Rafael Freitas Reale

Romildo Martins
email: romildo@ifba.edu.br

Da S Bezerra

Joberto Sérgio
email: joberto@unifacs.br

Barbosa Martins

Analysis of Bandwidth Allocation Models Reconfiguration Impacts

Keywords: BAM, Dynamic Network Management, MAM, RDM, AllocTC-Sharing, BAM Reconfiguration

Bandwidth Allocation Models (BAMs) are configured in MPLS/ DS-TE networks by setting Bandwidth Constraints (BCs) for traffic classes. Effectively, BAMs define how bandwidth (resource) is used and shared by network applications. Proposed BAMs like MAM -Maximum Allocation Model, RDM -Russian Dolls Model, G-RDM -Generalized RDM and AllocTC-Sharing aim to optimize the use of bandwidth for links with different bandwidth allocation characteristics and sharing. As such, the adoption of different BAMs and/or changes in the traffic profile may result in different traffic allocation patterns and distinct operational behaviors for the network. This article analyses the impact of BAM reconfiguration in relation to network parameters and characteristics such as link utilization, preemptions and flow blocking for different traffic scenarios. Authors argue that the creation of a mechanism for smoothing the BAMs reconfiguration process may result in benefits for network operation in terms of flow preemption and / or blocking.

Introduction and Motivation

Bandwidth Allocation Models (BAMs) are used by network managers in MPLS / DS-TE networks by configuring Bandwidth Constraints (BCs) for the purpose of defining rules and limits for bandwidth utilization by traffic classes (TCs) (set of applications) (F. Le Faucheur, 2005b) . These models effectively defined as bandwidth (resource) are obtained and shared by applications.

The BAM configuration for models like MAM, RDM, RDM-G and AllocTC-Sharing, among others, depends of a network traffic evaluation process typically done by the network manager. This evaluation process considers aspects such as the set of applications mapped to specific traffic classes (TC), the priorities assigned to each traffic class (TC) and defined application SLAs (Service Level Agreements) and QoS (Quality of Service) parameters.

Once a specific BAM is setup and configured for the network, it will typically have a static and optimized behavior in relation to link utilization, preemptions and blocked LSPs. In brief, configuration assumes a "traffic profile" and results in a "network behavior pattern".

In most cases, the evaluation and selection of BAM to be used in a network is a complex process and network managers often re-evaluate their choice in case of network problem and/or need of further optimization [START_REF] Jennings | Towards autonomic management of communications networks[END_REF] [START_REF] Strassner | An autonomic architecture to manage ubiquitous computing networks and applications[END_REF] [START_REF] Samaan | Towards autonomic network management: an analysis of current and future research directions[END_REF]. As such, the evaluation of bandwidth allocation models configuration could be eventually made by an autonomic framework capable of analyzing the network current state and its requirements (SLA/QoS).Based on this evaluation, the autonomic framework may infer the most appropriate BAM and adequate configuration. A proposal for a framework with autonomic features considering the BAM switching and optimization context presented is discussed in [START_REF] Reale | A Bandwidth Allocation Model Provisioning Framework with Autonomic Characteristics[END_REF].

This paper main motivation is to investigate the impact of BAM on-the-fly bandwidth constraints (BCs) reconfiguration in DS-TE networks. The main focus is to identify advantages, disadvantages and challenges while reconfiguring BAMs under different traffic scenarios.

In order to develop a management analysis scenario it is proposed that two basic management approaches could be used for BC reconfiguration: the Hard Approach and the Soft Approach.

The Hard Approach (HA) management premise is to reconfigure the network mandatorily. In other words, the HA management approach assumes that the network is supposed to be reconfigured immediately, independently of its current state and assumes any consequences to applications and related traffic. As such, switching with HA approach may lead to hurting BAM principles and get negative impacts on network parameters like preemption.

The Soft Approach (SA) management premise is to reconfigure the network smoothly. In doing that, the SA management approach preserves BAM principles and enforces the reduction of negative impacts on network evaluation parameters.

This article is organized as follows: we start with a brief analysis of most currently used and discussed BAMs (MAM, RDM, G-RDM and AllocTC-Sharing) in relation to BCs reconfiguration. Following that, we analyze the Hard and Soft management approaches using simulation, in order to investigate the impacts of on-the-fly BAM reconfiguration.

Bandwidth Allocation Models (BAMs) -A Brief Overview

The state of the art of the bandwidth allocation models is focused on the research for new models with different strategies for allocation and resource sharing [START_REF] Adami | G-RDM: a New Bandwidth Constraints Model for DS-TE networks[END_REF] (Le Faucheur, 2005a) (Le Faucheur, 2005c) [START_REF] Reale | Routing in DS-TE Networks with an Opportunistic Bandwidth Allocation Model[END_REF] [START_REF] Tong | Bandwidth Management for Supporting Differentiated Service Aware Traffic Engineering[END_REF][START_REF] Neto | A RDM-like Bandwidth Management Algorithm for Traffic Engineering with DiffServ and MPLS Support[END_REF]. The resource allocation strategy used by bandwidth allocation models makes use of three types of resource allocation (Figure 1):

1. Private Resource (Private) -In this alternative for resource allocation, the resource is exclusive of a Traffic Class (TC);

2. Sharing "high to low" (high-to-low -HTL) † -In this alternative, the bandwidth allocated to higher priority TCs and not in use can be temporarily allocated to TCs with lower priority; and

3. Sharing "low to high" (low-to-high -LTH) ‡ -In this alternative, the bandwidth allocated to lower priority TCs and not in use can be temporary allocated as loans to higher TCs priority.

The Maximum Allocation Model (MAM) is presented in Le Faucheur (2005c) and [START_REF] Tong | Bandwidth Management for Supporting Differentiated Service Aware Traffic Engineering[END_REF] and, in this model, each traffic class (TC) is configured by the network administrator to use a certain amount of link bandwidth (resource). This resource is allocated on demand to applications belonging to the traffic classes (TCs).

The MAM effectively isolates traffic classes (TC) and there isn't sharing of bandwidth between applications belonging to different classes. As shown in Figure 1, it can be stated that MAM only uses the strategy of allocating private resources.

The Russian Dolls Model (RDM) is presented in Le Faucheur (2005a) and Pinto [START_REF] Neto | A RDM-like Bandwidth Management Algorithm for Traffic Engineering with DiffServ and MPLS Support[END_REF] and, in this model, TCs with the highest values are hierarchically superior to TCs with lower values (priority scheme). The general reasoning involved in the operation of the RDM model is derived from the basic model of priorities used by network applications. In this case, the bandwidth not used by higher priority applications may be temporarily used by lower priority applications.

The RDM is an effective evolution of the MAM model and introduced the first attempt to share resources between TCs and hence LSPs. In effect, the RDM allows the sharing of high-priority applications unused bandwidth by low priority applications. According to previous classification for main resource allocation strategies, the RDM model uses only high-to-low (HTL) sharing.

The G-RDM model is basically a hybrid model combining allocation strategies "high to low" from RDM proposal with private resources strategy proposed by MAM (Le Faucheur, 2005c). The overall operation result is a hybrid model MAM / RDM with a limited amount of resource sharing for applications allocated to different traffic classes.

The AllocTC-Sharing model uses an opportunistic strategy for allocating resources (bandwidth) [START_REF] Reale | Routing in DS-TE Networks with an Opportunistic Bandwidth Allocation Model[END_REF]. AllocTC-Sharing allows two resource allocation strategies simultaneously: sharing high-to-low (HTL) and sharing low-to-high (LTH). The "high-to-low" allocation style is equivalent to the RDM model. The "low-to-high" bandwidth allocation style allows high priority classes temporarily allocate unused bandwidth reserved mainly for low priority classes in the form of loans. In this strategy, one considers the "preemption" less priority applications for priority applications where there is a need for bandwidth by priority applications.

BAM

‡ In this strategy, the concept of "loan" is the temporary use of bandwidth allocated to lower priority TCs by priority LSPs. "Devolution" is the premature closure of a priority LSP established by loan.

Computer networks are mostly dynamic, asymmetric, and multipath and typically have limited resources in terms of bandwidth. Bandwidth Allocation Models focus on resource utilization but, when managed individually, have a static behavior that is accordance with the dynamic, multi-path and asymmetric behavior of most networks. As such it is expected that different BAMs could be used depending on the traffic profile currently applied to the network.

In this section, we address design issues in order to support on-the-fly reconfiguration of BCs and operating results for two management strategies: Soft and Hard Approaches.

The MAM model admits no bandwidth sharing between traffic classes (TC) (neither "high-to-low" nor "low-to-high") and, as such, does not implement any preemption mechanism.

