Christophe Finot
email: christophe.finot@u-bourgogne.fr

Fang Feng

Yanne Chembo

Stefan Wabnitz

Gain sideband splitting in dispersion oscillating fibers

Keywords: Dispersion oscillating fiber, modulation instability, four-wave mixing

Introduction

Modulation instability (MI) is a nonlinear process that has been widely investigated in various fields of physics including plasma, hydrodynamics and optics, to cite a few. In the presence of a high power continuous wave (CW), MI leads to the emergence and amplification of gain sidebands in the wave spectrum. In nonlinear fiber optics, such a process has been demonstrated in fibers with anomalous, constant group velocity dispersion (GVD) [START_REF] Tai | Observation of modulational instability in optical fibers[END_REF], as well as in normal GVD fibers by enabling the fulfillment of the nonlinear phase-matching condition through either fourth order dispersion [START_REF] Pitois | Experimental observation of a new modulational instability spectral window induced by fourth-order dispersion in a normally dispersive single-mode optical fiber[END_REF], birefringence or a multimodal structure [3; 4]. More recently, a renewed experimental and theoretical interest in MI studies has been stimulated by the availability of fibers presenting a longitudinal and periodic modulation of their dispersion properties [START_REF] Droques | Experimental demonstration of modulation instability in an optical fiber with a periodic dispersion landscape[END_REF]. Indeed, thanks to the periodic dispersion landscape, which leads to quasi-phasematching (QPM) of the nonlinear four-wave mixing (FWM) process, MI sidebands can be observed even in the regime of normal average GVD of a dispersion-oscillating optical fiber (DOF) [6; 7; 8]. Recent experimental works have confirmed the QPM-induced MI process in the normal GVD regime of microstructure DOF around 1 m [START_REF] Droques | Experimental demonstration of modulation instability in an optical fiber with a periodic dispersion landscape[END_REF], as well as of non-microstructured highly nonlinear DOF at telecom wavelengths [9; 10].

To date, the role of the amplitude of the GVD oscillations on the MI spectrum of a DOF remains largely unexplored. In this work, we present a systematic study of the dependence of various sidebands which emerge at the output of a DOF, as the amplitude of the dispersion variations grows progressively larger. We unveil the emergence of new sidebands, as well as their unexpected splitting in sub-sidebands. As we shall see, the emergence of new sidebands may be qualitatively described by extending to a continuum set of frequencies the analytical theory that was established in [START_REF] Droques | Dynamics of the modulation instability spectrum in optical fibers with oscillating dispersion[END_REF] for a discrete set of sidebands. We also present a detailed study of the influence of optical losses on the profile and splitting of MI sidebands in a DOF.

Model and situation under investigation

The evolution of the optical field  in an optical fiber can be described by the nonlinear Schrödinger equation (NLSE) that includes both the Kerr nonlinearity  and second-order

dispersion  2 2 2 2 2 | | 0 2 2 i i z t                 (0)
In the last section of this paper, we will also discuss the influence of linear losses that are included through the coefficient  (negative and positive values of  leading to distributed amplification and losses respectively). In Eq. (1) we did not include higher-order dispersion terms or Raman scattering. Nevertheless, we checked that these effects do not have a noticeable influence on the MI spectral dynamics that we are going to describe.

MI induced by the longitudinal variations of chromatic dispersion has been theoretically

investigated before in a wide range of configurations, ranging from sinusoidal profiles with a spatial period of a few tens of meters [5; 9], up to dispersion-managed systems with periods of several kilometers [6; 12; 13; 14]. We consider in this contribution the specific case of DOF whose parameters are inspired by the previous numerical works which were carried out in the context of the transmission of high-speed telecommunication signals in dense dispersion managed links [START_REF] Fatome | Practical design rules for single-channel ultra high-speed dense dispersion management telecommunication systems[END_REF]. The fabrication of such a DOF is fully consistent with existing drawing techniques [START_REF] Provost | Dispersion-Managed Fiber with low chromatic dspersion slope[END_REF]. More precisely, the stepwise dispersion profile over one spatial period  of the fiber under study is provided by the map

z z                    (0)
where  2av is the average dispersion of the fiber, and  2ampS is half of the peak-to-peak amplitude of the dispersion variation (the second order dispersion  2 being related to the dispersion parameter D by D = -2 c  2 /  2 ,  being the wavelength and c the light velocity). Using a Fourier decomposition, it is possible to decompose the profile (2) into a series of sinus functions having spatial frequencies multiple to 1/. For the sake of simplicity, and since analytical solutions exist for this case, we concentrate here on the fundamental Fourier component, so that we may reduce the stepwise dispersion profile (1) to the following sinusoidal variation

