Eric Jolivet
email: eric.jolivet@univ-tlse1.fr

Alfonso Ávila Robinson
email: aavilarobinson@icems.kyoto-u.ac.jp

Hiroatsu Nohara
email: hiroatsu.nohara@univ-amu.fr

Shintaro Sengoku
email: sengoku@mot.titech.ac.jp

Accounting for the knowledge dynamics process of a science-based innovation

Introduction

The evolution, change and transformation of scientific and technological knowledge has been a prevalent topic in the field of innovation research. Due to the higher frequency and intensity of the waves of change disrupting the current economic system, this has been a particularly pressing need for technologies of an emerging nature. Over the years, different conceptualizations have been used to approach the dynamics of technological innovations. A review of the literature shows that these conceptualizations share in common the following three aspects (Van Lente, 2010): a) a framework clustering the heuristics and routines guiding the search processes of innovation, b) trajectories along which these search processes are channeled, and c) time-dependent phases of development. Table 1 provides a summary of some of these approaches.

Reference

Framework Trajectory Phases Foster (1986), [START_REF] Christensen | Exploring the limits of the technology S -curve. Part I: component technologies[END_REF] -S-curve birth, growth, maturity, and demise [START_REF] Abernathy | Patterns of industrial innovation[END_REF] dominant design -fluid pattern, transition pattern, and specific pattern [START_REF] Dosi | Technological paradigms and technological trajectories[END_REF] technological paradigm technological trajectory pre-paradigmatic, postparadigmatic [START_REF] Kuhn | The structure of scientific revolutions[END_REF] scientific paradigm -pre-paradigmatic science, normal science, crisis, revolution [START_REF] Geels | Typology of sociotechnical transition pathways[END_REF] sociotechnical regimes transition pathways interactions across three levels: regimes, niches, and landscape [START_REF] Sahal | Technological guideposts and innovation avenues[END_REF] technological guideposts innovation avenues -Table 1. Theoretical background of the present study.

This study builds on the work of [START_REF] Abernathy | Patterns of industrial innovation[END_REF], later refined by [START_REF] Anderson | Technological discontinuities and dominant designs: a cyclical model of technological change[END_REF] and [START_REF] Benner | Exploitation, exploration and process management: The productivity dilemma revisited[END_REF]. In the Abernathy and Utterback model (A-U model) of technological change, the patterns of innovation are divided into three stages: fluid, transition, and specific stages. The fluid or ferment stage is characterized by the multiplication of product designs and alternative routes in an explorative dynamics. In this stage, variety is created. In the specific stage, the deepening of a established route takes place through incremental and cumulative efforts within the framework of exploitative dynamics. Here, efficiency and economic performance dominate. Of interest for this study is the mechanism through which an formative industry enters into a more mature stage. For [START_REF] Abernathy | Patterns of industrial innovation[END_REF], this transition goes through a process of standardization initiated by the advent of a 'dominant design'. For industry players, the 'dominant design' signals that investments should now be focused in deepening and building up from this first product version. Given their embryonic nature, emerging technologies still lack such social and cognitive rules guiding their research and development (van Merkerk and van Lente, 2008). Actors in emerging fields, van Merkerk and van Lente (2008) argue, rely on speculation and promises. For them, this emerging phase is characterized by negotiation and experimentation. These processes lie at the heart of standardization.

In particular, this paper addresses the study of the standardization phase for sciencebased innovations. Over the years, [START_REF] Abernathy | Patterns of industrial innovation[END_REF]'s model (A-U model) has been discussed and refined at length by scholars. These research efforts have mainly focused on classical industry cases, such as the automotive industry [START_REF] Lee | The era of incremental change in the technology innovation life cycle: An analysis of the automotive emission control industry[END_REF], the float glass industry [START_REF] Uusitalo | Revisiting the case of float glass: Understanding the industrial revolution through the design envelope[END_REF], the semiconductor industry [START_REF] Funk | Systems, components and technological discontinuities: the case of the semiconductor industry[END_REF], among others. Far too little attention, however, has been paid to the use of the 'A-U model' on emerging science-based industries. It is well-known that the strong sectoral specificities of innovation (Pavitt, 1984;[START_REF] Malerba | Sectoral systems of innovation and production[END_REF] prevent researchers from extrapolating across sectors. Thus, there is a need for assessing the commonalities and specificities, in regards to standardization, between science-based and industry/businessbased innovation. For the case of science-based innovations, knowledge dynamics appear to be more networked and multi-level than in industry-based innovations [START_REF] Benner | Exploitation, exploration and process management: The productivity dilemma revisited[END_REF]Zuker et al 2002).

With this in mind, the empirical case of induced pluripotent stem cells (iPS cells) is used in this study. iPS cells are a novel, science-intensive technology first discovered in 2006 by a group of Japanese researchers at Kyoto University. The value chain of iPS cells encompasses a complex and interrelated set of technologies [START_REF] Roberts | The global intellectual property landscape of induced pluripotent stem cell technologies[END_REF]. This value chain runs from the upstream generation of iPS cells through the use of reprogramming technologies to their downstream conversion into other cells and their use in application domains as different as drug screening, toxicity testing, cell therapies, etc. [START_REF] Sengoku | Redefining the concept of standardization for pluripotent stem cells[END_REF]. Within this formative biomanufacturing system, the production of high-quality iPS cells at a high volume and low cost is a crucial requirement for the commercialization of iPS cell-derived products [START_REF] Silva | Generating iPSCs: Translating Cell Reprogramming Science into Scalable and Robust Biomanufacturing Strategies[END_REF]. By focusing on the alternative methods for the generation of iPS cells, this study addresses the following questions: What are the dynamics of knowledge evolution followed by science-based innovations as iPS cell generation technologies? In particular, how are the trajectories and paradigms of the dominant technological approaches selected? Moreover, What is the structure and evolution of the networks surrounding this knowledge? Is the structure conductive to domination? Besides this, the reasons for choosing a living cell-based process to the present study are: i) this a novel case of a science-based innovation, ii) this technology is expected to give rise to a wide range of innovations including medical and pharmaceutical uses, and iii) the dynamics of knowledge transfer and innovation have hardly been studied systematically to date. This paper is structured as follows. Section 2 provides a description of the case study under focus and the materials and methods used in this research. Section 3 continues with the results of this study. Section 4 concludes with discussions and conclusions.

Research design

The case and scope In order to test these research questions in an empirical manner we selected stem cells, particularly induced pluripotent stem (iPS) cells, as a case of novel biomedical technology [START_REF] Sengoku | Redefining the concept of standardization for pluripotent stem cells[END_REF]. As a pluripotent stem cell species, iPS cells have the ability to proliferate indefinitely and to differentiate into almost any other cell type in the body. iPS cells were first discovered in 2006 by Shinya Yamanaka and colleagues at Kyoto University. Since their discovery, iPS cells have been regarded as a promising, potentially disruptive technology for the fields of drug discovery and development and regenerative medicine [START_REF] Sengoku | Redefining the concept of standardization for pluripotent stem cells[END_REF]Barfoot et al, 2013). Given these potentials and the possibility to circumvent the ethical and political debates surrounding embryonic stem (ES) cells, the field of iPS has experienced an exponential growth in the recent years (Barfoot et al, 2013).

Different approaches have been proposed to generate pluripotent stem cells from mature cells (Table 2). Generally, these approaches are referred as nuclear reprogramming, as they attempt to reverse mature cells into a state of pluripotency. In particular, iPS cells are generated by the introduction of defined genetic or chemical factors, also defined as transcription or inducing factors. Over the years, different methods have been demonstrated to generate iPS cells. Basically, these iPS cell reprogramming methods differ in terms of the 'vehicle' used to deliver these pluripotency-related transcription factors into the cell nuclei. These 'reprogramming vehicles' could be as different as DNA or RNA molecules, protein, nanomaterials, bacteria, among others. Later sections provide a more detailed description of the different iPS cell reprogramming methods. [START_REF] Hochedlinger | Nuclear reprogramming and pluripotency[END_REF][START_REF] Qi | Nuclear reprogramming and induced pluripotent stem cells: a review for surgeons[END_REF].

Material and method

To capture the dynamics of knowledge evolution in the field of iPS cells, this study retrieved scientific publications from the Thomson Reuters' Web of Science (WoS) database. In order to obtain the iPS cell-related publications, a broad search query was used: TS=((induc* NEAR/25 pluripoten* NEAR/25 stem) OR ((IPS OR IPSC) AND (stem NEAR/5 cell*))), where * is a wildcard. A total of 2, 283 publications were collected, which were manually evaluated in order to identify those documents relevant for the field of iPS cells. A total of 1,535 articles and conference proceedings published from 2006 to 2012 were collected after this procedure. Subsequently, the set of iPS cell-relevant publications were evaluated to select those publications directly related to the reprogramming of iPS cells. By this, it is meant only those publications focusing on the development or enhancement of an iPS cell generation technology were selected for this study. At the end, 581 publications were found to be directly related to this subfield of iPS cell research. Each of these publications were assessed to define the specific reprogramming method(s) involved in their studies. This was done through the evaluation of the titles, abstracts, and/or experimental/materials sections of these publications. For this purpose, as shown in Table 3, a taxonomy of iPS reprogramming methods was developed in this study.

Table 3. Taxonomy of delivery systems for iPS cell generation/reprogramming.

The construction of the taxonomy of reprogramming methods relied on an exhaustive review of the technical literature and additional sources of information, such as on-line resources and expert advice. There appears to be a consensus among researchers about the classification of iPS cell reprogramming approaches. Two trajectories are crucial for discerning among the different reprogramming methods are the nature of the reprogramming vehicle and its integration/excisibility. The former relates to the usage or not usage of viruses as carriers for the delivery of the iPS cell-related genes. Relatedly, the latter refers to the deleterious integration of material into the genome of the host cell. Different problems for integrating genetic materials: cell death, residual expression and reactivation of reprogramming factors, immunogenicity, uncontrolled silencing of transgenes, and insertational mutagenesis (Hu, 2014). For the case of excisable vectors, the integrated materials can be removed from the genome through the use of specific enzymes.

The collected 581 publications together with their respective allocated reprogramming approach(es) formed the basis for the results to be presented in the following sections. To test the research questions proposed in this study, a series of bibliometric methods and social network approaches, supported by a series of interviews with domain experts, were devised (Table 4). For the case of the dynamics of knowledge creation, a series of longitudinal bibliometric analyses were conducted to trace the trajectories of knowledge accumulation for each iPS cell generation/reprogramming method. Additionally, citation networks were built with the software CitNetExplorer (van Eck and Waltman, 2014) in order to evaluate the evolution of this sub-field of research and their underpinning reprogramming paradigms (de [START_REF] Nooy | Exploratory social network analysis with Pajek[END_REF]. To obtain a clearer visualization of the citation networks, this study set a threshold of eight or more citations on the network nodes. Due to the restrictions of the software, up to 100 nodes were able to be displayed on the network. Figure 1 presents the general longitudinal trends in the total number of publications collected for the field of iPS cells as a whole and those related to iPS cell reprogramming approaches.

On Figure 1a (left), it can be readily seen that the year 2010 marked the transition of the field of iPS cells from an upstream-to a downstream-dominated path of knowledge accumulation. By downstream-dominated knowledge paths, it is referred to activities such as the differentiation of iPS cells into other types of body cell or the use of these differentiated iPS cells into any of the potential application domains. This does not imply that the imminent problems revolving around the search processes in the field of iPS cell reprogramming have already been solved. They are still highly relevant. However, in terms of the number of publications, iPS cell reprogramming appears to be outweighted by downstream activities. Figure 1b (right) shows that knowledge in this subfield has been mainly accumulated in viral-based reprogramming approaches. This is not surprising, as these approaches were used in the pioneering studies of the field of iPS cells. Nevertheless, as shown in Figure 1b, a series of alternative reprogramming approaches have been demonstrated in the recent years. This is directly related to the on-going efforts aiming at the development of the most suitable and efficient methods of iPS cell generation [START_REF] Roberts | The global intellectual property landscape of induced pluripotent stem cell technologies[END_REF].