Considering MAM, if a traffic (TC) has available bandwidth and the manager wants to assign it to another TC, there is no design and/or operational impact. It is just a management decision. The reconfiguration process, in turn, may consider distinct approaches:

1. MAM reconfiguration with Soft Approach (SA) -in this case, it should be necessary to create a monitoring mechanism in order to track LSPs setup and closure in order to keep track of the bandwidth constraints and availability. Based on this information, the reconfiguration management process could be executed smoothly and preserve stablished LSPs.

2. MAM reconfiguration with Hard Approach -in this case, LSPs could be preempted abruptly in order to reconfigure the network to a new set of bandwidth constraint.

In both solutions it is required an implementation effort (monitoring and LSP preemption) in order to implement the adopted management approach. It is important to notice that the HA effectively contradicts the MAM basic premise of reserving bandwidth for applications.

The RDM model allows low-priority applications to benefit from available bandwidth not allocated by high priority applications. In this model existing available bandwidth in TCs that will reduce the size of BC will not have any operational impact.

In RDM, low saturated TCs to may reduce their BC bandwidth on behalf of other TCs by using a preemption mechanism in order to make the necessary adjustments (eventual LSP teardown). It is necessary in this case to verify that the internal implementation of the algorithms counters is updated consistently. The main reconfiguration problem occurs when we need to reconfigure a high priority saturated TC in order to assign bandwidth for a low priority TC. In this situation we may have the following management approaches:

1. RDM reconfiguration with Soft Approach -in this case, it is necessary to create a monitoring mechanism in order to keep track of LSPs setup and teardown. A bandwidth availability control is necessary to guarantee any smooth reconfiguration.

2. RDM reconfiguration with Hard Approach -in this case, the existing preemption mechanism can be used to terminate established LSP belonging to the saturated high priority TC. For all preempted LSPs the bandwidth constraint is adjusted and reconfigured.

The hard reconfiguration approach will effectively hurt the RDM model by allowing a low priority LSP force a teardown of high priority LSP. The soft reconfiguration approach, in turn, will smoothly depend on LSPs teardown closing time.

It is important to note that the RDM model reconfiguration differs from the MAM since it only occurs when we need to reduce the bandwidth configured for BC (bandwidth constraints) of a saturated high priority TC.

The AllocTC-Sharing model supports both styles of sharing concomitantly: "high-to-low" and "low-to-high". The AllocTC-sharing enables the use of available bandwidth in both directions. Thus, the AllocTC-sharing model implementation has mechanisms for preemption "high-to-low" and devolution (Preemption "low to high"). In terms of reconfiguration, there are the following approaches for AllocTC-Sharing:

1. AllocTC-Sharing reconfiguration with Hard Approach -In this case, it is not necessary any additional implementation. It is only necessary to check the implementation of the internal counters to ensure bandwidth consistency after reconfiguration of BCs.

2. AllocTC-Sharing reconfiguration with Soft Approach -In this case, it is necessary to create a monitoring and control mechanism in order to keep track of LSPs setup and teardown and to adjust bandwidth constraint availability.

As an operational effect of the reconfiguration with Hard Approach, we can have a high priority LSP being teardown or being counted as a lower loan TC and, as such, it loses its intrinsic priority with reconfiguration.

In summary, the management of bandwidth constraints reconfiguration for allocation models MAM, RDM and AllocTC-Sharing occurs transparently when available TC bandwidth is transferred among TCs regardless of its priority. Major BAM reconfiguration issues can occur when a saturated TC is required to provide extra bandwidth to any other TC.

In the three allocation models evaluated (MAM, RDM and AllocTC-Sharing), the reconfiguration with Hard Approach may imply in unwanted operating results which corresponds in effect to have LSPs being abruptly teardown. In MAM and RDM models, the reconfiguration with Hard Approach requires additional implementation (monitoring and control preemption facility) in relation to models actual implementation.

The reconfiguration with Soft Approach requires a management plan to coordinate the reconfiguration. This plan makes adjustments gradually observing the teardown of LSPs that release resources (bandwidth). The operational effects of this approach point to a better preservation of the operational characteristics of the BAMs, but need some execution time in order to converge to a new reconfiguration pattern of the BAM considered.

We argue that the two approaches (HA and SA) are important for most networks (having distinct traffic profiles) and should be implemented in order to guarantee the choice between them is a managerial issue.

In the following section, it will be presented the evaluation of the Hard Approach vs. the Soft Approach for the RDM model using a traffic scenario where a saturated high priority TC needs to be reconfigured in order to release bandwidth for a of lower priority TC (worst case).

BAM Reconfiguration -Evaluation and Analysis

In this section, BAM reconfiguration is evaluated with the main objective of exemplifying and validating different operational behaviors discussed for the reconfiguration with Hard Approach and Reconfiguration with Soft Approach.

In terms of evaluation scenario, the RDM model was chosen since it is largely referenced in the literature and also because it's operational behavior can be mapped to other models.

The network topology used intends to demonstrate the discussed limiting traffic scenarios and included a traffic source (S1) and a destination (D) interconnected through a link of 622 Mbps with two classes of traffic (Figure 2.): TC0 -low priority applications and TC1 -high priority applications. Bandwidth Constraints (BCs) are reconfigured in three phases (Table 2):

1. In the first phase, simulation starts with high priority LSPs (TC1) BC1 set to 60% of the link (372.20 Mbps) and the network is saturated and stabilized with this configuration.

2. In the second phase, simulation starts with the network saturated and after 450 LSPs being created, BC1 is reduced by 20% leading to a TC1 available bandwidth of 248.8 Mbps (40%). The overall scenario was simulated with 05 random seeds and the results are presented as the mean value obtained (confidence interval 95%) [START_REF] Macdougall | Simulating Computer Systems Techniques and Tools[END_REF].

Finally in phase

The parameters evaluated in the simulation were: link utilization (by TC and link), preemptions and LSP blocking.

Simulated Characteristics of RDM (BAM) Reconfiguration

The main difference between the Hard and the Soft management approaches is reflected in the number of preemptions resulting from RDM reconfigurations. As can be seen in Figure 3, when the Hard Approach is adopted, the number of preemptions increases significantly reconfiguration simulated by phase changes. This is due to the fact that LSPs that exceed TC1 BC are terminated immediately (abruptly). The Soft Approach leads to a smoother operational behavior of the network and results in having a significantly smaller number of preemptions for TC1 (Figure 5). TC0 behavior is conventional or, in other words, RDM-like (Figure 4).

Figure 1 :

 1 Figure 1: BAMs allocation strategy

Figure 2 :

 2 Figure 2: Network Topology used in the simulation In terms of the simulation, we force the saturation of the network by making LSPs setups to exceed I number LSPs teardowns.t As such, simulation will lead to a scenario of high saturation and competition between TCs: 1. LSPs arrival requests: modeled exponentially with a mean of 2 s; 2. LSPs duration: modeled exponentially -average of 250 seconds, causing saturation of the link; 3. LSP band: evenly distributed between 5 Mbps and 15 Mbps; 4. Stop Criteria -1000 LSPs.

Figure 3 :Figure 4 :

 34 Figure 3: Number of Preemptions (total)

Figure 5 :

 5 Figure 5: Link utilization Figure 6 illustrates link load by TCs. The reconfiguration with Hard Approach immediately reallocates TC0 bandwidth for all phases. It can also be observed the immediate drop in TC1 use and TC0 immediate growth. In the reconfiguration with Soft Approach, the bandwidth availability for TC1 and TC0 is adjusted gradually.

Table 1 :

 1 BAMs operating characteristics (MAM, RDM and ALLOCTC -SHARING) †

	Operational	MAM	RDM AllocTC-
	Characteristics			Sharing
	Sharing from "high to low"	No	Yes	Yes
	Sharing from "low to high"	No	No	Yes
	Bandwidth utilization with high volume of traffic (low priority)	Low	High	High
	Bandwidth utilization with high volume of traffic (high priority)	Low	low	High
	Isolation between TCs	High	Medium	Low

Table 2 :

 2 Simulation phases -bandwidth constraints (BCs) by traffic class (TC)

* We gratefully acknowledge financial support from FAPESB (Fundo de Amparo à Pesquisa do Estado da Bahia -Brazil) to this project.

Table 3 presents a summary of preemptions, blocking and link utilization for SA and HA reconfiguration approaches. In general, the Hard Approach results in a greater number of preemptions for TC1 but, in turn, results in a smaller number of blockings and greater average link utilization.

The reconfiguration Soft Approach respects the intrinsic priority of the LSPs (TC1) by waiting LSPs teardown. The number of preemptions is reduced and is slightly greater. Network managers may adopt different BAMs as a possible management response to changes in their network traffic profile. As discussed in section III of this paper, distinct BAMs allocation behaviors will lead to different traffic allocation patterns with distinct operational parameters such as preemption, link utilization and LSP (flow) blocking. Considering then that managers will have to switching among BAMs, this paper has analyzed and evaluated the impact of these parameters (behavior) during BAMs reconfiguration process.