  2 22 ( ) sin 2 / av amp z z        (0)
where  2amp is the corresponding amplitude of the dispersion variation:

 2amp = 4  2amps /  .
Let us consider here a DOF with a Kerr nonlinearity of =2 W -1 .km -1 , a spatial period of the dispersion oscillation =1 km, and a dispersion average value of D av = -0.5 ps/km/nm at the wavelength  0 =1550 nm. The fiber is pumped by a CW with the average power P=0.75 W at  0 .

The NLSE (0) is numerically solved by the standard split-step Fourier algorithm including a weak input white noise seed, and the results are averaged over 24 shots.

In the presence of sinusoidal longitudinal GVD variations, QPM of FWM or MI leads to the appearance of resonant gain sidebands, whose angular frequency shift relative to the pump can be analytically predicted as follows [6] :

P L J                             (0)
where J p is the Bessel function of order p. Note that Eqs.(4-5) are derived from Eqs.(1-3) in a perturbation limit, namely whenever | 2amp / 2av |<<1 [7; 17]. Therefore it is particularly interesting to study the domain of their validity in situations where the amplitude of the dispersion oscillations is equal or even much larger than the average dispersion. Note that indeed dispersion managed transmission systems almost always operate in the so-called strong management regime, namely the condition | 2amp / 2av |>>1 holds, which is precisely the opposite of the domain of validity of the perturbation theory that permits to derive Eqs.(4-5).

Influence of the amplitude of the dispersion fluctuation

We start our study by investigating the influence of the amplitude of dispersion fluctuations  2amp

(or the corresponding D amp ) on the MI spectrum which recorded after 12 spatial periods, i.e., a propagation distance of 12 km. Results are plotted in Fig. 1 for three levels of the amplitude of GVD oscillations, namely D amp =0.5, 1.7 and 3.7 ps/km/nm, see subplots (a), (b) and (c), respectively. Our motivation in varying the amplitude of the dispersion oscillations rather than other parameters (such as the average dispersion or the longitudinal period, for example) is that a modification of the dispersion amplitude  2amp should in principle have no influence, on the basis of Eq. (0), on the position of the MI sidebands.

Indeed, for a relatively low level of dispersion fluctuations, that is D amp = 0.5 ps/km/nm (case (a) of Fig. 1), we observe the generation of unequally spaced and narrow spectral sidebands, whose position is in qualitative agreement with the analytical predictions of Eq. (0) (see dashed vertical lines). The possibility of observing QPM-induced MI in such a situation, where a large set of spectral lines is generated, was experimentally confirmed in the work of Droques et al [START_REF] Droques | Experimental demonstration of modulation instability in an optical fiber with a periodic dispersion landscape[END_REF].

However, for an increasing level of dispersion amplitude oscillations (e.g., D amp = 1.7 ps/km/nm, case (b) of Fig. 1), we observe two main features that lead to a very different structure of the output MI spectrum. We first notice from Fig. 1(b) that some of the spectral lines (for example, the lines corresponding to p = 2 or p = 5) have disappeared from the MI spectrum.

Moreover, we may also point out in Fig. 1(b) the development of a new set of regularly spaced sidebands with a broader bandwidth. The first feature is explained by the evolution of the gain coefficient according to Eq. (0) which leads, for some values of the argument, to the annihilation of the sideband amplitude, as it was already confirmed experimentally in [START_REF] Droques | Dynamics of the modulation instability spectrum in optical fibers with oscillating dispersion[END_REF]. On the other hand, the second feature is linked to FWM between the pump wave and the first QPM sideband, and the subsequent cascading of the FWM process, as it was previously discussed and demonstrated experimentally in [START_REF] Finot | Competing four-wave mixing processes in dispersion oscillating telecom fiber[END_REF]. For an even higher amplitude of the dispersion oscillations, i.e. D amp = 3.7 ps/km/nm (case (c) in Fig. 1), we observe instead of a single gain sideband, the unexpected emergence of a pair of sidebands around the angular frequency  1 . The emergence of the new pair of sidebands is not explained by nonlinear mixing between the various other sidebands, but it is an intrinsic property of the MI gain spectrum of Eqs (1,3) in the strong dispersion management regime.

