larger class of spaces where one can talk about them. A substantial class is that of (measurable) spaces endowed with Markov processes. As particular interesting cases, we have random walks on groups, and graphs endowed with their natural "simplicial" Markov chains. Recall here that for a function f , its Laplacian ∆f is the function:

∆f (x) = 1 ν(x) y∼x f (y) -f (x)
where y ∼ x means y is adjacent to x, and ν(x) is the valency of x.

• In this paper, we will always deal with complex valued functions φ : X → C, where X is a Riemannian manifold or a graph.

1.1.2. Holomorphic functions. Even in the most classical case, there are some and in fact deep differences between harmonicity and holomorphy. Maybe, the silliest one is that the (usual) product of two holomorphic functions is holomorphic, but this is not the case of harmonic functions! This suggests to define holomorphic functions just by forcing invariance under multiplication:

Date: March 29, 2007. Definition 1.1. Let φ : X → C, where X is a Riemannian manifold or a graph. We say that φ is holomorphic if both φ and its square φ 2 are harmonic.

Recall the classical formula for functions on Riemannian manifolds: ∆(f g) = f ∆g + g∆f + ∇f, ∇g (it is remarkable that this is also valid in the case of graphs, for a natural definition of the gradient ∇).

Therefore, if X = R 2 ( = C), φ = f + √ -1g is holomorphic in the sense of the definition above, iff, ∇f = ∇g , and ∇f, ∇g = 0. This exactly means that φ is conformal, or to use a more precise terminology, it is semi-conformal which means that φ may have singular points (i.e. where dφ = 0). 1.1.3. Some remarks.

• A mapping φ : C → C is holomorphic, according with our definition, iff φ is holomorphic or anti-holomorphic, in the usual sense.

Actually, an individual anti-holomorphic mapping (on any complex manifold) can be restored to become holomorphic. However, when considering all holomorphic and anti-holomorphic mappings together, it is a nuisance to separate them. Indeed that is exactly the counterpart of non-linearity in the equation ∆φ = ∆φ 2 = 0. In the general case unlike that of complex manifolds, this does not split into two linear equations corresponding to holomorphy and anti-holomorphy.

However, for the sake of simplicity, we will keep our term "holomorphic" as it is defined, since in fact, we will always deal with individual functions.

• In the classical case of a mapping φ : C → C, conformality of φ, implies conformality and hence harmonicity for all powers φ N , for all integer N , that is, for all N , the equation ∆φ = . . . = ∆φ N = 0 ( * ) N is satisfied (this has an obvious meaning for N = ∞). More generally:

Theorem 1.2.

• For any Riemannian manifold X, we have,

( * ) 2 =⇒ ( * ) ∞
That is, a holomorphic function satisfies all ( * ) N for any N .

• For any N > 2, there is a graph X and a holomorphic function on it, which satisfies ( * ) N but which does not satisfy ( * ) N +1 .

• For a graph X, of finite valency, there is a finite m = m(X), such that any function satisfying ( * ) m is constant (and hence satisfies trivially ( * ) ∞ ).

1.1.4. A f urther motivation : Dirichlet problem. Consider the simplest Dirichlet problem for bounded harmonic functions defined on the unit disc D of C. Classical harmonic analysis theory allows one to define a boundary value isomorphism :

φ ∈ h ∞ (D) → ∂ ∞ φ ∈ L ∞ (∂D)
from the Hardy space of bounded harmonic functions on the disc, to the space of bounded measurable functions on the circle.

The Hardy space H ∞ (D) of holomorphic (in the usual sense) bounded functions, is a part of h ∞ (D), but its image in L ∞ (S 1 ) by the boundary value mapping, is very hard to explicit. In analytic words, this certainly reduces to an invariance by Hilbert transform, but this cannot help in understanding how holomorphic functions are more "regular" than harmonic ones. It is singularly suggesting to bring out a criterion, ensuring that a function ψ ∈ L ∞ (S 1 ) is the boundary value of a holomorphic function! Therefore, our motivation from the Dirichlet problem viewpoint, is that we are introducing here an "abstract" notion of holomorphy, which may help to understand the true nature of holomorphic functions. On the other hand, this notion of holomorphy for functions on X, allows one to specify a class of bounded measurable functions on its Poisson boundary, which are more regular than the others. 1.1.5. A more general framework. Actually, one can take any ring F , and consider F -valued functions defined on X. In the case of graphs, it is meaningful to speak of harmonicity. As for Riemannian manifolds, one just needs the ring to be topological, since the Laplacian can be defined via its infinitesimal mean value property. Definition 1.3. Let X be a Riemannian manifold or a graph, and F a topological ring. A mapping φ : X → F is called F -holomorphic, iff φ and φ 2 are harmonic.

We have just rewritten the same definition given in the case F = C. Our goal is to point out some examples that seem exciting, and deserve attention. We mention here the case when F is, the quaternionic field (over the reals), and the matrix algebras M n (R) or M n (C)... Remark 1.4. As an example, in another direction, consider the D'Alembertian operator on R 2 , φ = ∂ ∂x ∂ ∂y φ, where φ : R 2 → R 2 . See the second R 2 as a ring (in fact an R-algebra) which is the product of R by itself. Suppose φ is a solution of φ = φ 2 = 0. It then follows, essentially (up to a switch of factors...), that φ(x, y) = (f (x), g(y)). This means that φ is a conformal mapping of the Minkowski space (R 2 , dxdy).

Remark 1.5. The previous example suggests to define a wave equation on graphs. Recall at this stage, that more generally than the canonical Markov chain on a graph, there are Markov chains with arbitrary non-equiprobable transition functions, or equivalently, the graph has weighted edges. This is also equivalent to endow the graph with an adapted metric, which is just given by these weights. Now, the idea is to allow weights to take negative values. Analogously to the term "pseudo-Riemannian", one may call such graphs "pseudo-metric" As a diffusion operator, one obtains something like a D'Alembertian, with more complicated properties. They seem well adapted and connected to the dynamics on the graph. It is particularly interesting to define, amongst pseudo-metric graphs, a notion of Lorentzian ones: there is only one negative direction which corresponds to the "time". 1.2. Harmonic morphisms. The discussion above corresponds essentially to our chronological motivation in bringing out this notion of holomorphy. We then looked for possible correlated notions in the geometric literature. It was difficult to conclude if we weren't, hopefully, intuitively guided to harmonic morphisms. After, having a look on this wide theme, we discovered that our holomorphic functions on Riemannian manifolds, are nothing but harmonic morphisms with values in R 2 . This is however obscure, and not clear to extract, even, in standard celebrated references in this theory. In any case, motivation of harmonic morphisms are never stated as ours here. On the other hand, in the discrete case, that is for graphs, a theory of harmonic morphisms is not yet completely established. Maybe, the problem comes from the choice, to see Riemannian manifolds and graphs as objects of the same category or not. Anyway, let's anticipate and say that our holomorphic functions can not be seen as harmonic morphisms from graphs to R 2 . 1.2.1. Riemannian case. Recall that for Riemannian manifolds, a harmonic morphism is a mapping which preserves sheaves of local harmonic real functions. In other words, φ : X → Y is a harmonic morphism, if for any real harmonic function f defined on an open subset of Y , f • φ is harmonic. 1.2.2. Graph case. Obviously in this definition, one can let X and Y , to be, both, or just one among then, a graph (instead of a Riemannian manifold). In particular, one can speak of harmonic morphisms X → C, where X is a graph. 1.2.3. A difference. Harmonic morphisms in the discrete case do not behave as nicely as in the classical case. Here is one difference between the two situations, with respect to our notions of holomorphy.