It was assumed two possible management approaches in order to facilitate the analyses of the network parameters behavior. In general, it was indicated that the reconfiguration process with Hard Approach, necessary in networks needing rapid response to network traffic changes will potentially impact applications by increasing the number of preemptions.

In turn, the reconfiguration process with Soft Approach leads to a smoother impact in terms of preemptions and does maintain the normal characteristics of the BAM. SA main drawback is the required convergence time that is directly dependent on the LSP teardown procedure.

Both reconfiguration approaches do require an additional level of intervention in the network beyond the BAM implementation itself (monitoring and action facilities).


Olga Bernardi

Anna Florio

Existence of Lipschitz continuous Lyapunov functions strict outside the strong chain recurrent set

The aim of this paper is to study in detail the relations between strong chain recurrence for flows and Lyapunov functions. For a continuous flow on a compact metric space, uniformly Lipschitz continuous on the compact subsets of the time, we first make explicit a Lipschitz continuous Lyapunov function strict -that is strictly decreasing-outside the strong chain recurrent set of the flow. This construction extends to flows some recent advances of Fathi and Pageault in the case of homeomorphisms; moreover, it improves Conley's result about the existence of a continuous Lyapunov function strictly decreasing outside the chain recurrent set of a continuous flow. We then present two consequences of this theorem. From one hand, we characterize the strong chain recurrent set in terms of Lipschitz continuous Lyapunov functions. From the other hand, in the case of a flow induced by a vector field, we establish a sufficient condition for the existence of a C 1,1 strict Lyapunov function and we also discuss various examples. Moreover, for general continuos flows, we show that the strong chain recurrent set has only one strong chain transitive component if and only if the only Lipschitz continuous Lyapunov functions are the constants. Finally, we provide a necessary and sufficient condition to guarantee that the strong chain recurrent set and the chain recurrent one coincide.

2 The function L T : definition and properties

Introduction

Let φ = {φ t } t∈R be a continuous flow on a compact metric space (X, d). In the paper [START_REF] Easton | Chain transitivity and the domain of influence of an invariant set[END_REF][Page 96], Robert Easton introduced the notion of strong chain recurrence.

DEFINITION. (Strong chain recurrence)

(i) Given x, y ∈ X, ε > 0 and T > 0, a strong (ε, T )-chain from x to y is a finite sequence (x i , t i ) i=1,...,n ⊂ X × R such that t i ≥ T for all i, x 1 = x and setting x n+1 = y we have

n i=1 d(φ t i (x i ), x i+1 ) < ε. (1)
(ii) A point x ∈ X is said to be strong chain recurrent if for all ε > 0 and T > 0 there exists a strong (ε, T )-chain from x to x. The set of strong chain recurrent points is denoted by SCR(φ).

(iii) The points x, y ∈ SCR(φ) belong to the same strong chain transitive component of SCR(φ) if for any ε > 0 and T > 0 there exist a strong (ε, T )-chain from x to y and a strong (ε, T )-chain from y to x.

This notion sharpens the one of chain recurrence, in which it is only required

d(φ t i (x i ), x i+1 ) < ε (2)
for all i. Chain recurrent sets are both easily seen to be closed and invariant, see e.g. [START_REF] Akin | The general topology of dynamical systems[END_REF][Pages 12-13, 71 and 109-110], [START_REF]The gradient structure of a flow. I[END_REF][Theorem 3.3B], [START_REF] Zheng | Attractor, chain recurrent set and limit set of flow[END_REF][Page 248] and [START_REF] Bernardi | A Conley-type decomposition of the strong chain recurrent set, Ergodic Theory and Dynamical Systems[END_REF][Proposition 3 .1]. Furthermore, strong chain recurrent set (strictly) contains the set of non-wandering points and it is (strictly) contained in the chain recurrent set, that is N W(φ) ⊆ SCR(φ) ⊆ CR(φ), see for example [START_REF] Zheng | Attractor, chain recurrent set and limit set of flow[END_REF][ Theorem 2.4] and [START_REF]The gradient structure of a flow. I[END_REF][Theorem 3.3B].

The study of the intimate relations between chain recurrent sets and Lyapunov functions comes from the celebrated paper [START_REF] Conley | Isolated invariant sets and the Morse index[END_REF] by Charles Conley and has had recent important advances by Albert Fathi and Pierre Pageault in [START_REF] Fathi | Aubry-Mather theory for homeomorphisms[END_REF] and [START_REF] Pageault | Lyapunov functions: a weak KAM approach[END_REF]. Before recalling these results, we need to premise the notions of Lyapunov function, neutral set and first integral.

DEFINITION. (Lyapunov function, neutral set and first integral)

A function f : X → R is a Lyapunov function for φ if f • φ t ≤ f for every t ≥ 0. The neutral set of a Lyapunov function f is

N (f ) := {x ∈ X : ∃t > 0 such that f (φ t (x)) = f (x)}. A function f : X → R is a first integral for φ if f • φ t = f for every t ∈ R.
In particular, f is a first integral if and only if N (f ) = X. We refer to Lemma 1.5 in [START_REF] Abbondandolo | Chain recurrence, chain transitivity, Lyapunov functions and rigidity of Lagrangian submanifolds of optical hypersurfaces[END_REF] for a characterization of Lyapunov functions and first integrals in the case of a flow induced by a locally Lipschitz continuous vector field.

In the seminal paper [START_REF] Conley | Isolated invariant sets and the Morse index[END_REF], Conley described the structure of the chain recurrent set in terms of attractors and their "complementary repellers" and -as an outcome-he proved the theorem below. Some authors refer to this result as the "fundamental theorem of dynamical systems", see e.g. [START_REF] Norton | The fundamental theorem of dynamical systems[END_REF] and [START_REF] Franks | Notes on Chain Recurrence and Lyapunov Functions[END_REF] for an instructive treatment of the matter. THEOREM. [START_REF] Conley | Isolated invariant sets and the Morse index[END_REF] Let φ : X × R → X be a continuous flow on a compact metric space. Then there exists a continuous Lyapunov function u : X → R for φ such that (i) N (u) = CR(φ).

(ii) If x, y ∈ CR(φ), then u(x) = u(y) if and only if x and y belong to the same chain transitive component of CR(φ).

(iii) u(CR(φ)) is a compact nowhere dense subset of R.

Conley called u complete Lyapunov function for φ, see Definition 5.2 of Section 4. In particular, point (i) of the previous theorem is an either/or statement about what can happen: parts of the space are either chain recurrent or "gradient-like". In the same year -see [START_REF] Easton | Chain transitivity and the domain of influence of an invariant set[END_REF][Proposition 3]-Easton connected the notion of strong chain recurrence to the property for the corresponding Lipschitz first integrals to be constants. Easton's contribution represented the first step towards the study of the relations between strong chain recurrence and Lyapunov functions. For a homeomorphism g on a compact metric space, Fathi and Pageault in [START_REF] Fathi | Aubry-Mather theory for homeomorphisms[END_REF] and [START_REF] Pageault | Lyapunov functions: a weak KAM approach[END_REF] presented a new variational point of view to face the study of recurrent sets and Lyapunov functions. Their techniques are very different from Conley's original ones and are inspired by Fathi's work in weak KAM theory, see [START_REF] Fathi | The Weak KAM Theorem in Lagrangian Dynamics[END_REF]. Indeed, thanks to the arbitrariness of the parameter ε > 0 involved in both the definitions of recurrence, Fathi and Pageault equivalently described recurrent points as minima of appropriate functionals defined on the space of finite sequences of points. In particular, bearing in mind formulae (1) and ( 2), for strong chain recurrent points the functional is the sum of the amplitudes of the jumps; for chain recurrent points, the functional is the maximum of the amplitudes of the jumps. For a discrete dynamical system, if u is a Lyapunov function for g, that is u • g ≤ u, the neutral set of u is

N (u) := {x ∈ X : u(g(x)) = u(x)}.
We remind two of their fundamental results. On one hand, they provided a new proof of the above Conley's theorem in the framework of discrete-time dynamical systems. On the other hand, they proved the next THEOREM. [START_REF] Fathi | Aubry-Mather theory for homeomorphisms[END_REF]. Let g : X → X be a homeomorphism on a compact metric space. Then (i) There exists a Lipschitz continuous Lyapunov function u for g such that N (u) = SCR(g).

(ii) SCR(g) = N (f ), where the intersection is taken over all Lipschitz continuous Lyapunov functions f for g.

(iii) SCR(g) has a unique strong chain transitive component if and only if the only Lipschitz continuous Lyapunov functions for g are the constants.