Social dynamics

This section describes the dynamics of change among the communities involved in the development of the field of iPS cell reprogramming approaches. For that purpose, a series of two-mode networks were built to relate the different iPS cell reprogramming approaches with the affiliated co-authoring organizations of the collected publications (Figure 3). Three time periods are displayed in this figure : 2006-2009, 2010-2012, and 2006-2012. In these networks, the location of a particular organisation denotes their patterns of usage of iPS cell reprogramming methods. By analysing the spread of nodes in terms of their geographical region, it can be seen that Asian organizations tend to focus on either retroviral or lentiviral approaches. In contrast, American organizations, and partly European organizations, tend to make use of both approaches. Moreover, American organizations appear to be more willing to make use of alternative iPS cell reprogramming methods, such as plasmid-, transposon-, chemical-, RNA-, or protein-based approaches. Exceptions are some Japanese organizations, such as Riken or Kyoto University. This closely resonates to the strong downstream competences of American and European organizations compared to their Asian counterpars. Figure 3. Two-mode networks on organization and delivery method over time.

An alternative way to trace the dynamics of knowledge accumulation is through the construction and analysis of citation networks. As described above, the iPS cell reprogramming-related publications together with their list of references were used to construct the citation network shown in Figure 4. For this purpose, the software CitNetExplorer was used. In this network, nodes are arranged by their publication year on the y-axis. Moreover, the network nodes are colored according to the type of reprogramming approach used. As these networks include the whole list of cited references, those nodes not included in the list of iPS cell reprogramming-related documents are colored in light blue in Figure 4. In line with the previous discussions, the pathways of knowledge accumulation are dominated by viral-based reprogramming approaches. From the years 2009-2010, alternative reprogramming approaches have been demonstrated, but their influence is still not significant enough for them to build their own specific paths of knowledge evolution.

Discussion

The question of knowledge creation dynamics Our observation shown in Figure 3 confirmed the separation of research institutuions according to differeces in delivery methods, and that this tendency is particular with methods located in the periphery in the map. For instance, there have been few overlaps of research insitutions across methods using a Sendai virus, a transposon, chemicals and proteins, whereas most of these insititutions have been interrelated for viral-baesd methods. We also observed that the emergence of these peripheral methods does not mean a takeover from the central ones although those at the periphery such as transposon or chemicals hold substantial superiority in a clinical usage (Sengoku, in press). Theser results strongly suggest that the agreement to form a dominant design has never been a general trend but limited inside each of representative academic groups.

These results are identical to observations in precedent studies. which show a clear separation through the innovation process even in the identical methodological category, in particular, three groups which had succeeded the generation of human iPS cells [START_REF] Roberts | The global intellectual property landscape of induced pluripotent stem cell technologies[END_REF]. Each of these cell lines thus trajectories were protected by an individual set of patents, produced at a commercial base by an individual affiliated biotech firms, and translated into clinic by a part of the individual academic group.

The question of knowledge content codification and incorporation: technology

Development was taken place individually with very few cooperative interactions for integration, however, there are scientific conditions which affect each of these actors to participate in a coding and institutionalization process with institutional ream.

Figure 4 shows knowledge interrelations in the course of the development reprogramming methods as citing-cited networks, where the integration of precedent additional references was achieved by Takahashi's study on the discovery of the iPS cell in 2006 [START_REF] Takahashi | Induction of pluripotent stem cells from mouse embryonic and adult fibroblast cultures by defined factors[END_REF]. This study was widely diversified by following studies across regions, and notably, the development process the intensive interralations of knowledge through citing-cited compling interactions. This result implies that the condition in terms of intensive knowledge transfer at the scientific research level has made representative groups compartmentalise with one another and concentrated into a specific methodology, resulting in the existence of mulitipl designs for the iSPC reprogramming.

The question of social dynamics on which dominant designs are grounded

The present observation is strongly supporting the projection of scientific research networks of cooperation and also constraints, which is well comprehended to be the nature of science-based industry. As shown in Figure 3 and 4, we confired a multi-layer situation of cooperation and competiton: separation in methodological development (figure 3) whereas intensive interrelation through the citation of scientific papers in the knowledge creation (figure 4), and competition in the product development and marketing [reference].

This feature has a good accordance to previous studies. One representative case is a study on a constraint between collaboration and competition (Bubela at al, 2010). Their bibliometric research showed that technological development at the atenting level negatively impacts collaboration patterns in scientific research, suggesting the co-existence of collaboration and commercialisation in this case of science-driven industry.

Contribution

Science-based innovations have increasingly contributed to the renewal and creation of novel businesses and industries recently. Classical models of technological dynamics and evolution have however been designed mainly on the basis of industry-based innovations. Thus our question: how do these classical tools apply to science-based innovations? How can we adapt them to make them more accurate instruments for managers?

In this paper we applied the classical Abernathy-Utterback Model, later refined by Tushman on the case of a novel science-based biomedical industry, iPS cells. We found the distinction between explorative and exploitative dynamics of knowledge accurate, but the classical models needed to be adapted to more multi-level and networked environment along three dimensions of source, mechanism, and the effects of standardisation.

Acknowledging for these findings and differences from classical industry cases led us to extend the Abernathy-Utterback model into the development of both science-based and classical industries.

Practical implication, limitation and future perspectives

Throughout the present study we obtained the following practical implications  The overall framework set by Abernathy-Utterback model holds when it comes to analysing science-based industry development as we retrieved the exploratory versus exploitative staging in the innovation journey of the iPS cells. The model was, however, much less accurate in the crucial phase of standardization and transition in the science-based case. In the regard, 'classical industries' and contemporary 'sciencebased industries' differ along the three dimensions -technological sourcing, the mechanism and the effect of standardisation.

 In science-based cases, the horizontal and inter-organizational emergence of standards and technological trajectory is further complicated by a vertical and interinstitutional imperative of bridging the two world of science and industry.

We also found there are limitations not to be addressed in the present research. First, the bibliometric analysis was limited to publications, not fully expanded to other indicators. Particularly, patent issues appear to be more closely related to the present discussion. The authors can argue that the analysis based on the scientific articles constitutes a rough indicator of model competition or evolution, as the patent issues in the life sciences are tightly linked to new knowledge in the scientific articles. This however has to be verified in a later study.

Second, In the present paper we does not distinguish the usage of iPS cells. As mentioned previously, iPS cells has a technology platform potential which can be applied not for therapeutic uses, but also basic research and drug discovery purposes. Considering the characteristics and the pros/cons of each reprogramming method, the choice of methods may be affected with the orientation of usages, in which quality standards and the production process could vary according to one or another. This type of moves should be integrated in a following study.

Figure 1 .

 1 Figure 1. The cumulative number of publications related to the iPSC reprogramming: a) total number of publications, and b) by type of delivery system.

Figure 2

 2 Figure 2 assesses the proportions of publications across the different viral-based reprogramming methods over the years. The pioneering retroviral approaches used by Yamanaka and collegues appear to be gradually giving way to alternative viral-based methods.Here, lentiviral, Sendai viral, and adenoviral-based reprogramming methods stand out. This shift is highly related to the potentially non-integrating nature, i.e. the viral vectors do not integrate into the genome, of these viral approaches. As such, these approaches appear to be attracting the interest of researchers as potential processes for the generation of clinical-grade iPS cells.

Figure 2 .

 2 Figure 2. The number of publications over time by the type of viral vectors.

Figure 4 .

 4 Figure 4. A citation network of publications related to the iPSC reprogramming.

Table 2 .

 2 Representative approaches for nuclear reprogramming

	Methods	Experimental approach	Machanistic insights
	Nuclear transfer	Reproducible cloning: functional test	Allows epigenetic changes to be
		for reprogramming to totipotency	distinguished from genetic changes
		Somatic cell nuclear transfer: efficient
		deviation of genetical matched ES
		cells with normal potency
	Cell fusion	Nuclear reprogramming of somatic	Allow study of genetics of
		genome in hybrids generated with	reprogramming
		pluripotent cells
	Cell extract	Exposure of somatic nuclei or	Allows biochemical and kinetic
		permeabilised cells to extracts from	analysis of reprogramming
		oocytes or pluripotent cells
	Cell explanation	Explanation in nuclear selects for	Allows study of genetics of
		pluripotent, reprogrammed cells	reprogramming
	Direct	Generation of pluripotent cells by the	Cells generated are autologous to
	reprogramming by	introduction of defined genetic or	donour
	defined factors	chemical factors	Technically straightforward

Acknowledgement

This research was supported by the Japanese Ministry of Education, Culture, Sports, Science and Technology and the Japan Society for the Promotion of science (a grant-in-aid for Scientific Research B, grant no. 26301022). In creating this manuscript, we also received valuable assistance from Dr Yoshimi Yashiro, an Associate Professor of Center for iPS Cell Research (CiRA). For all of this assistance, we express our deepest gratitude and appreciation.

Topic

Research questions Research Methods

Knowledge creation dynamics

What are the dynamics of knowledge evolution for science-based innovations? How are the trajectories and paradigms of dominant theories selected? On the other hand, two-mode networks were constructed to evaluate the dynamics of the groups enaged in the scientific development of the field of iPS cell reprogramming/generation. These networks relate the different delivery systems with the co-authoring organizations of the relevant publication sub-set. A threshold equal or greater than three was set on the network edges. Moreover, three periods of time are studied in this analysis: 2006-2009, 2010-2012, and 2006-2012. The results of both research methods were validated through the conduction of a series of interviews with experts in the field of iPS cell generation/reprogramming.

Result

This section presents the results of the two analyses of this study: dynamics of knowledge creation and social dynamics. A total of 1,537 publications relevant for the field of iPS cells between 2006-2012 were identified. From these, as previously mentioned, 581 (around 38%) were found to be directly related to the reprogramming of iPS cells.


J I Zhuchao

STRUCTURE OF JULIA SETS FOR POST-CRITICALLY FINITE ENDOMORPHISMS ON P 2

Keywords: Mathematics Subject Classification. 37F15, 37F50, 32U40 Post-critically finite endomorphism, Julia set, laminar current, Fatou disk. 1

Let f be a post-critically finite endomorphism (PCF map for short) on P 2 , let J1 denote the Julia set and let J2 denote the support of the measure of maximal entropy. In this paper we show that: 1. J1 \ J2 is contained in the union of the (finitely many) basins of critical component cycles and stable manifolds of sporadic super-saddle cycles. 2. For every x ∈ J2 which is not contained in the stable manifold of a sporadic super-saddle cycle, there is no Fatou disk containing x. Here sporadic means that the super-saddle cycle is not contained in a critical component cycle. Under the additional assumption that all branches of P C(f ) are smooth and intersect transversally, we show that there is no sporadic super-saddle cycle. Thus in this case J1 \ J2 is contained in the union of the basins of critical component cycles, and for every x ∈ J2 there is no Fatou disk containing x.

As consequences of our result: 1.We answer some questions of Fornaess-Sibony about the non-wandering set for PCF maps on P 2 with no sporadic super-saddle cycles. 2. We give a new proof of de Thélin's laminarity of the Green current in J1 \ J2 for PCF maps on P 2 . 3. We show that for PCF maps on P 2 an invariant compact set is expanding if and only if it does not contain critical points, and we obtain characterizations of PCF maps on P 2 which are expanding on J2 or satisfy Axiom A.