To verify this statement, we may write the perturbed CW solution of the NLSE (1,3) as

  ( , ) ( , ) exp zt P u z t iPz     
, where we suppose |u| 2 <<P, so that we obtain the linearized

equation for u(Z,T)   2 * () 2 Z TT dZ iu u u u     . ( 6
)
Here we defined  2 =L nl /L d , where the nonlinear length L nl =1/(P), the dispersion length

L d =t 0 2 /| 2av
  2 2 () 2 Z i i a d Z a a b      , (7a)   2 2 () 2 Z i i b d Z b a b       . ( 7b
)
These equations are equivalent to a single, second-order ODE or generalized Mathieu equation for the Stokes sideband a(z) [START_REF] Abdullaev | Modulational instability in optical fibers with variable dispersion[END_REF]. In the general case, the resulting periodic ODE cannot be solved analytically for an arbitrary strength of the dispersion oscillation: the parametric resonance conditions (4) being strictly valid only in the limit of small oscillations. However, the corresponding linear stability analysis can still be rigorously carried out numerically by the Floquet theory, which is analogous to Bloch wave theory in solid state physics [18; 19].

By defining the solution vector of [START_REF] Abdullaev | Modulational instability in optical fibers with variable dispersion[END_REF] as s=(a,b), and choosing the two fundamental initial conditions s 1 (Z=0)=(1,0) and s 2 (Z=0)=(0,1), one readily obtains from the solution of [START_REF] Abdullaev | Modulational instability in optical fibers with variable dispersion[END_REF].

at Z=' the principal solution matrix S=[s 1 (') t , s 2 (') t ] (where t denotes the vector or matrix transpose). According to Floquet's theorem, the eigenvalues of S, or Floquet's multipliers =exp( F +i), such that ||>1 yield the linear instability of the CW with respect to the growth of sidebands with frequency detuning . Since the scattering matrix after an integer number n of periods is simply S n , one may relate the Floquet's multipliers to the usual MI gain G after one period of the dispersion oscillation by the relation G=2 F /'. Figure 2 compares the MI gain spectra computed from the Floquet's stability theory of Eqs. [START_REF] Abdullaev | Modulational instability in optical fibers with variable dispersion[END_REF], with the results of the numerical solutions of Eq.( 1) as reported in Fig. 1, showing the MI gain after n=12 periods of the dispersion oscillation. As can be seen, there is an excellent agreement between the numerics and the Floquet's stability analysis for all values of the dispersion oscillation amplitudes. Clearly, in In order to get a better overview of the role of the dispersion oscillation amplitude on the MI spectrum, we carried a more systematic study, which is summarized in Fig. 3 From Fig. 3, we may observe a non-monotonic evolution of the gain as a function of the dispersion oscillation amplitude. Thus we have studied in more details the evolution of the gain at the discrete set of QPM wavelengths which are predicted by Eq. (0). The corresponding results are summarized in Fig 4, where we compare the analytical predictions of Eq. (0) with the gain that is evaluated from numerical simulations at  1 and  2 . Several points can here be stressed.

First, Fig. 4 provides the confirmation of the explanation for the behavior which is observed in the case of Fig. 1(b): for D amp = 1.7 ps/km/nm, the gain of the second sideband vanishes. Whereas the gain of the first sideband reaches its maximum value, thus stimulating cascaded FWM between the pump and this sideband.

From Fig. 3(a), we may also notice that for D amp around 3 ps/km/nm, the gain induced by the FWM between the 1 st QPM band and the pump overlaps with the second QPM sideband, thus explaining the discrepancy between numerics and analytical theory (4) for the position of the second sideband as it is observed in Fig. 3. Next, we may observe from Fig. 4 that, for the case reported in Fig. 1(c), both numerics and analytic formula [START_REF] Droques | Experimental demonstration of modulation instability in an optical fiber with a periodic dispersion landscape[END_REF] predict that the gain at the first sideband position  1 vanishes. An important conclusion that we can draw from Fig. 4 is that the agreement between the numerical simulations and the analytical predictions (4-5) is excellent over the whole range of dispersion oscillation amplitudes under study. Therefore, we may conclude that the observation in the case of Fig. 1(c) of two sidebands on either side of the angular frequency  1 is not in contrast with the perturbation theory analytical results of Eqs. (0) and (0).