We refer respectively to Theorem 4.4, Corollary 4.5 and Theorem 4.8 in [START_REF] Pageault | Lyapunov functions: a weak KAM approach[END_REF] for the proofs of the points (i), (ii) and (iii) above. More recently, an adaptation of Fathi and Pageault's techniques has led to the proof of point (ii) of the previous theorem also for a flow which is Lipschitz continuous for every t ≥ 0, uniformly for t on compact subsets of [0, +∞), see [START_REF] Abbondandolo | Chain recurrence, chain transitivity, Lyapunov functions and rigidity of Lagrangian submanifolds of optical hypersurfaces[END_REF][Theorem 2.2]. However, in [START_REF] Abbondandolo | Chain recurrence, chain transitivity, Lyapunov functions and rigidity of Lagrangian submanifolds of optical hypersurfaces[END_REF] it is not constructed a single Lyapunov function whose neutral set coincides with the strong chain recurrent set. We finally underline that, more recently, Fathi and Pageault in [?] give a criterion for the approximation of a Lyapunov function by a smooth one and also show that it is possible to obtain a smooth version of Conley's fundamental theorem for flows. This paper intends to examine in depth the relations between strong chain recurrence and Lyapunov functions in the case of flows. First, we prove this improvement of point (i) of Conley's original result: THEOREM 1. Let φ : X × R → X be a continuous flow on a compact metric space (X, d), uniformly Lipschitz continuous on the compact subsets of [0, +∞). Then there exists a Lipschitz continuous Lyapunov function u : X → R for φ such that

N (u) = SCR(φ).
The proof of this theorem combines together variational and dynamical methods. In particular, we start by constructing -by accurately adapting Fathi and Pageault's techniques-a function which strictly decreases along the flow out of a closed set containing SCR(φ) and definitively (that is, for t ≥ T ) in time. We refer to Proposition 3.1 for the precise statement. Thereafter, by using some ideas coming from the original proof of Conley's theorem, we modify this function in order to obtain the desired Lyapunov function, that is strictly decreasing outside SCR(φ) for any t > 0. See Lemma 4.1, Lemma 4.2 and the consequent Theorem 4.1. We proceed by discussing the main consequences of the above theorem. From one hand, we give an alternative -i.e. constructive-proof of Theorem 2.2 in [START_REF] Abbondandolo | Chain recurrence, chain transitivity, Lyapunov functions and rigidity of Lagrangian submanifolds of optical hypersurfaces[END_REF]: COROLLARY 1. Let φ : X × R → X be a continuous flow on a compact metric space (X, d), uniformly Lipschitz continuous on the compact subsets of [0, +∞). Then

SCR(φ) = N (f ).
where f is any Lipschitz continuous Lyapunov function for φ.

From the other hand, we discuss a sufficient condition in order to establish the existence of a C 1,1 strict Lyapunov function for a flow induced by a vector field: COROLLARY 2. Let (M, g) be a C ∞ closed connected Riemannian manifold. Let V : M → T M be a C k vector field, k ≥ 2, inducing the flow φ. If SCR(φ) = CR(φ) then there exists a C 1,1 Lyapunov function u : M → R for φ such that N (u) = SCR(φ).

See respectively Corollary 4.1 and Corollary 4.2. In particular, the proof of the second corollary uses existence and regularity results for sub-solutions of the so-called Mañé Hamiltonians. We then pass to discuss some cases -Examples 4.1 and 4.2-where the hypothesis SCR(φ) = CR(φ) of this corollary is not satisfied and any sub-solutions of weak KAM theory provide first integrals, while the Lyapunov function previously constructed is not a first integral. We also remark that the converse implication of this corollary does not in general hold true.

To be thorough, in Definition 5.2 we remind the milder notion of pseudo-complete Lyapunov function and in Proposition 5.3 we prove a necessary and sufficient condition for the strong chain recurrent and the chain recurrent sets to be equal:

PROPOSITION. Let φ : X × R → X be a continuous flow on a compact metric space (X, d).

Then, SCR(φ) = CR(φ) if and only if there exists a pseudo-complete Lyapunov function u : X → R for φ such that u(SCR(φ)) is totally disconnected.

We finally analyze the relations between Lipschitz continuous Lyapunov functions and the strong chain transitive components of SCR(φ). As a final outcome -see Theorem 5.1-we show that point (iii) of Fathi and Pageault's theorem still holds in the case of a continuous flow: THEOREM 2. Let φ : X × R → X be a continuous flow on a compact metric space (X, d). SCR(φ) has a unique strong chain transitive component if and only if the only Lipschitz continuous Lyapunov functions for φ are the constants.

Let φ : X × R → X be a continuous flow on a compact metric space (X, d). The next definitions are the continuous-time versions of the ones introduced by Fathi and Pageault for homeomorphisms, see [START_REF] Pageault | Lyapunov functions: a weak KAM approach[END_REF][Chapter 2, Section 3] and [START_REF] Fathi | Aubry-Mather theory for homeomorphisms[END_REF][Section 2.1]. We also underline that their settings have recently been extended by Ethan Akin and Jim Wiseman in [START_REF] Akin | Chain recurrence for general spaces[END_REF] both to relations and to uniform spaces. Let x, y ∈ X. For any T > 0, we indicate by

C T (x, y) the set of chains C = (x i , t i ) i=1,...,n ⊂ X × R from x 1 = x to x n+1 = y such that t i ≥ T for all i. The cost of going from x to y through a chain C ∈ C T (x, y) is given by l T (C) := n i=1 d(φ t i (x i ), x i+1 ).
Moreover, for any T > 0, we define the non-negative function

L T : X × X → [0, +∞) L T (x, y) := inf{l T (C) : C ∈ C T (x, y)}. (3)
In the next proposition, we summarize some useful facts about the function L T . We refer to [START_REF] Pageault | Lyapunov functions: a weak KAM approach[END_REF][Proposition 3.1] and [START_REF] Fathi | Aubry-Mather theory for homeomorphisms[END_REF][Proposition 2.1] for analogous results in the case of a homeomorphism.

Proposition 2.1. Let φ : X × R → X be a continuous flow on a compact metric space (X, d).

For any fixed T > 0, the following properties hold:

(i) For any x, y, z ∈ X, L T (x, y) ≤ L T (x, z) + L T (z, y). (ii) For any x ∈ X, L T (x, φ t (x)) = 0 ∀t ≥ T. (iii) For any x, y, z ∈ X, |L T (x, y) -L T (x, z)| ≤ d(y, z).
As a consequence, for any fixed x ∈ X, the function z → L T (x, z) is 1-Lipschitz continuous.

(iv) For any fixed x ∈ X, the function z → L T (z, x) is upper semicontinuous.

Proof. For points (i), (ii) and (iii) we refer to Lemma 2.4 in [START_REF] Abbondandolo | Chain recurrence, chain transitivity, Lyapunov functions and rigidity of Lagrangian submanifolds of optical hypersurfaces[END_REF].

(iv) By definition of L T , for any z ∈ X and ε > 0, from the definition of infimum, there exists a chain

C ε ∈ C T (z, x) such that l T (C ε ) < L T (z, x) + ε 2 .
Let t 1 = t 1 (z) ≥ T be the first time of the chain C ε . Since φ t 1 is a homeomorphism, for any ε > 0 there exists δ = δ(ε) > 0 so that

d(z, y) < δ ⇒ d(φ t 1 (z), φ t 1 (y)) < ε 2 .
Let us consider the chain Cε ∈ C T (y, x) obtained from C ε by substituting the first point with y. Hence, if d(z, y) < δ, we have that

L T (y, x) ≤ l T ( Cε ) ≤ d(φ t 1 (z), φ t 1 (y)) + l T (C ε ) < ε + L T (z, x).
For any T > 0, we define the subset A T of X as

A T := {x ∈ X : L T (x, x) = 0}. (4)
Since C T (x, y) is contained in C T (x, y) when T ≥ T , the function T → L T (x, y) is monotonically increasing for any pair (x, y) ∈ X × X and A T ⊆ A T . We notice that

SCR(φ) = T >0 A T . (5)
The next proposition characterizes the points of the strong chain recurrent set in terms of the functions L T .

Proposition 2.2. Let φ : X × R → X be a continuous flow on a compact metric space (X, d).

(i) If x ∈ SCR(φ), then L T (φ t (x), x) = L T (x, φ t (x)) = 0 for any T > 0 and t ∈ R.
(ii) If for any T > 0 there exists a time

t = t(T ) ∈ R such that L T (φ t (x), x) = L T (x, φ t (x)) = 0, then x ∈ SCR(φ).
In particular, if x ∈ SCR(φ) then x and φ t (x) belong to the same strong chain transitive component of SCR(φ) for any t ∈ R.