1. Introduction 1.1. Background. Let f : P 2 → P 2 be a holomorphic endomorphism of degree ≥ 2, where P 2 is the complex projective plane. The first Julia set J 1 is defined as the locus where the iterates (f n ) n≥0 do not locally form a normal family, i.e. the complement of the Fatou set. Let T be the dynamical Green current of f , defined by T = lim n→+∞ d -n (f n ) * ω F S , where ω F S is the Fubini-Study (1,1) form on P 2 . The Julia set J 1 coincides with Supp (T ), and the self intersection measure µ = T ∧ T is the unique measure of maximal entropy of f . See Dinh-Sibony [START_REF] Dinh | Dynamics in several complex variables: endomorphisms of projective spaces and polynomial-like mappings[END_REF] for background on holomorphic dynamics on projective spaces.

We define the second Julia set to be J 2 = Supp µ. From the definitions we know that J 2 ⊂ J 1 . By Briend-Duval [START_REF] Briend | Exposants de Liapounoff et distribution des points périodiques d'un endomorphisme de CP k[END_REF], J 2 is contained in the closure of the set of repelling periodic points. However contrary to the situation in dimension one there may exists repelling periodic point outside J 2 . A major problem in holomorphic dynamics is to investigate the structure of J 1 \ J 2 . A promising picture is that J 1 \ J 2 is foliated (in some appropriate sense) by holomorphic disks D along which ( f n | D ) n≥0 is a normal family. Such disks are called Fatou disks. The dynamical Green current T is called laminar in some open set Ω if it expresses as an integral of integration currents over a measurable family of compatible holomorphic disks (which means that these disks have no isolated intersections) in Ω. These disks are automatically Fatou disks. Let σ T = T ∧ ω be the trace measure of T, which is a natural reference measure on J 1 . If T is laminar in Ω, then for σ T a.e. x ∈ Ω there exists a germ of holomorphic Fatou disk D containing x. De Thélin proved in [START_REF] De | Sur la laminarité de certains courants[END_REF] and [START_REF] De | Un phénomène de concentration de genre[END_REF] that T is laminar in J 1 \ J 2 for post-critically finite endomorphisms on P 2 . We note that by the works of Dujardin in general T is not necessarily laminar in J 1 \ J 2 [START_REF] Dujardin | A non-laminar dynamical Green current[END_REF], but a related weaker result holds [START_REF] Dujardin | Fatou directions along the Julia set for endomorphisms of CP k[END_REF].

A holomorphic endomorphism f on P k , k ≥ 1 is called post-critically finite (PCF for short) if the post-critical set

P C(f ) := n≥1 f n (C(f ))
is an algebraic subset of P k , where C(f ) := x ∈ P k : Df (x) is not invertible is the critical set.

In dimension 1 , this coincides with the usual definition of PCF maps on the Riemann sphere P 1 . PCF maps play an important role in one-dimensional complex dynamics, mainly because the remarkable topological classification theorem of Thurston [START_REF] Douady | A proof of thurston's topological characterization of rational functions[END_REF]. PCF maps are still of interest in higher dimension, and their dynamics have been investigated by many authors. Here are some results which are useful for our purpose:

1. The Fatou set of a PCF maps on P 2 is the union of the basins of super-attracting cycles (Fornaess-Sibony [START_REF] Erik | Complex dynamics in higher dimensions[END_REF], Ueda [START_REF] Ueda | Critical orbits of holomorphic maps on projective spaces[END_REF] and Rong [START_REF] Rong | The Fatou set for critically finite maps[END_REF]).

2. J k = P k for strictly PCF maps on P k (k = 2 by Jonsson [START_REF] Jonsson | Some properties of 2-critically finite holomorphic maps of P 2[END_REF], general k by Ueda [START_REF] Ueda | Critically finite holomorphic maps on projective spaces[END_REF]), our main result can be seen as a generalization of Jonsson's result.

3. The eigenvalues of periodic points of PCF maps on P 2 are either 0 or larger than 1 (Le [START_REF] Van Tu | Fixed points of post-critically algebraic endomorphisms in higher dimension[END_REF]).

The dynamics of PCF maps on P k , k ≥ 2 were further studied by Ueda [START_REF] Ueda | Critically finite holomorphic maps on projective spaces[END_REF] and Astorg [START_REF] Astorg | Dynamics of post-critically finite maps in higher dimension[END_REF]. Moreover, interesting examples of PCF maps were constructed by Crass [START_REF] Crass | A family of critically finite maps with symmetry[END_REF], Fornaess-Sibony [START_REF] Erik | Critically finite rational maps on P 2[END_REF] and Koch [START_REF] Koch | Teichmüller theory and critically finite endomorphisms[END_REF].

1.2. Basins of critical component cycles. In this paper we investigate the dynamics on the Julia sets J 1 and J 2 for PCF maps on P 2 . Let f be a PCF map on P 2 . Recall that the critical set C(f ) is an algebraic curve. We call an irreducible component Λ of C(f ) periodic if there exists an integer n ≥ 1 such that f n (Λ) = Λ. Such an irreducible component is called a periodic critical component. There are finitely many periodic critical components. The set Λ, f (Λ), . . . , f n-1 (Λ) is called a critical component cycle. Similarly, a critical point x satisfying f n (x) = x for some n ≥ 1 is called a periodic critical point. The set x, f (x), . . . , f n-1 (x) is called a critical point cycle. Since f n and f have the same Julia sets, to investigate the structure of J 1 \J 2 we may assume that all periodic critical components are invariant.

An important observation is that in P 2 , any invariant critical component Λ is an attracting set. By definition an attracting set Λ in P 2 is an invariant compact subset such that there is a neighborhood U of Λ satisfying f (U ) ⊂⊂ U, and Λ = n≥1 f n (U ). The open set U is called a trapping region of Λ. The attracting basin B(Λ) of an attracting set Λ is by definition the set n≥0 f -n (U ), where U is a trapping region of Λ. Equivalently B(Λ) is the set of points attracted by Λ, i.e. B(Λ) = x ∈ P 2 : dist (f n (x), Λ) → 0, as n → +∞ .

An attracting basin in P 2 is always disjoint from J 2 , except when Λ = P 2 , see [START_REF] Taflin | Attracting currents and equilibrium measures for quasi-attractors of P k[END_REF] Proposition 1.1 for a proof. Fornaess and Sibony [START_REF] Erik | Complex dynamics in higher dimensions[END_REF] proved that for any fixed Riemannian metric on P 2 , if Λ is an invariant critical component, then for x ∈ P 2 , when dist (x, Λ) → 0, we have

dist(f (x), Λ) = o(dist(x, Λ))
It follows that for ǫ > 0 sufficiently small, the ǫ -neighborhood

U ǫ = x ∈ P 2 : dist(x, Λ) < ǫ satisfies f (U ǫ ) ⊂⊂ U ǫ , and Λ = n≥1 f n (U ǫ ).
Then Λ is an attracting set with trapping region U ǫ . Fornaess and Sibony [START_REF] Erik | Complex dynamics in higher dimensions[END_REF] also showed that an invariant irreducible curve in P 2 has genus 0 or 1. Bonifant, Dabija and Milnor showed that an elliptic curve can not be an attracting set [START_REF] Bonifant | Elliptic curves as attractors in P 2 Part 1: Dynamics[END_REF], so the invariant critical component Λ must be a (possibly singular) rational curve. Now let f be a PCF map on P 2 with an invariant critical component Λ. Let π : Λ → Λ be the normalization of Λ, then f restricted to Λ lifts to a map f from Λ to itself. By [START_REF] Jonsson | Some properties of 2-critically finite holomorphic maps of P 2[END_REF], f is a PCF map on P 1 of degree ≥ 2. Let J(f ) be the Julia set of f, and let ν be the unique measure of maximal entropy of f . Then ν = π * ( ν) is an invariant measure on P 2 . It follows that Supp ν = π(J(f )), and ν is a hyperbolic measure of saddle type. Daurat showed that the Green current T is laminar in the basin B(Λ) and subordinate to the stable manifolds x∈ Supp ν W s (x). See [START_REF] Daurat | Hyperbolic saddle measures and laminarity for holomorphic endomorphisms of P 2 (C)[END_REF] for the proof and for more details (it is easy to verify in our case the trapping region U ǫ satisfies conditions (Tub) and (SJ) in Daurat's paper when ǫ sufficiently small). See also for the case when Λ is totally invariant. 1.3. Main results. At this stage we get a nice description of the dynamics in the attracting basins of critical component cycles for PCF maps on P 2 . In this paper we will show that except maybe for a possibly "small" set (the stable manifolds of sporadic super-saddle cycles), every point in J 1 \ J 2 is indeed contained in the basin of a critical component cycle, and for every point in J 2 , there is no Fatou disk passing through it. Recall that a fixed point is called supersaddle if it has one 0 eigenvalue and one eigenvalue with modulus larger than 1. A sporadic super-saddle fixed point is by deninition a super-saddle fixed point which is not contained in a critical component cycle. We will see in Lemma 2.9 that there are at most finitely many sporadic super-saddle cycles. The following is our main result.

Theorem 1.1. Let f be a PCF map on P 2 of degree ≥ 2, then:

(1) J 1 \ J 2 is contained in the union of the basins of critical component cycles and stable manifolds of sporadic super-saddle cycles.

(2) For every x ∈ J 2 which is not contained in the stable manifold of a sporadic super-saddle cycle, there is no Fatou disk containing x.

Here are some remarks of Theorem 1.1. First, for general holomorphic endomorphisms on P 2 , there can exist an invariant compact set E ⊂ J 2 with positive entropy such that for every x ∈ E there exists a Fatou disk containing x, see Taflin [START_REF] Taflin | Blenders near polynomial product maps of C 2[END_REF] Theorem 1.6.

Second, if f has no critical component cycle, by Theorem 1.1 and the fact that the Green current T put no mass on pluri-polar set, we conclude that the stable manifolds of sporadic super-saddle cycles must be contained in J 2 , and we have J 1 = J 2 . However if f has critical component cycles, we can not easily conclude that the stable manifolds of sporadic supersaddle cycles are contained in J 2 by the previous argument. In fact the following phenomenon can happen: there exist a positive closed current S with continuous local potential in P 2 and two disjoint open sets Ω 1 , Ω 2 such that Supp(S| Ω 1 ) and Supp(S| Ω 2 ) intersect along a subvariety. A similar phenomenon appears in a dynamical context in Dujardin-Favre [START_REF] Dujardin | Distribution of rational maps with a preperiodic critical point[END_REF] Example 6.13, see also Dujardin [START_REF] Dujardin | Cubic polynomials: A measurable view of parameter space[END_REF] Section 5.1. Their example is the bifurcation current in the parameter space of cubic polynomials.

Third, to the author's knowledge, there are no sporadic super-saddle cycles for all known examples of PCF maps on P 2 . It is interesting to know whether there exists a PCF map on P 2 carrying a sporadic super-saddle cycle. If the answer is yes, it is also interesting to know whether the stable manifold of this sporadic super-saddle point is contained in J 2 or J 1 \ J 2 . However, under the additional assumption that all branches of P C(f ) are smooth and intersect transversally, the answer of this question is no.

Theorem 1.2. Let f be a PCF maps on P 2 of degree ≥ 2 such that all branches of P C(f ) are smooth and intersect transversally, then every super-saddle cycle is contained in a critical component cycle, that is, sporadic super-saddle cycles do not exist.