Theorem 1.6. Let φ : X → C be a mapping.

• If X is a Riemannian manifold, then φ is holomorphic, iff, φ is a harmonic morphism.

• There is a graph X having a holomorphic function φ : X → C which is not a harmonic morphism.

In fact, a non-constant harmonic morphism is an open mapping (see §2). In particular a harmonic morphism from a 1-dimensional objet (for instance a graph) to C is constant. The second part of the theorem above means that there are non-constant holomorphic functions on some graphs. 1.3. Content of the article, Rigidity in the discrete case. Our first aim is to introduce this point of view of holomorphy, which we find sufficiently motivated as we said above, and it is at least amusing! This point of view is obscure in the literature related to harmonic morphisms. For example, (a) Jacobi's problem, a precursor problem in the theory, since it exactly asks for a classification of global harmonic morphisms R 3 → R 2 , is expressed as a partial differential equation, but different from our formulation of holomorphic mappings (from R 3 → C). This problem was solved only recently [START_REF] Baird | Bernstein theorems for harmonic morphisms from R 3 and S 3[END_REF]. Other more "elementary" proofs were obtained afterwards (see for instance [START_REF] Duheille | Une preuve probabiliste élémentaire d'un résultat de P[END_REF]). We anticipate here to announce our "elementary" proof which will appear in a forthcoming paper.

Also, as a contribution of the paper is the fact that the concept of holomorphy in the discrete case seems to be new. We were convinced of this for a long time, and after parts of the present paper were written, we discovered (in special circumstances) that the terminology "discrete holomorphy" was already used. Indeed in [START_REF] Mercat | Holomorphie discrète et modèle d'Ising[END_REF] (and related papers) the author introduced a notion of holomorphy for forms and functions on graphs. Inspired by Hodge theory, the definition is global, and linear. It is therefore different from our notion, yet, possible analogies are interesting to find.

1.3.1. Content. Instead of a systematic study of discrete holomorphy in the present article, we focus on some particular examples which illustrate various subtleties.

We will give some attention to the simplest non-trivial cases : the 3-valenced tree T 3 and its (1-dimensional) "dual graph" T r 3 (see §6). The spaces of holomorphic functions have finite dimension (which is of course not the case of harmonic functions), and endowed with the beautiful action of the automorphism group. 1.3.2. Random dynamics versus Poisson kernel. In fact, holomorphic functions are "essentially" determined when given on some finite subsets of vertices. There is a "holomorphic dynamics" which allows one to calculate the holomorphic functions progressively on the other vertices. However, there is at each step some finite choice to consider, in some sense, one has a kind of "random dynamics".

One can conclude some general philosophy. In the case of harmonic functions, one goes to (space) infinity, and reproduce objects by a Poisson integral. In the holomorphic case, a random holomorphic dynamics of finite subsets of the space itself allows one to reproduce the function.

We do not claim our tentative to formulate all these notions is optimal. We find however exciting (and funny) the beauty of these structures and their connections with many natural notions, as correspondences ([2]), ...

Let us now summarize some obtained results.

1.3.3. The 3-valenced tree T 3 . The case of T 3 is the simplest non-trivial one. Many (amusing) facts will be said about it. The following theorem summarizes results obtained along §4.

Theorem 1.7. Let, φ ∈ Hol * (T 3 ), that is, a non-constant holomorphic function on T 3 . Then:

(1) φ sends (the vertices of ) T 3 onto (the vertices of ) a hexagonal tiling τ of the Euclidean plane.

(2) In fact φ is the universal covering of the hexagonal tiling. In particular:

• φ is locally injective, that is, it is injective on any (closed) ball of radius 1 in T 3 . More precisely, φ is conformal (see Remark 4.1 for definition).

• the image of a geodesic of T 3 is a locally injective walk on the hexagonal tiling. It could be periodic, but generally goes to infinity.

-Let H α,β (T 3 ), for α = β be the space of holomorphic functions taking the values α and β on two adjacent vertices. Let H be the subgroup of Aut(T 3 ) fixing (individually) these vertices, then:

(3) The action of H on H α,β (T 3 ) is simply transitive. In particular, the group and the space are homeomorphic (in particular both are compact). (4) The same is true for the product action of the group SG × H on the space Hol * (T 3 ), where SG is the similarity group of C. The previous product group and space are therefore homeomorphic. (5) In contrast, the (transitive) action of SG×Aut(T 3 ) on Hol * (T 3 ) is not free.

The stabilizer of a point (i.e. a non-constant holomorphic function) is a discrete group Γ acting freely on T 3 , which is nothing but the fundamental group of the hexagonal tiling.

One sees from this how are rigid holomophic functions in comparison to harmonic ones. In particular, the space of holomorphic functions has finite (topological) dimension (exactly 4), whereas that of harmonic has infinite dimension.

Also, one sees relationship between holomorphy and random walk on the hexagonal lattice, in fact, only partial random walks are involved, those, called here locally injective.

1.3.4. The graph T r 3 . Here, a holomorphic function is determined once given on any triangle, but with some discrete indetermination, leading to a holomophic dynamics on the space of triangles. Theorem 1.8. A holomorphic function on T r 3 is determined once given on any one of its triangles. There is however some discrete indetermination in the extending process of the holomorphic function, leading to a holomophic random dynamics in the space of triangles.

• Correspondence. Precisely, this holomorphic dynamics is apprehended in the correspondence S which is a 3-dimensional (complex) quadric defined in C 3 × C 3 (endowed with coordinates (p, e, f ; x, y, z)) by:

e + f -y + (-y + z) = 0 e 2 + f 2 + y 2 + (-y + z) 2 = 0 p -(x + y) = 0
• Random dynamics. In other words, S is the graph of a holomorphic multivalued transformation M : (p, e, f ) ∈ C 3 → (x, y, z) ∈ C 3 , which encodes the diffusion process of a holomorphic function.

• Action of Aut(T r 3 ). Two holomorphic functions are equal up to composition by an element of Aut(T r 3 ), iff, they take same values on vertices of two triangles of T r 3 .

