Khaled Gaaloul
email: khaled.gaaloul@sap.com

Philip Miseldine
email: philip.miseldine@sap.com

Franc ¸ois Charoy

Towards Proactive Policies supporting Event-based Task Delegation

Keywords: Workflow, Delegation, Access Control, Policy

Delegation mechanisms are receiving increasing interest from the research community. Task delegation is a mechanism that supports organisational flexibility in the human-centric workflow systems, and ensures delegation of authority in access control systems. In this paper, we consider task delegation as an advanced security mechanism supporting policy decision. We define an approach to support dynamic delegation of authority within an access control framework. The novelty consists of reasoning on authorisation dependently on task delegation events, and specifies them in terms of delegation policies. When one of these events changes, our access policy decision may change proactively implying dynamic delegation of authority. Existing work on access control systems remain stateless and do not consider this perspective. We highlight such limitations, and propose a task delegation framework to support proactive enforcement of delegation policies.

I. INTRODUCTION

Many of the complex day to day applications in large organisations are conducted using workflow management systems. Workflow systems automate the management and coordination of organisational or business processes. A workflow typically comprises a set of coordinated activities, known as tasks. We currently observe a move away from predefined strict workflow modelling towards approaches supporting flexibility on the organisational level. One specific approach is that of task delegation [START_REF] Gaaloul | A Secure Task Delegation Model for Workflows[END_REF]. We define task delegation to support organisational flexibility in the human-centric workflow systems, and to ensure delegation of authority in access control systems.

Typically, organisations establish a set of security policies, that regulate how the business process and resources should be managed [START_REF] Atluri | Supporting conditional delegation in secure workflow management systems[END_REF]. While a simple policy may specify which user can be assigned to execute a task, a complex policy may specify additional authorisation constraints supporting delegation. Any mechanism that is used to support task delegation is based on workflow specifications and user authorisation information. Delegation authorisation constraints are defined as events on the control-, data-and task assignment layers of a workflow [START_REF] Gaaloul | A Secure Task Delegation Model for Workflows[END_REF]. Hence, secure task delegation implies the presence of a fixed set of delegation events specifying delegation of authority and enforcing access control policies.

Most of the work done in the area of security constraints and access rights requirements does not treat delegation in sufficient details. On one hand, existing work in the domain of role based-access control are relatively coarse-grained and lack of flexibility [START_REF] Seitz | Policy administration control and delegation using xacml and delegent[END_REF]. On the other hand, existing work on authorisation decision making remain stateless and do not consider dynamic enforcement of policies [START_REF] Chadwick | Adding support to xacml for dynamic delegation of authority in multiple domains[END_REF] and [START_REF] Bertino | Specifying and enforcing access control policies for xml document sources[END_REF]. At present, responses arising from access control requests are stateless such that a response is given to a particular request which is valid and true only at the time the request is made. If, however, this response changes due to a policy adaptation, no mechanism currently exists that allows the new response to be conveyed to the original requestor proactively. Such a mechanism is vital for supporting dynamic delegation of authority. When delegating a task, often the reasoning behind this is dependent on transient conditions (events). When one of these conditions changes during execution, our access policy decision may change: what was once acceptable for the delegatee to access, may not be suitable once the delegation is revoked. We do believe that delegation events define dynamic constraints for authorisation decision making that should not be neglected in advanced security mechanisms supporting task delegation.

The contribution of the paper is the definition of a dynamic delegation of authority approach to support proactive policies in access control systems. First, we motivate our work by introducing a real world scenario supporting delegation within a workflow. We then identify specific events-based task delegation model that would enforce policy changes. The novelty consists of separating the various aspects of delegation with regards to users, tasks and events, and specifies them in terms of delegation policies. Defined policies will regulate delegation control flow within a process while ensuring dynamic delegation of authority. Further, we present existing access control framework for authorisation decision making and discuss their functionalities and limitations. Finally, we present our approach that permits proactive enforcement of delegation policies, and discuss its integration into existing access control systems.

The remainder of this paper is organised as follows. Section 2 presents an e-government case study and motivates a delegation scenario. In section 3 we define a task delegation model, and present the security requirements for our approach. Section 4 presents our delegation framework for access control systems. Section 5 discusses related work. In section 6 we conclude and outline several topics of potential future work.

II. CONTEXT AND PROBLEM STATEMENT

A. Motivating Example

To understand the motivation of our research, we present a real world scenario supporting task delegation. Mutual Legal Assistance (MLA) defines a workflow scenario involving national authorities of two European countries regarding the execution of measures for protection of a witness in a criminal proceeding. Here we describe the MLA process part in the Eurojust organisation A, where a Eurojust member (Prosecutor) receives the request of assistance from the Europol member in order to process it and send it to the concerned authority in country B (see Figure 1). User Alice member of role Prosecutor is assigned to execute the MLA process in Eurojust A. Activities that are part of the process are represented as tasks.

We applied the Business Process Modeling Notation (BPMN) to the MLA process. BPMN has emerged as a standard notation for capturing business processes, especially at the level of domain analysis and high-level systems design [START_REF]The Workflow Management Coalition. Process Definition Interface XML Process Definition Language[END_REF]. BPMN supports so called artifacts that enrich the process model by information entities that do not affect the underlying control flow and define extension points to add additional information to the model. We provided an extension for security semantics to leverage the specification of access control security policies, by adding authorisation constraints artifact regarding users authorised to perform a task. In this scenario, the task "Translate Documents" T3 is originally only accessible by Prosecutor Alice, a fact defined in the workflow policy. This task is a long-running task and is expected to take 5 working days to complete. Alice is unavailable to execute this task due to illness, and will delegate it to User Bob. Bob is a member of role Assistant and is a subordinate to Prosecutor in the organisation hierarchy. Task delegation is a suitable approach to support organisational flexibility and to ensure alternative scenarios [START_REF] Gaaloul | A Secure Task Delegation Model for Workflows[END_REF].

A policy can be defined as a level of defining access to task resources. We define an authorisation policy P for the MLA process. During delegation, the policy P is updated so that User Bob is now allowed to complete task T3. As such, Alice and Bob are here the delegator and the delegatee, respectively. Bob claims the task, and issues an access control request, is granted access, and executes the task. After two days, Alice interrupts her sick leave and returns to work. Once again, Alice is able to claim the task. Due to qualification considerations, it is decided that Alice should complete the task, and that Bob should revoke his actions, and free the task. The policy P need to be updated to reflect that only Alice has access to the task. As such, the original request made by Bob would now evaluate to a deny decision for access.

In traditional access control frameworks no mechanism exists that would alert User Bob to this fact automatically. Accordingly, it is not possible to revoke a previous response given to an access control request. Moreover, a manual review of the current access control rights and task executions is costly, labor intensive, and prone to errors. With a proactive mechanism, when the policy changes to reflect delegation events, the delegatee will be informed proactively. This inquires the need to support specific interactions on the access control architecture that they run on. Specific interactions are meant to be task delegation events that would be automatically captured, and conveyed back to the requestor for appropriate actions.

B. Discussion

We leverage a role-based delegation model to support human-centric interactions in the context of long-running tasks [START_REF] Barka | Framework for role-based delegation models[END_REF]. We assume that task execution is atomic and delegation authority is exclusively granted to the delegatee. Therefore, delegation criteria such as cascaded and/or partial are not considered in this paper [START_REF] Zhang | A rule-based framework for rolebased delegation and revocation[END_REF]. On a delegation event, we define a task delegation relation as follows:

Definition 1. We define a task delegation relation RD = (T,u 1 ,u 2 ,DC), where T is the delegated task, u 1 the delegator, u 2 the delegatee, and DC the delegation constraints.

Delegation constraints refer to the right of delegating accordingly to the workflow policy. For instance, DC is based on the role hierarchy (RH) definition in Eurojust, where the Assistant Bob is a subordinate to the Prosecutor Alice. In addition, DC refers to a temporary delegation, where a delegation period must be defined. Bob is not allowed to exceed T3 deadline (5 working days). We define the delegation relation for T3: RD = (T3,Alice,Bob,(RH,5 days)).

Proactive policy enforcement mechanism is vital for supporting delegation in long-running tasks. This inquires the need to support specific interactions based on delegation events that would be automatically captured, and conveyed back to the requestor for appropriate actions within the access control system. At present, we can enforce delegation access rights via policy adaptation (i.e. permitting the delegatee to perform the delegated tasks). Subsequently, we update the RD specification in the workflow policy once a delegation event is triggered. It consists of adding a new policy authorisation constraint for the delegated user. If this constraint changes due to a policy adaptation (e.g., a task revocation event), a new response need to be conveyed to the delegator proactively.

III. TASK DELEGATION POLICIES BASED-EVENTS

In this section, our concern is to identify events that would enforce policy changes. First, we present our task delegation model, and define transitions as events ruling task life cycle. We then specify from defined events delegation policies that would change during task execution. We argue that task delegation implies dynamic delegation of authority: when one of these events occurs, our access policy decision may change.

A. Task Delegation Model (TDM)

In recent work we defined a task delegation model (TDM) based on task life cycle specifications in the workflow management coalition [START_REF] Gaaloul | A Secure Task Delegation Model for Workflows[END_REF], [START_REF]Workflow Management Coalition Terminology and Glossary[END_REF]. Figure 2 depicts a UML state diagram that illustrates the life cycle of our TDM in the form of a state transition diagram from the time that a task is created through to final completion, cancellation or failure. It can be seen that there are series of potential states that comprise this process. A task, once created, is generally assigned to a user. The assigned user can choose to start it immediately or to delegate it. Delegation depends on the assignment transition, where the assigned user has the authority to delegate the task to a delegatee in order to act on his behalf. Intermediate events define controlled delegation within a workflow (e.g., delegate, cancel, revoke). For instance, the delegator might want to cancel. Our TDM would then go back to the previous state (Assigned state). The delegation control flow behaviour remains internal according to the task model, where Completed, Cancelled and Failed are the final states.

B. Security Requirement Statements

We define delegation transitions as events ruling delegation behaviour. We enriched TDM with additional events supporting delegation requirements such as pull/push mode and grant/transfer kind. The internal behaviour based-events may be a source to a policy change, thereby introducing advanced security requirements in access control enforcement systems. For instance, events like accept or revoke inquire access control enforcement supporting delegatee privileges. From the TDM, we analyse security requirements that need to be taken into account to define delegation policies basedevents (see Figure 2).

• Delegation mode: It defines how delegation request is issued. Pull mode assumes that a delegator has at his disposal a pool of delegatees to be selected to work on his behalf. Push mode assumes that a delegator is waiting for an acceptance from a potential delegatee. Derived events from push mode are accept, cancel and revoke. • Delegation kind: It may be classified into grant or transfer [START_REF] Crampton | Delegation in rolebased access control[END_REF]. A grant delegation model allows a delegated access right (privileges) to be available for both delegator and delegatee. As such, the delegator is still having the control to validate or revoke the task, and the delegatee to execute it. However, in transfer delegation models, the ability to use a delegated access right is transferred to the delegatee; in particular, the delegated access right is no longer available to the delegator. There is no validation required and the task is terminated (complete/fail) by the delegatee.