Proof. (i) Let x ∈ SCR(φ), T > 0 and t ∈ R be fixed and take N = N (T, t) ∈ N such that t ≤ N T . Since x ∈ SCR(φ), L (N +1)T (x, x) = 0. This means that for any ε > 0 there exists a chain

C ε = (x i , t i ) i=1,...,n ∈ C (N +1)T (x, x) such that l (N +1)T (C ε ) ≤ ε.
Let consider the following chain Cε ∈ C T (φ t (x), x) obtained from C ε :

Cε = ((φ t (x), t 1 -t), (x 2 , t 2 ), . . . , (x n , t n )) .
Then it clearly holds that

0 ≤ L T (φ t (x), x) ≤ l T ( Cε ) = l (N +1)T (C ε ) ≤ ε.
From the arbitrariness of ε > 0, we deduce that L T (φ t (x), x) = 0. Since the above argument can be repeated for any fixed x ∈ SCR(φ), T > 0 and t ∈ R, we conclude that if x ∈ SCR(φ) then L T (φ t (x), x) = 0 for any T > 0 and t ∈ R.

For a fixed x ∈ SCR(φ), T > 0 and t ∈ R, we now prove the other equality. If t ≥ T , the fact that L T (x, φ t (x)) = 0 corresponds exactly to property (ii) of Proposition 2.1. Consequently, let t < T and choose N = N (T, t) ∈ N such that T < N |t|. Moreover, observe that by the invariance of the strong chain recurrent set, φ t (x) ∈ SCR(φ). This means that, for any ε > 0 there exists a chain

C ε = (x i , t i ) i=1,...,n ∈ C (N +1)|t| (φ t (x), φ t (x)) such that l (N +1)|t| (C ε ) ≤ ε.

Let us consider the following chain

Cε ∈ C T (x, φ t (x)) obtained from C ε : Cε = ((x, t 1 + t), (x 2 , t 2 ), . . . , (x n , t n )) , so that 0 ≤ L T (x, φ t (x)) ≤ l T ( Cε ) = l (N +1)|t| (C ε ) ≤ ε.
From the arbitrariness of ε > 0, we conclude that L T (x, φ t (x)) = 0. Since the above argument holds for any fixed x ∈ SCR(φ), T > 0 and t < T , we conclude that if x ∈ SCR(φ) then L T (x, φ t (x)) = 0 for any T > 0 and t ∈ R. (ii) Conversely, for a point x ∈ X, let assume that for any T > 0 there exists a time t = t(T ) ∈ R such that

L T (φ t (x), x) = L T (x, φ t (x)) = 0
Thanks to property (i) of Proposition 2.1, we have that

0 ≤ L T (x, x) ≤ L T (x, φ t (x)) + L T (φ t (x), x) = 0.
From the arbitrariness of T > 0, we deduce that x ∈ SCR(φ).

Construction of Lipschitz Lyapunov functions definitively strict

Through the whole Sections 3 and 4, we assume that φ : X × R → X is a continuous flow on a compact metric space, uniformly Lipschitz continuous on every compact subset of [0, +∞). This means that for any T > 0 there exists M T > 0 such that

d(φ t (x), φ t (y)) ≤ M T d(x, y) ∀t ∈ [0, T ]. (6)
The above Lipschitz regularity assumption is surely satisfied by the flow of a Lipschitz continuous vector field on a compact manifold. Moreover, we precise that we can make a measurable choice of the function s → M s , for instance

M s := max sup t∈[0,s], x =y d(φ t (x), φ t (y)) d(x, y) , 1 . (7)
Let K := {x 1 , . . . , x j , . . .} be a countable dense subset of X; such a set exists since X is a compact metric space. For any T > 0, we define the function

u T : X → R u T (x) := j∈N 1 2 j L T (x j , x). (8)
We underline that this technique is the same of Fathi and Pageault, see [START_REF] Fathi | Aubry-Mather theory for homeomorphisms[END_REF][Proposition 2.3] and [START_REF] Pageault | Lyapunov functions: a weak KAM approach[END_REF][Chapter 2, Proposition 4.1]. Clearly, the function u T is bounded. Indeed, for any x ∈ X, it holds that

|u T (x)| ≤ j∈N 1 2 j L T (x j , x) ≤ j∈N 1 2 j d(φ T (x j ), x) ≤ 2 diam(X)
where diam(X) = max{d(x, y) : x, y ∈ X} < +∞. In the next proposition we summarize the properties of u T .

Proposition 3.1. Let φ : X × R → X be a continuous flow on a compact metric space (X, d), uniformly Lipschitz continuous on compact subsets of [0, +∞). For any fixed T > 0, the following properties hold:

(i) u T is 2-Lipschitz continuous.
(ii) u T is definitively a Lyapunov function for φ, that is

u T (φ t (x)) ≤ u T (x)
for any x ∈ X and t ≥ T.

(iii) u T is definitively strict outside A T 2 , that is u T (φ t (x)) < u T (x) for any x ∈ X \ A T 2
and t ≥ T.

Proof. (i) Thanks to property (iii) of Proposition 2.1, for any x, y ∈ X it holds that

|u T (y) -u T (x)| ≤ j∈N 1 2 j |L T (x j , y) -L T (x j , x)| ≤ 2d(x, y)
(ii) By point (i) of Proposition 2.1, for any x, y ∈ X we have that

u T (y) -u T (x) = j∈N 1 2 j (L T (x j , y) -L T (x j , x)) ≤ 2L T (x, y)
For y = φ t (x) and t ≥ T , we obtain

u T (φ t (x)) -u T (x) ≤ 2L T (x, φ t (x)) = 0
where -in the last equality-we use property (ii) of Proposition 2.1.

(iii) Arguing by contradiction, let us suppose that u T is not definitively strict outside A T

2

. This means that there exist z ∈ X \ A T 2 and s ≥ T such that

u T (φ s (z)) -u T (z) = j∈N 1 2 j (L T (x j , φ s (z)) -L T (x j , z)) = 0. (9)
Since from properties (i) and (ii) of Proposition 2.1, it holds

L T (x j , φ t (x)) -L T (x j , x) ≤ 0 ∀x ∈ X, j ∈ N and t ≥ T, hypothesis (9)
equals to

L T (x j , φ s (z)) -L T (x j , z) = 0 ∀j ∈ N. (10)
Let m ∈ N be fixed. In the sequel we prove that there exists x j(m) ∈ K such that

L T (x j(m) , φ s (z)) < 1 m . ( 11
)
Notice that, since K is a dense sequence in X, it is always possible to take x j(m) ∈ K such that

d(x j(m) , z) < 1 mM 2T
. By hypothesis [START_REF] Bernardi | A Conley-type decomposition of the strong chain recurrent set, Ergodic Theory and Dynamical Systems[END_REF], this implies

d(φ t (x j(m) ), φ t (z)) < 1 m ∀t ∈ [0, 2T ]. (12)
We now explicitly exhibit a chain C ∈ C T (x j(m) , φ s (z)) such that

l T (C) < 1 m .
From one hand, if s ∈ [T, 2T ] then C = (x j(m) , s). Indeed, by [START_REF] Fathi | Aubry-Mather theory for homeomorphisms[END_REF], we have

l T (C) = d(φ s (x j(m) ), φ s (z)) < 1 m .
From the other hand, if s > 2T , let consider the chain C ∈ C T (x j(m) , φ s (z)) defined as follows:

C = (x j(m) , T ), (φ T (z), s -T ) .
By [START_REF] Fathi | Aubry-Mather theory for homeomorphisms[END_REF] again, it holds [START_REF] Fathi | On the Hausdorff dimension of the Mather quotient[END_REF] immediately follows. Consequently -see (10)-we obtain:

l T (C) = d(φ T (x j(m) ), φ T (z)) + d(φ s (z), φ s (z)) < 1 m + 0 = 1 m . Since L T (x j(m) , φ s (z)) ≤ l T (C) for any C ∈ C T (x j(m) , φ s (z)), inequality (
L T (x j(m) , z) < 1 m . ( 13
)
By using previous inequality, we are now going to conclude that L T 2 (z, z) = 0. Given an arbitrary chain C ∈ C T (x j(m) , z) from x j(m) to z: C = (x j(m) , t 1 ), (y 2 , t 2 ), . . . , (y n , t n ) , let us consider the chain C ∈ C T 2 (z, z) obtained by C and defined as follows:

C = z, T 2 , φ T 2 (x j(m) ), t 1 - T 2
, (y 2 , t 2 ) , . . . , (y n , t n ) .