• Orbital structure. On the space of marked triangles C × (C 2 -{0}), consider the relation:

∆ ∼ ∆ ′ ⇐⇒ ∃ φ holomorphic such that ∆ ′ = φ(∆).
Then ∼ is an equivalence relation.

• Projectivization. By homogeneity, we get a correspondence in CP 1 × CP 1 , and an equivalence relation in CP 1 .

(It turns out that the equivalence relation ∼, or its projective variant, are not defined by group actions).

These two theorems do not answer many other remaining questions on the subject of holomorphic functions on these special two spaces T 3 and T r 3 . Actually, in this introductory article, we could not go deeply in analysis : the phenomenology by itself is rich enough, and formulations are not straightforward! Preliminary simulations promise a beautiful and strong dynamics. 1.3.5. Other graphs, non-rigid situations. The results for T 3 allow one to understand holomorphic functions for any 3-valenced graph. This is discussed in §7, where also remarks are made, giving evidences to a rigidity phenomenon in the case of the Cayley graph of Z 2 . In contrast, one observes abundance of holomorphic functions for trees of valency ≥ 4. The study of these trees will be continued in §8, where we formulate the "conjugate part problem".

Some properties of harmonic morphisms, Proofs in the Riemannian case

We recall in what follows, fundamental properties of harmonic morphisms in the context of Riemannian manifolds. They are due to B. Fuglede and T. Ishihara (independently) [START_REF] Fuglede | Harmonic morphisms between Riemannian manifolds[END_REF][START_REF] Ishihara | A mapping of Riemannian manifolds which preserves harmonic functions[END_REF]. Let φ : X → Y be a smooth mapping between two Riemannian manifolds. Then φ is a harmonic morphism, iff, φ is harmonic and horizontally semi-conformal. The latter means that, if φ is not constant, then for almost all x, if H x denotes the normal space of ker D x φ, then:

D x φ : H x → T φ(x) Y is a conformal isomorphism (on the complementary negligible set where D x φ = 0).
It then follows in particular that a harmonic morphism is an open mapping.

As an example if dim X = 1, and dim Y = 2, then the harmonic morphism φ is constant.

Remark 2.1. This fact can also be proved (in an essentially similar way) when X is merely a graph (say, a 1-dimensional object more general than manifolds). This explains the comment after Theorem 1.6.

Consider now a mapping φ : X → C, where X is a Riemannian manifold. Assume φ is a harmonic morphism. The mappings z → z N are harmonic on C. By definition, φ N is harmonic for any N , in particular φ is holomorphic.

Assume now that φ is holomorphic. From the discussion following Definition 1.1, it follows that φ is horizontally semi-conformal (as defined above). ¿From the above characterization, φ is a harmonic morphism.

Summarizing, φ is a harmonic morphism, iff, ∆φ = ∆φ 2 = 0, iff, ∆φ N = 0 for all N .

This proves the Riemannian parts in Theorems 1.2 and 1.6 Finally, let us say that the fact that harmonic morphisms satisfy a quadratic equation, or equivalently that ( * ) 2 =⇒ ( * ) ∞ , is stated through standard papers like [START_REF] Fuglede | Harmonic morphisms between Riemannian manifolds[END_REF]. However, it was just used as an intermediate step of proofs.

Generalities on graphs

We start with a general local study, and fix notations. Let s 0 be a vertex of X, with valency n: s 1 , . . . , s n are the adjacent vertices of s 0 . The formel difference s is 0 denotes the oriented edge joigning s 0 to s i .

Let φ be a (complex) function on X, with z i = φ(s i ). The oscillation of φ along the edge s is 0 is δ i = z iz 0 . The gradiant of φ at s 0 is the vector -→ δ = (z 1z 0 , . . . , z nz 0 ) ∈ R n . (Actually the values {z i } are not ordered and therefore, -→ δ is an element of the symmetric quotient of R n . However this precision does not matter in our analysis here, and for the sake of simplicity, we keep our notations above).

The function φ is harmonic at s 0 if its gradiant at s 0 has a vanishing divergence (i.e. arithmetic mean): (z iz 0 ) = δ i = 0. Fact 3.1. The function φ is holomorphic, iff, both its mean oscillation and quadratic mean oscillation vanish at any vertex:

δ i = δ 2 i = 0
In fact, more generally, for a harmonic function φ, the mean quadratic oscillation equals the mean oscillation of its square φ 2 . Proof. At s 0 , the mean oscillation of φ and φ 2 are respectively:

δ i = (z i -z 0 ) and (z 2 i -z 2 0
). Assume φ is harmonic at s 0 , then, z i = nz 0 (n is the valency at s 0 ). Thus,

δ 2 i = (z i -z 0 ) 2 = z 2 i +nz 2 0 -2z 0 z i = z 2 i +nz 2 0 -2nz 2 0 = (z 2 i -z 2 0 )
We will also need the following lemma. Its proof reduces to the possibility of solving in C, a system of a linear and a quadratic equations, on two unkowns. Lemma 3.2. Let n ≥ 3 and δ 1 , ..., δ n-2 ∈ C be given. Then there exist a pair {δ n-1 , δ n }, unique up to switch, of (complex) solutions of

( n-2 i=1 δ i ) + δ n-1 + δ n = ( n-2 i=1 δ 2 i ) + δ 2 n-1 + δ 2 n = 0.
In particular, to fix notations for future use in § §4 and 6:

Fact 3.3.
For given e (resp. e and f ) there is {u, v} a unique pair up to switch, such that:

e + u + v = 0 e 2 + u 2 + v 2 = 0 ( resp. e + f + u + v = 0 e 2 + f 2 + u 2 + v 2 = 0 )
4. Dynamical and ergodic study for the 3-valenced tree T 3

We investigate here holomorphic functions on T 3 , the (bi-infinite) tree of valency 3. However, because of technical difficulties (due to the fact that T 3 can not be well ordered, that is endowed with an orientation inducing an orientation on any geodesic), we will start by considering intermediate cases.

4.1. The smallest tree, the tripod. This is a (finite) tree Y with 4 vertices, O, O ′ , A, B, all of valency 1, except O, which has valency 3 (all the others are extremal).