• Delegation of authority: It permits to a delegator to assign a subset of his assigned privileges to a delegatee who currently do not possess the required authorisation to execute the task. For instance, delegate is an event that will trigger task delegation, thereby updating a policy to enforce access control for a delegatee. • Access control enforcement: It permits dynamic policy enforcement. For instance, revoke implies the revocation of delegated privileges where the delegator will take the control back on his assigned task and, therefore, cancel the previous policy decision.

Definition 2. We define a policy P = (target,rule,C), where target defines where a policy is applicable, rule is a set of rules that defines the policy decision result, and C the policy constraints set that validates the policy rule. The specified events define the condition to validate the policy decision effect. An event change may inquire a policy decision change. In the following, we classify delegation events and identify the relationship between delegation events, delegation criteria and policies decision change (see Table I). We do believe that events such as accept or validate are a part of delegation policies and have a direct impact on delegated authority, thereby inquiring dynamic enforcement of policies. We present two examples to explain Table I: Example 1: validate event is defined in both push and pull modes. It supports grant delegation, where a delegatee need to wait for the validation from the delegator. This event will enforce a policy change and terminate authorisation policy for the delegatee. The validation leads to the completion of the task, and the revocation of the delegated privileges. For instance, Bob work is validated by Alice and then his delegated authority is no more valid in the policy. Example 2: fail event is defined in both push and pull modes. It supports transfer delegation, where a delegatee terminates the task by himself without validation. Defined policy will take effect until the termination of the task during transfer delegation, where no new updates are required since all the task privileges are transferred to the delegatee.

Returning to the example, we can observe a dynamic policy enforcement during delegation. Initially, T3 is delegated to Bob and the delegation policy for T3: P D = (RD,permit,(RH, 5 days,delegate)). In the meanwhile, User Alice is back to work before delegation is done and would cancel what was performed by User Bob so far. Alice is once again able to claim the task and will cancel the policy effect (permit) for Bob. The event revoke will be updated in the policy, and a notification (deny) is then conveyed back to Bob for appropriate actions. Thus, the delegation policy for T3 need to be updated and P D = (RD,deny,revoke).

IV. A SECURE FRAMEWORK FOR TASK DELEGATION

In this section, we develop a framework to support secure task delegation. We present a modular architecture ensuring dynamic delegation of authority and show how proactive policy decisions will be implemented on existing access control frameworks (ACF). In the context of delegation, when a request is issued, it is stored along with details of how to inform the requestor (the delegatee) if the policy decision to the request changes. When a policy is changed, previous requests are re-evaluated, and the requestor is informed that his access rights have changed. To support this approach, we propose an extension to an abstract ACF architecture that permits proactive enforcement of policies, based on delegation events that would alter previous policy decisions (see policies changes identified in Table I).

A. Architecture Overview

We describe the main components of the task delegation framework supporting proactive policy enforcement. We detail what parts were changed in which way and what the new extended architecture looks like (see Figure 3).

• Policy Manager: It allows an administrator to define policies. Through a graphical user interface, the administrator can navigate through the policy document, select document elements (e.g., targets, authorisation rules, obligations), and specify values for selected elements. For instance, an administrator defines an authorisation policy P with decision permit on target task T3 for subject Alice with role Prosecutor. In the context of delegation, we assume that a delegator is allowed to administrate policies, thereby defining delegation rules. Delegation policies will embed delegatees attributes for authentication and authorisation purposes. • ACF: An access control framework (ACF) is defined as a set of software components that accept requests to access resources, analyse these against policies representing actual access rights to resources, and return a response based on this analysis. To illustrate the original architecture of an ACF without extension, a request is issued by the requestor, which is received by the Receiver component in ACF. This is then sent to the Analyser component that queries policies stored in a policy database. A response is generated by the Responder component, which defines a decision (permit, deny, or notapplicable) that is sent back to the requestor. It should be noted, that the above appears asynchronous for the requestor; they provide the request, and a response is produced. • Dynamic Policy Enforcement: It defines our approach to support proactive policy decisions. We extend the ACF architecture with additional components related to policy database. When the Receiver receives a request, it sends this to a Request Database that stores this request. A Policy Adaptation Listener component polls the Policy Database and sends an event to a Re-evaluator component when a policy has changed (see delegation events that change policy in Table I). This queries the Request Database to retrieve the previous requests made, and sends this to the Analyser component for re-evaluation. The Analyser then sends back a new response to the Reevaluator, which queries the Request Database to see if this is different to the response given to the request being analysed. If this is a different response, the Invocation Manager component invokes the "contact point" provided by the requestor (and stored in the Request object) with the new response.

B. Architecture Requirements

On an architectural level, as requests are required to be re-evaluated upon a policy change, a storage mechanism of previous requests and the given response is needed. If a previous request is re-evaluated and a different response to the one stored is produced, the ACF must inform the requestor of the new result. Thus, a mechanism must exist that triggers a reevaluation when it detects a policy change. These effects of the policy change would be automatically captured, and conveyed back to the requestor for appropriate actions. In addition, an invocation component is needed that actually marshals this information to the requestor.

On a language level, the approach would require new constructs to describe the invocation method that the ACF can use to contact the requestor. As such, this acts as a "contact point" for the requestor. In a service-oriented architecture (SOA), this contact point could consist of the endpoint of a service that could be invoked (see the Invoke Web Service in Figure 3). Subsequently, all access policies must be centralised and referenced by the SOA architecture which is protected. We give an SOA a single point of access and we let the services register with our ACF. Since services are essentially black boxes, we define how to contact them and to sort out what it means when policy changes (i.e., proactive policies).

On a technical level, the Policy Manager generates policies and subsequently embed credentials attributes for authentication and authorisation purposes [START_REF] Gaaloul | A Secure Task Delegation Model for Workflows[END_REF]. Credentials providers such as certification of authorities issue digital certificates to the requestor in order to compute his request by the ACF. At this stage, the Receiver component acts as a policy enforcement point to perform access control by making decision request and enforcing decisions. For instance, an X.509 Attribute Certificate (AC) is issued to the delegatee for authentication and authorisation purposes. AC will ensure integrity, protection and non-repudiation through a digital signature. The Receiver gets his attributes certificate and checks his permissions afterwards. The retrieved attributes are validated against the policy (e.g., subject attributes, validity time). Once the delegatee has been successfully authenticated he will attempt to perform specified actions on task resources. At each attempt, the Receiver passes the access request to the Analyser to decide. Decisions results (permit, deny, or notapplicable) are then sent via the Responder.

A new re-evaluation of a policy defines new AC for further request with regards to policy changes. For instance, a revocation implies the cancellation of the issued AC for the delegatee previously. Currently, techniques like temporary certificates or certificate revocation list (CRL) are time-based, and, therefore, do not fulfill event-based requirements. As a first solution, a service is invoked to contact the delegatee and based on the mutual agreement between delegation principals, appropriate actions will take place (e.g., the cancellation of his work and the log off from the system).

V. RELATED WORK

The eXtensible Access Control Markup Language (XACML) was developed in order to provide a uniform way of specifying access control policies in XML [START_REF]eXtensible Access Control Markup Language (XACML v2.0). Standard, Organization for the Advancement of Structured Information Standards (OASIS)[END_REF]. Policies comprising Rules, possibly restricted by Conditions, may be specified and targeted at Resources, Subjects and Actions. Resources, Subjects, Actions and Conditions are matched with information in an authorisation request context using attribute values and a rich set of value-matching functions. The outcome or Effect of a policy evaluation may be Permit, Deny, NotApplicable or Indeterminate. The current XACML standard does not provide explicit support for task delegation.

Chadwick et al. [START_REF] Chadwick | Adding support to xacml for dynamic delegation of authority in multiple domains[END_REF] proposed a solution based on the XACML conceptual and data flow models supporting dynamic delegation of authority. The approach describes a new conceptual entity working alongside the XACML policy decision point in the authorisation decision making. The proposed architecture offers a flexible and dynamic way to manage users credentials and assert them in the remote credential providers, however this is not enough to support dynamic delegation of authority. We do believe that delegating a task requires more effort and involves additional specifications related to task delegation events. Our TDM provides the means to determine faithfully delegation polices in-line, thereby ensuring proactive policy decisions when events such as revoke and validate are triggered during task execution.

Seitz et al. [START_REF] Seitz | Policy administration control and delegation using xacml and delegent[END_REF] investigated how an authorisation management system based on XACML can be extended to use flexible delegation mechanisms. They developed a separate policy administration point component called Delegent that specifies allowed modifications on different elements of an XACML policy for different users. This work was later extended in a product called Axiomatics to support delegent authorisation system [START_REF] Axiomatics | Delegent authorisation system. OASIS XACML InterOp Demo[END_REF]. Authors investigate the administration side of the policy while overriding access control in XACML. Administrating delegation policies is, however, stateless and lacks of reactivity to support policy change when delegating a task. We need a reactive approach to reflect events change and support dynamic enforcement of policies.

PERMIS [START_REF] Chadwick | The permis x.509 role based privilege management infrastructure[END_REF] is a middleware authorisation framework, which focuses mainly on role based access control (RBAC) model. PERMIS XML policy can be extended to support additional conditions such as role assignment validity, delegation depth and target access clauses. The PERMIS framework does not provide direct support for bilateral exchange of policies and credentials to address privacy issue and trust. This is not enough to manage security for systems in which organisations are dynamically built with the collaboration of multiple independent organisations sharing their resources. This is especially the case when we consider authorisation decision making supporting delegation policies involving ad-hoc as well as process-based human interactions cross-organisations.

VI. CONCLUSION AND FUTURE WORK

Providing access control mechanisms to support dynamic delegation of authority via proactive policy enforcement, is a non-trivial task to model and engineer. In this paper we have presented problems and requirements that such a model demands, and have architected a solution based on existing access control frameworks (ACFs). We have also presented at a conceptual level the different components that are necessary to support dynamic policy enforcement. The motivation of this direction is based on real world processes from an egovernment case study, where a long-running task may support changes during delegation. Delegation policies may change according to specific events. We define the nature of events based on task, and describe their interactions with policies decisions. When relevant events occur, we define how they are detected and how to interact with them. In this context, we proposed an extension to an abstracted ACF architecture that permits delegation policies enforcement, thereby ensuring dynamic delegation of authority.

The next stage of our work is the implementation of our framework using XACML standard. We propose an extension to XACML specifications supporting task delegation requirements. We then integrate the proactive policy enforcer in the existing architecture. Future work will look also at enriching our approach with additional delegation constraints supporting historical records. Delegation history will be used to record delegation that have been made to address administrative requirements such as auditing.