From Fig. 3(a) and 4, we may also note that a similar situation also appears for dispersion oscillations with an amplitude around D amp = 6.7 ps/km/nm, where again the theory predicts a vanishing gain for the first sideband: the emergence of two new two sidebands is also observed in this case.

P L J                        (8)
The above formula is an extension of Eq. (0), which was initially aimed at describing the gain exclusively at the discrete set of the QPM frequencies, to a continuum range of frequencies.

Despite the use of Eq. ( 8) is not fully rigorous (as this formula was derived for = 1 only), we can see from Fig. 5 that it provides interesting insights in the existence of the two sidebands on either side of the angular frequency  1 . From the previous analysis of the case reported in Fig. 1(c), we may conclude that the existence of two new sub-sidebands on either side of the angular frequency  1 does not violate the prediction of both Eq. ( 0) and (0), which forecast to zero gain at  1 in this case. The emergence of these sidebands is thus directly linked to their splitting from the first QPM sideband, whose gain at  1 is precisely zero. Quite interestingly, the gain of the two sidebands in the vicinity of  1 can be approximated by simply extending the validity of Eq. (0) to a continuum range of frequencies.

Influence of optical losses/gain

In the second part of this article, we shall examine the influence of fiber linear gain or losses on the resulting output MI spectrum after 12 kilometers of propagation. For simplicity, we limit our analysis for the significant case where sideband splitting is observed, i.e., D amp = 3.7 ps/km/nm.

The impact of gain can be seen in Fig. 7(a): here we observe, for a linear gain of 0.05 dB per km, the amplification of the inner sub-sideband (i.e. the sideband located at frequencies below  1 ), whereas quite surprisingly the MI gain of the outer sub-sideband decreases (i.e. the sideband located at frequencies above  1 ). We also observe in Fig. 7(a) the emergence of a new FWM- induced sideband, located close to the second QPM sideband.

On the other hand when dealing with the impact of linear losses, we may notice that even with a level of losses as low as 0.05 dB per km (Fig

), the MI gain of the inner sideband experiences a dramatic decrease. To the contrary, the outer sideband undergoes a slight gain increase. For a further increase of the level of losses (i.e., 0.1 dB/km, Fig. 7(c)), the gain of the inner sideband drops down to a few dB only, whereas the gain of the outer sub-sideband does not change significantly. Let us point out that, contrary to several other studies which were earlier carried out with reference to dispersion managed transmission systems [START_REF] Ambomo | Critical behavior with dramatic enhancement of modulational instability gain in fiber systems with periodic variation dispersion[END_REF], we do not consider here a periodic amplification of the field. A more systematic study on the impact of linear loss on the MI spectrum is summarized on Fig. 8(a): these results confirm that losses have a very different impact on inner and outer subsidebands. To further highlight the radically different evolution of the gain for the two sidebands as a function of the linear gain/loss coefficient, we have reported the respective behavior in Fig. 8(b). As it can be seen, the inner sideband is enhanced by the presence of linear gain, but it is extremely sensitive to an even low level of losses. To the contrary, the outer sub-sideband suffers from the presence of linear gain; whereas and its amplitude is increased by the presence of a moderate amount of losses. For the usual levels of fiber losses (>0.2 dB/km), the inner sidebands cannot be clearly distinguished, so that only the outer sub-sidebands will be observed in practice. The different dependence on gain and loss of the two sub-sidebands may be qualitatively explained by the fact that the angular frequency of the resonant sidebands predicted by Eq. (0) are power dependent, higher powers favoring lower values of  1 . A more complete picture of the influence of the pump level on the detailed evolution of the first sideband is provided on In order to get slightly more quantitative insights on the impact of loss or gain, we clearly cannot rely on Eq. ( 8) as the overall shape provided by this expression is not power dependent.