Let φ be a function on Y , and denote by e, u, v its associated oscillations at O,

that is e = φ(O ′ ) -φ(O), u = φ(A) -φ(O) and v = φ(B) -φ(O).
Suppose φ is holomorphic at O, then,

e + u + v = e 2 + u 2 + v 2 = 0.
Fact 3.3 says that u and v are completely determined, up to order, when e is given. More precisely, let j = e 2 3 πi be a cubic root of unity, then, necessarily, up to order, The natural "geometric" way to embed Y in the Euclidean plane is to imagine the edges OO ′ , OA and OB, having the same length and making equal angles at O, which must be 2π 3 . We see from above, that a holomorphic mapping φ is then conformal (the target space C is of course endowed with its Euclidean structure). 4.2. Rooted tree. Let now A be an infinite tree of valency 3, with a root R. So, all vertices have valency 3, except R which has valency 2. A model of it is given by the free monoid on two letters {a, b}, the root R corresponds to the void word. The vertices are words on a and b.

u = je, v = j 2 e For instance, φ(A) -φ(O) = α(φ(O) -φ(O ′ )) for some α ∈ {-j, -j 2 }. Thus, φ(A) = φ(O) + α(φ(O) -φ(O ′ )) Also, φ(A) = φ(O ′ ) + (φ(O) -φ(O ′ )) + α(φ(O) -φ(O ′ ). B A O O'
Instead of A, we consider an extended tree A O ′ O , obtained by gluing an oriented segment O ′ O to A, where O is identified with R. Now, the root becomes O ′ .

If φ is a function given at O ′ and O, then one extends it holomorphicaly step by step, applying the rule described for Y. To formulate this, orient A O ′ O naturally, in such a way that O ′ is a source, and for every S a word on {a, b}, the edges Sa and Sb are positively oriented.

If, φ is already defined on S ′ and S, so that S ′ S is an oriented edge, then,

φ(SX) = φ(S) + α S (X)(φ(S) -φ(S ′ )), X ∈ {a, b}
where α S : {a, b} → {-j, -j 2 } is a bijection.

Therefore, φ is completely determined by, the "randonm variables" α S , associated to any S (a word on a and b).

φ(S n ) = φ(O ′ ) + w + α 0 w + (α 1 α 0 )w + . . . + (α n-1 α n-2 . . . . .α 0 )w,
where This Φ can be characterized among holomorphic mappings by the fact that it preserves "2-dimensional" (or planar) orientation of the tree. To define it, imagine the tree naturally embeded in an oriented Euclidean plane. This induces an "orientation" on the tree, that is a choicee of positive angle between Sa, and Sb, for any S ∈ A. Here, we decree that ∠(Sa, Sb) is positive. In this case Φ actually preserves the orientation. It will be seen just below how to obtain all holomorphic mappings by means of Φ, and then that only Φ preserves orientation.

α i = α Si .
Summarizing these nice properties of Φ:

Fact 4.2. Φ is holomorphic, (planar) orientation preserving, and locally injective, that is, at any vertex, Φ is injective on its immediate neighbourhood (i.e. on the Y around it), and is conformal (as defined in Remark 4.1). On the other hand, this action is free. To see this, one verifies that Φ has a trivial stabilizer, that is no γ except the identity satisfies : Φ • γ = Φ. To prove it, one sees that, such a γ must fix both a and b. Therefore, at the next step, that is for words of length 2, the only non-trivial possible action of γ is to induce a transposition on {aa, ab} or {ba, bb}. This means Φ takes a unique value on one on these sets, which contradicts the fact that Φ is locally injective (Fact 4.2). Now, the formal proof of the freedom of the action can be performed by induction.

Simply-transitive action of

Both H α,β and Aut(A O ′ O ) are compact (when endowed with the compact-open topology, although the elements of H α,β are not bounded!). They are in fact topologically Cantor sets. One checks that: γ → Φ • γ is a homoeomorphism. 4.2.4. Action of the similarity group. The similarity group SG of C acts (by the left) on the space of holomorphic functions for any space. They are mappings : z → θz + t, where θ, t ∈ C, θ = 0.

In our case, let Hol = Hol(A O ′ O ) be the space of holomophic functions. This is a locally compact space (for the compact-open topology). Let Hol * be the set of non-constant functions, that is, Hol * = Hol -C. Equivalently, Hol * is the (disjoint) union of all the H α,β , for α, β ∈ C, α = β. Then, SG permutes the spaces H α,β . In fact, a non-trivial element of SG can preserve no individual H α,β . On the other hand, Aut(A O ′ O ) acts by preserving each H α,β . One then consider the produt action of SG × Aut(A O ′ O ). From previous developments, we get: Let H α,β (T 3 ), for α = β, be the space of holomorphic functions taking values α and β on two adjacent vertices. The action of Aut(A) × Aut(A) on it is simply transitive. The same is true for the action of the group SG × Aut(A) × Aut(A) on the space Hol * (T 3 ), which are therefore homeomorphic.

In contrast, the (transitive) action of Aut(T 3 ) on Hol * (T 3 ) is not free. The stabilizer of a point is a (canonical) discrete group Γ acting freely on the tree T 3 .

4.4.

Hexagonal tiling. To given α and β = α + w, one associates a hexagonal tiling of R 2 (=C), with α as a vertex, and w as an edge (trough α). The fundamental tile is the regular hexagon with vertices α, α + w, α + wjw, α + wjw + j 2 w, α + wjw + j 2 ww, α + wjw + j 2 ww + jw.

The tiling τ α,β is obtained by reflections along its edges of the fundamental hexagon. Up to similarity, we can restrict ourselves to τ = τ 0,1 .

From §4.2.1, we get: Proposition 4.5. Let, φ ∈ Hol * (T 3 ), that is a non-constant holomorphic function on T 3 . Then, up to rescaling by a similarity of C, we have:

• φ sends (the vertices of ) T 3 onto (the vertices of ) τ .

• In particular, for a geodesic (injective path) C = S 0 , S 1 , . . . , S n , . . . in T 3 , the image, c = s 0 , . . . , s n = φ(S n ), . . . is a path of τ .

-The path c is locally injective, that is s i = s i+1 = s i+2 .

• Moreover, because of the local injectivity of φ, a ramification of paths in T 3 induces a ramification in τ . More exactly, if two geodesics C = S 1 , . . . , S n , . . . and C ′ = S ′ 1 , . . . , S ′ n , . . . bifurcate at i, that is, S j = S ′ j , for j < i, and S i = S ′ i , then, the same is true for their images, that is, s i = s ′ .

• Summarizing: φ : T 3 → τ is the universal covering, and the discrete group Γ in Proposition 4.4 is the fundamental group of τ Locally injective random walk on τ . Let A be the set of oriented edges of the fundamental hexagon of τ , which are labelled u, v, w, and then (necessarily) -u, -v, -w.