Fig. 1 .

 1 Fig. 1. MLA delegation scenario

Fig. 2 .

 2 Fig. 2. Task delegation model

  A delegation policy is a policy P D = (target D ,rule D ,C D ), where target D = RD, rule D ⊆ rule and C D ⊂ C and C D = CD events.

Fig. 3 .

 3 Fig. 3. Architectural extensions supporting delegation policies


V Grines
email: vgrines@yandex.ru

F Laudenbach
email: francois.laudenbach@univ-nantes.fr

O Pochinka
email: olga-pochinka@yandex.ru

Self-indexing energy function for Morse-Smale diffeomorphisms on 3-manifolds

Keywords: Mathematics Subject Classification: 37B25, 37D15, 57M30 Morse-Smale diffeomorphism, Morse-Lyapunov function, Heegaard splitting

The paper is devoted to finding conditions to the existence of a self-indexing energy function for Morse-Smale diffeomorphisms on a 3manifold M 3 . These conditions involve how the stable and unstable manifolds of saddle points are embedded in the ambient manifold. We also show that the existence of a self-indexing energy function is equivalent to the existence of a Heegaard splitting of M 3 of a special type with respect to the considered diffeomorphism.

Introduction

Let M n be a smooth closed orientable n-manifold. A diffeomorphism f : M n → M n is called a Morse-Smale diffeomorphism if its nonwandering set Ω(f ) consists of finitely many hyperbolic periodic points (Ω(f ) = P er(f )) whose invariant manifolds have mutually transversal intersections. D. Pixton [START_REF] Pixton | Wild unstable manifolds[END_REF] defined a Lyapunov function for a Morse-Smale diffeomorphism f as a Morse function 1 ϕ : M n → R such that ϕ(f (x)) < ϕ(x) when x is not a peri-odic point and ϕ(f (x)) = ϕ(x) when it is. Such a function can be constructed in different ways2 (see for instance [START_REF] Meyer | Energy functions for Morse-Smale systems[END_REF]).

If ϕ is a Lyapunov function for a Morse-Smale diffeomorphism f , then any periodic point of f is a critical point of ϕ (see lemma 2.1). The opposite is not true in general since a Lyapunov function may have critical points which are not periodic points of f . Then Pixton [START_REF] Pixton | Wild unstable manifolds[END_REF] defined an energy function for a Morse-Smale diffeomorphism f as a Lyapunov function ϕ such that the critical points of ϕ coincide with the periodic points of f and proved the following results.

• For any Morse-Smale diffeomorphism given on a surface there is an energy function.

• There is an example of a Morse-Smale diffeomorphism on S3 which has no energy function.

If p is a periodic point of period k p for the Morse-Smale diffeomorphism f : M n → M n then the stable manifold is W s (p) = {x ∈ M n | f mkp (x) → p when m → +∞}; the unstable manifold is W u (p) = {x ∈ M n | f mkp (x) → p when m → -∞}. The point p is said to be a sink (resp. source) when dim W u (p) = 0 ( resp. dim W u (p) = n). The point p is called a saddle point when dim W u (p) = 0, n. A stable (resp. unstable) separatrix of the saddle point p is a connected component of W s (p) \ p (resp. W u (p) \ p).

Let us recall that a Morse-Smale diffeomorphism f : M n → M n is called gradient-like if for any pair of periodic points x, y (x = y) the condition W u (x)∩ W s (y) = ∅ implies dim W s (x) < dim W s (y). When n = 3, a Morse-Smale diffeomorphism is gradient-like if and only if the two-dimensional and onedimensional invariant manifolds of its different saddle points do not intersect 3 .

Let f : M n → M n be a gradient-like diffeomorphism. Then, it follows from [START_REF] Smale | Differentiable dynamical systems[END_REF] (theorem 2.3), that the closure l of any one-dimensional unstable separatrix ℓ of a saddle point σ is homeomorphic to a segment which consists of this separatrix and two points: σ and some sink ω. Moreover, l is everywhere smooth except, maybe, at ω. So the topological embedding of l may be complicated in a neighborhood of the sink. According to [START_REF] Artin | Some wild cells and spheres in three-dimensional space[END_REF], ℓ is called tame (or tamely embedded) if there is a homeomorphism ψ : W s (ω) → R n such that ψ(ω) = O, where O is the origin and ψ( l \ σ) is a ray starting from O. In the opposite case ℓ is called wild.

In the above mentioned Pixton's example, the non-wandering set of f : S 3 → S 3 consists of exactly four fixed points: one source α, two sinks ω 1 , ω 2 , one saddle σ whose one unstable separatrix ℓ 1 is tamely embedded and the other ℓ 2 is wildly embedded (see fig. 1). Later, the class G 4 of diffeomorphisms on S 3 with such a nonwandering set was considered in [START_REF] Ch | Knots as topological invariant for gradient-like diffeomorphisms of the sphere S 3[END_REF], where it was proved that, for every diffeomorphism f ∈ G 4 , at least one separatrix ℓ 1 is tame. It was also shown that the topological classification of diffeomorphisms from G 4 is reduced to the embedding classification of the separatrix ℓ 2 . Hence it follows that there exist infinitely many diffeomorphisms from G 4 which are not topologically conjugate.

According to Pixton, if the separatrix ℓ 2 is wildly embedded, the Morse-Smale diffeomorphism f ∈ G 4 has no energy function. The present paper is devoted to finding conditions to the existence of a self-indexing energy function (in the sense of definition 1.2 below) for Morse-Smale diffeomorphisms on 3manifolds.

The first and third authors thank grant RFBR No 08-01-00547 of the Russian Academy for partial financial support.

Formulation of the results

If ϕ is a Lyapunov function of a Morse-Smale diffeomorphism f : M n → M n then any periodic point p is a maximum of the restriction of ϕ to the unstable manifold W u (p) and a minimum of its restriction to the stable manifold W s (p) (see lemma 2.1). If these extremums are non-degenerate then the invariant manifolds of p are transversal to all regular level sets of ϕ in some neighborhood U p of p. This local property is useful for the construction of a (global) Lyapunov function. So we introduce the following definitions.

Definition 1.1 A Lyapunov function ϕ : M n → R for a Morse-Smale diffeo- morphism f : M n → M n is called a Morse-Lyapunov function if
any periodic point p is a non-degenerate maximum of the restriction of ϕ to the unstable manifold W u (p) and a non-degenerate minimum of its restriction to the stable manifold W s (p).

Theorem 1 Among the Lyapunov functions of a Morse-Smale diffeomorphism f those which are Morse-Lyapunov form a residual set in the C ∞topology.

If p is a critical point of a Morse function ϕ : M n → R then, according to the Morse lemma (see, for example, [START_REF] Milnor | Morse theory[END_REF]), in some neighborhood V (p) of p there is a local coordinate system x 1 , . . . , x n , named Morse coordinates, such that x j (p) = 0 for each j = 1, n and ϕ reads ϕ

(x) = ϕ(p) -x 2 1 -. . . -x 2 q + x 2 q+1 + . . . + x 2 n
, where q is the index ϕ at p4 . It is convenient to deal with self-indexing Morse function for which ϕ(p) = q.

If ϕ is a Lyapunov function for a Morse-Smale diffeomorphism f : M n → M n then q = dim W u (p) for any periodic point p of f (see [START_REF] Pixton | Wild unstable manifolds[END_REF], lemma on p. 168). The next definition follows from S. Smale [START_REF] Smale | On gradient dynamical systems[END_REF] who introduced a similar one for gradient-like vector fields.

Definition 1.2 A Morse-Lyapunov function ϕ is called a self-indexing energy function when the following conditions are fulfilled:

1) the set of the critical points of function ϕ coincides with the set P er(f ) of the periodic points of f ;

2) ϕ(p) = dim W u (p) for any periodic point p ∈ P er(f ).

Sometimes we shall speak of a self-indexing energy function even when it is only defined on some domain N ⊂ M n , meaning that the above conditions hold only for points x ∈ N such that f (x) ∈ N. In the next results we only deal with 3-dimensional manifolds.

Let f : M 3 → M 3 be a gradient-like diffeomorphism, ω be a sink of f and L(ω) be the union of all unstable one-dimensional separatrices of saddles which contain ω in their closure. The collection L(ω) is tame if there is a homeomorphism ϕ : W s (ω) → R 3 such that ϕ(ω) = O, where O is the origin and ϕ( l \ σ) is a ray starting from O for any separatrix ℓ ∈ L(ω). In the opposite case the set L(ω) is wild. Notice that the tameness of each separatrix ℓ ∈ L(ω) does not imply the tame property of L(ω). In [START_REF] Debrunner | A mildly wild imbedding of an n-frame[END_REF] there is an example of a wild collection of arcs in R 3 where each arc is tame. Using this example and methods of realization of Morse-Smale diffeomorphisms suggested in [START_REF] Ch | Knots as topological invariant for gradient-like diffeomorphisms of the sphere S 3[END_REF] and [START_REF] Ch | Classification of Morse-Smale diffeomorphisms with the finite number of heteroclinic orbits on 3-manifolds[END_REF], it is possible to construct a gradient-like diffeomorphisms on S 3 having a wild bundle L(ω).

If L(ω) consists of exactly one separatrix ℓ then the tame property is equivalent to the existence of a smooth 3-ball B ω ⊂ W s (ω) such that ℓ∩∂B ω consists of exactly one point (it follows from a criterion in [START_REF] Harrold | A characterization of tame curves in three-space[END_REF]). Thus we give the following definition.

Definition 1.3 We say that L(ω) is almost tamely embedded in M 3 if there is a smooth closed 3-ball B ω ⊂ W s (ω) with ω ∈ int B ω such that ℓ ∩ ∂B ω consists of exactly one point for each separatrix ℓ ⊂ L(ω).
If α is a source, there is a similar definition for L(α). We say that the union L of the one-dimensional separatrices is almost tamely embedded in M 3 if L(ω) and L(α) are so for each sink and source.

Theorem 2 If a Morse-Smale diffeomorphism f : M 3 → M 3 has a selfindexing energy function then it is gradient-like and the set L of onedimensional separatrices is almost tamely embedded.

We would like to understand what conditions could be added to the almost tame embedding property to obtain sufficient conditions for the existence of a self-indexing energy function.

Let f : M 3 → M 3 be a Morse-Smale diffeomorphism. Let us denote by Ω + (resp. Ω -) the set of all sinks (resp. sources), by Σ + (resp. Σ -) the set of all saddle points having one-dimensional unstable (resp. stable) invariant manifolds, by L + (resp. L -) the union of the unstable (resp. stable) onedimensional separatrices. We set A(f

) = Ω + ∪ L + ∪ Σ + , R(f ) = Ω -∪ L -∪ Σ - and L = L -∪ L + . By construction, A(f ) (resp. R(f )) is a connected set
which is an attractor (resp. a repeller)5 of f . We set

g(f ) = |Σ + ∪ Σ -| -|Ω + ∪ Ω -| + 2 2 ,
where |.| stands for the cardinality. We will denote by H the set of Morse-Smale diffeomorphisms f : M 3 → M 3 with the following properties:

1) f is gradient-like;

2) the set L of one-dimensional separatrices of f is almost tamely embedded in M 3 ;

3)

M 3 \ (A(f ) ∪ R(f )) is diffeomorphic to S g(f ) × R where S g(f )
is an orientable surface of genus g(f ) 6 .