Therefore, to evaluate roughly the influence of  on the MI spectrum, we used the following approximate approach: we consider the output gain spectra (in dB) after n periods of the dispersion oscillation as the sum of the previously computed gain spectra experienced over a single spatial period for a power at the input of the period that increases/decreases exponentially with propagation length:

    1.. 0 ,, B F dB d in i GG Pe         . (9)
Resulting spectra for values of  similar to Fig. 7 are plotted in Fig. 9(b). A pretty good agreement with the numerical integration of Eq. ( 2) is observed, which confirms our qualitative explanation. A satisfactory agreement is also reached in Fig. 8(b), so that we may conclude that Eq. 9 reproduces well the trends which are obtained from the full and rigorous simulation of the NLSE (1). 7 : results from numerical integration of Eq. (0) (grey lines) are compared with results from Eq. 9 (black line).

It also proves interesting to study the longitudinal evolution of the MI gain of the two sub-sidebands in the presence of either linear gain or loss. The corresponding results are summarized in Fig. 10, and confirm that the previous conclusions remain valid at other values of the propagation distance. We note however here a limit of the approach based on Eq. 9: being based on a sum of positive terms, the gain predicted by Eq. 9 can intrinsically only increase upon propagation. Therefore, this is not possible to reproduce with Eq.( 9) the short scale fluctuations of the gain. Moreover, instead of the non-monotonic evolution of gain with distance recorded for the inner sideband for a loss of 0.1 dB/km (see Fig. 10(a1)), with the approach provided by Eq.( 9)

we observe a plateau (see Fig. 10(b1)). predicted by Eq. 9 (b1,b2).

Discussion and conclusion

In this work we have studied the evolution of the modulation instability spectrum in an optical fiber with periodically varying dispersion, as a function of the dispersion oscillation amplitude.

Existing theory for DOFs predicts, in addition with the well-known modulation instability that is observed in fibers with constant dispersion, the emergence of a discrete, infinite set of parametric resonance sidebands both in the normal and in the anomalous dispersion regime. However these parametric resonance sidebands are obtained in a perturbative limit of relatively small dispersion oscillations. Hence there is in principle no guarantee about the survival and existence of parametric resonances in DOFs with relatively large amplitude dispersion oscillations, or in the so-called strong dispersion management regime.

As a matter of fact, we found that even in the limit of dispersion oscillation with an amplitude much larger than the average fiber dispersion, the predictions of the perturbative analysis generally hold surprisingly well. Nevertheless, in situations where the standard perturbative analysis predicts that the parametric sidebands have a vanishing amplitude, we observed an unexpected emergence of new, higher order sidebands which results from the splitting of the corresponding original parametric sideband. Our numerical results are in good quantitative agreement with a Floquet stability analysis of four-wave mixing in a DOF.

Furthermore, we have also shown that linear gain or loss may have a dramatic influence in reshaping the relative amplitude of each of the new sidebands.

From the application point of view, the observed sideband splitting may be considered as generating a double-sideband, suppressed carrier modulation format (where by carrier we mean the original first order parametric resonant sideband). Moreover, analogous phenomena are expected to occur in other types of quasi-phase-matched nonlinear interactions, for example in quadratic materials. From a fundamental viewpoint, the present sideband splitting is likely to be a universal feature of strongly parametrically driven nonlinear dynamical systems, such as for example the splitting of Arnold's tongues [START_REF] Paar | Intermingled fractal Arnold tongues[END_REF] .

p

   . More recently, it has been shown that the gain experienced by the p th sidebands after a propagation length L can be predicted by the formula[START_REF] Droques | Dynamics of the modulation instability spectrum in optical fibers with oscillating dispersion[END_REF] :

Figure 1 :

 1 Figure 1 : Evolution of the output spectra recorded after 12 km of propagation for different values of the amplitude of the dispersion fluctuation: D amp = 0.5 ps/km/nm, 1.7 ps/km/nm and 3.7 ps/km/nm, subplots (a), (b) and (c) and respectively. The vertical dashed lines represent the predictions from Eq. (4).