By [START_REF] Fathi | Aubry-Mather theory for homeomorphisms[END_REF], we have

L T 2 (z, z) ≤ l T 2 ( C) = d(φ T 2 (z), φ T 2 (x j(m) )) + d(φ t 1 (x j(m) ), y 2 ) + • • • + d(φ tn (y n ), z) < 1 m + l T (C).
Since L T (x j(m) , z) < 1 m , by taking the infimum over all possible chains in C T (x j(m) , z), we have

L T 2 (z, z) ≤ 1 m + 1 m = 2 m .
From the arbitrariness of m ∈ N, we obtain

L T 2 (z, z) = 0. Since z ∈ X \ A T 2
, previous equality gives the desired contradiction.

Existence of Lipschitz Lyapunov functions with SCR(φ) as neutral set

For a continuous flow φ : X × R → X satisfying hypothesis (6), this section is devoted to prove the existence of a Lipschitz continuous Lyapunov function which is strict outside SCR(φ). The proof -see Theorem 4.1-is preceded by two technical lemmas aiming to overcome the fact that the function (8) constructed in Proposition 3.1 is only definitively a strict Lyapunov function for φ. We then proceed to discuss some consequences of Theorem 4.1 and to give various examples. For any T > 0, we start by defining

ũT : X → R ũT (x) := max s∈[0,T ] u T (φ s (x)) (14)
and

ūT : X → R ūT (x) := 1 M T +∞ 0 e -s M s ũT (φ s (x))ds. ( 15
)
where the function s → M s has been defined in [START_REF] Conley | Isolated invariant sets and the Morse index[END_REF]. We remark that the above definitions are inspired by the ones given by Conley in the proof of the so-called fundamental theorem of dynamical systems, see [START_REF] Conley | Isolated invariant sets and the Morse index[END_REF][Chapter II, Section 5, Page 33] and [Chapter II, Section 6, Page 39].

The main properties of functions ( 14) and ( 15) are presented in the next two lemmas.

Lemma 4.1. Let φ : X × R → X be a continuous flow on a compact metric space (X, d), uniformly Lipschitz continuous on compact subsets of [0, +∞). For any fixed T > 0, the following properties hold:

(i) ũT is 2M T -Lipschitz continuous.
(ii) ũT is a Lyapunov function for φ.

(iii) ũT is definitively strict outside A T

2

, that is ũT (φ t (x)) < ũT (x) for any x ∈ X \ A T 2 and t ≥ T.

Proof. Thanks to property (i) of Proposition 3.1 and assumption (6) on the flow, for any x, y ∈ X ũT (y) -ũT (x) = max

s∈[0,T ] u T (φ s (y)) -max s∈[0,T ] u T (φ s (x)) ≤ max s∈[0,T ] (u T (φ s (y)) -u T (φ s (x))) ≤ ≤ 2 max s∈[0,T ] d(φ s (y), φ s (x)) ≤ 2M T d(x, y).
We conclude by exchanging the role of x and y.

(ii) By property (ii) of Proposition 3.1, we know that

u T (φ τ (x)) ≤ max s∈[0,T ] u T (φ s (x))
∀x ∈ X and τ ≥ 0.

By taking the maximum for τ ∈ [t, T + t], we obtain

max τ ∈[t,T +t] u T (φ τ (x)) ≤ max s∈[0,T ] u T (φ s (x)) = ũT (x) Since now max τ ∈[t,T +t] u T (φ τ (x)) = max s∈[0,T ] u T (φ s+t (x)) = ũT (φ t (x)),
the thesis immediately follows.

(iii) By property (iii) of Proposition 3.1, for any x ∈ X \ A T 2 and t ≥ T , we have

ũT (φ t (x)) = max s∈[0,T ] u T (φ s+t (x)) < u T (x) ≤ max s∈[0,T ] u T (φ s (x)) = ũT (x).
Lemma 4.2. Let φ : X × R → X be a continuous flow on a compact metric space (X, d), uniformly Lipschitz continuous on compact subsets of [0, +∞). For any fixed T > 0, the following properties hold:

(i) ūT is 2-Lipschitz continuous.
(ii) ūT is a Lyapunov function for φ.

(iii) ūT is strict outside A T 2 for any t > 0.

Proof. (i) By property (i) of Lemma 4.1 and assumption (6) on the flow, we immediately obtain that for all x, y ∈ X

ūT (y)-ū T (x) = 1 M T +∞ 0 e -s M s (ũ T (φ s (y)) -ũT (φ s (x))) ds ≤ 2 +∞ 0 e -s M s d(φ s (y), φ s (x))ds ≤ 2d(y, x).
We conclude by exchanging the role of x and y.

(ii) The statement is a direct consequence of property (ii) of Lemma 4.1. Indeed, for any x ∈ X and t ≥ 0, it clearly holds

ūT (φ t (x)) -ūT (x) = 1 M T +∞ 0 e -s M s (ũ T (φ s+t (x)) -ũT (φ s (x))) ds ≤ 0 (iii) By definition, ūT (φ t (x)) -ūT (x) = 1 M T +∞ 0 e -s M s (ũ T (φ s+t (x)) -ũT (φ s (x))) ds.

Let us introduce

s := max{s ∈ [0, +, ∞] : ũT (φ s (x)) = ũT (x)}
and notice that -by property (iii) of Lemma 4.1s ∈ [0, T ). Consequently, for any x ∈ X \ A T 2 and t > 0, we have ũT (φ s+t (x)) < ũT (x) = ũT (φ s(x)).

By the previous strict inequality, we deduce that

ūT (φ t (x)) -ūT (x) < 0 ∀x ∈ x ∈ X \ A T 2
and t > 0.

We finally prove Theorem 4.1. Let φ : X × R → X be a continuous flow on a compact metric space (X, d), uniformly Lipschitz continuous on compact subsets of [0, +∞). Then

u : X → R x → u(x) := n∈N 1 2 n ūn (x) (16)
is a Lipschitz continuous Lyapunov function for φ such that N (u) = SCR(φ).

Proof. The Lipschitz continuity of u is a direct consequence of property (i) of Lemma 4.2. Indeed, for any x, y ∈ X, we have

u(y) -u(x) = n∈N 1 2 n (ū n (y) -ūn (x)) ≤ 4d(x, y)
and we conclude by exchanging the role of x, y. Moreover, from property (ii) of Lemma 4.2, it holds

u(φ t (x)) -u(x) = n∈N 1 2 n (ū n (φ t (x)) -ūn (x)) ≤ 0 ∀x ∈ X and t ≥ 0.
This means that u is a Lyapunov function for φ. Finally, let x ∈ X \ SCR(φ) and t > 0. We first notice that the strong chain recurrent set for φ -see ( 5)-can be equivalently expressed as SCR(φ)

= n∈N A n 2 . As a consequence, if x ∈ X \ SCR(φ) = X \ n∈N A n 2 = n∈N X \ A n 2 , then there exists n ∈ N such that x ∈ X \ A n 2
. By property (iii) of Lemma 4.2, we then have ūn (φ t (x)) -ūn (x) < 0 and therefore

u(φ t (x)) -u(x) = n∈N n =n 1 2 n (ū n (φ t (x)) -ūn (x)) + 1 2 n (ū n(φ t (x)) -ūn (x)) < 0.
This proves that u is strict outside SCR(φ), that is N (u) ⊆ SCR(φ). However, by Proposition 1.6 in [START_REF] Abbondandolo | Chain recurrence, chain transitivity, Lyapunov functions and rigidity of Lagrangian submanifolds of optical hypersurfaces[END_REF], the neutral set of every Lipschitz continuous Lyapunov function for φ contains the strong chain recurrent set of φ, so SCR(φ) ⊆ N (u). We then conclude that N (u) = SCR(φ).

The above theorem implies the following characterization of the strong chain recurrent set in terms of Lipschitz continuous Lyapunov functions.

Corollary 4.1. Let φ : X × R → X be a continuous flow on a compact metric space (X, d), uniformly Lipschitz continuous on the compact subsets of [0, +∞). Then,

SCR(φ) = f ∈L(φ) N (f ).
where L(φ) denotes the set of all Lipschitz continuous Lyapunov functions for φ.

Proof. By Proposition 1.6 in [START_REF] Abbondandolo | Chain recurrence, chain transitivity, Lyapunov functions and rigidity of Lagrangian submanifolds of optical hypersurfaces[END_REF], the neutral set of every Lipschitz continuous Lyapunov function for φ contains the strong chain recurrent set of φ, that is

SCR(φ) ⊆ f ∈L(φ) N (f ).
Since -by Theorem 4.1-the function u defined in ( 16) is such that SCR(φ) = N (u), the thesis immediately follows.

We stress that the corollary above has been already proved in [START_REF] Abbondandolo | Chain recurrence, chain transitivity, Lyapunov functions and rigidity of Lagrangian submanifolds of optical hypersurfaces[END_REF][Theorem 2.2]. However, in [START_REF] Abbondandolo | Chain recurrence, chain transitivity, Lyapunov functions and rigidity of Lagrangian submanifolds of optical hypersurfaces[END_REF] the inclusion ⊇ is obtained by an accurate adaptation of Fathi and Pageault's techniques. While, in our proof, it is an immediate consequence of the existence of the Lyapunov function [START_REF] Pageault | Lyapunov functions: a weak KAM approach[END_REF].