Consider Ω = A N . Let X i : ω = (ω i ) ∈ Ω → ω i ∈ C. Consider (partial) sums S i (ω) = j=i j=0 X j (ω)
. A sequence S 0 , S 1 , . . . S n is supposed to represent a path of τ . However, not all ω's are allowed. For instance, if S 0 (ω) = u, then, in order that ω represent a walk in τ , X 1 (ω) must equals -u, v or -w. In general ω belongs to a subshift of finite type defined by means of an incidence matrix A τ , with entries 0 and 1: a sequence ω is allowed, iff, A τ ωiωi+1 = 1, for all i ∈ N. Now, being furthermore a locally injective path in τ , leads to another subshift Σ τ inj . The new forbidden subsequences are of the form y, -y, where, y ∈ A. If A = {u, v, w, -u, -v, -w} is identified, preserving order with {1, . . . , 6}, then the incidence matrix of Σ τ inj is:

        0 1 0 0 0 1 1 0 1 0 0 0 0 1 0 1 0 0 0 0 1 0 1 0 0 0 0 1 0 1 1 0 0 0 1 0        
• Ergodic theory of the subshift Σ τ inj , growth of random variables S n , are straightforwardly related to growth of holomorphic functions on T 3 .

Proof of the graph case in Theorems 1.2 and 1.6

We have the following straightforward generalization of Lemma 3.2 Fact 5.1. Let δ 1 , δ 2 be given. Consider the set of equations on unknowns δ 3 , ..., δ n :

n i=3 δ p i + (δ p 1 + δ p 2 ) = 0,
for p ∈ {1, . . . , N }, where N is an integer.

If N ≤ n -2, then a solution exists. It is unique up to permutation (of the δ i , i ≥ 3) if N = n -2. In opposite, if N > n -2, then, in order that solutions exist, δ 1 and δ 2 must vanish, and the solution in this case is trivial :

δ i = 0, ∀i.
This yields as a corollary the claim that a holomorphic function which satisfies ( * ) N where N is the (maximal) valency of the graph X is constant. In particular, in the case of T 3 only constant functions satisfy ( * ) 3 , so all other holomorphic functions satisfy ( * ) 2 but not ( * ) 3 .

A harmonic morphism X → C, must satisfy ( * ) ∞ . Only constant functions are so (say, assuming the graph has a finite valency).

This proves all the discrete content of Theorems 1.2 and 1.6.

N -holomorphic functions.

The previous fact suggests to introduce N -holomorphic functions as those satisfying ( * ) N . For instance a theory of ( * ) N holomorphic functions on T N +1 , the tree of valency N + 1 may be developed in a same way as holomorphic functions on T 3 . The situation is a little bit complicated. For instance, instead of the hexagonal tiling, we get a picture with other kinds of piecewise lines.

(See the following figures in the cases N = 2, 3, 4, 5)

Figure 5 : N-holomorphic functions 6. The graph T r 3

The graph T r 3 , is a kind of 3-valenced tree, where vertices are replaced by triangles, which explains our notation here.

More precisely T r 3 is a homogeneous graph with valency 4 and (minimal) cycles of length 3.

It is obtained from a countable family of triangles, where, to each triangle is glued exactly another one at each vertex.

It can also be considered as the "dual graph" of the tree T 3 , that is, the graph which vertices are the edges of T 3 , and two of them are joined by an edge (in T r 3 ) if the have a common vertex (in T 3 ). A concrete model is obtained by taking middles of the edges of T 3 and drawing segments joining middles of adjacent edges.

In the sequel, we will see, analogously to the case of T 3 , that a holomorphic function, is constructed following some dynamics, when it is given at vertices of some triangle. As in the case of T 3 , a nuisance but also a richness of the dynamics comes from its non-deterministic character, that is, the switch choice. However, with respect to this point, the case of T r 3 is highly more complicated. Our attempt here is just to roughly study this dynamics, which we think is worthwhile to investigate, at least because it is natural! 6.1. Dynamics on triangles.

6.1.1. Starting-up. Let OAB be a triangle in T r 3 , and OCD and adjacent one, with a common vertex O. If φ is holomorphic, and is given at O, A and B, then using Fact 3.3 in a same way as in the case of T 3 , one deduces the values of φ at C and D, up to a switch.

Let's introduce notations which will be kept along this §. We denote:

p = φ(O), e = φ(A) -φ(O), f = φ(B) -φ(O), u = φ(C) -φ(O), v = φ(D) -φ(O)
Here p, e and f are given. We infer from Fact 3.3, that u and v can be computed, but up to a switch. Indeed, this reduces to solving the following equation on u and v, where e and f are supposed known:

u + v + (e + f ) = u 2 + v 2 + (e 2 + f 2 ) = 0
In other words, after elementary algebraic manipulation: Fact 6.1. u (or equivalently v) is a solution of the second order equation:

u 2 + (e + f )u + (e 2 + f 2 + ef ) = 0
(In the sequel, we will choose, arbitrary solutions, ψ 1 (e, f ) and ψ 2 (e, f ), see below). Therefore, we are led to study the transformation:

M : (p, e, f ) → (p -u, -u, -u + v)
The difficulty comes of course from that this is not an univalued transformation. Indeed M is constructed by means of the 2-valued mapping:

I : (e, f ) ∈ C 2 → (u, v) ∈ C 2
In order to get univalued mappings, we choose, in an arbitrary way, solutions ψ 1 , ψ 2 : C 2 → C as in Fact 6.1. Actually, at this stage, we do not mind on any regularity of these mappings., but the only "algebraic" condition is that, this gives rise to an involution, that is, ψ = (ψ 1 , ψ 2 ) is an involution. Therefore, we get two "sections" for M :

(e, f ) ∈ C 2 → (u, v) ∈ C 2 .
It is an involution (as a 2-valued mapping). One way to let this being a "standard" mapping, is to consider the quotient space C 2 /Z 2 , where Z 2 = Z/2Z acts by (e, f ) → (f, e). This gives rise to a well defined (continuous) involution

I of C 2 /Z 2 .
In order to see a simple picture of this, it is useful to projectivize all things. In this case, the Z 2 action on CP 1 becomes a π rotation, around, north and south poles, say. The quotient space is still, topologically, a 2-sphere.

It is suggestive to think about I as a holomorphic mapping of the "orbifold" C 2 /Z 2 (or equivalently CP 1 /Z 2 ). This seems to not be so right, since for instance, I does not preserve the singular locus of orbifolds. Anyway, the true dynamics which interests us, that is, M 1 and M 2 , are not compatible with tacking this quotient.

Maybe, the induced mapping C 2 → C 2 /Z 2 , could be thought, more naturally, as holomorphic. However, having a different source and target spaces, deprives one to perform dynamics (i.e. to iterate). 6.1.4. Catastrophe. The "singular locus" of I is the set of (e, f ), where equation 3.3 has a double root, that is u = v. It consists of the two complex lines S and N in C 2 generated by (1

-i √ 2, 1 + i √ 2) and (1 + i √ 2, 1 -i √ 2
). In the projective plane CP 1 , this corresponds to two points, {S, N }, say. One then hopes that ψ 1 and ψ 2 (introduced in §6.1.2) can be continuously defined outside S ∪ N , or equivalently on CP 1 -{S, N }. It turns out that this is not possible : despite ψ 1 = ψ 2 along CP 1 -{S, N }, the real line bundle that they define (the fiber above (e, f ) is the real line joining ψ 1 (e, f ) and ψ 2 (e, f )) is not (topologically) trivial, i.e. non-orientable (on CP 1 -{S, N }).