Theorem 3 If a Morse-Smale diffeomorphism f : M 3 → M 3 belongs to H then it has a self-indexing energy function.

It follows from [START_REF] Grines | New relation for Morse-Smale systems with trivially embedded one-dimensional separatrices[END_REF] that if the set of one-dimensional separatrices is tamely embedded (that is, L(ω) and L(α) are tame for each sink ω and source α) then

M 3 \ (A(f ) ∪ R(f )) is diffeomorphic to S g(f )
× R and M 3 admits a Heegaard splitting 7 of genus g(f ). Thus we get the next result.

Corollary 1.4 If the set L of one-dimensional separatrices of a gradient-like diffeomorphism f : M 3 → M 3 is tamely embedded, then f has a self-indexing energy function.

The next theorem gives necessary and sufficient conditions to the existence of a self-indexing energy function by means of special Heegaard splittings of M 3 . We also need the following definition.

Definition 1.5 Let D be a subset of M n . It is said to be f -compressed when f (D) is contained in the interior of D.
Theorem 4 A gradient-like diffeomorphism f : M 3 → M 3 has a self-indexing energy function if and only if M 3 is the union of three domains with mutually disjoint interiors, M 3 = P + ∪ N ∪ P -, satisfying the following conditions.

1) P + (resp. P -) is a f -compressed (resp. f -1 -compressed) handlebody of genus g(f ) and A(f ) ⊂ P + (resp. R(f ) ⊂ P -);

2) W s (σ + ) ∩ P + (resp. W u (σ -) ∩ P -) consists of exactly one twodimensional closed disk for each saddle point σ + ∈ Σ + (resp. σ -∈ Σ -);

3) there is a diffeomorphism q : S g(f ) × [0, 1] → N such that q(S g(f ) × {t}), t ∈ [0, 1] bounds an f -compressed handlebody.

Remark 1.6 Observe that condition 2) implies that the 1-dimensional separatrices are almost tamely embedded. Indeed, if thin neighborhoods of the disks P + ∩ W s (σ + ), σ + ∈ Σ + , are removed from P + , one gets a union of balls whose boundaries fulfill definition 1.3. 6 Notice that items 1) and 2) do not imply item 3). In section 5 there is an example of a gradient-like diffeomorphism on M 3 = S 2 × S 1 whose set of one-dimensional separatrices is almost tamely embedded and such that M 3 \ (A(f ) ∪ R(f )) is not a product. 7 Let us recall that a three-dimensional orientable manifold is called a handlebody of a genus g ≥ 0 if it is obtained from a 3-ball by an orientation reversing identification of g pairs of pairwise disjoint 2-discs in its boundary. The boundary of such a handlebody is an orientable surface of genus g. A Heegaard splitting of genus g ≥ 0 for a manifold M 3 is a representation of M 3 as the gluing of two handlebodies of genus g by means of some diffeomorphism of their boundaries. Their common boundary after gluing, a surface of genus g in M 3 , is called a Heegaard surface.

M n → M n . Then 1) -ϕ is Lyapunov function for f -1 ; 2) if p is a periodic point of f then ϕ(x) < ϕ(p) for every x ∈ W u (p) \ p and ϕ(x) > ϕ(p) for every x ∈ W s (p) \ p;
3) if p is a periodic point of f then p is a critical point of ϕ.

Proof:

1) It follows from the definition that ϕ(x) ≤ ϕ(f -1 (x)) for any x and the equality only holds for the periodic points. By multiplying this inequality by -1 we get the wanted inequality.

2) As the unstable manifold of a periodic point p for f coincides with the stable manifold of p for f -1 it is enough to prove items 2) and 3) only for W u (p). Let x ∈ W u (p) \ p. It follows from the definition of the unstable manifold of a periodic point that lim

m→∞ f -mkp (x) = p; hence lim m→∞ ϕ(f -mkp (x)) = ϕ(p). We have ϕ(x) < ϕ(f -kp (x)) < . . . < ϕ(f -mkp (x)) < . . . and, hence, ϕ(x) < ϕ(p).
3) Let us assume that p is a regular point of ϕ. Thus the level set ϕ -1 (ϕ(p)) is (n -1)-manifold. It follows from point 2) that T p W u (p) and T p W s (p) must be tangent to ϕ -1 (ϕ(p)). This is impossible because T p W u (p) and

T p W s (p) generate T p M n . ⋄ Denote Ox 1 . . . x q = {(x 1 , . . . , x n ) ∈ R n | x q+1 = . . . = x n = 0} and Ox q+1 . . . x n = {(x 1 , . . . , x n ) ∈ R n | x 1 = . . . = x q = 0}. Lemma 2.2 Let f : M n → M n be a Morse-Smale diffeomorphism and O(p) be a periodic orbit of period k p and dim W u (p) = q. Then there exist a Morse- Lyapunov function ϕ O(p) defined on some neighborhood U O(p) of O(p) and, for each r ∈ O(p), Morse coordinates x 1 , . . . , x n for ϕ O(p) near r such that: 1) ϕ O(p) (O(p)) = q; 2) (W u (r) ∩ U O(p) ) ⊂ Ox 1 . . . x q and (W s (r) ∩ U O(p) ) ⊂ Ox q+1 . . . x n . Proof: As O(p) is a hyperbolic set then, for each r ∈ O(p), there is a splitting of the tangent space T r M n as a direct sum T r M n = T r W u (r) ⊕ T r W s (r) such that Df r (T r W u (r)) = T f (r) W u (f (r)) and Df r (T r W s (r)) = T f (r) W s (f (r)).
Moreover, there is a metric • on M n such that for some λ, 0 < λ < 1, we have:

Df -1 (v u ) ≤ λ v u , Df (v s ) ≤ λ v s , for each v u ∈ E u and v s ∈ E s , where E u = r∈O(p) T r W u (r) and E s = r∈O(p) T r W s (r) 8 .
Let us define ϕ :

E u ⊕ E s → R by the formula ϕ(v u , v s ) = q -v u 2 + v s 2 .
Let us check that ϕ(Df

(v u , v s )) < ϕ(v u , v s ) for all non-zero v u ∈ E u or v s ∈ E s . Indeed, ϕ(Df (v u , v s ))-ϕ(v u , v s ) = -Df (v u ) 2 + Df (v s ) 2 + v u 2 -v s 2 ≤ -1 λ 2 v u 2 + λ 2 v s 2 + v u 2 -v s 2 ≤ -( 1 λ 2 -1) v u 2 -(1 -λ 2 ) v s 2 < 0 for all non-zero v u ∈ E u and v s ∈ E s .
Identify a small neighborhood U O(p) of O(p) with a neighborhood of the zero-section of E u ⊕ E s by a diffeomorphism which maps the local unstable (resp. stable) manifold into E u (resp. E s ). For every

v = (v u , v s ) ∈ U O(p) we have f (v u , v s ) = Df (v u , v s ) + o(v). Therefore ϕ(f (v u , v s )) < ϕ(v u , v s ) for all non-zero (v u , v s ) ∈ U O(p) if
(f ), c > 0 and v(x) : M n → R be a C ∞ -function such that 0 ≤ v(x) ≤ c, v(x) ≡ c on U and v(x) ≡ 0 out of U. One checks that ϕ c := ϕ + v.ϕ P er(f ) is a Morse-Lyapunov function when c is a small enough positive constant. Indeed, if p is a periodic point, ϕ| W s (p) has a minimum at p (lemma 2.1) and ϕ P er(f ) | W s (p) has a non-degenerate minimum. Hence ϕ c | W s (p)
has a non-degenerate minimum at p for any c > 0 9 . Similarly,

ϕ c | W u (p) has a
8 Such a metric is called Lyapunov metric, see, for example, [START_REF] Katok | Introduction to the Modern Theory of Dynamical Systems[END_REF]). 9 As p is a critical point for functions ϕ| W s (p) (x) and ϕ P er(f

) | W s (p) (x) then they have forms ϕ| W s (p) (x) = ϕ| W s (p) (p) + Q 1 (x) + P 1 (x) and ϕ P er(f ) | W s (p) (x) = ϕ P er(f ) | W s (p) (p) + Q 2 (x) + P 2 (x), where Q 1 (x), Q 2 (x)
are quadratic forms and P 1 (x), P 2 (x) satisfy to condition lim

x →0 Pi(x) x 2 = 0, i = 1, 2. As p is a minimum for ϕ| W s (p) (x) then Q 1 (x) ≥ 0 for x = p, as p is a non-degenerate minimum for ϕ P er(f ) | W s (p) (x) then Q 2 (x) > 0 for x = p and, hence,
non-degenerate maximum at p. As T p M n is the direct sum of T p W u (p) and T p W s (p), then ϕ c is non-degenerate at p. Each point in P er(f ) is a nondegenerate critical point of ϕ c , for any c > 0. Moreover, since the sum of two Lyapunov functions is a Lyapunov function, there is some open neighborhood U of P er(f ) on which ϕ c is a Morse-Lyapunov function for any c > 0.

Besides, there is a positive ε such that, for every x ∈ U ∩ f -1 ( U ), one has ϕ(x) > ϕ(f (x)) + ε. Thus, whatever ϕ P er(f ) is, there exists a small c so that ϕ c fulfills the Lyapunov inequality for every To prove this theorem we need the two next lemmas. Proof: Assume the contrary: a Morse-Smale diffeomorphism f : M n → M n has a self-indexing energy function ϕ : M n → R and it is not gradient-like. Then there are points x, y ∈ P er(f

x ∈ U ∩ f -1 ( U). If ϕ c is a Morse function then the proof is finished. If ϕ c is not a Morse function then a last C ∞ -approximation of ϕ c ,
) (x = y) such that W u (x) ∩ W s (y) = ∅ and dim W s (x) ≥ dim W s (y). Put dim W u (x) = k, dim W u (y) = m and z ∈ W u (x) ∩ W s (y). As n -k = dim W s (x) ≥ dim W s (y) = n -m then k ≤ m. According to lemma 2.1, ϕ(z) < ϕ(x) = dim W u (x) = k, ϕ(z) > ϕ(y) = dim W u (y) = m, hence, k > m. This is a contradiction. ⋄ Lemma 3.2 If a Morse-Smale diffeomorphism f : M 3 → M 3
has a selfindexing energy function ϕ then the union of its one-dimensional separatrices are almost tamely embedded in M 3 .