We note that the assumption that all branches of P C(f ) are smooth and intersect transversally are satisfied by examples constructed by Crass [START_REF] Crass | A family of critically finite maps with symmetry[END_REF], Fornaess-Sibony [START_REF] Erik | Critically finite rational maps on P 2[END_REF] and Koch [START_REF] Koch | Teichmüller theory and critically finite endomorphisms[END_REF], in fact in these examples P C(f ) are union of projective lines. A direct corollary of Theorem 1.1 and 1.2 is the following.

Corollary 1.3. Let f be a PCF maps on P 2 of degree ≥ 2 such that all branches of P C(f ) are smooth and intersect transversally, then (1) J 1 \ J 2 is contained in the union of the basins of critical component cycles [START_REF] Bedford | Dynamics of regular polynomial endomorphisms of C k[END_REF] There is no Fatou disk containing x for every x ∈ J 2 .

1.4. Some corollaries of the main results. In [START_REF] Erik | Some open problems in higher dimensional complex analysis and complex dynamics[END_REF], Fornaess and Sibony asked several questions about the non-wandering set Ω(f ) of a holomorphic endomorphism on P 2 . The following are the questions: Q1: Is Ω(f ) the closure of periodic points and Siegel varieties? (Here a Siegel variety is an irreducible analytic set X of positive dimension such that there exists a subsequence {n j } such that f n j | X → Id.) Q2: Is the closure of the repelling periodic points open in Ω(f )? Q3:

Is J 2 open in Ω(f )? Q4: Describe Ω(f ) \ J 2 .
As a corollary of Theorem 1.1, we can give answers to the above four questions for PCF maps on P 2 with no sporadic super-saddle cycles.

Theorem 1.4. Let f be a PCF map on P 2 of degree ≥ 2. Then J 2 is the closure of repelling periodic points. If we further assume that there is no sporadic super-saddle cycle, then Ω(f ) is the closure of periodic points, and Ω(f ) \ J 2 is the union of super-attracting cycles together with ∪(C ∩ J 1 ), where C ranges over the set of critical component cycles, in particular J 2 is open in Ω(f ).

Next, as a direct corollary of Theorem 1.1, we obtain a new proof of the laminarity of the Green current in J 1 \ J 2 for PCF maps on P 2 , which was first proved by de Thélin [START_REF] De | Un phénomène de concentration de genre[END_REF].

Corollary 1.5 (de Thélin). Let f be a PCF map on P 2 of degree ≥ 2. Then the Green current T is laminar in

J 1 \ J 2 .
We note that Fornaess and Sibony have studied the non-wandering set and the problem of laminarity of the Green current for s-hyperbolic holomorphic endomorphisms on P 2 (s-hyperbolic means Axiom A plus some technical conditions), see [START_REF] Erik | Hyperbolic maps on P 2[END_REF].

Finally, we obtain a charactrization of expanding invariant compact sets for PCF maps on P 2 . By using both Theorem 1.1 and the methods in the proof of Theorem 1.1, we obtain the following result.

Theorem 1.6. Let f be a PCF map on P 2 of degree ≥ 2 and let K be an invariant compact set. Then K is expanding if and only if K does not contain critical points.

We note that for rational maps on P 1 , an invariant compact set in the Julia set is expanding if it does not intersect the closure of the post-critical set. It is an open problem that whether the same is true for holomorphic endomorphisms on P k , i.e. whether an invariant compact set in J k is expanding if it does not intersect the closure of the post-critical set. See Maegawa [START_REF] Maegawa | On Fatou maps into compact complex manifolds[END_REF] for a discussion of this problem. Theorem 1.6 answer this question for PCF maps on P 2 .

As corollaries of Theorem 1.6 and Theorem 1.1, we obtain characterizations of PCF maps on P 2 which are expanding on J 2 or satisfy Axiom A.

Corollary 1.7. Let f be a PCF map on P 2 of degree ≥ 2. Then f is expanding on J 2 if and only if every critical component of f is preperiodic to a critical component cycle, and f is Axiom A if and only if f is expanding on J 2 , and for every critical component cycle C, C ∩ J 1 is a hyperbolic (saddle) set.

1.5. Outline of the paper. The structure of this paper is as follows. Section 2 is devoted to some preliminaires. In particular we recall Ueda's results about Fatou maps and the normality of backward iterates of holomorphic endomorphisms on P k , k ≥ 1.

In section 3 we prove Theorem 1.2 and the following theorem Theorem 1.8. Let f be a PCF map on P 2 of degree ≥ 2, then J 2 is the closure of the set of repelling periodic points.

Here is a comment of Theorem 1.8. We note that for a general holomorphic endomorphism on P 2 , repelling periodic point may not be contained in J 2 . Indeed there exist examples possessing isolated repelling points outside J 2 , see [START_REF] Erik | Dynamics of P 2 (examples)[END_REF] and [START_REF] John | Superattractive fixed points in C n[END_REF]. The proof of Theorem 1.8 is a rather quick consequence of a result of Ueda [START_REF] Ueda | Critical orbits of holomorphic maps on projective spaces[END_REF].

In section 4 we prove Theorem 1.1, by using the previous results. The proof is divided into two steps. First, we prove that there exists a Fatou disk containing x for every x ∈ J 1 \ J 2 which is not contained in the basin of critical component cycle. Second, we prove that if there is a Fatou disk containing x for some x ∈ J 1 , then x must be contained in the basin of a critical component cycle or in the stable manifold of a sporadic super-saddle cycle. Theorem 1.1 is a combination of these two steps.

In section 5 we prove Theorem 1.4, Corollary 1.5, Theorem 1.6 and Corollary 1.7.

In section 6 we discuss some open problems about the existence of sporadic super-saddle periodic points (for PCF maps on P 2 ) and the possible generalization of Theorem 1.1 to higher dimension.

Acknowledgements. I would like to thank my advisor Romain Dujardin for his advice, help and encouragement during the course of this work. I also would like to thank Xavier Buff, Henry de Thélin, Van Tu Le, Jasmin Raissy, Matteo Ruggiero, Weixiao Shen and Gabriel Vigny for useful discussion.

Preliminaires

In this section we recall some results of Ueda that we will use later. We start with the following definitions of Ueda [START_REF] Ueda | Critical orbits of holomorphic maps on projective spaces[END_REF] Definition 4.5 and [32] Section 1.

Definition 2.1. Let f be a holomorphic endomorphism on P k of degree ≥ 2. Let Z be a complex analytic space. A holomorphic map h :

Z → P k is called a Fatou map if {f n • h} n≥1 is a normal family. A Fatou disk D ⊂ P k is an image of a non-constant Fatou map φ : D → P k , where D is the unit disk.
Note that with this definition, a Fatou disk may be singular.

The following fundamental result about Fatou maps can be found in Ueda [START_REF] Ueda | Critically finite holomorphic maps on projective spaces[END_REF] Proposition 2.1.

Let π : C k+1 \{0} → P k be the canonical projection. Given a holomorphic map φ : X → P k , we say a holomorphic map Φ :

X → C k+1 \ {0} is a lift of φ if π • Φ = φ holds.
For a homogeneous regular polynomial endomorphism F on C k , we define the dynamical Green function as

G(z) := lim n→∞ 1 d n log F n (z) , for z ∈ C k .
Theorem 2.2. Let X be a complex analytic space. Let φ : X → P k be a holomorphic map. Then the following are equivalent.

(1) φ is a Fatou map.

(2) f j • φ contains a locally uniformly convergent subsequence.

(3

) If V is an open set of X and Φ V : V → C k+1 \ {0} is a holomorphic lift of φ| V , then G • Φ V is pluriharmonic on V . (4) For any x ∈ X, there exists a neighborhood V of x and Φ V : V → C k+1 \ {0} a holomorphic lift of φ| V , such that G • Φ V is identically 0 on V .
Next we introduce the conception of points with bounded ramification, first introduced by Ueda in [START_REF] Ueda | Critical orbits of holomorphic maps on projective spaces[END_REF] Definition 4.5.

Definition 2.3. Let f be a holomorphic endomorphism on P k of degree ≥ 2. A point q is said to be a point of bounded ramification if the following conditions are satisfied:

(1) there exists a neighborhood W of q such that P C(f ) ∩ W is an analytic subset of W .

(2) there exists an integer m such that for every j > 0 and every p ∈ f -j (q), we have that ord f j , p ≤ m.

In the case k = 2, we have the following characterization of points of bounded ramification for PCF maps on P 2 . (See Ueda [START_REF] Ueda | Critical orbits of holomorphic maps on projective spaces[END_REF] Lemma 5.7.) Lemma 2.4. Let f be a PCF map on P 2 of degree ≥ 2. Then the points with unbounded ramification are the union of critical component cycles and critical point cycles.

Next we introduce the following abstract result of Ueda. (See [START_REF] Ueda | Critical orbits of holomorphic maps on projective spaces[END_REF] Lemma 3.7 and Lemma 3.8.) Lemma 2.5. Let X be a complex manifold and D an analytic subset of X of codimension 1. Let x ∈ X and let X 0 be a compact neighborhood of x such that the pair (X 0 , X 0 ∩ D) can be triangulated, i.e. there exist simplicial complex K and its subcomplex L whose underlying spaces are X 0 and X 0 ∩ D. Let W be the open star of x with respect to the complex K. Then for every integer m ≥ 0, there exists an irreducible normal complex space Z and η : Z → W holomorphic such that:

(1) η is m-universal, in the sense that for every

D ∩ W -branched holomorphic covering h : Y → W (i.e. the ramification locus of h is contained in D ∩ W ) with sheet number ≤ m, there exists a holomorphic map γ : Z → Y such that h • γ = η. (2) If h : Y → W is a D ∩ W branched holomorphic covering, then h -1 (x) is a single point.
Specializing to holomorphic endomorphisms on P k , we get the following corollary: Corollary 2.6. Let f be a holomorphic endomorphism on P k of degree ≥ 2. Let x ∈ P k be a point with bounded ramification. Then for every neighborhood W of x satisfying the assumption in Lemma 2.5, there exists an irreducible normal complex space Z, and η :

Z → W a P C(f ) ∩ W branched holomorphic covering map such that if W n denotes a connected component of f -n (W ), then there exists a holomorphic map g n : Z → W n such that f n • g n = η, i.e. the following diagram is commute. Z W W n η gn f n
The map g n constructed above can be seen as a kind of inverse branch of f n .

Proof. Since x has bounded ramification, there exists m ≥ 0 such that ord (f n , x n ) ≤ m for every x n ∈ f -n (x) and n ≥ 0. Take X = P k and D = P C(f ) in Lemma 2.5. Then for W satisfying the assumption in Lemma 2.5, there exists an irreducible normal complex space Z, and η : Z → W a P C(f ) ∩ W branched holomorphic covering map, satisfy the two conclusion of Lemma 2.5. Let W n denote a connected component of f -n (W ), then by Lemma 2.5 (2), W n ∩ f -n (x) contains a single point, thus f n : W n → W has sheet number ≤ m. By Lemma 2.5 (1), there exists a holomorphic map g n : Z → W n such that f n • g n = η. Thus the proof is complete.

The sequence {g v } defined in Corollary 2.6 is in fact normal, and any limit map of {g v } is also a Fatou map, by the following result of Ueda ([32] Theorem 2.4 ).

Lemma 2.7. Let f be a holomorphic endomorphism on P k of degree ≥ 2. Let Z be a complex analytic space and h : Z → P k a holomorphic map, for every integer n let g n : Z → P k be the holomorphic map such that f n • g n = h. Then {g n } is a normal family. Further more if φ is a limit map of a subsequence of {g n } , then φ is a Fatou map.

The following lemma is implicitly contained in [START_REF] Ueda | Critically finite holomorphic maps on projective spaces[END_REF] Theorem 4.2, which will be important in our proof of the main results.