6.2. Random dynamics. The graph T r 3 can be encoded as follows. We have a central triangle ∆ 0 . We denote by 1, 2, 3 its vertices. Any triangle ∆ in T r 3 is obtained by specifying a vertex i which indicated the "direction" of ∆, together with a word of the free monoid F 2 on two letters {a, b} (which may be empty ∅). Therefore, any ∆ = ∆ 0 is encoded by (l, i), where l ∈ F 2 , and i ∈ {1, 2, 3}.

For instance, in the direction 1, we have: ∆ 0 , (∅, 1), ((a, 1), (b, 1)), ((aa, 1), (ba, 1), (ab, 1), (bb, 1)), . . .

(b,3)

Figure 7

Actually, one must deal with marked triangles. For instance, the central (geometric) triangle ∆ 0 gives rise to three marked triangles (∆ 0 , i), where i ∈ {1, 2, 3} indicates the vertex.

Each triangle (l, i) gives rise to 3 marked triangles, one for each vertex. Firstly, a marked triangle (l, i, 0), where the marked vertex is that sheared by its (previous) generating triangle. Now, imagine T r 3 naturally embed in an oriented Euclidean plane. This induces an orientation (i.e. an order on the edges) of the marked triangle (l, i, 0). We get therefore two marked triangles (l, i, -) and (l, i, +).

A generic marked triangle will be denoted ∆ and its underlying (non-marked) triangle is denoted ∆.

One can speak of c∆, for c a letter (∈ {a, b}). Also, one defines, in a natural way, c∆ = c(l, i, ǫ) = (cl, i, ǫ), for ǫ ∈ {0, -, +} a sign. In other words, the action of F 2 is trivial on the marking.

Exactly as in §4.2, we have: Proposition 6.3. Let φ be a holomorphic function on T r 3 . For any triangle ∆,

there is α ∆ , a bijection {a, b} → {M 1 , M 2 }, such that, φ(c∆) = α ∆ (c)(∆)
, where c is a letter ∈ {a, b}. The same action is induced on marked triangles, that is, φ(c∆) = α ∆ (c)(∆).

6.2.1. Special holomorphic functions. As in the case of T 3 , some holomorphic functions appear special. They are obtained, in a deteministic way, that is all the α ∆ are the same, for example, α ∆ (a) = M 1 , α ∆ (b) = M 2 (for any ∆). Note however that this depends on the choice of M 1 and M 2 , or equivalently that of ψ 1 and ψ 2 ( §6.1.2) 6.2.2. Action of Aut(T r 3 ). Let Aut 0 denote the stabilizer of ∆ 0 in Aut(T r 3 ), the automorphism group of T r 3 . Applying elements of Aut allows one to restore all randomness in the construction of holomorphic functions, that is the choice of the α ∆ . From it one infers that, two holomorphic functions taking the same values on ∆ 0 are equivalent up to composition in Aut 0 . Now, if one considers the full action of Aut(T r 3 ), one gets: Fact 6.4. Two holomorphic functions φ and φ ′ are equivalent in Aut(T r 3 ), iff, they take the same values on two triangles ∆ and ∆ ′ 6.2.3. Space of triangles. Orbital structure. The space of marked (oriented) triangles is T = {(p, e, f ) ∈ C 3 , (e, f ) = (0, 0)} (as was said previously, degenerate triangles, i.e. with e = 0 or f = 0, are also considered). We have seen, how holomorphic functions induce, a kind of random dynamics on T . Consider the relation ∼ defined by

∆ ∼ ∆ ′ ⇐⇒ ∃φ holomorphic, ∆, ∆ ′ triangles of T r 3 , ∆ = φ(∆), ∆ ′ = φ(∆ ′ )
In other words, thanks to the Fact above: ∆ ∼ ∆ ′ , iff, there is φ holomorphic, γ ∈ Aut(T r 3 ), such that ∆ and ∆ ′ are images of (the central triangle) ∆ 0 by φ and φ • γ, respectively.

Previous developments show that ∼ is a well defined equivalence relation on T (despite randomness, and arbitrary choose of ψ 1 , ψ 2 ...).

It is worthwhile to investigate ergodic theory of this relation! The marked vertex of the triangle can be identified under this action with 0 ∈ C. It remains to consider pairs (e, f ) ∈ C 2 -(0, 0), up to homothety. This is exactly CP 1 , which is interpreted as the space of conformal triangles (allowed to be degenerate).

In particular, ∼ passes to an equivalence relation ≈ on CP 1 . Let us observe here that an adapted parallel construction in the case of T 3 , yields an equivalence relation ≈ which is trivial. Here, our ≈ is far from being so: not only the dynamics on triangles is not trivial, but even their conformal type is strongly transformed. 6.3. Language of correspondences. Denote (x, y, z) = (p-u, -u, -u+v). Then, (p, u, v) = (x + y, -y, -y + z). Therefore, the "graph" of the multi-valued transformation M , is the subset of (p, e, f ; x, y, z) ∈ C 3 × C 3 , such that:

e + f -y + (-y + z) = 0 e 2 + f 2 + y 2 + (-y + z) 2 = 0 x -(p + y) = 0
The space of solutions S, is a 3-dimensional quadric contained in a 4-dimensional linear subspace. We have surjective induced projection π 1 : S → C 3 and π 2 : S → C 3 . (Actually, instead of C 3 , one may better consider the space of triangles T , and S ⊂ T × T ). The projection π 1 and π 2 are branched converings. With all these objects, we are bringing out a natural situation leading to the useful notion of correspondences (see [START_REF] Bullett | Dynamics of holomorphic correspondences[END_REF] for fundamental references). The point is to consider the multi-valued dynamics: ∆ → π 2 (π -1 1 (∆)). , --3(e 2 + f 2 ) -2ef) and gets, as example the following two pictures : Therefore any holomorphic function on X gives rise to a holomorphic function on T 3 , which is therefore fully understood thanks to Theorem 1.7. We straightforwardly deduce from it the following: Theorem 7.1. Let X be a graph of valency 3. If X is the hexagonal tiling, then any holomorphic function X → C is the restriction of a similarity.

If X has a cycle of length ≤ 5, then, any holomorphic function on X is constant (here we tacitly assume X is connected). In fact, in general, holomorphic functions are trivial unless X covers the hexagonal tiling.

As example, the cayley graph of P SL(2, Z) has no non-constant holomorphic functions.

Figure 9 : Cayley graph of P SL(2, Z) 7.0.4. Cayley graph of Z n . Consider the cayley graph X n of Z n (associated to the canonical basis together with its opposite), that is, the standard lattice in R n .

For n = 1, harmonic functions are affine : n → a + bn and holomorphic functions are constant.