Q 1 (x) + c • Q 2 (x) > 0 for x = p.
It follows from reducibility of positive-definite quadratic form to sum of squares of all coordinates that

Q 1 (x) + c • Q 2 (x) is non-degenerate. Thus ϕ c | W s (p)
has non-degenerate minimum at the point p. Proof: We shall only give a proof10 for a sink. Let ω be a sink of period k ω , L(ω) be the union of all unstable one-dimensional separatrices whose closure contains ω. According to lemma 3.1, f is gradient-like and, hence, for any separatrix ℓ ∈ L(ω) its closure l consists of ℓ ∪ {ω, σ ℓ }, where σ ℓ is a saddle point of f . As ϕ is a Morse-Lyapunov function then in some neighborhood U ℓ of σ ℓ equipped with Morse coordinates we have ϕ(x) = 1x 2 1 + x 2 2 + x 2 3 and W u (σ ℓ ) is transversal to the regular level sets of ϕ in U ℓ . Let U be the union of the U ℓ 's for ℓ ⊂ L(ω).

Let k be an integer so that f k leaves each ℓ invariant. Since the action of f k on ℓ is discrete it has a fundamental domain [a ℓ , b ℓ ] ⊂ U, hence transversal to the level sets of ϕ. More precisely, ϕ| [a ℓ ,b ℓ ] has no critical points and 1 = ϕ(σ ℓ ) > ϕ(a ℓ ) > ϕ(b ℓ ). We may choose all the a ℓ 's in the same level set of ϕ. We will show that the level set of ϕ| W s (ω) containing the a ℓ 's is a 2-sphere crossing ℓ at a ℓ only.

As ϕ is a self-indexing energy function for f then ω is the unique critical point of ϕ| W s (ω) and its index is 0. Moreover, in some neighborhood V (ω) of ω, equipped with Morse coordinates, ϕ reads ϕ(x) = x 2 1 + x 2 2 + x 2 3 and hence every regular level set of ϕ near ω is a smooth 2-sphere which bounds a smooth 3-ball containing ω. According to the Morse theory 11 , for any value c ∈ (0, 1), ϕ -1 (c) ∩ W s (ω) is also a smooth 2-sphere S c ω which bounds a smooth 3-ball

B c ω ⊂ W s (ω) such that ω ∈ f k (B c ω ) ⊂ int B c ω .
We choose c = ϕ -1 (a ℓ ), a value which does not depend on ℓ (see figure 2).

Assume that there is one point y = a ℓ in ℓ ∩ S c ω ; certainly y belongs to the interval (a ℓ , ω) in ℓ. We have y = f mk (x) for some x ∈ [a ℓ , b ℓ ] and some positive integer m. By the Lyapunov property we have c = ϕ(y) < ϕ(x) < ϕ(a ℓ ) = c, which is a contradiction. ⋄ Thus, theorem 2 follows directly from lemmas 3.1 and 3.2. The next lemma is useful for the proof of the necessary conditions in theorem 4 (see lemma 3.4).

Lemma 3.3 For any Morse

-Smale diffeomorphism f : M 3 → M 3 |Σ -| -|Ω -| + 1 = |Σ + | -|Ω + | + 1 = g(f ).
Proof: According to [START_REF] Shub | Homology theory and dynamical systems[END_REF], a Morse-Smale diffeomorphism induces in all homology groups isomorphisms whose eigenvalues are roots of unity. Thus there is an integer k such that f k leaves P er(f k ) fixed, f k | W u (p) preserves the orientation of W u (p) for any point p ∈ P er(f k ) and 1 is the only eigenvalue of the isomorphism induced by f k on homology. Applying the Lefschetz formula to

f k yields p∈P er(f k ) (-1) dim W u (p) = 3 i=0 (-1) i t i ,
where t i is the trace of the map induced by f k on the i-th homology group H i (M, R). By assumption on k, t i coincides with the i-th Betti number and the alternating sum of the t i 's is the Euler characteristic, which is 0 since M is an odd-dimensional, closed oriented manifold. So we get

|Σ -|-|Ω -| = |Σ + |-|Ω + |. ⋄ Lemma 3.4 If a Morse-Smale diffeomorphism f : M 3 → M 3 has a self- indexing energy function ϕ then M 3
is the union of three domains with mututally disjoint interiors, M 3 = P + ∪N ∪P -, satisfying the following conditions. 1) P + (resp. P -) is a f -compressed (resp. f -1 -compressed) handlebody of genus g(f ) and A(f

) ⊂ P + (resp. R(f ) ⊂ P -); 2) W s (σ + ) ∩ P + (resp. W u (σ -) ∩ P -) consists of exactly one two- dimensional closed disk for each saddle point σ + ∈ Σ + (resp. σ -∈ Σ -);
3) there is a diffeomorphism q : S g(f ) × [0, 1] → N such that q(S g(f ) × {t}), t ∈ [0, 1] bounds an f -compressed handlebody.

Proof: 1) For 0 < ε < 1, we set

P + ε = ϕ -1 ([0, 1 + ε]).
According to the Morse theory, it is obtained by gluing |Σ + | 1-handles 12 to an union of |Ω + | 3-balls. Moreover, it is connected since M itself is connected and any generic path in M, whose end points are in P + ε , may be pushed by the gradient flow of ϕ into P + ε . Therefore P + ε is a handlebody of genus |Σ + | -|Ω + | + 1, that is g(f ) according to lemma 3.3. As ϕ is a Lyapunov function, P + ε is f -compressed. By definition of a self-indexing energy function and lemma 2.1, ϕ(A(f )) = [0, 1] and, consequently A(f ) ⊂ int P + ε . Similarly, using the diffeomorphism f -1 , we get that

P - ε = ϕ -1 ([0, 2 -ε]
) is a handlebody of genus g(f ) which f -1 -compressed and contains R(f ) in its interior.

2) As ϕ is a Morse-Lyapunov function then, for a small enough ε 0 ∈ (0, 1 2 ), the handlebodies P + = P + ε 0 and P -= P - ε 0 satisfy the following:

P + ∩ W s (σ + ) (resp. P -∩ W u (σ -)) consists of exactly one two-dimensional disk D σ + (resp. D σ -) for any σ + ∈ Σ + (resp. σ -∈ Σ -).
3) We take N = ϕ -1 ([1 + ε 0 , 2ε 0 ]). As ϕ has no critical points on N and is constant on each boundary component, N satisfies to condition 3). ⋄

4 Construction of a self-indexing energy function for a gradient-like diffeomorphism in dimension 3

In this section, the considered manifold is 3-dimensional and f : M 3 → M 3 is a gradient-like Morse-Smale diffeomorphism whose set L of all 1-dimensional separatrices are almost tamely embedded.

Auxiliary lemmas

Lemma 4.1 Let ω be a sink of period k ω of f and L(ω) be the set of the 1dimensional separatrices ending at ω. Then there exists a smooth closed 3-ball

B ⊂ W s (ω), ω ∈ int B, such that: 1) B is f kω -compressed;
2) for any ℓ ⊂ L(ω) the sphere S = ∂B crosses ℓ at one point a ℓ only and transversely.

Proof: For simplicity we make k ω = 1. By definition, there exists a closed ball B 0 ⊂ W s (ω) whose boundary S 0 meets condition 2). If S is an embedded sphere in W s (ω) then B(S) will denote the ball it bounds. If S meets condition 2), then ω ∈ int B(S).

Let m be the first integer such that f k (S 0 ) ∩ S 0 = ∅ for all k > m. For any x ∈ S 0 we choose a compact neighborhood K x of x in W s (ω) such that f (K x )∩K x = ∅; it exists since S 0 avoids the fixed point. From the family of the K x 's we extract a finite covering K 1 , . . . , K p of S 0 . By a usual transversality theorem ( [START_REF] Hirsch | Differential Topology[END_REF], chap. 3), we may approximate S 0 in the C ∞ topology by a sphere having the following property: f (S 0 ∩ K 1 ) is transversal to S 0 . A next approximation allows one to get such a transversality along K 1 ∪ K 2 , and so on. Condition 2) is kept when approximating. Thus, in what follows, we may assume that S 0 itself is transversal to its successive images f (S 0 ), . . . , f m (S 0 ). In the next step, we are going to modify S 0 into S 1 which still fulfills condition 2) and such that f k (S 1 ) ∩ S 1 = ∅ for all k ≥ m. Iterating this process will yield the wanted sphere S. Indeed, as f (S) is disjoint from S and ω is an attractor, we must have f (S) ⊂ int B(S), which means that B(S) is f -compressed.

Assume first m = 1 (that is, f k (S 0 ) ⊂ int B 0 for all k ≥ 2) and denote Σ = f (S 0 ). Each intersection curve γ in S 0 ∩ Σ bounds a disk D ⊂ Σ. We choose γ to be innermost in the sense that the interior of D contains no intersection curves. Then the curve γ bounds a singular disk D ′ ⊂ S 0 such that D ∪ D ′ is an embedded 2-sphere homotopic to zero in W s (ω) \ {ω}. We notice that D and D ′ have the same number (0 or 1) of intersection points with any one-dimensional separatrix ℓ, since Σ, as S 0 does, also satisfies condition 2). We define S ′ 0 as the sphere obtained from S 0 by removing the interior of D ′ , gluing D along γ, pushing so that S ′ 0 avoids D ∪ D ′ , and smoothing (S ′ 0 still meets condition 2)). Notice that the intersection curves of f -1 (S 0 ) ∩ S 0 are in bijection by f with the intersection curves of f (S 0 ) ∩ S 0 . It will be useful to perform the above construction with an innermost disk D ⊂ f -1 (S 0 ) instead of D ⊂ f (S 0 ). In both cases, we have to check:

(i) f k (S ′ 0 ) ⊂ B(S ′ 0 ) for all k ≥ 2 (which is equivalent to f -k (S ′ 0 ) ∩ S ′ 0 = ∅); (ii) there are less intersection curves in f (S ′ 0 ) ∩ S ′ 0 than in f (S 0 ) ∩ S 0 . Point (ii) is not always true; it depends on the position of D with respect to B 0 . But we shall prove that there always exists an innermost disk D, in f (S 0 ) or in f -1 (S 0 ), such that (ii) is satisfied.

Case 1: D ⊂ f (S 0 ) and D ∩int B 0 = ∅. Forgetting the pushing-smoothing, for k > 1 we have

f k (S ′ 0 ) ⊂ f k (S 0 ) ∪ f k (Σ) = f k (S 0 ) ∪ f k+1 (S 0 ) ⊂ int B 0 ⊂ B(S ′ 0 ), hence (i) holds. We also have f (S ′ 0 ) ∩ S ′ 0 ⊂ (f (S 0 ) ∪ f 2 (S 0 )) ∩ S ′ 0 ⊂ f (S 0 ) ∩ S ′ 0 , as f 2 (S 0
) lies in the interior of B 0 and D in its exterior. Hence, (ii) holds after pushing-smoothing.