Let now (M, g) be a C ∞ closed connected Riemannian manifold and V : M → T M be a C k vector field, k ≥ 2, inducing the flow φ : R × M → M . Denote by v x the norm of an element v ∈ T x M relatively to the metric g and introduce the Mañé Hamiltonian

H V : T * M → R, H V (x, p) = 1 2 p 2 x + p(V (x)).
We indicate by A V the projected Aubry set associated to H V . Since the constant functions are solutions of the Hamilton-Jacobi equation

H V (x, d x v) = 0, the Mañé critical level is c(H V ) = 0.
From one hand, it holds that A V ⊆ CR(φ), see e.g. [START_REF] Mañé | Lagrangian flows: the dynamics of globally minimizing orbits[END_REF][Section I, Theorem V], [START_REF] Pageault | Lyapunov functions: a weak KAM approach[END_REF][Introduction, Section 3, Proposition 3.2] and [START_REF]On the Conley decomposition of Mather sets[END_REF][ Corollary 2]. From the other hand, we know that there exists a

C 1,1 critical sub-solution u : M → R of H V (x, d x v) = 0 which is strict outside A V , see [4][Lemma 7]
. In particular, d x u(V (x)) ≤ 0 and therefore u is a C 1,1 Lyapunov function for φ. Since u is strict outside A V , it holds N (u) ⊆ A V . Furthermore, recall that for any x ∈ A V and any sub-solution v 1 2

d x v 2 x + d x v(V (x)) ≤ 0, (17)
d x v does not depend on v (see [START_REF] Sorrentino | Lecture notes on Mather's theory for Lagrangian systems[END_REF][Proposition 5.1.23]). Consequently, since the constant functions are (sub-)solutions of H(x, d x v) = 0, we have d x v = 0 on A V . In particular, this implies

d x u(V (x)) = 0 for all x ∈ A V or equivalently A V ⊆ N (u). Summarizing, u is a C 1,1 Lyapunov function for φ with N (u) = A V .
Therefore, by the previous corollary, SCR(φ) ⊆ A V . From both inclusions, we conclude that in general

SCR(φ) ⊆ A V ⊆ CR(φ). (18)
Corollary 4.2. If SCR(φ) = CR(φ) then there exists a C 1,1 Lyapunov function such that N (u) = SCR(φ).

Proof. If SCR(φ) = CR(φ) then necessarily SCR(φ) = A V = CR(φ) and the C 1,1 critical sub- solution u : M → R of H V (x, d x u) = 0 which is strict outside A V gives the desired Lyapunov function with N (u) = SCR(φ).
Next basic Example 4.1 shows two different smooth flows on T 1 = R/Z, having the following properties:

(i) For the first flow, the hypothesis of Corollary 4.2 holds and therefore it exists a differentiable Lyapunov function which is strict outside the strong chain recurrent set.

(ii) For the second flow, we have that SCR(φ) A V = T 1 . In such a case, even if every subsolution given by weak KAM theory provides a first integral for φ, the dynamical system admits a differentiable Lyapunov function strict outside SCR(φ).

Example 4.1. On the circle T 1 = R/Z equipped with the standard quotient metric, consider the dynamical systems of Figures 1 and2, where the bold line and the arrows denote respectively fixed points and the direction of the flow φ.

Relatively to the dynamical system of Figure 1 Another case where sub-solutions of weak KAM theory provide first integrals while the Lyapunov function of Theorem 4.1 is not a first integral, is discussed in the next Example 4.2. On T 2 = R 2 /Z 2 endowed with the standard quotient metric, consider the flow φ associated to the vector field (see Figure 3):

V (x, y) = (f (x), 1) (19)
where

f (x) = cos(4πx) + 1 x ∈ 0, 1 4 ∪ 3 4 , 1 0 x ∈ 1 4 , 3 4
In such a case, the strong chain recurrent set coincides with the set of periodic points, that is

SCR(φ) = 1 4 , 3 4 × T 1
and CR(φ) = T 2 . Moreover -by points (i) of Theorem 4.8 and (i) of Lemma 4.14 in [START_REF] Fathi | On the Hausdorff dimension of the Mather quotient[END_REF] againthe projected Aubry set associated to the Mañé Hamiltonian

1 2 (p 2 1 + p 2 2 ) + f (x)p 1 + p 2 is A V = CR(φ) = T 2 .
Consequently, every sub-solutions given by weak KAM theory is a first integral for φ while function [START_REF] Pageault | Lyapunov functions: a weak KAM approach[END_REF] We finally remark that the converse implication of Corollary 4.2 does not in general hold true. For example, consider the dynamical system of Figure 2. In such a case, it is easy to construct a smooth Lyapunov function which is strict outside SCR(φ) = F ix(φ). However, SCR(φ) CR(φ) = T 1 . A necessary and sufficient condition for the strong chain recurrent and chain recurrent sets to be equal is proved in Proposition 5.3 of the next section.

Strong chain transitivity and Lipschitz Lyapunov functions

In this section, φ : X × R → X is a continuous flow on a compact metric space (X, d).

We first introduce L T -dominated and L-dominated functions and then we explain their relation with Lipschitz continuous Lyapunov functions and the strong chain recurrent set. Next definitions and results are the continuous-time versions of the ones appearing in [START_REF] Pageault | Lyapunov functions: a weak KAM approach[END_REF][Chapter 2, Section 4] and [START_REF] Fathi | Aubry-Mather theory for homeomorphisms[END_REF][Section 2.2] for homeomorphisms.

(i) A function f : X → R is said to be L T -dominated if there exists K T > 0 such that f (y) -f (x) ≤ K T L T (x, y) ∀x, y ∈ X. (ii) A function f : X → R is said to be L-dominated if it is L T -dominated for any T > 0.
The proof of the next proposition is an easy adaptation for flows of the proof of Lemma 4.3 in [START_REF] Pageault | Lyapunov functions: a weak KAM approach[END_REF].

Proposition 5.1. A Lipschitz continuous function f : X → R is L T -dominated if and only if it is definitively a Lyapunov function for φ, that is f (φ t (x)) ≤ f (x)
for any x ∈ X and t ≥ T.

Proof. From one hand, let f : X → R be a L T -dominated, Lipschitz continuous function for φ. Thanks to point (ii) of Proposition 2.1, for any fixed x ∈ X and t ≥ T it holds

f (φ t (x)) -f (x) ≤ K T L T (x, φ t (x)) = 0,
that is f is a definitively Lyapunov function.

On the other hand, let f : X → R be a K-Lipschitz continuous, definitively Lyapunov function for φ. Fixed x, y ∈ X, let C = (x i , t i ) i=1,...,n ∈ C T (x, y) be a chain from x to y. Then

f (x i+1 ) -f (x i ) ≤ f (x i+1 ) -f (φ t i (x i )) ≤ Kd(φ t i (x i ), x i+1 ) ∀i = 1, . . . , n.
By adding all the inequalities, we conclude that

f (y) -f (x) = n i=1 (f (x i+1 ) -f (x i )) ≤ K n i=1 d(φ t i (x i ), x i+1 ) = K l T (C).
More precisely, by considering the infimum over all possible chains in C T (x, y) we have

f (y) -f (x) ≤ KL T (x, y) ∀x, y ∈ X ( 20
)
where K is any Lipschitz constant for f . Corollary 5.1. A Lipschitz continuous function f : X → R is L-dominated if and only if it is a Lyapunov function for φ.

In the sequel, we denote by L 1 T (φ) the set of 1-Lipschitz continuous functions f : X → R such that f (φ t (x)) ≤ f (x) for any x ∈ X and t ≥ T . The next lemma is the continuous-time version of Proposition 4.6 in [START_REF] Pageault | Lyapunov functions: a weak KAM approach[END_REF].

Lemma 5.1. For any x ∈ SCR(φ) and y ∈ X, we have

L T (x, y) = sup f ∈L 1 T (φ) f (y) -f (x). (21)
Proof. By the previous Proposition 5

.1, if f ∈ L 1 T (φ) then f is L T -dominated; moreover, see inequality (20), it results that K T = 1. This means that f (w) -f (z) ≤ L T (z, w) ∀z, w ∈ X and consequently L T (z, w) ≥ sup f ∈L 1 T (φ) (f (w) -f (z)) ∀z, w ∈ X.
In order to prove the other inequality, let us fix x ∈ SCR(φ) and define f x (•) := L T (x, •). We notice that the function f x (•) ∈ L 1 T (φ). Indeed:

f x (z) -f x (y) = L T (x, z) -L T (x, y) ≤ d(y, z) ∀y, z ∈ X and f x (φ t (y)) -f x (y) = L T (x, φ t (y)) -L T (x, y) ≤ L T (y, φ t (y)) = 0
for any y ∈ X and t ≥ T . Moreover, it holds that

L T (x, y) = L T (x, y) -L T (x, x) = f x (y) -f x (x) because L T (x, x) = 0. As a consequence, L T (x, y) ≤ sup f ∈L 1 T (φ) (f (y) -f (x))
∀x ∈ SCR(φ) and y ∈ X.