Fig. 2

 2 Fig.2 the peaks resulting from multiple FWM cascading of the MI sidebands with the pump in Fig. 1(b) are absent from the MI spectrum computed with Floquet's theory, as cascading effects

Figure 2 :

 2 Figure 2 : Anti-Stokes sideband spectrum as Fig.1. The results from the Floquet linear stability analysis of the NLSE with periodic dispersion (black lines) are compared with the results of the numerical integration of Eq. (1) (grey lines)

  figure, several different behaviors can be clearly observed. Up to D amp = 1 ps/km/nm, the

Figure 3 :

 3 Figure 3 : Evolution of the output MI spectrum after a distance of 12 kilometers according to the level of dispersion fluctuations. (a) Results obtained for a sinusoidal dispersion modulation. (b) Results obtained for a square dispersion modulation. The position of the sidebands predicted from Eq. (0) is plotted with vertical dashed lines. The spectra corresponding to cases of Fig. 1 are highlighted by dashed horizontal lines.

Figure 4 :

 4 Figure 4 :Evolution with D amp of gain at the frequencies predicted by Eq. (0) for the first and second QPM sidebands (black and grey colors respectively). Results obtained from numerical simulations (solid lines) are compared with analytical predictions from Eq. (0) (circles and squares). Positions of the three cases reported in Fig.1are marked with dashed vertical lines.

Figure 5 :

 5 Figure 5 : Magnification of the MI gain spectrum around the first QPM sideband vs. D amp . Numerical results (a) are compared with the analytical results (b) predicted by Eq. (8) and by the Floquet linear stability analysis (c). Blue dashed vertical lines indicates the angular frequency  1 as predicted from Eq. (0) and horizontal lines highlight the values of dispersion fluctuations that are then used in Fig.6.

Figure 6 :

 6 Figure 6 : Output spectrum after 12 kilometers of propagation obtained from numerical simulations (black line), from analytical equation (grey line) and from the Floquet linear stability analysis (grey circles): (a) D amp = 3.7 ps/km/nm; (b) D amp = 6.7 ps/km/nm.

Figure 7 :

 7 Figure 7 : Evolution of the output spectra after 12 kilometers of propagation for different levels of losses or gain (solid black lines): (a) 0.05 dB/km of gain. (b) 0.05 dB/km of loss. (c) 0.1 dB/km of loss. Results obtained without losses are plotted with grey solid lines.

Figure 8 :

 8 Figure 8 : (a) Evolution of the output spectrum after 12 kilometers of propagation according to the level of linear losses/gain. (b) Evolution of the parametric gain experienced by inner and outer sidebands (grey and black lines or symbols, respectively) according to the level of linear losses/gain. The numerical results (solid lines) are compared with results based on Eq. 9 (dots).

Fig. 9 (

 9 Fig. 9(a), where the Floquet-based linear stability analysis (carried out without loss) is used to

Figure 9 :

 9 Figure 9 : (a) Evolution of the first sideband after 12 km of propagation according to the input power of the cw and as predicted by the of Floquet linear stability analysis without any loss or gain. The dashed blue line represents the central frequency of the sideband as predicted by Eq. (0), whereas the dotted horizontal line stands for the operating power that has been considered in previous section. (b) Details of the first sideband for the same levels of loss as Fig.7: results from numerical integration of Eq. (0) (grey lines) are compared with results from Eq. 9 (black line).

Figure 10 :

 10 Figure 10 : Longitudinal evolution of the gain for the inner and outer sub-sidebands (panels (a1,b1) and (a2,b2) respectively) for different levels of attenuations/losses: -0.05, 0, 0.05 and 0.1 dB/km, solid black, dotted black, dark grey and light grey lines respectively. Results from numerical simulations (a1,a2) are compared with the results

  |, and t 0 =1 ps is a reference time unit, so that Z=z/L nl and T=t/t 0 . Moreover,

	d(Z)=[1+sin(2Z/')] with = 2amp /| 2av |, and '=/L nl . By writing now the solution of (6)
	as the sum of Stokes and anti-Stokes sidebands, u(Z,T)=a(Z)exp{iT)+b * (Z) exp{-iT), one
	obtains the two coupled linear ordinary differential equations (ODEs) with periodic coefficients
	for a(z) and b(z)

Acknowledgements

We thank Julien Fatome, Stéphane Pitois, Kamal Hammani and Alexej Sysoliatin for stimulating discussions. We acknowledge the financial support of the Conseil Regional de Bourgogne (Pari Photcom), the funding of the Labex ACTION program (ANR-11-LABX-01-01), and the Italian Ministry of University and Research (grant no. 2012BFNWZ2).