Lemma 2.8. Let f be a holomorphic endomorphism on P k of degree ≥ 2. Let K be compact connected subset of P k such that {f n | K } contains a uniformly convergent subsequence and K is not a single point . let φ be the limit map of the uniformly convergent subsequence of

f n | K . Then (1) If φ is a constant map, then x := φ(K) is a point of unbounded ramification.
(2) If φ is not a constant map and there exists x ∈ φ(K) which is a point of bounded ramification. Then there exists a Siegel variety Y containing x, i.e. an irreducible analytic set of positive dimension such that there is a subsequence

{n j } such that f n j | Y → Id.
Proof. The proof is contained in the proof of [START_REF] Ueda | Critically finite holomorphic maps on projective spaces[END_REF] Theorem 4.2, except at one point one should replace the branched covering map ξ : X → P k in [START_REF] Ueda | Critically finite holomorphic maps on projective spaces[END_REF] by the branched covering η : Z → W , where W is a neighborhood of x, and η : Z → W is the branched covering constructed in Corollary 2.6. The idea is to construct locally uniformly convergent subsequences in the following style: g j : Z → P k such that f n j • g j = η. The property of the limit map of well-chosen {g j } will either give a contradiction of (1), or give the desired Fatou map of (2).

We end this section by showing the following finiteness result. Lemma 2.9. Let f be a PCF map on P 2 of degree ≥ 2. Then the set of critical point cycles is a finite set.

Proof. For every critical point cycle, there is a periodic point x in this cycle such that x ∈ C(f ). Let V be the union of periodic components in P C(f

), then x ∈ V ∩ C(f ). It is clear that every irreducible curve in V ∩ C(f ) is contained in a critical component cycle. So if x is not contained in a critical component cycle, then x is contained in the dimension 0 subvariety of V ∩ C(f ).
Since this dimension 0 part is just a union of finite number of points, there are at most finitely many critical point cycles which are not contained in a critical component cycle.

Location of periodic points.

Let f be a PCF map on P 2 of degree ≥ 2. In this section we prove Theorem 1.2 and Theorem 1.8. By Briend-Duval [START_REF] Briend | Exposants de Liapounoff et distribution des points périodiques d'un endomorphisme de CP k[END_REF], J 2 is contained in the closure of the set of repelling periodic points. Thus to prove J 2 is the closure of the set of repelling periodic points, we only need to prove that repelling periodic points are contained in J 2 . We start with a definition. Definition 3.1. Let f be a PCF map on P 2 of degree ≥ 2. A fixed point x 0 is called repelling if all eigenvalues of Df at x 0 have modulus larger than 1. A fixed point x 0 is called supersaddle if Df at x 0 has one 0 eigenvalue and one eigenvalue with modulus larger than 1. A fixed point x 0 is called super-attracting if Df at x 0 has only 0 eigenvalues.

Note that by the result of Le [START_REF] Van Tu | Fixed points of post-critically algebraic endomorphisms in higher dimension[END_REF], for PCF maps on P 2 every periodic point is either repelling, super-saddle or super-attracting.

3.1. Repelling points. We first recall the following fundamental result, for the proof see Sibony [START_REF] Sibony | Dynamique des applications rationnelles de P k[END_REF] Corollaire 3.6.5. Proposition 3.2. Let f be a holomorphic endomorphism on P k of degree ≥ 2. Then x ∈ J k if and only if for every neighborhood U of x,

P k \ ∞ n=0 f n (U ) is a pluri-polar set.
Now we can prove Theorem 1.8.

Theorem 3.3. Let f be a PCF map on P 2 of degree ≥ 2, then every repelling periodic point belongs to J 2 .

Proof. Our argument concerns the backward iterates around a fixed point. This kind of argument has already appeared in Fornaess-Sibony [START_REF] Erik | Dynamics of P 2 (examples)[END_REF]. Without loss of generality we may assume x 0 is fixed. Since x 0 / ∈ C(f ), by Lemma 2.4, x 0 is a point of bounded ramification. We are going to show that for every neighborhood U of x 0 , we have P 2 \ P C(f ) ⊂ ∞ n=0 f n (U ). Thus since P C(f ) is algebraic, by Proposition 3.2 we will get x 0 ∈ J 2 .

Let y ∈ P 2 \ P C(f ) be an arbitrary point. Let W be a neighborhood of x 0 such that y ∈ W and W satisfies the condition in Corollary 2.6. It can be achieved, by first joining x 0 and y by a smooth embedded (real) curve, and let W be a sufficiently thin tubular neighborhood of this curve such that ∂W is smooth. Let m ≥ 0 such that ord (f n , x n ) ≤ m for every x n ∈ f -n (x 0 ) and n ≥ 0. Let η : Z → W a P C(f ) ∩ W branched holomorphic covering map as in Corollary 2.6. For n ≥ 0, let W n denote the connected component of f -n (W ) containing x 0 . Then by Corollary 2.6 we can define holomorphic map g n : Z → W n such that f n • g n = η. By Lemma 2.7, {g n } is a normal family. We are going to show that actually g n converges to the fixed point x 0 . Since x 0 is repelling, there exists a small neighborhood Ω ⊂ W such that g n converges to x 0 uniformly on η -1 (Ω). Now let φ be any limit map of some subsequence of {g n }. Since φ is constant on an open set η -1 (Ω), φ is constant on Z. Thus any limit map of some subsequence of {g n } is the constant map z → x 0 . This implies that g n converges to the fixed point x 0 . In particular if z 0 satisfies η(z 0 ) = y, we have g n (z 0 ) → x 0 when n → +∞. Now let U be an arbitrary neighborhood of x 0 . Since g n (z 0 ) → x 0 , there exists

N > 0 such that g N (z 0 ) ∈ U . Since f N • g N = η we get y ∈ f N (U ). Since y ∈ P 2 \ P C(f ) is arbitrary we get P 2 \ P C(f ) ⊂ ∞ n=0 f n (U )
. By the arbitrariness of U , we have x 0 ∈ J 2 , which completes the proof.

3.2. Super-saddle points. Let f be a PCF map on P 2 of degree ≥ 2. We investigate the location of super-saddle cycles of f. The following is an equivalent formulation of Theorem 1.2. Theorem 3.4. Let f be a PCF map on P 2 of degree ≥ 2, Let x 0 be a super-saddle fixed point in P 2 . If the branches of P C(f ) are smooth and intersect transversally at x 0 , then x 0 is contained in a critical component cycle.

Proof. In the following we let C x 0 (f ) denote the union of critical components containing x 0 and let

P C x 0 (f ) := ∞ n=1 f n (C x 0 (f )).
It is easy to observe that P C x 0 (f ) = P C x 0 (f n ) for every n ≥ 1. Then up to an iterate of f, we may assume that all periodic components of P C x 0 (f ) are invariant, and every component of P C x 0 (f ) is mapped to an invariant component by at most one iterate. It remains to show that x 0 is contained in an invariant component of C x 0 (f ). We argue by contradiction. Assume x 0 is not contained in an invariant component of C x 0 (f ), then there exists V an invariant component of P C x 0 (f ) such that V ⊂ C x 0 (f ). We first show that V = P C x 0 (f ). We argue by contradiction, assuming V = P C x 0 (f ), then there exists a neighborhood U of x 0 such that f :

U → f (U ) is a V -branched covering. Then, since x 0 ∈ C x 0 (f ), f -1 (V ) contains a component of C x 0 (f ). Since f -1 (V ) also contains V ,
we deduce that f -1 (V ) is singular at x 0 . We recall the following result of Ueda [START_REF] Ueda | Critical orbits of holomorphic maps on projective spaces[END_REF] Lemma 3.5.

Lemma 3.5. Let f : U 1 → U 2 be a V -branched holomorphic covering, where U 1 , U 2 are complex manifolds and V is a codimension 1 analytic subset of U 2 . Suppose that x 0 ∈ U 1 is a singular point of f -1 (V ), then f (x 0 ) is a singular point of V .
Coming back to our situation, letting U = U 1 and f (U ) = U 2 in above lemma, we know that x 0 is a singular point of V . This is impossible, since by our assumption V is smooth at x 0 . Thus there must exist a component

V 1 of P C x 0 (f ) such that V 1 = V.
Let V ′ = f (V ), then by our assumption V ′ is invariant. We recall the following result of Le [START_REF] Van Tu | Fixed points of post-critically algebraic endomorphisms in higher dimension[END_REF] Proposition 5.5. Lemma 3.6. Let f : (C 2 , 0) → (C 2 , 0) be a proper holomorphic germ and let Σ 1 , Σ 2 be two irreducible germs at 0 such that

Σ 1 = Σ 2 , f (Σ 1 ) = Σ 2 and f (Σ 2 ) = Σ 2 . If Σ 2 is
smooth at 0 then the eigenvalue of Df at 0 are 0 and λ where λ is the eigenvalue of

D 0 f | T 0 Σ 2 .
Coming back to our situation, take V = Σ 2 (resp. V ′ = Σ 2 ) in above lemma, since x 0 is super-saddle, we get that the eigenvalue of f restricted to V (resp. V ′ ) at x 0 must be repelling. By our assumption V ′ and V intersect transversally, the only possible case that x 0 being super-saddle is when V ′ = V . Thus we must have f (V 1 ) = V by our assumption that every component of P C x 0 (f ) is mapped to an invariant component by one iterate. To summarize, there exists

V 1 ⊂ P C x 0 (f ) such that V 1 = V , and every component of P C x 0 (f ) is mapped to V by one iterate (in particular, f (V 1 ) = V ).
We do a local coordinate change such that x 0 = (0, 0), V = {x = 0} ∩ U, where U is a small neighborhood of (0, 0). By [START_REF] Van Tu | Fixed points of post-critically algebraic endomorphisms in higher dimension[END_REF] Proposition 5.5, 0 is a repelling point of f | V . Let λ be the eigenvalue of f | V at 0, |λ| > 1. Thus after a linear coordinate change fixing V , the expression of f in this coordinate has the following form (3.1) f (x, y) = (G(x, y), λy + ax + H(x, y)),

Where G(x, y) = O x 2 , y 2 , x is a factor of G and H(x, y) = O |x| 2 , |y| 2 .
Since V 1 is a smooth curve that intersects {x = 0} transversally, we may assume V 1 = {y = φ(x)} for some holomorphic function φ. We do a local coordinate change x ′ → x y ′ → yφ(x). In this coordinate the expression of f has the same form as in (3.1), and V 1 = {y = 0}. In the following we work in this coordinate.

By our assumption that every component of P C x 0 (f ) is mapped to an invariant component by one iterate, there exists a critical component

C such that f (C) = V 1 . Thus C satisfies the equation (3.2) λy + ax + H(x, y) = 0.
By the implicit function theorem, C is a smooth curve that intersects with {x = 0} transversally. We let C = {y = ψ(x)} for some holomorphic function ψ.

By direct calculation, the Jacobian of f is

Jac(f ) = ∂G ∂x λ + ∂H ∂y - ∂G ∂y a + ∂H ∂x .
Since C is in the critical set of f, ψ satisfies the following equation

(3.3) ∂G ∂x (x, ψ(x)) λ + ∂H ∂y (x, ψ(x)) - ∂G ∂y (x, ψ(x)) a + ∂H ∂x (x, ψ(x)) = 0.
Take differential of x in the both sides of (3.2) we get

(3.4) λψ ′ (x) + a + ∂H ∂x (x, ψ(x)) + ∂H ∂y (x, ψ(x))ψ ′ (x) = 0.
Combining (3.3) and (3.4) we get

λ + ∂H ∂y (x, ψ(x)) ∂G ∂x (x, ψ(x)) + ∂G ∂y (x, ψ(x))ψ ′ (x) = 0. Since λ + ∂H ∂y (x, ψ(x)) = 0 in a neighborhood of 0, ψ satisfies ∂G ∂x (x, ψ(x)) + ∂G ∂y (x, ψ(x))ψ ′ (x) = 0.
This implies that ψ satisfies G(x, ψ(x)) = 0 for every x. Then by the expression (3.1), we must have f (C) ⊂ V. Since f (C) = V 1 , we have f (C) = (0, 0), which is a contradiction since f is a locally finite to one map. Thus x 0 is contained in an invariant critical component, and the proof is complete. Corollary 3.7. Let f be a PCF map on P 2 of degree ≥ 2 such that all branches of P C(f ) are smooth and intersect transversally, then every periodic point in J 2 is repelling.