Previous Lemma 5.1 leads to prove the following results for the set L(φ) of all Lipschitz continuous Lyapunov functions for φ. The corresponding version for homeomorphisms is given by Corollary 4.7 and Theorem 4.8. of Chapter 2 in [START_REF] Pageault | Lyapunov functions: a weak KAM approach[END_REF]. Proposition 5.2. Any f ∈ L(φ) is constant on every strong chain transitive component of SCR(φ). Moreover, if x, y ∈ SCR(φ) belong to different strong chain transitive components of SCR(φ), then there exists a function f ∈ L(φ) such that f (x) = f (y).

Proof. Given f ∈ L(φ) with Lipschitz constant K f > 0, it clearly holds that f K f ∈ L 1 T (φ) for any T > 0. Let now x, y ∈ SCR(φ) be in the same strong chain transitive component of SCR(φ), that is L T (x, y) = 0 = L T (y, x) for all T > 0. By equality (21), one has 0 = L T (x, y) = sup Since now -by Proposition 2.2-if z ∈ SCR(φ) and s ∈ R then z and φ s (z) belong to the same strong chain transitive component of SCR(φ), we conclude that f (x) = f (φ s(x)) = f (x) and f (y) = f (φ ŝ(y)) = f (y).

As a consequence, we have proved that f ∈ L(φ) is such that f (x) = f (y).

The proof of the next theorem is essentially the same of Theorem 4.8 in Chapter 2 in [START_REF] Pageault | Lyapunov functions: a weak KAM approach[END_REF] and it is omitted. We notice that Fathi and Pageault's statement is formulated by using the so-called d-Mather classes, corresponding exactly to the strong chain transitive components of the strong chain recurrent set for a given homeomorphism.

Theorem 5.1. Let φ : X × R → X be a continuous flow on a compact metric space (X, d). SCR(φ) has a unique strong chain transitive component if and only if the only Lipschitz continuous Lyapunov functions for φ are the constants.

In the fundamental theorem of dynamical systems recalled in the introduction (see [START_REF] Conley | Isolated invariant sets and the Morse index[END_REF][Chapter II]), Conley made explicit a so-called complete Lyapunov function for φ, whose properties are listed below. (iii) f (CR(φ)) is a compact nowhere dense subset of R.

We notice that the function given by Theorem 4.1 cannot be in general assumed to be complete with respect to SCR(φ). For example, look at the dynamical system of Figure 2. In such a case, every fixed point is a strong chain transitive component of SCR(φ) = F ix(φ) but it is not possible to construct a continuous Lyapunov function such that properties:

  , the hypothesis of Corollary 4.2 is satisfied since SCR(φ) = CR(φ) = F ix(φ) and it clearly exists a C 1,1 (also C ∞ ) Lyapunov function whose neutral set is the strong chain recurrent set. Otherwise, in the case of the dynamical system of Figure2, SCR(φ) = F ix(φ) and CR(φ) = T 1 . Moreover -see point (i) of Theorem 4.8 and point (i) of Lemma 4.14 in[START_REF] Fathi | On the Hausdorff dimension of the Mather quotient[END_REF]-the projected Aubry set associated to the corresponding Mañé Hamiltonian coincides with CR(φ) = T 1 . Consequently, every sub-solutions given by weak KAM theory is a first integral but -according to Theorem 4.1it clearly exists a Lipschitz continuous (even smooth) Lyapunov function strict outside F ix(φ).

Figure 1 :

 1 Figure 1: First dynamical system of Example 4.1.

Figure 2 :

 2 Figure 2: Second dynamical system of Example 4.1.

4 , 3 4 × T 1 .Figure 3 :

 4413 Figure 3: The vector field (19).

Definition 5 . 1 .

 51 (Dominated function)

  that f (y) = f (x). Conversely, let x, y ∈ SCR(φ) be in different strong chain transitive components of SCR(φ). Exchanging the role of x, y if needed, this means that there exists a time T > 0 such that L T (x, y) > 0. By using equality (21) again, we conclude that there exists a function f ∈ L 1 T (φ) such that f (y) = f (x). Let us now definef (z) := max s∈[0,T ] f (φ s (z))which is in L(φ). Moreover, there exist s = s(x) ∈ [0, T ] and ŝ = ŝ(y) ∈ [0, T ] such that f (x) = f (φ s(x))and f (y) = f (φ ŝ(y)).

Definition 5 . 2 .

 52 (Complete Lyapunov function) Let φ : X × R → X be a continuous flow on a compact metric space (X, d). A continuous Lyapunov function f : X → R for φ is called complete if:

  (i) N (f ) = CR(φ).

(

  ii) If x, y ∈ CR(φ), then f (x) = f (y) if and only if x and y belong to the same chain transitive component of CR(φ).

Acknowledgements. O. Bernardi has been supported by the project CPDA149421/14 of the University of Padova. O. Bernardi and A. Florio acknowledge the support of G.N.F.M. The authors also thank the anonymous referee for the careful reading of the manuscript and his/her many insightful comments and suggestions.

(i) N (f ) = SCR(φ) and (ii) If x, y ∈ SCR(φ), then f (x) = f (y) if and only if x and y belong to the same strong chain transitive component of SCR(φ) simultaneously hold. In particular, if we change the setting from chain recurrence to strong chain recurrence, the next notion of pseudo-complete Lyapunov function is crucial. This definition was first introduced (at least at our knowledge) by Katsuya Yokoi for homeomorphisms (see [START_REF] Yokoi | On strong chain recurrence for maps[END_REF][Definition 5.1] and also [START_REF] Wiseman | The generalized recurrent set and strong chain recurrence, Ergodic Theory and Dynamical Systems[END_REF][Section 6]).

Indeed, from Theorem 4.1 and the first part of Proposition 5.2, we immediately deduce the next Corollary 5.2. Let φ : X × R → X be a continuous flow on a compact metric space (X, d), uniformly Lipschitz continuous on compact subsets of [0, +∞). Then there exists a pseudocomplete Lyapunov function for φ.

We finally remark that the notion of pseudo-complete Lyapunov function is useful to discuss the condition for the strong chain recurrent and chain recurrent sets to be equal (see [START_REF] Yokoi | On strong chain recurrence for maps[END_REF][Theorem 5.3] for the case of a homeomorphism).

Proposition 5.3. Let φ : X × R → X be a continuous flow on a compact metric space (X, d).

Then, SCR(φ) = CR(φ) if and only if there exists a pseudo-complete Lyapunov function u : X → R for φ such that u(SCR(φ)) is totally disconnected.

Proof. Let SCR(φ) = CR(φ). By Conley's fundamental theorem (see [START_REF] Conley | Isolated invariant sets and the Morse index[END_REF][Chapter II, Section 6.4]), there exists a complete Lyapunov function u for φ. Clearly, u is also a pseudo-complete Lyapunov function for φ such that u(SCR(φ)) = u(CR(φ)) is totally disconnected. Conversely, let u : X → R be a pseudo-complete Lyapunov function for φ such that u(SCR(φ)) is totally disconnected. Without loss of generality, we suppose that u(x) ≥ 0 for any x ∈ X. In order to show that CR(φ) ⊆ SCR(φ), we remind that the next conditions are equivalent:

(ii) There exists an attractor K such that x / ∈ K but ω(x) ⊆ K.

See [START_REF] Conley | Isolated invariant sets and the Morse index[END_REF][Chapter II, Section 6.2, Page 37] for this equivalence. Let x / ∈ SCR(φ) and t > 0 such that u(φt(x)) < u(x). Since u(SCR(φ)) is totally disconnected, there exists r 0 > 0 such that u(φt(x)) < r 0 < u(x) and r 0 / ∈ u(SCR(φ)). We define U := {y ∈ X : u(y) ∈ [0, r 0 )}. Since u is a Lyapunov function for φ, we have that K := ω(U ) ⊆ U , which means that K is an attractor. Moreover, x / ∈ K but, since φt(x) ∈ U , ω(x) ⊆ K. From the equivalence of points (i) and (ii) recalled above, we conclude that x / ∈ CR(φ). This prove that CR(φ) ⊆ SCR(φ) and therefore CR(φ) = SCR(φ).