Case 1': D ⊂ f -1 (S 0 ) ∩ B 0 . The proof of (i) and (ii) in this case is similar to the previous one in replacing the positive iterates of f by the negative iterates.

Case 2: D ⊂ f (S 0 ) ∩ B 0 and D ∩ f 2 (B 0 ) = ∅. We have f 2k (S ′ 0 ) ⊂ B(f 2 (S 0 )), hence disjoint from S ′ 0 for all k > 0. Similarly, f 3 (S ′ 0 ) lies in f 3 (B 0 ) ⊂ int (f (B 0 ) ∩ B 0 ), thus it does not intersect D and (i) holds. Before pushing-smoothing, we have f (S ′ 0 ) ⊂ f (S 0 ) ∪ f (D). As f (D) lies in f 2 (B 0 ), we have f (S ′ 0 ) ∩ S ′ 0 ⊂ f (S 0 ) ∩ S ′ 0 .
Hence, pushing decreases the number of intersection curves and (ii) holds.

Case 2': D ⊂ f -1 (S 0 ) ∩ int f -2 (B 0 ) and D ∩ int B 0 = ∅. By using the negative iterates of f one proves that points (i) and (ii) hold.

Case 3: D ⊂ f (S 0 ) ∩ B 0 and D ∩ f 2 (B 0 ) = ∅. We look at the intersection curves of D with f 2 (S 0 ) and choose one of them, α, which is innermost on D: α = ∂d with d ⊂ D. There is a unique disk d ′ on f 2 (S 0 ) such that the embedded sphere d ∪ d ′ does not surround ω. There are two subcases: (a) d ⊂ f 2 (B 0 ) and (b) d ⊂ int B 0 \ int f 2 (B 0 ). When (a), f -2 (d) meets the condition of case 1' and, when (b), it meets the condition of case 2'. In both subcases, points (i) and (ii) hold for this innermost disk. Finally, in any case it is possible to reduce the number of intersection curves of S 0 with its image, keeping condition 2).

Repeating this process yields S 1 , a sphere meeting condition 2) and such that f (S 1 ) ∩ S 1 = ∅, (which implies that f (B(S 1 )) is f -compressed).

When m > 1, the end of the proof goes as follows. We introduce g r = f 2 r . For r big enough, we have g k r (S 0 ) ∩ S 0 = ∅ for all k ≥ 2. According to what we just explained, after changing S 0 into another sphere S 1 we get g r (S 1 ) ∩ S 1 = ∅. This amounts to decrease r by 1: g k r-1 (S 1 ) ∩ S 1 = ∅ for all k ≥ 2. Recursively, we find a ball satisfying both required conditions. ⋄

We now consider the orbit O(ω). We just found a ball B ⊂ W s (ω) such that B lies in the interior of f -kω (B). We choose a sequence

B = B 0 ⊂ B 1 ⊂ . . . ⊂ B kω-1 ⊂ f -kω (B ω ) with mutually disjoint boundaries. Set B O(ω) = kω-1 j=0 f j (B j ). It is clearly f -compressed. Lemma 4.2 For each j = 0, k ω -1, let B j ⊂ W s (f j (ω)) be a ball centered at f j (ω). The union B = kω-1 j=0 f j (B j
) is assumed to be f -compressed. Then there is a self-indexing energy function ϕ : B → R for f having ∂B as a level set.

Proof: According to lemma 2.2, there is an open neighborhood U of O(ω), U ⊂ B, and a self-indexing energy function ϕ O(ω) : U → R for f . A level set of ϕ O(ω) whose value is positive and small is the union of k ω copies of 2-spheres. For each j = 0, k ω -1, we choose a smooth 3-ball Q j in U, centered at f j (ω), with boundary G j such that G = j G j is a level set of ϕ O(ω) . We denote

Q = j Q j , S j = ∂B j , S = j S j .
In changing G by a small isotopy we may assume that S is transversal to

f -k (G) for all k ∈ N. Then S ∩ ( k∈N f -k (G))
consists of a finite family C of closed curves. We have two cases (1) C = ∅, (2) C = ∅.

In case (1), N will denote the least integer such that f N (B) ⊂ int Q. We have two subcases (1a) N = 1 and (1b) N > 1. We first consider (1a). It is known that the domain B \ int Q is diffeomorphic to a union of k ω copies of S 2 × [0, 1] (see [START_REF] Hirsch | Differential Topology[END_REF], chap. 8 13 ). Hence there is a smooth function ϕ :

B → R extending ϕ O(ω) | Q and having no critical point in B\int Q. We claim that ϕ is a self-indexing energy function for f | B . Indeed, for x ∈ Q\O(ω), ϕ(f (x)) < ϕ(x) as it is true for ϕ O(ω) . When x ∈ B \ Q, we have ϕ(f (x)) < ϕ(∂Q) < ϕ(x).
Let us consider case (1b) and set B = f N -1 (B). By the construction B is fcompressed and satisfies to condition (1a). Hence there is a self-indexing energy function φ :

B → R for f | e B .
For any x ∈ B, we define ϕ(x) = φ(f N -1 (x)). It is easy to check that ϕ is the required function.

Let us consider case [START_REF] Ch | Knots as topological invariant for gradient-like diffeomorphisms of the sphere S 3[END_REF]. A curve C ∈ C is said innermost on S if C bounds a disk D C ⊂ S whose interior contains no intersection curves from C. Consider such an innermost curve. We have

C ⊂ f -k C (G) for some integer k C and f k C (C) is an innermost curve on f k C (S). There is a unique disk E C in G which is bounded by f k C (C) such that the sphere f k C (D C ) ∪ E C is homotopic to zero in W s (O(ω)) \ O(ω). We define G ′ = (G \ E C ) ∪ f k C (D C ). It bounds Q ′ ⊂ W s (O(ω)), a union of 3-balls which contains O(ω). The fact that C is an innermost curve implies f (Q ′ ) ⊂ int Q ′ .
There are two occurrences: (2a) 2a) and G in case (2b) is a level set of some self-indexing energy function defined respectively on f -1 ( Q) and on Q. By the construction the number of curves in S ∩ ( k∈N f -k ( G)) is less than in C. We will repeat this process until getting a union of 3-balls Q which is f -compressed and whose boundary G does not intersect f k (S) for any k. Then we are reduced to case (1) and the lemma is proved. ⋄

f (Q) ⊂ Q ′ ⊂ Q and (2b) Q ⊂ Q ′ ⊂ f -1 (Q). In both cases there is a smooth approximation, Q of Q ′ such that Q ⊂ int Q ′ in case (2a) and Q ′ ⊂ int Q in case (2b); Q is still f -compressed. Set G = ∂ Q. According to item (1a), f -1 ( G) in case (

A nice neighborhood of the attractor A(f ) (or the repeller R(f ))

Let f : M 3 → M 3 be a gradient-like diffeomorphism whose 1-dimensional separatrices are almost tame. Let us construct a "nice" neighborhood of the Each connected component U σ of U Σ + , σ ∈ Σ + , is endowed with Morse coordinates (x 1 , x 2 , x 3 ) as in the conclusion of lemma 2.2:

ϕ Σ + (x 1 , x 2 , x 3 ) = 1 -x 2 1 + x 2 2 + x 2 3
, the x 1 -axis is contained in the unstable manifold and the (x 2 , x 3 )-plane is contained in the stable manifold.

According to lemma 4.1, there exists an f -compressed domain B + , made of |Ω + | balls, which is a neighborhood of Ω + and such that each separatrix ℓ ∈ L + intersects ∂B + in one point a ℓ only. Due to the λ-lemma14 (see, for example, [START_REF] Palis | On Morse-Smale dynamical systems[END_REF]), replacing B + by f -n (B + ) for some n > 0 if necessary, we may assume that ∂B + are transverse to the regular part of the level set C := ϕ -1 Σ + (1) and each of the intersections C ∩ ∂B + consists of 2|Σ + | circles. Due to lemma 4.2 there is a self-indexing energy function ϕ B + :

B + → R with a level set ∂B + . For ε > 0 set B + ε = ϕ -1 B + ([0, ϕ B + (∂B + ) -ε]) and
Figure 4: A pair of nice neighborhoods (P + , P -)

E ε = (B + \ int (B + ε )) ∩ {1 -ε ≤ ϕ Σ + ≤ 1 + ε}.
We choose ε > 0 such that:

1) ∂B + and ∂B + ε are transverse to the level sets ϕ -1

Σ + (1±ε), f (B + ) ⊂ int B + ε and (B + \ int (B + ε )) \ {ϕ Σ + < 1 -ε} is a product; 2) ϕ(f -1 (E ε )) > 1 + ε (it is possible as ϕ(f -1 (C \ Σ + )) > 1).
We introduce H + , the closure of {x ∈ U Σ + | x / ∈ B + , ϕ Σ + (x) ≤ 1 + ε} (see figure 3). By construction there is a smoothing P + of B + ∪ H + such that: 15 , that is g(f ).

-P + is f -compressed; -P + is connected (see, for example, [3], lemma 1.3.3)); -P + is a handlebody of genus |Σ + | -|Ω + | + 1
We call P + a nice neighborhood of the attractor A(f ) (see figure 4). Making a similar construction for f -1 we obtain a nice neighborhood P -of the repeller R(f ), which is also a handlebody of genus g(f ) (lemma 3.3).

Construction of a self-indexing energy function on

P + and P - Denote d ε the part of the level set ϕ -1 Σ + (1 -ε) belonging to U Σ + \ int B + ε ). By construction d ε is the union of 2|Σ + | disks. Denote D ′ ε the union of disks in ∂B + ε such that ∂d ε = ∂D ′ ε .
We form S, a union of spheres, by removing the interiors of the D ′ ε from ∂B + ε and gluing the d ε . Denote B(S) the union of balls bounded by S and containing Ω + . We check that B(S) is f -compressed. Indeed, it is true for

B + ε . Moreover, f (d ε ) does not intersect d ε nor ∂B + ε \ D ′ ε .
The first intersection is empty as ϕ Σ + is a Lyapunov function and d ε lies in a level set of it. The second one is empty as

ϕ Σ + (d ε ) = 1 -ε ≤ ϕ Σ + (x), for any x ∈ U Σ + , x ∈ ∂B + ε \ D ′ ε .
Let K be the domain between ∂P + and S. We define a function ϕ + : K → R whose value is 1 + ε on ∂P + , 1ε on S, coinciding with ϕ Σ + on K ∩ H + and without critical points outside H + . This last condition is easy to satisfy as the domain in question is a product cobordism. With all the informations that we have on the image of f , it is easy to check that ϕ + is a Lyapunov function. Indeed, it is obvious for the points of K which are not in H + . Suppose that x ∈ K ∩ H + . Then we have two possibilities: a) f (x) ∈ H + ; b) f (x) / ∈ H + . In the first case the conclusion follows from the Lyapunov property of ϕ Σ + . In the second case we are going to show that f (x) ∈ {ϕ Σ + < 1 -ε} and then the conclusion also holds. Suppose on the contrary that f (x) / ∈ {ϕ Σ + < 1 -ε}. Then f (x) belongs to the domain E ε . But it follows from the choice of E ε that f -1 (E ε ) does not intersect H + . We get a contradiction.