Identify X 2 with Z 2 = Z + iZ embeded in C. One easily verifies that, only if k ≤ 3, the function z → z k is harmonic on X 2 . On the other hand , the mappings z → az + b, and similarly z → az + b, are holomorphic. It is very likely that these are all holomorphic functions on X 2 ? 7.1. Non-rigidity, Trees of higher valency. For a tree T of valency ≥ 4, and two adjacent vertices s 0 and s 1 , on which a function φ takes values z 0 and z 1 , there is at least 1 degree of freedom in extending holomorphically, progressively φ to neighbouring vertices. Assume, to simplify notation, the valency equals 4 and let s 2 , s 3 and s 4 the other neighbours of s 0 . The associated values z i = φ(s i ), for i = 2, 3, 4, satisfy:

i=4 i=2 δ i = -δ 1 i=4 i=2 δ 2 i = -δ 2 1
where δ i = z iz 0 . Only δ 1 is given, thus, we have 2 equations and 3 unknowns, and hence, a 1-dimensional complex space of solutions. 7.1.1. Bounded holomorphic functions. We will now observe that with this freedom, we can choose φ to be bounded (and non-constant). (Even not formally stated previously, this is not possible for T 3 and T r 3 ). To do this, we show that, there is a positive real number r < 1, such that, for any δ 1 , the above equations admit solutions satisfying |δ i | ≤ r|δ 1 |, for i = 2, 3, 4. (With this, one constructs a holomorphic function φ having an exponentially decreasing oscillation along edges, and hence bounded). By homogeneity, it suffices to consider the case δ 1 = -1. To get a solution in this case, we write δ 2 = r 1 e iθ , δ 3 = δ2 = r 1 e -iθ and δ 4 = r 2 real and positive. We claim that equations: 2r 1 cos θ + r 2 = 1, 2r 2 1 cos 2θ + r 2 2 = -1 admit solutions with 0 < r 1 , r 2 < 1. This is done by elementary analysis around r 1 = r 2 = 1, and θ = π 2 .

Remark 7.2. As was suggested in §5, a theory parallel to that of T 3 , may be developed for a tree T N of valency N , if one considers "(N -1)-holomorphic functions". This would surely give beautiful pictures.

The conjugate part Problem

Let us recall the problem. We consider a real function f on a graph X, and we ask, when (and how) it has a conjugate part, that is another real function g, such that φ = f + ig is holomorphic? Actually, we have finally give up to systematically investigate this problem in the present article, since of unexpected subtleties. Only partial results will be exposed here. To mention a rough conclusion, we say that also here a random dynamics is in order, and as it is natural to expect, the problem has some cohomological flavour. 8.0.2. Conditions. It turns out that the same conditions, stated in the Riemannian case, as explained after Definition 1.1, hold in the discrete case. For any vertex s, g satisfies:

∇ s f = ∇ s g , ∇ s f, ∇ s g = 0, ∇ s g, (1, . . . , 1) = 0
The last condition expresses the fact that g is harmonic. If s has valency n, these conditions mean exactly: ∇ s g must belong to C f s , a (n -3)-sphere of radius ∇ s f in R n . From this "freedom", say when C f has higher dimension, one may ask if f is allowed to have more than one conjugate part (up to constant). It turns out this is possible, for instance in the case of the (homogeneous) tree of valency 4. Notice however, that because of the norm gradient equality, the space of conjugates (of a given f ) is compact (up to constants). 8.0.3. Non-existence examples. On the other hand, on the same tree (of valency 4), there are harmonic functions with no conjugate part. For an example (see Figure 10), consider A a vertex, B, C, D and E its neighbours. Consider the sub-tree A consisting of A, B, C, D, E and the 3 remaining vertices of each B and C.

Take f to be everywhere 0 on A, except on D and E where it takes two opposite non vanishing values, say 1 and -1.

Suppose a conjugate g exists. Adding a constant, we can assume g(A) = 0. The norm gradient equality implies that, like f , g vanishes on all neighbours of B and C.

On the other hand, writing the conjugate part equations on A, we get solutions which must vanish on D and E, but not on B and C, which is a contradiction. Observe here that f can be easily extended (not uniquely) to a harmonic function on all the tree.

One may believe the "degeneracy" of f , that is the vanishing of many of its oscillations and gradients, was the responsible on non-existence of its conjugate part. This is not true, all harmonic functions sufficiently near f have no conjugate part as well. Indeed, having a conjugate part is a closed condition. This follows from the compactness mentioned above (due the the norm gradient equality). From the classification of holomorphic functions on T 3 , one infers that a harmonic function has conjugate part if and only if its gradient has a constant norm (throughout T 3 ) (Actually, this corollary is equivalent to the classification itself). Our principal existence result is that this condition, norm constancy of the gradient, is sufficient for trees of higher valences. This is, by no means, a necessary condition, but seems nevertheless, natural, and gives an interesting class of examples.

Theorem 8.1. Let f be a harmonic on a (homogeneous) tree X, of (finite) valency ≥ 4, and assume its gradient has a constant norm, say, ∇ s f = 1, ∀s ∈ X. Then f has a conjugate part. There is (ν -3) degrees of freedom (at each vertex) in choosing this conjugate part, where ν is the valency. The space of conjugates of f is compact.

Proof. Choose O ∈ X, and for r ∈ N, let B r be the (closed) ball centred at O of radius r (i.e. the set of vertices that can be reached from O by a sequence of at most r edges). Consider also the sphere S r , the boundary of B r .

Trees are characterized by the following "strict convexity" property of B r : if s ∈ S r , then, only one neighbour p(s) of s lies in B r , in fact in B r -S r . All the others are outside B r , in fact they belong to S r+1 . Furthermore a vertex of S r+1 is adjacent to exactly one vertex of S r .

Assume g is defined on B r , and let us show how to extend (with some freedom) it to B r+1 . Let s ∈ S r . Then, g is already defined on exactly one neighbour p(s) ∈ S r-1 of s.

In order, to extend g to other neighbours (∈ S r+1 ), we just have to give ∇ s g. Actually, exactly one component, say a 1 , of ∇ s g is already given, that corresponding to the edge p(s)s. The gradient ∇ s g must belong to the sphere C f s . The question reduces then to the fact that this sphere C f s contains a vector with first projection a 1 (again to be precise, by first projection, we mean that corresponding to the edge p(s)s). The proof of the theorem will be completed thanks to the following lemma. Its significance is that, the spheres C f s and C f p(s) have exactly the same projection on the coordinate corresponding to the edge p(s)s. where α = n-1 nδ 2 1 . In particular, this depends only on δ 1 .

Before proving it let us make some remarks about this lemma. Firstly, it was stated in the case δ = 1, but, obviously, a general formula is available for any constant norm.