Proof. Let x 0 ∈ J 2 be a periodic point. By Le [START_REF] Van Tu | Fixed points of post-critically algebraic endomorphisms in higher dimension[END_REF], x 0 is either repelling, super-saddle or super-attracting. Since super-attracting periodic points belong to the Fatou set, x 0 is not super-attracting. By Theorem 3.4, super-saddle periodic points are contained in critical component cycles, in particular they are not in J 2 . So x 0 must be repelling.

Structure of the Julia sets.

In this section we prove Theorem 1.1. We will first prove that there exists a Fatou disk containing x for every x ∈ J 1 \ J 2 which is not contained in a basin of critical component cycle (Theorem 4.5). Second we prove that if there is a Fatou disk containing x for x ∈ J 1 , then x must be contained in the basin of a critical component cycle or in the stable manifold of a sporadic super-saddle cycle (Theorem 4.8). Then Theorem 1.1 is a combination of these two theorems. We start with several lemmas. Lemma 4.1. Let f be a PCF map on P 2 of degree ≥ 2. Let x 0 ∈ P k and let v be a subsequence of integers such that x v → y and y is of bounded ramification, where x v = f v (x 0 ). Let W be a neighborhood of y and η : Z → W a holomorphic covering as in Corollary 2. [START_REF] Crass | A family of critically finite maps with symmetry[END_REF].

Let W v denote the connected component of f -v (W ) containing x 0 . Let g v : Z → W v such that f v • g v = η.
Assume g v converges to a constant map, then x 0 ∈ J 2 and y ∈ J 2 Proof. We first prove y ∈ J 2 . We take W = B(y, r) for sufficiently small r. Let Z ′ = η -1 (B(y, r/2)). Let N large enough such that x v ∈ B(y, r/2) when v ≥ N. Since g v converges to a constant map and Z ′ ⊂⊂ Z, we have diam [START_REF] Dinh | Dynamics in several complex variables: endomorphisms of projective spaces and polynomial-like mappings[END_REF] Theorem 2.22, there exists a fixed point of f v-N in W . Letting r → 0, we get that y is approximated by periodic points. By the result of Le [START_REF] Van Tu | Fixed points of post-critically algebraic endomorphisms in higher dimension[END_REF], every periodic point of a PCF map f on P 2 is repelling, super-saddle or super-attracting. Since y is not in a critical component cycle, and since there are only finitely many super-attracting periodic points and super-saddle periodic points outside critical component cycles (see Lemma 2.9), y is approximated by repelling periodic points. By Theorem 3.3, y ∈ J 2 . Let z = η -1 (y). The convergence of g v to x 0 implies g v (y) → x 0 , thus there is a sequence of preimage {y v } of y such that y v converges to x 0 . By the backward invariance of J 2 , we conclude that x 0 ∈ J 2 . Lemma 4.2. Let f be a PCF map on P 2 of degree ≥ 2. Let x 0 ∈ P 2 and let v be a subsequence of integers such that x v → y and y is of bounded ramification, where x v = f v (x 0 ). Let W be a neighborhood of y and η : Z → W a holomorphic covering as in Corollary 2. [START_REF] Crass | A family of critically finite maps with symmetry[END_REF]

g v (Z ′ ) → 0. Let v large enough such that W ′ := f N (g v (Z ′ )) ⊂⊂ W. Then f v-N : W ′ → W is a polynomial-like map, By
. Let W v denotes the connected component of f -v (W ) containing x 0 . Let g v : Z → W v such that f v • g v = η.
Assume g v converges to a non-constant map φ, then there exists a Fatou disk containing x 0 .

Proof. Let W = B(y, r) for small r. Let M = φ(Z), we will show that M contains a Fatou disk containing x 0 . Let N large enough such that

x v ∈ W when v ≥ N. Then for v ≥ N, there exist z v ∈ Z ′ such that g v (z v ) = x 0 . Let z = η -1 (y), it is clear that z v → z when v → +∞. Let v → +∞ in the equation g v (z v ) = x 0 we get φ(z) = x 0 , then x 0 ∈ M . By Lemma 2.7, φ : Z → P 2 is a Fatou map, by definition, this implies that { f n | M } n≥1 is a normal family. Let D ⊂ Z be a holomorphic disk containing z such that φ is not a constant map when restricted to D, then φ(D) ⊂ M is a Fatou disk containing x 0 .
Lemma 4.3. Let f be a PCF map on P 2 of degree ≥ 2. Let x 0 ∈ J 1 such that x 0 is not contained in the attracting basin of a critical component cycle nor contained in the stable manifold of a super-saddle cycle, then there exists a subsequence v of positive integers such that x v = f v (x 0 ) → y, where y is a point of bounded ramification.

Proof. There are at most finitely many critical point cycles, which are not contained in the critical component cycles (see Lemma 2.9). We denote this finite set by E. We show that if x 0 ∈ J 1 satisfying that x 0 is not contained in the attracting basins of critical component cycles and ω (x 0 ) contains only points of unbounded ramification, then x 0 is contained in the stable manifold of a sporadic super-saddle cycle. If x 0 satisfies the above assumption, by Lemma 2.4 we know that ω (x 0 ) ⊂ E. We recall the following basic property of ω -limit set: Lemma 4.4. Let X be a compact metric space, let f : X → X be a continuous map, and let g : ω(x 0 ) → ω(x 0 ) be the restriction of f on the ω-limit set of x 0 , then there is no non-trivial open subset U of ω(x 0 ) such that g(U ) ⊂ U .

Proof. Bowen ([4] Theorem 1) proved the above lemma for homeomorphisms, but the proof also holds for non-invertible maps. For the completion we give a proof here. Assume by contradiction that such open subset U exists, let Y := ω(x 0 ). By our assumption 2ǫ := dist(Y \U, g(U )) > 0. Choose 0 < δ < ǫ such that dist(x 1 , x 2 ) < δ implies dist(f (x 1 ), f (x 2 )) < ǫ for every x 1 , x 2 ∈ X. Now it is clear that there is N > 0 such that dist(f n (x 0 ), Y ) < δ when n > N (otherwise ω(x 0 ) will be strictly larger than Y ). Pick M ≥ N such that dist(f M (x 0 ), g(U )) < ǫ and dist(f M (x 0 ), y) < δ for some y ∈ Y . Then dist(g(U ), y) < 2ǫ, which implies y ∈ U . Then

dist(f M +1 (x 0 ), g(U )) ≤ dist(f M +1 (x 0 ), g(y)) < ǫ. Inductively for all m ≥ M we have dist(f m (x 0 ), g(U )) < ǫ. This implies Y ∩ (Y \ U ) = ∅, which is a contradiction.
Coming back to our situation, since ω(x 0 ) is a finite set, the only possibility is that ω (x 0 ) contains a single periodic cycle. Since x 0 ∈ J 1 and ω(x 0 ) contains no points of bounded ramification, this cycle must be a super-saddle cycle. Thus x 0 is contained in the stable manifold of this super-saddle cycle. It follows that if x 0 is not contained in the attracting basin of a critical component cycle nor contained in the stable manifold of a super-saddle cycle, ω (x 0 ) has non-empty intersection with the set of points of bounded ramification. This completes the proof. Now we can prove the following theorem. Theorem 4.5. Let f be a PCF map on P 2 of degree ≥ 2. If x ∈ J 1 \ J 2 which is not contained in the basin of a critical component cycle, then there is a Fatou disk D containing x.

Proof. Suppose first x 0 is contained in the stable manifold of a super-saddle cycle. Then there exists an embedded holomorphic disc D containing x 0 such that D coincides with the local stable manifold at x 0 , and it is clear that {f n |D} n≥1 is normal. Now suppose x 0 satisfies the assumptions of the theorem, and x 0 is not contained in the stable manifold of a supersaddle cycle. By Lemma 4.3, we can choose a subsequence v of positive integers such that x v = f v (x 0 ) → y, and y is of bounded ramification. Consider a neighborhood W of y and η : Z → W the holomorphic covering as in Corollary 2.6. Let W v denotes the connected component of

f -v (W ) containing x 0 . Let g v : Z → W v such that f v • g v = η
. By Lemma 2.7 {g v } is a normal family. By passing to some subsequence, we may assume g v converges to a holomorphic map φ. φ can not be a constant map, since otherwise by Lemma 4.1, x 0 ∈ J 2 , which is a contradiction. Thus φ is a non-constant map. By Lemma 4.2, there is a Fatou disk containing x 0 , which completes the proof.

Before proving Theorem 4.8, we need some lemmas. Lemma 4.6. Let f be a PCF map on P 2 of degree ≥ 2. Let x ∈ J 1 which is not contained in the basin of a critical component cycle, nor in the stable manifold of a super-saddle cycle. Assume moreover that there is a Fatou disk X containing x. Then there is a point y ∈ ω(x) such that there is a Siegel variety containing y.

Proof. We choose y ∈ ω(x) such that y is not a super-saddle periodic point. Then by Lemma 2.4, y is a point of bounded ramification. Let φ be a limit map of {f n | X } such that φ(x) = y. Then by Lemma 2.8, there is a Siegel variety X containing y. Lemma 4.7. let f be a PCF map on P 2 of degree ≥ 2. Let X be a Siegel variety, then

X ⊂ P C(f ).
Proof. Assume by contradiction that X ⊂ P C(f ). By passing to an iterate of f we may assume X is contained in an invariant component V of P C(f ). Let π : V → V be the normalization of V, and let f be the lift of f on V . Then by Jonsson [START_REF] Jonsson | Some properties of 2-critically finite holomorphic maps of P 2[END_REF] V = P 1 or a torus, and f is a PCF map on P 1 or an expanding torus map. We have the following commutative diagram

V V V V f π π f Let U ⊂ π -1 (X)
be an open set which is contained in the unramification locus of π. Then U is a Siegel variety in V . It is well-known that a PCF map on P 1 or a expanding torus map can not have such a Siegel variety, a contradiction.

Next we can prove Theorem 4.8, thus completes the proof of Theorem 1.1.

Theorem 4.8. Let f be a PCF map on P 2 of degree ≥ 2. If x ∈ J 1 such that there is a Fatou disk containing x. Then x must be contained in the basin of a critical component cycle, or in the stable manifold of a super-saddle cycle.