According to lemma 4.2, there is an extension of ϕ + to B(S), which is a self-indexing Morse-Lyapunov function. Finally, we get the desired self-indexing energy function on P + .

As the set R(f ) is an attractor of f -1 and P -is a nice neighborhood of R(f ), it is possible to construct (as above) a self-indexing energy function φ-: P -→ R of f -1 . It follows from lemma 2.1 that the function ϕ -: P -→ R given by the formula ϕ -(x) =φ-(x) + 3 is a self-indexing energy function of f on P -.

Proof of theorem 3

The main assumption, which is not a necessary condition, is the following:

M 3 \ (A(f ) ∪ R(f )) is diffeomorphic to the product S g(f ) × R. Let us denote S ± = ∂P ± . It is easy to arrange that ϕ -(S -) > ϕ + (S + ). First assume ( * ) S -∩ ( k>0 f -k (S + )) = ∅.
Let m be the first positive integer such that f m (S -) ⊂ int P + . If m = 1, as N is a product, there exists a smooth function ϕ : N → R without critical points which extends ϕ + ∪ ϕ -: P + ∪ P -→ R. It is a Lyapunov function as f (N) ⊂ P + and f -1 (N) ⊂ P -. If m > 1, the surfaces f -1 (S + ), f -2 (S + ), . . . , f -m+1 (S + ) are mutually "parallel", that is: two by two they bound a product cobordism, diffeomorphic to S g(f ) × [0, 1] (see for instance theorem 3.3 in [START_REF] Grines | New relation for Morse-Smale systems with trivially embedded one-dimensional separatrices[END_REF]). Therefore they subdivide N in product cobordisms and there exists a function ϕ extending ϕ ± on P ± , without critical points on N and having f -1 (S + ), . . . , f -m+1 (S + ) as level sets. One easily checks that such a ϕ is a self-indexing energy function for f . We now explain the end of the proof, that is, how to reduce oneself to ( * ). Without loss of generality we may assume that S -is transversal to k>0 f -k (S + ), which implies that there is a finite family C of intersection curves.

We are going to describe (as in lemma 4.2) a process decreasing the number of intersection curves by an isotopy of P + among handlebodies which are f -compressed; they will be all equipped with a self-indexing energy function which is constant on the boundary.

For simplifying the statement of the next lemma, we use the following definition. Definition 4.3 Let S be a proper bicollared embedded surface in a 3-manifold W (proper meaning ∂S ⊂ ∂W when ∂S = ∅). One says that S is incompressible in W if any simple curve γ in S, which bounds an embedded disk in W starting on one side of S along its boundary, is homotopic to zero in S, and hence bounds an embedded disk in S 16 .

For instance, our S + and S -are incompressible in N. Lemma 4.4 1) Any 2-sphere which is embedded in N, P + or P -bounds a ball there. 2) S + (resp. S -) is incompressible in P + \ A(f ) (resp. P -\ R(f )).

3) S + is incompressible in N ∪ (P -\ R(f )). 4) Both S + and S -and their images by

f k , k ∈ Z, are incompressible in M \ (A(f ) ∪ R(f )).
Proof: 1) As each of the considered domains embeds into R 3 , every embedded sphere bounds a 3-ball (generalized Schönflies theorem 17 [4]).

2) Let γ be a simple curve in S + which bounds a disk δ in P + \ A(f ). In changing δ by a small isotopy we may assume that δ is transverse to W s (Σ + ); so δ ∩ W s (Σ + ) consists of a finite family I of arcs with end points in γ and a finite family L of closed curves. Each arc α ∈ I, after some isotopy in W s (Σ + ) pushing α into S + , indicates a way of decomposing γ as a connected sum γ 1 #γ 2 of two simple curves of S + bounding disks in V . Of course, if the conclusion of 2) in lemma 4.4 holds for both curves γ 1 and γ 2 , it also does for γ. Finally, one reduces oneself to consider the case when γ ∩ W s (Σ + ) = ∅ (that is, I = ∅); thus, γ can be thought of as a curve in ∂B + where B + , a union of 3-balls, is obtained from P + by cutting along the disks P + ∩ W s (Σ + ). Similarly, by cut-and-paste, it is possible to remove from L the closed curves which bound disks in W s (Σ + ) \ Σ + .

First, assume that L is empty; in other words, δ ⊂ B + . Thus, there are two disks d ′ and d ′′ in ∂B + which is bounded by γ. According to item 1), δ divides one component of B + into two balls B ′ and B ′′ , with d ′ ⊂ B ′ and d ′′ ⊂ B ′′ . Since δ ∩ A(f ) is empty and each component of B + contains exactly one connected component of

B + ∩ A(f ), one of B ′ or B ′′ is disjoint from A(f ). If it is B ′ , that means that d ′ is a disc in the boundary of P + . Moreover B ′ is a ball in P + \ A(f ).
In order to finish the proof, we have to consider the case when L is not empty, but made of curves which bound disks in W s (Σ + ) each one having one saddle point in its interior. Let c be such a curve which is innermost in δ; it bounds a disk d c ⊂ W s (Σ + ) and a disk δ c ⊂ δ. Let σ ∈ Σ + be the saddle point in d c and set D(σ) = P + ∩ W s (σ). After smoothing and a small isotopy, d c ∪ δ c gives rise to an embedded 2-sphere S which is disjoint from W s (Σ + ) and, hence, lies in a connected component B 0 of B + . As S intersects A(f ) in one point exactly, both separatrices of σ must enter two different connected components of B + , one being B 0 and the other being denoted B 1 . Then D(σ) decomposes P + as a connected sum P + = P 0 #P 1 , with P j ⊃ B j for j = 0, 1.

The sphere S bounds a 3-ball B in B 0 , but, since there is some separatrix of σ which enters B without getting out, B must contain one sink ω 0 ; moreover, since δ c avoids A(f ), ω 0 is in the closure of no other separatrix. Hence, P 0 is a ball. If γ ⊂ ∂P 0 , there is nothing to do; so, assume γ ⊂ ∂P 1 . Since ∂P 0 is a 2-sphere, it is equivalent that γ bounds a disk in ∂P + or in ∂P 1 . This allows us to ignore W s (σ) ∪ W s (ω 0 ). Repeating this process we are reduced to the case when L is empty.

3) As N is a product, S + and S -are clearly incompressible in N. One looks at δ, a disk in N ∪ P -whose boundary lies in S + , and at its intersection curves with S -. Using 2) and the innermost curve techniques, one reduces to the case δ ⊂ N and the conclusion follows. 4) For proving the statement for S + , we take δ, an embedded disc in M \ (A(f ) ∪ R(f )) with boundary in S + and a collar of ∂δ transverse to S + . Then, in general position, we have finitely many intersection curves in int δ ∩ (S + ∪ S -). By using 2) and 3) one eliminates successively all intersection curves. Finally, δ lies in N or P + \ A(f ) and the conclusion follows. ⋄ Lemma 4.4 allows us to remove all intersection curves which are homotopic to zero in S + or, equivalently, in f k (S -), k > 0 (as in the proof of lemma 4.2). We recall m, the largest integer such that f m (S -) ∩ S + = ∅. Let F - be a connected component of f m (S -) ∩ N. Since f m (S -) ∩ S -= ∅, we have ∂F -⊂ S + . We claim that F -in incompressible in N. Indeed, if δ is a disk in N with boundary γ ⊂ F -, according to 3) in lemma 4.4, γ is homotopic to zero in f m (S -). As none of the components of ∂F -is homotopic to zero, γ is homotopic to zero in F -.

Therefore, according to F. Waldhausen (corollary 3.2 in [START_REF] Waldhausen | On irreducible 3-manifolds which are sufficiently large[END_REF]), there is some surface F + ⊂ S + diffeomorphic to F -, with ∂F + = ∂F -, and F + ∪F -bounds a domain ∆ in N, which, up to smoothing of the boundary, is diffeomorphic to F -× [0, 1]. We then change S + to S ′ by removing the interior of F + and gluing F -. After a convenient smoothing, this surface S ′ has less intersection curves with k f k (S -) than S + . By construction, it bounds a handlebody P ′ which is isotopic to P + and f -compressed; moreover P ′ carries a self-indexing Lyapunov function. Arguing recursively, we are reduced to case ( * ). In this final recursive argument, once lemma 4.4 is proved, it is no longer usefull that P + intersects W s (Σ + ) along disks. This finishes the proof of theorem 3.

Proof of theorem 4

The necessary condition of theorem 4 is yielded by lemma 3.4. For proving that it is sufficient, we have to construct a self-indexing energy function on M 3 under the assumptions of theorem 4 that we recall now. The diffeomorphism f : M 3 → M 3 is a gradient-like diffeomorphism and M 3 is the union of three domains with mututally disjoint interiors, M 3 = P + ∪ N ∪ P -, satisfying the following conditions. 1) P + (resp. P -) is a f -compressed (resp. f -1 -compressed) handlebody of genus g(f ) and A(f

) ⊂ P + (resp. R(f ) ⊂ P -); 2) W s (σ + ) ∩ P + (resp. W u (σ -) ∩ P -) consists of exactly one two- dimensional closed disk for each saddle point σ + ∈ Σ + (resp. σ -∈ Σ -);
3) there is a diffeomorphism q : S g(f ) × [0, 1] → N such that q(S g(f ) × {t}), t ∈ [0, 1] bounds an f -compressed handlebody.

Our assumption makes P + (resp. P -) very close to a nice neighborhood of A(f ) (resp. R(f )) in the sense of 4.2. If we remove from P + a thin neighborhood of the f -compressed union of disks P + ∩ W s (Σ + ), we get a f -compressed domain B + , union of |Ω + | balls, such that ∂B + intersects each separatrix ℓ ∈ L + in one point only. Adding H + to it (as in 4.2), we get a new handlebody, we still denote P + , which is a genuine nice neighborhood of A(f ). We perform a similar change on P -and the complement remains a product. We can construct self-indexing energy functions ϕ + : P + → R and ϕ -: P -→ R as in 4.3 which are constant on their respective boundaries. Let us denote S ± = ∂P ± . It is easy to arrange that ϕ -(S -) > ϕ + (S + ). Finally we can extend ϕ ± to N due to condition 3) of the theorem.