We also notice that a similar result is true for n = 3, the projection in this case consists of two points : ± 2 3δ 2 1 . The theorem itself is true for T 3 , in fact as was said above, it is optimal in this case. Observe however, that the theorem does not extend to the (trivial) case of the 2-valenced tree T 2 (≈ Z).

Finally, observe that, our second example of a harmonic function without conjugate part (see Fig. 10) corresponds to the case where α = 0 (i.e. δ 1 = √ 3/2 and δ 2 = δ 3 = δ 4 = -√ 3/6).

Proof. Consider the mapping π : a → a 1 . The projection of the gradient of π on the hyperplane H = {(x 1 , ..., x n ) ∈ R n , x i = 0} is collinear to v = (1n, 1, ..., 1), and so if a is a critical point of π restricted to the submanifold C, then there exist α, β so that a = αv + βδ. ¿From the fact that a is orthogonal to δ, we infer that : β = -α < v, δ >. But, since δ i = 0, we have < v, δ >= -nδ 1 . Finally, we have a = α((1n) + nδ 2 1 , 1 + nδ 1 δ 2 , ..., 1 + nδ 1 δ n ). Thus, v-< v, δ > δ 2 = n(n -1nδ 2 1 ), and hence, a 2 1 = n-1 nδ 2 1 . Remark 8.3. (Canonical conjugate?) The conjugate part problem is sub-determined: solutions are not unique (when they exist). It is interesting to define ways of selecting unique solutions. Maybe, by means of a companion variational problem?

Figure 1 :

 1 Figure 1 : The graph T r 3

Figure 2 :

 2 Figure 2 : The tripod Remark 4.1. (Conformality)The natural "geometric" way to embed Y in the Euclidean plane is to imagine the edges OO ′ , OA and OB, having the same length and making equal angles at O, which must be 2π 3 . We see from above, that a holomorphic mapping φ is then conformal (the target space C is of course endowed with its Euclidean structure).

Figure 3 :

 3 Figure 3 : Rooted tree 4.2.1. Chain rule. If O ′ = S -1 , O = S 0 , . . . S n is a geodesic, i.e. a simple path in A O ′ O , and w = φ(O)φ(O ′ ), then,

4. 2 . 2 .

 22 Planar orientation, Canonical example. For two given different complex numbers α and β, there is a canonical holomorphic function Φ with Φ(O ′ ) = α and Φ(O) = β. It is determined by taking a same α S for all S, defined by: α S (a) = -j, α S (b) = -j 2

  Aut. Let H α,β be the space of holomorphic functions on A O ′ O with prescribed (and distinct) values φ(O ′ ) = α and φ(O) = β. Let Aut(A) be the automorphism group of A or equivalently A O ′ O . It fixes O ′ and O, and for any S, there are elements of the group which exchange Sa and Sb. On the other hand, this exchange was the only one freedom in determining the holomorphic function, step by step, as explained above. This can be translated into the fact that, the right action by composition of Aut(A O ′ O ) on H α,β , is transitive.

Fact 4 . 3 .Proposition 4 . 4 .

 4344 The product action of SG × Aut(A O ′ O ) on Hol * is simply transitive. Consequently, Hol * is homeomorphic to SG × Aut(A O ′ O ).4.3.Full tree T 3 . The full (bi-infinite) tree T 3 is obtained by gluing A O ′ O and A OO ′ along their respective edges O ′ O and OO ′ . The group Aut(A) × Aut(A) can be viewed as the subgroup of Aut(T 3 ) fixing the edge O ′ O (i.e. each of its extremities). With a little bit analysis, one gets: An element of Hol * (T 3 ), that is a non-constant holomorphic function on T 3 is locally injective, more precisely it is conformal (as defined in Remark 4.1).

  Figure 4 : Hexagonal tiling 4.4.1. Locally injective random walk on τ . Let A be the set of oriented edges of the fundamental hexagon of τ , which are labelled u, v, w, and then (necessarily)-u, -v, -w.Consider Ω = A N . Let X i : ω = (ω i ) ∈ Ω → ω i ∈ C. Consider (partial) sums S i (ω) =

Figure 6 : Two triangles 6 . 1 . 2 .

 6612 Figure 6 : Two triangles

M 1 :Fact 6 . 2 . 6 . 1 . 3 .

 162613 (p, e, f ) → (pψ 1 (e, f ), -ψ 1 (e, f ), -ψ 1 (e, f ) + ψ 2 (e, f )), M 2 : (p, e, f ) → (pψ 2 (e, f ), -ψ 2 (e, f ), -ψ 2 (e, f ) + ψ 1 (e, f )) Summarising: If the image of the triangle (O, OA, OB) by φ is (p, e, f ), then the image of (C, CO, CD) and (D, DO, DC) are, in order, either, M 1 (p, e, f ), M 2 (p, e, f ), or inversely M 2 (p, e, f ), M 1 (p, e, f ). The involution. An important part of dynamics is contained in the 2-valued mapping, I :

6. 2 . 4 .

 24 Action of the similarity group, Conformal triangles. All involved dynamical notions I, M 1 , M 2 , ∼... commute with the action of the similarity group SG of C.

Figure 8 :

 8 Figure 8 : Images of balls in T r 3 of radius 3 and 5 by holomophic functions

Figure 10 :

 10 Figure 10 : No conjugate part 8.0.4. An existence result.From the classification of holomorphic functions on T 3 , one infers that a harmonic function has conjugate part if and only if its gradient has a constant norm (throughout T 3 ) (Actually, this corollary is equivalent to the classification itself). Our principal existence result is that this condition, norm constancy of the gradient, is sufficient for trees of higher valences. This is, by no means, a necessary condition, but seems nevertheless, natural, and gives an interesting class of examples.

Lemma 8. 2 .

 2 Let n ≥ 4, δ = (δ 1 , . . . , δ n ) ∈ R n \ {0}, δ i = 0, δ 2 i = 1. Consider the sphere C = {a = (a 1 , ..., a n ) ∈ R n , a i δ i = 0, a 2 i = 1, a i = 0}Then the image of C by the projection on the first factor of R n is the segment [-α, α]

  6.3.1. Projective version.As above, one can see what happens for conformal triangles by taking quotient under the action of the similarity group. More precisely, we consider the quotient of (C × (C 2 -0)) × (C × (C 2 -0)) by the product action of SG × SG (recall that SG is the group of similarities of C). We get CP 1 × CP 1 , and inside it, corresponding to S, a quadric C, which thus determines a correspondence of CP 1 .

	6.3.2. Growth, Simulation. As was said, we restrict ourselves in this case to formu-
	lation rather than a systematic study of the dynamics. In order to have an idea,
	one can implement the multi-valuated mapping : M (p, e, f ) = ( 2p+(e+f )-√ -3(e 2 +f 2 )-2ef 2 (e+f )-√ -3(e 2 +f 2 )-2ef , 2