Proof. Our argument borrows some ingredients in Ueda's paper [START_REF] Ueda | Critically finite holomorphic maps on projective spaces[END_REF] and Le's paper [START_REF] Van Tu | Fixed points of post-critically algebraic endomorphisms in higher dimension[END_REF]. We argue by contradiction. Suppose x is not contained in the basin of a critical component cycle, nor in the stable manifold of a super-saddle cycle. Then by Lemma 4.6, there is a point y ∈ ω(x) such that there is a Siegel variety X containing y. By Lemma 4.7 we may assume

X ∩ P C(f ) = ∅ (shrink X if necessary).
Let Z be the universal covering space of P 2 \ P C(f ), and let η :

Z → P 2 \ P C(f ) be the universal covering map. Let z 0 ∈ Z satisfying η(z 0 ) ∈ φ(X). Recall that φ = lim j→∞ f n j | X .
By passing to a subsequence we may assume k j := n j+1 -n j → +∞. Let W be a neighborhood of η(z 0 ) such that W ∩ P C(f ) = ∅, and let Z 0 be a neighborhood of z 0 such that η : Z 0 → W is a biholomorphism. Let x 0 ∈ X satisfying f n j (x 0 ) → η(z 0 ). Let W j be the connected component of f -k j (W ) containing f n j (x 0 ). Let g j : Z 0 → W j be the holomorphic map such that f k j • g j = η as in Corollary 2.6. Since Z is simply connected, there is no obstruction to extend g j as a globally defined map on Z. In the following g j will denote this globally defined map. By Lemma 2.7 {g j } is a normal family. By shifting to a subsequence we may assume {g j } converges locally uniformly to a holomorphic map g.

Let V := η -1 (X) ∩ Z 0 . We claim that g = η on V. To show this, let z ∈ V be an arbitrary point, we are going to show g(z) = η(z). Let x ∈ X satisfying f n j (x) → η(z). Since for j large enough we have f n j+1 (x) ∈ W , we get f n j (x) ∈ W j . Let z j+1 ∈ Z 0 such that g j (z j+1 ) = f n j (x). By the equation f k j • g j = η we have that η(z j+1 ) = f n j+1 (x). Since f n j+1 (x) → η(z), we get that z j+1 → z. Now let j → +∞ in the equation g j (z j+1 ) = f n j (x) we get that g(z) = η(z). Thus g = η on V.

The equation g = η defines a closed analytic set Y ⊂ Z. Since V ⊂ Y , Y has dimension one. Let Y 0 be an irreducible component of Y 0 of pure dimension 1. By Lemma 2.7 g is a Fatou map, thus η| Y 0 is a Fatou map since g = η on Y 0 . Since Y 0 is a closed analytic set in Z and η is a covering map, the following property holds: for every

x ∈ P 2 \ P C(f ), there is a neighborhood U of x such that either η -1 (U ) ∩ Y 0 = ∅ or every connected component of η -1 (U ) ∩ Y 0 is relatively compact in Y 0 .
Let π : C 3 \ {0} → P 2 be the canonical projection. Let F be a homogeneous polynomial endomorphism on C 3 which induce f . It is clear that F is a PCF map on C 3 . Let G F be the Green function of F . Recall that η : Y 0 → P 2 is a Fatou map. By Theorem 2.2, there exists a holomorphic lift Φ :

Y 0 → C 3 \ {0} of η| Y 0 such that π • Φ = η and G F • Φ is identically zero on Y 0 . Since η(Y 0 )∩P C(f ) = ∅ we get that Φ(Y 0 )∩P C(F ) = ∅.
By the construction of Φ and by the fact that π is an open map, the following property holds: for every

x ∈ C 3 \ P C(F ), there is a neighborhood U of x such that either Φ -1 (U ) ∩ Y 0 = ∅ or every connected component of Φ -1 (U ) ∩ Y 0 is relatively compact in Y 0 . Let Y 1 be the smooth part of Y 0 , then Y 1 is a Riemann surface. Let S := Φ(Y 0 \ Y 1 ), then S is a discrete set in C 3 \ P C(F ). Since G F • Φ is identically zero on Y 1 , Φ(Y 1 )
is contained in a bounded domain in C 3 . This implies that Y 1 is a hyperbolic Riemann surface, i.e. there exists a universal covering map κ : D → Y 1 , where D is the unit disk. Since κ is a covering map, the following property holds:

for every x ∈ C 3 \ (P C(F ) ∪ S), there is a neighborhood U of x such that either (Φ • κ) -1 (U ) ∩ D = ∅ or every connected component of (Φ • κ) -1 (U ) ∩ D is relatively compact in D (⋆).

We consider the map Φ • κ : D → C 3 . The following idea of using Fatou-Riesz's theorem was first appeared in Le [START_REF] Van Tu | Fixed points of post-critically algebraic endomorphisms in higher dimension[END_REF]. As we mentioned before Φ • κ(D) is contained in a bounded domain in C 3 . By the well known Fatou-Riesz's theorem (see Milnor [START_REF] Milnor | Dynamics in One Complex Variable[END_REF] Theorem A.3), for Lebesgue a.e. θ ∈ [0, 2π), the radial limit

τ θ = lim r→1 - Φ • κ(re iθ ) exists for some τ θ ∈ C 3 .
We next show that for every limit τ θ , τ θ ∈ P C(F ) ∪ S. Assume by contradiction that τ θ / ∈ P C(F ) ∪ S. Let U be a small neighborhood of τ θ , then by property (⋆), (Φ • κ) -1 (U ) is a disjoint union of relatively compact open set in D. Then for every R ∈ (0, 1) there exists a r satisfying R < r < 1 such that re iθ / ∈ (Φ • κ) -1 (U ) (by the connectedness of the interval [R, 1]). This contradicts the fact that τ θ = lim r→1 -Φ • κ(re iθ ). Thus τ θ ∈ P C(F ) ∪ S.

Since S is a countable set, there must exist a positive Lebesgue measure set E ⊂ [0, 2π) such that for every θ ∈ E either lim r→1 -Φ • κ(re iθ ) ∈ P C(F ) or lim r→1 -Φ • κ(re iθ ) = s for an element s ∈ S. We will show that these two cases are both impossible, then we will get a contradiction.

We first treat the case that for every θ ∈ E, lim r→1 -Φ • κ(re iθ ) ∈ P C(F ). Let P be a homogeneous polynomial that defines P C(F ), i.e. Second we treat the case that for every θ ∈ E, lim r→1 -Φ • κ(re iθ ) = s, where s ∈ S. Let P 1 , P 2 , P 3 be three linear polynomial in C 3 that define s, i.e. {s} = x ∈ C 3 : P i (x) = 0, i = 1, 2, 3 .

Then for every i we get that P i • Φ • κ is a bounded holomorphic function on D such that the radial limit lim

r→1 - P • Φ • κ(re iθ ) = 0
for every θ ∈ E, where E has positive Lebesgue measure. By Fatou-Riesz's theorem, for every i we get that P i • Φ • κ is identically zero on D. This implies Φ • κ(D) = {s} which imples Φ(Y 1 ) = {s}, contradicts the fact that Y 1 has dimension 1. This completes the proof.

We end this section by showing the sporadic super-saddle cycle is contained in J 2 . Lemma 4.9. Let f be a PCF map on P 2 of degree ≥ 2, then every sporadic super-saddle cycle is contained in J 2 .

Proof. Let x be a super-saddle cycle which is not contained in a critical component cycle. Passing to an iterate of f we may assume every x is fixed and every periodic component of P C(f ) is invariant. Let V be an invariant component of P C(f ) containing x, by our assumption V ⊂ C(f ). By Lemma 3.6, the eigenvalue λ of Df | TxV satisfies |λ| > 1. Let π : V → V be the normalization of V, and let f be the lift of f on V . Let x ∈ V such that π(x) = x. Since V is smooth at x, such a x is unique. Then x is a repelling fixed point of f . Since V is either P 1 or a torus and f is either a rational function of degree ≥ 2 or an expanding torus map, x can be approximated by a sequence of periodic points {ŷ n } ⊂ V such that y n = π(ŷ n ) is in the smooth part of P C(f ). Then by the result of Le [START_REF] Van Tu | Fixed points of post-critically algebraic endomorphisms in higher dimension[END_REF], y n is a repelling periodic point of f for every n. By Theorem 1.8, y n ∈ J 2 . Since y n → x we get x ∈ J 2 as desired.

Some corollaries of the main results

In this section we prove Theorem 1.4, Corollary 1.5, Theorem 1.6 and Corollary 1.7.

5.1. Non-wandering set for PCF maps on P 2 . Before proving Theorem 1.4, we prove the following lemma. Lemma 5.1. Let f be a PCF map on P 2 of degree ≥ 2. Let C be an invariant critical component. Then Ω(f ) ∩ C is the union of super-attracting cycles and J 1 ∩ C, moreover periodic points are dense in Ω(f ) ∩ C.

Proof. Let π : C → C be the normalization of V, and let f be the lift of f on C. Then C = P 1 , and f is a rational function of degree ≥ 2, see [START_REF] Erik | Complex dynamics in higher dimensions[END_REF]. Moreover by Jonsson [START_REF] Jonsson | Some properties of 2-critically finite holomorphic maps of P 2[END_REF], f is also a PCF map. We have the following commutative diagram.

C C C C f π π f
It is clear that if p is a periodic point of f , then π(p) is a periodic point of f. Let J( f ) be the Julia set of f . Since periodic points of f are dense in J( f ), periodic points of f are dense in π(J( f ). We claim that J 1 ∩ C = π(J( f ). By the above discussion π(J( f ) is contained in the closure of periodic points. These periodic points must be super-saddle. Thus π(J( f ) ⊂ J 1 ∩C. To show J 1 ∩C ⊂ π(J( f ), it is equivalent to show that for every y in the Fatou set of f , π(y) is in the Fatou set of f . Since f is a PCF map, y is contained in the basin of a super-attracting cycle of f . For every super-attracting fixed point z of f , π(z) is a fixed point of f . It is clear that π(z) is not repelling, since π(z) ∈ C(f ). It is also clear that π(z) is not super-saddle, since otherwise π(z) is a smooth point of C (see Le [START_REF] Van Tu | Fixed points of post-critically algebraic endomorphisms in higher dimension[END_REF] Proposition 5.4), and z must be a repelling point of f . Thus π(z) must be a super-attracting fixed point of f . Thus π(y) is in the basin of π(z), this implies π(y) is in the Fatou set of f . Thus J 1 ∩ C = π(J( f ) as desired. Since π(J( f ) is contained in the closure of periodic points, J 1 ∩ C is contained in Ω(f ). Since the Fatou set of f is the union of basins of super-attracting cycles, we get that Ω(f ) ∩ C is exactly the union of super-attracting cycles and J 1 ∩ C. This implies periodic points are dense in Ω(f ) ∩ C. Now we can prove Theorem 1.4. Recall the statement. Theorem 5.2. Let f be a PCF map on P 2 of degree ≥ 2. Then J 2 is the closure of repelling periodic points. If further assumed there is no sporadic super-saddle cycle, then Ω(f ) is the closure of periodic points and Ω(f ) \ J 2 is the union of super-attracting cycles together with ∪(C ∩ J 1 ), where C ranges over the set of critical component cycles, in particular J 2 is open in Ω(f ).

Proof. By Theorem 1.8, J 2 is the closure of the set of repelling periodic points. In the following we assume there is no sporadic super-saddle cycle. First we show that Ω(f ) is the closure of periodic points. Let x ∈ Ω(f ) such that x / ∈ J 2 . By Theorem 1.1 and the fact that Fatou set are only super-attracting basins, either x is contained in the basin of a super-attracting cycles, or x is contained in the basin of a critical component cycle. In the first case it is clear that x must be a super-attracting periodic point. In the second case, x must be contained in a critical component cycle, thus x is contained in the closure of periodic points by Lemma 5.1. Again by Lemma 5.1, for every critical component C we have Ω(f ) ∩ C is the union of super-attracting cycles and J 1 ∩ C. Thus Ω(f ) \ J 2 is the union of super-attracting cycles together with ∪(C ∩ J 1 ), where C ranges over the set of critical component cycles. 5.2. Laminarity of the Green current for PCF maps on P 2 . Recall the statement of Corollary 1.5.

Corollary 5.3 (de Thélin). Let f be a PCF map on P 2 of degree ≥ 2. Then the Green current T is laminar in J 1 \ J 2 .