Example

In this section we construct an example of a Morse-Smale diffeomorphism f on M 3 = S 2 × S 1 possessing an energy function and such that M 3 \ (A(f ) ∪ R(f )) is not a product. More precisely we prove next proposition. Proposition 5.1 There exists a Morse-Smale diffeomorphism f : S 2 × S 1 → S 2 × S 1 with the following properties:

1) the non-wandering set Ω(f ) consists of four hyperbolic fixed points: ω is a sink, σ + and σ -are saddles of respective indices 1 and 2, α is a source, hence f is gradient-like diffeomorphism for which

A(f ) = W u (σ + ) ∪ {ω}, R(f ) = W s (σ -) ∪ {α} and g(f ) = 1 (with notation introduced after theorem 2); 2) M 3 \ (A(f ) ∪ R(f )) is not diffeomorphic to the product T 2 × R (here T 2

is the two-dimensional torus) and f satisfies the conditions of theorem 4 (hence it possesses an energy function).

Proof: We first define f + on a 3-ball B + as the homothety centered at ω of ratio 1/2. Let A + be the closure of B + \ f + (B + ); it is a fundamental domain for f + | B + . Let d + 1 , d + 2 be two disjoint disks in ∂B + , with respective centers a + 1 , a + 2 , which are used as attaching disks for a 1-handle

H + ∼ = [-1, +1] × D 2
, where D 2 is a 2-disk. We have to extend f + to P + = B + ∪ H + so that the point {0} × {0} of H + is a hyperbolic fixed point of index 1, with the core of H + as the local unstable manifold and the meridian disk ∆ + = {0}×D 2 of H + as the stable manifold (take for instance f + | ∆ + as being the 1/2-contraction). This extension will be essentially determined once we define the embedding

f + : ([-1, -1 2 ] ∪ [ 1 2 , 1]) × D 2 → A + , that is: a pair of disjoint tubes in A + joining 1 2 d + 1 , 1 2 d + 2 to f + (d + 1 ), f + (d + 2
) respectively. We describe below the cores of these tubes.

For them, we choose a so-called string link C + formed with a pair of disjoint arcs (c + 1 , c + 2 ) in A + , each one joining a + i , i = 1, 2 to its image f + (a + i ). The following properties are required: i) the pair (A + , C + ) is not a product (S 2 \ {x, y}) × [0, 1]; ii) there exists an involution I + : (A + , C + ) → (A + , C + ) permuting both boundary components of A such that I + | ∂B + = f + .

An example of such a string link there is shown on figure 5 for which involution I + is mirror image with respect to middle sphere of A + . By construction, f + is a compression of P + with two hyperbolic fixed points, a sink ω and a saddle σ + of index 1. The unstable manifold of σ + consists of the core of H + 1) The domain W + is not a product T 2 × [0, 1].

2) There is an involution J + : W + → W + which permutes both boundary components such that J + | ∂P + = f + .

Proof:

1) We can see f + (P + ) as the tubular neighborhood of a closed curve κ + in P + which intersects ∆ + , the meridian disk of P + , in one point only, namely σ + . By cutting P + along ∆ + , we get a 3-ball Q + ∼ = B + and a relative knot κ ′+ = κ + ∩ Q + which consists of the union of c + 1 , c + 2 and an unknotted arc in f + (∂B + ) joining f + (a + 1 ) to f + (a + 2 ). If we cut f + (P + ) along ∆ + , we get a tubular neighborhood of κ ′+ . It is easy to prove that condition i) on the chosen string link C + is equivalent to the following i'):

i') there is no embedded disk in Q + whose boundary consists of κ ′+ and one arc in ∂Q + .

Assume that W + is a product. Then there exists a 2-annulus R + with one boundary component in ∂P + and the other consisting of κ + . By usual techniques, the intersection R + ∩ ∆ + can be reduced to an arc joining both boundary components of R + . Thus, cutting R + along ∆ + yields a disk in Q + whose existence is forbidden by i').

2) Let N + be a tubular neighborhood of C + in A + which is invariant by I + . The end fibers of N + consist of the disks d + 1 , d + 2 and their images by f + .

Another description of W + is the following. We remove the interior of N + (that is, the open tubular neighborhood) and, along ∂N + ∼ = S 1 × [0, 1] × {-1, 1}, we glue H ′+ := S 1 ×[0, 1]×[-1, 1]. In this description of W + , the restriction of f + to ∂P + ∩H + is the "identity" of S 1 ×{0}×[-1, 1] → S 1 ×{1}×[-1, 1]. On the other hand, the involution I + , restricted to ∂N + is conjugate to Id| S 1 ×τ , where τ is the standard involution of the interval [0, 1]. Therefore, I + extends to H ′+ as an involution J + which is the "identity" from ∂P + ∩H ′+ ∼ = S 1 ×{0}×[-1, 1] to f + (∂P + ) ∩ H ′+ ∼ = S 1 × {1} × [-1, 1]. Finally J + = f + on ∂P + . ⋄

Now, let us consider the quotient W + /f + and the natural projection p + : W + → W + /f + . By the above construction T + = p + (∆ + ∩ W + ) is a 2-torus. Let V + ∼ = T + × [-1, 1] be a tubular neighborhood of T + in W + /f + and ĥ : W + /f + → W + /f + be a diffeomorphism such that ĥ preserves p + , ĥ = id outside of int V + and ĥ(T + ) ∩ T + = ∅. Then the lift h : W + → W + of ĥ preserves ∂P + , commutes with f + and h(∆ + ∩ W + ) ∩ ∆ + = ∅.

We now finish the construction of our example. We consider a new copy P -of P + . We glue them by h • J + , viewed as a diffeomorphism W -→ W + where W -is the copy of W + in P -. Let f -: P -→ P -be the copy of f + . Hence, our ambient manifold is M 3 = P -∪ h•J + P + and the diffeomorphism f : M 3 → M 3 is defined by f | P + = f + and f | P -= (f -) -1 ; one easily checks that both definitions fit together. Our f is a Morse-Smale diffeomorphism as the unstable manifold of σ -avoids the stable manifold of σ + .

The repeller R(f ) is the attractor of f -, that is the copy of A(f ) in P -. By changing P -into f -1 (P -), then P + and f -1 (P -) are no longer overlaping; they only have a common boundary and the assumptions of theorem 4 are satisfied. This example is the desired one as M 3 \ (A(f ) ∪ R(f )) is not the product T 2 × R. Indeed, if it would be a product, then W + itself should be one, contradicting the preceding lemma. ⋄

Figure 1 :

 1 Figure 1: Pixton's example

  relatively to U , makes it a Morse-Lyapunov function everywhere. ⋄ 3 Necessary conditions to the existence of a self-indexing energy function Theorem 2 If a Morse-Smale diffeomorphism f : M 3 → M 3 has a self-indexing energy function then it is gradient-like and the set L of onedimensional separatrices is almost tamely embedded.

Lemma 3 . 1

 31 If a Morse-Smale diffeomorphism f : M n → M n has a selfindexing energy function ϕ then it is gradient-like.

Figure 2 :

 2 Figure 2: One-dimensional separatrix and self-indexing energy function

Figure 3 :

 3 Figure 3: Construction of a nice neighborhood

Figure 5 :

 5 Figure 5: String link

  this neighborhood is chosen small enough. Hence ϕ is the desired function.

⋄ ¿From lemma 2.2 we deduce the following genericity theorem.

Theorem 1 Among the Lyapunov functions of a Morse-Smale diffeomorphism f those which are Morse-Lyapunov form a residual set in the C ∞ -topology.

Proof: We recall that a property is said generic when it is shared by all points in a residual subset (i.e. a set which is a countable intersection of dense open subsets). Let us show that in the set of Lyapunov functions for f , the set of Morse-Lyapunov functions for f is open and dense; hence, it is residual. Here, our property is clearly open. Let us show it is dense. Let us consider ϕ, a Lyapunov function for the Morse-Smale diffeomorphism f . In some open neighborhood U of P er(f ) take any function ϕ P er(f ) which is a Morse-Lyapunov function of f ; it exists according to lemma 2.2. Let U ⊂ int U be a closed neighborhood of P er

In 1978 C. Conley[START_REF] Conley | Isolated Invariant Sets and Morse Index[END_REF] proved the existence of a continuous Lyapunov function (that is a function which strictly decreases along orbits outside the chain recurrent set and is constant on components of the chain recurrent set) for any flow (or homeomorphism) given on a compact manifold. This fact was named later the Fundamental Theorem of dynamical systems (see, for example,[START_REF] Robinson | Dynamical systems: stability, symbolic dynamics and chaos[END_REF], theorem 1.1, p. 404). Notice, that for Morse-Smale diffeomorphisms the chain recurrent set is exactly the non-wandering set and components of the chain recurrent set are the periodic

orbits.[START_REF] Ch | Classification of Morse-Smale diffeomorphisms with the finite number of heteroclinic orbits on 3-manifolds[END_REF] Let us remark that the two-dimensional invariant manifolds of different saddle points of a gradient-like diffeomorphism may have a non empty intersection, namely along the so-called heteroclinic curves.

The number of negative eigenvalues of the matrix ∂ 2 ϕ ∂xi∂xj (p) is called the index of the critical point p.

A compact set A ⊂ M n is an attractor of a diffeomorphism f : M n → M n if there is a neighborhood V of the set A such that f (V ) ⊂ V and A = n∈N f n (V ). A set R ⊂ M n is called a repeller of f if it is an attractor of f -1 .

It is mainly the proof of proposition 2 in Pixton's article[START_REF] Pixton | Wild unstable manifolds[END_REF], except that our conclusion is slightly stronger than his; we also take advantage of the fact that our energy function is generic.

If ϕ -1 [a, b] is compact and does not contain critical points, then ϕ -1 (-∞, a] is diffeomorphic to ϕ -1 (-∞, b] (see, for example, Theorem 3.1 in[START_REF] Milnor | Morse theory[END_REF]).

A 3-dimensional 1-handle is the product of an interval with a 2-disc. The gluing is made along the top and bottom disks (see Section 3 in[START_REF] Milnor | Morse theory[END_REF]).

It is proved in this chapter that any smooth embedding of a ball into the interior of the standard ball is isotopic to a round ball -a result of J. W. Alexander, 1923. It is also proved that the isotopy extends as an ambient isotopy; hence the claim about the complement of a ball follows.

The λ-lemma claims that f -n (∂B + ) ∩ U σ tends to {x 1 = 0} ∩ U σ in the C 1 topology when n goes to +∞.

By the Mayer-Vietoris exact sequence β 0β 1 = |Ω + | -|Σ + |, where β 0 , β 1 are the Betti numbers of P + . Take account of that β 0 = 1 and β 1 is the genus of P + .

It is well known by topologists that this definition is equivalent to the fact that the inclusion S ֒→ W induces an injection of fundamental groups; but we do not use this deep result.

In[START_REF] Brown | A proof of the generalized Schoenflies theorem[END_REF] M. Brown proved the topological statement. A smooth version of this result is available in[START_REF] Cerf | Sur les difféomorphismes de la sphère de dimension trois (Γ 4 = 0)[END_REF].