Proof. Let E be the union of sporadic super-saddle cycles. Let W be the union of stable manifolds of super-saddle cycles in E. Then W is a pluri-polar set, and T put no mass on W (since the local potential of T is Hölder continuous, see [START_REF] Dinh | Dynamics in several complex variables: endomorphisms of projective spaces and polynomial-like mappings[END_REF]). By Theorem 1.1, T | J 1 \J 2 put full mass on the union of the basins of critical component cycles. By Daurat [START_REF] Daurat | Hyperbolic saddle measures and laminarity for holomorphic endomorphisms of P 2 (C)[END_REF], T is laminar in every basin of critical component cycle. Thus T is laminar in J 1 \ J 2 .

5.3.

Expanding sets for PCF maps on P . Recall the statement of Theorem 1.6. The following proof is an adaption of the argument in Maegawa [START_REF] Maegawa | On Fatou maps into compact complex manifolds[END_REF].

Theorem 5.4. Let f be a PCF map on P 2 of degree ≥ 2 and let K be an invariant compact set. Then K is expanding if and only if K does not contain critical points.

Proof. The only if part is obvious. In the following we prove K is expanding if K does not contain critical points. To prove that K is an expanding set, it is equivalent to prove the following limit tends to infinity:

min x∈K min v∈TxP 2 ,|v|=1 |D(f n )(v)| → +∞, n → +∞.
Assume by contradiction that the limit does not go to infinity. Then there exists a constant C > 0 and a sequence of points {x m } ⊂ K and vectors {v m } ⊂ T K P 2 such that |v m | = 1 and |D(f nm ) xm (v m )| < C, where {n m } is a sequence of integers. By shifting to a subsequence we may assume f nm (x m ) → x ∈ K. Since K is invariant and K ∩ C(f ) = ∅, by Lemma 2.4 we know that x is a point of bounded ramification. Let W be a neighborhood of x and let η : Z → W be the branched covering constructed in Corollary 2.6. Let W m be the connected component of f -nm (W ) containing x m . Let g m : Z → W m be the holomorphic map such that f nm • g m = η. Then by Lemma 2.7 by shifting to a subsequence we may assume {g m } locally uniformly converges to a Fatou map g. Since K is an invariant compact set and K ∩ C(f ) = ∅, by Theorem 1.1 we get K ⊂ J 2 . Again by Theorem 1.1 and the fact that K ∩ C(f ) = ∅, there is no Fatou disk containing points in K. Thus g must be a constant map. Thus we have diam W m → 0. Since K ∩ C(f ) = ∅, W m ∩ C(f ) = ∅ for m large enough. Thus f nm : W m → W is a biholomorphism when m large. We can then define g ′ m : W → W m be the inverse map of f nm | Wm , and it easily follows that g ′ m converges locally uniformly to a constant map. This implies This contradicts (5.2). Thus the proof is complete.

5.4. PCF maps on P 2 that are expanding on J 2 or Axiom A. We first discuss the PCF maps on P 2 that are expanding on J 2 . Recall the first part of the statement of Corollary 1.7.

Corollary 5.5. Let f be a PCF map on P 2 of degree ≥ 2. Then f is expanding on J 2 if and only if every critical component of f is preperiodic to a critical component cycle.

Proof. We first show the if part. If every critical component of f is preperiodic to a critical component cycle, then C(f ) is contained in the basins of critical component cycles. Thus J 2 ∩ C(f ) = ∅ and J 2 is expanding follows form Theorem 1.6.

Next we show the only if part. Assume by contradiction that there is a critical component C which is not preperiodic to a critical component cycle. Let V be a periodic component of P C(f ) such that V ⊂ C(f ). Then by the argument in the proof of Lemma 5.1, there is a repelling periodic point x ∈ V . By Theorem 1.8 we know that x ∈ J 2 . Let y ∈ C(f ) such that f n (y) = x for some n > 0. Then y ∈ J 2 since J 2 is totally invariant. Thus J 2 ∩ C(f ) = ∅. Thus J 2 is not expanding, which is a contradiction. This completes the proof.

Next we introduce the notion of Axiom A for holomorphic endomorphisms on P k . (For the general definition for smooth endomorphisms on Riemannian manifold, see for instance [START_REF] Qian | Ergodic theory for Axiom A endomorphisms[END_REF].) Definition 5.6. Let f be a holomorphic endomorphism on P 2 of degree ≥ 2. Then f satisfies the Axiom A if the following two conditions hold:

(1) The non-wandering set Ω(f ) is hyperbolic.

(2) Ω(f ) is the closure of periodic points.

We give the following characterization of Axiom A for PCF maps on P 2 . Some interesting examples of Axiom A PCF maps can be found in Ueno [START_REF] Ueno | Dynamics of symmetric holomorphic maps on projective spaces[END_REF].

Corollary 5.7. Let f be a PCF map on P 2 of degree ≥ 2. Then f is Axiom A if and only if f is expanding on J 2 , and for every critical component cycle C, C ∩ J 1 is a hyperbolic (saddle) set.

Proof. By Theorem 1.4 we know that Ω(f ) is the closure of periodic points. Thus f is Axiom A if and only if f is hyperbolic on Ω(f ). By Theorem 1.4, Ω(f ) \ J 2 is the union of superattracting cycles and C ∩ J 1 , where C is a critical component cycle. Thus f is Axiom A if and only if f is expanding on J 2 , and for every critical component cycle C, C ∩ J 1 is a hyperbolic (saddle) set (since super-attracting cycle is automatically a unstable dimension 0 hyperbolic set).

Further discussion

In Lemma 2.9 we proved that if f is a PCF map on P 2 , then the set of sporadic super-saddle cycles is a finite set. However we do not know any examples of PCF maps on P 2 carrying such sporadic super-saddle fixed points. One may conjecture that such sporadic super-saddle fixed points actually can not exist for PCF maps on P 2 . Theorem 3.4 may be seen as an evidence supporting this conjecture.

It is natural to ask whether our main result Theorem 1.1 can be generalized to higher dimension. Let f be a PCF map on P k , one may conjecture that in nice enough cases, the picture might be the following:

(1) Let J m be the Julia set of order m. Then J m \ J m+1 is contained in the union of the attracting basins of m-dimensional periodic submanifolds, where these submanifolds are contained in the critical set.

(2) There is no Fatou disk containing x for every x ∈ J k .

At least two difficulties need to be overcome for proving this conjectural picture. The first is that the classification of points of bounded ramification is not known for higher dimensional PCF maps. In dimension 2 this classification is known by Lemma 2.4. A more serious problem is that, even if the classification of the points of bounded ramification is known, one can possibly prove that for x ∈ P k \ J k , ω(x) is contained in the set of points of unbounded ramification. But it is not possible to show (1), the structure of the Julia sets filtration, by the argument in Theorem 4.5 and 4.8, since in our argument the points of bounded ramification play an important role.

P

  C(F ) = x ∈ C 3 : P (x) = 0 . Then P • Φ • κ is a bounded holomorphic function on D such that the radial limit lim r→1 - P • Φ • κ(re iθ ) = 0 for every θ ∈ E, where E has positive Lebesgue measure. By Fatou-Riesz's theorem P • Φ • κ is identically zero on D. This implies Φ • κ(D) ⊂ P C(F ) which imples Φ(Y 1 ) ⊂ P C(F ), a contradiction.

  )| → 0, m → +∞,where U is a small neighborhood of x.By the definition of g ′ m (5.1) is equivalent to the following (5.2) minz∈Um min v∈Tz P 2 ,|v|=1 |D(f nm )(v)| → +∞, m → +∞,whereU m := f -nm (U ) ∩ W m .By our construction of {x m } and {v m }, it follows that x m ∈ U m for m large and|D(f nm ) xm (v m )| < C.


Cédric Montanier

Mathieu Fanuel

Hélène Rogniaux

David Ropartz

Anne-Marie Di Guilmi

Antoine Bouchoux

Immobilized enzymes at work: when surface density matters Context and Objective

come

Results

In nature, the plant-based organic carbon contained within plant cell walls is mainly recycled by the action of cellulolytic microorganisms, such as bacteria and fungi, which produce complex arrays of cell wall-degrading enzymes. Many of the enzymes that degrade plant cell wall polysaccharides are modular proteins, which contain single or multiple copies of both catalytic domain(s) and CBM(s). Thus, substrate is attacked by different enzymes acting together at different regions, and much more than sum of individual different enzymatic activities, synergism drives the efficiency of such system. Furthermore, in the case of the cellulosome, a large multi-component cell-bound structures, enzymes are localized at close distance to each-other. The benefits of this spatial proximity on the efficiency of the enzymatic reaction are still poorly understood. We investigate this question by using an in-house developed system, Jo-In, where enzymes are immobilized with controlled densities -therefore distances -that can be controlled precisely. The enzyme used is a xylanase that participates to the hydrolysis of plant cell wall polymers, the Xyn11A from Neocallimastix patriciarum. Our approach preserved the intrinsic activity of the enzyme, making the density of grafting the only parameters that is tuned1 .

cedric.montanier@insa-toulouse.fr Precise control of the distance between immobilized enzymes using Jo and In recombinant proteins 2,3 .

Confocal microsopy on beads coated with In-GFP Confocal microscopy using fluorescent long polymers

Immobilization does not affect enzyme activity; only distance does. Enzyme proximity and hydrolytic profile.

Overall, results show that xylanase molecules can be distanced from 9.5 to 64.4 nm center-to-center. Using small 4-nitrophenyl-ß-D-xylotrioside as substrate, no modification of the kinetic parameters is observed compare to the enzyme in solution. However, when long polymer beechwood xylan is used as substrate, kinetic parameters are affected with higher density of grafting. The product profile was analyzed by HPAEC-PAD and MALDI-TOF. Data indicate that immobilized enzyme product profiles are different from those produced by the enzymes dispersed in solution; the immobilized enzymes release more short oligo-saccharides and oligomers with average DP more homogenous. Our results question the relationship between spatial proximity and synergistic effect as encountered in the cellulosome.

  LISBP • Laboratoire d'Ingénierie des Systèmes Biologiques et des Procédés • Mise à Jour du poster : XX/XX/XXXXX • nom.prenom@insa-toulouse.fr Immobilized enzymes at work: when surface density matters Context and Objective Cédric Montanier 1 , Mathieu Fanuel 2 , Hélène Rogniaux 2 , David Ropartz 2 , Anne-Marie Di Guilmi 3 , Antoine Bouchoux 1 1-LISBP, Université de Toulouse, CNRS, INRA, INSA, Toulouse, France, 2-INRA, UR1268 Biopolymers Interactions Assemblies, F-44316 Nantes, France, 3-CEA, 18 route du panorama, 92265 Fontenay-aux-roses, France

  Specific activity using small chromogenic substrate pNP-X 3 a the equivalent concentration of enzyme that corresponds to 0.3% of beads volume fraction,as expressed in mg/L, b the volume fraction of beads that corresponds to 4.47 mg/L of InNpXyn11A, c the concentration of free InNpXyn11A in solution, as expressed in mg/LKinetic parameters using beecwodd xylan as long chain polymerConcentration of xylooligosaccharides X 1 to X 6 determined by HPAEC-PAD Average degree of polymerization of xylooligosaccharides (DP 3 to 30) released over the time detected by MALDI-TOF experiments (linear detector) with beads 2. The vertical bars are standard deviations and not error bars. They reflect the degree of size polydispersity of the released chains.Control experiments were performed with free enzymes in solution at equivalent concentration of beads.

-Montanier et al.,

2 -Izoré et al., 2010 3-Bonnet et al., 2017


Audrey De Jong

Oriol Roca

Claude Guérin

published or not. The documents may come