Oussama Ajbal

Fixed points of endomorphisms and relations between metrics in preGarside monoids

, it is proved that the fixed points submonoid and the periodic points submonoid of a trace monoid endomorphism are always finitely generated. We show that for finitely generated left preGarside monoids, that includs finitely generated preGarside monoids, Garside monoids and Artin monoids, the fixed and periodic points submonoids of any endomorphism are also finitely generated left preGarside monoids under some condition, and in the case of Artin monoids, these submonoids are always Artin monoids too. We also prove algebraically some inequalities, equivalences and non-equivalences between three metrics in finitely generated preGarside monoids, and especially in trace monoids and Garside monoids.

Introduction

Trace monoids, or equivalently partially commutative monoids, are monoids of a particular interest and have been widly studied. In particular, they are Artin Monoids. This explains why they are also called Right Angle Artin monoids (RAAM for short). In [START_REF] Rodaro | Fixed points of endomorphisms of trace monoids[END_REF][START_REF] Rodaro | Fixed points of endomorphisms of trace monoids[END_REF], the authors consider endomorphisms of trace monoids. They study the submonoid of fixed points of such an endomorphism and prove that it is finitely generated. They also caracterise those endomorphisms that are contractions relatively to some natural distance on trace monoids. In [START_REF] Crisp | Symmetrical subgroups of Artin groups[END_REF], Crisp obtained similar results for Artin monoids. He proved that the submonoid of fixed points is even finitely presented, but only in the special case of an isomorphism. Here, we aim to unified and extend both results to all Artin monoids and, more generally, to the larger classes of preGarside monoids.

Given a monoid M , we denote by End(M ) the endomorphism monoid of M . For ϕ ∈ End(M ), we say that x ∈ M is a fixed point of ϕ if ϕ(x) = x. If ϕ n (x) = x for some n ≥ 1, we say that x is a periodic point of ϕ. Let Fix(ϕ) (respectively Per(ϕ)) denote the submonoid of all fixed points (respectively periodic points) of ϕ. Clearly,

Per(ϕ) = n≥1 Fix(ϕ n ).
Our First result is the following:

Theorem 0.1.

(i) If M is a finitely generated left preGarside monoid, with an additive and homogeneous norm ν, and ϕ is in End(M ) such that the morphism π is well-defined, then Fix(ϕ) and Per(ϕ) are also finitely generated left preGarside monoids.

(ii) if M is an Artin monoid, and ϕ is in End(M ), then Fix(ϕ) and Per(ϕ) are also Artin monoids.

Similar result holds if one considers right preGarside monoids.

Two distances d 2 and d 3 on trace monoids have been introduced in [START_REF] Bonizzoni | About infinite traces[END_REF] and [START_REF] Kwiatkowska | A metric for traces[END_REF]. They were proved to be uniformaly equivalent in [START_REF] Kummetz | The topology of mazurkiewicz traces[END_REF]. In [START_REF] Rodaro | Fixed points of endomorphisms of trace monoids[END_REF], the authors caracterise those endomorphisms that are contractions relatively to d 2 . We will prove that in the general context of Artin monoids, distance d 2 and d 3 are no more uniformaly equivalent in general. Indeed, distance d 2 should be replace by an alternative one, that we denote by d 1 . We prove that d 1 is larger than the other two in the general case, and we will also prove that d 1 and d 3 are uniformly equivalent in the case of Artin monoids of spherical type.

Finally, replacing d 2 by d 1 , we extend [START_REF] Rodaro | Fixed points of endomorphisms of trace monoids[END_REF]Theorem 4.1], to all Artin monoids in Theorem 4.2 (we refer to the next sections for notations).

Theorem 0.2. Let M be an Artin monoid and ϕ be in End(M ).

(i) ϕ is a contraction with respect to d 1 ;

(ii) for all u, v ∈ M red , α(uv) = u ⇒ α(ϕ(uv)) = α(ϕ(u)).

Preliminaries

We start in this section by defining the monoids we will work with in this paper, namely left preGarside monoids and Artin monoids.

Consider a monoid M . It is said to be cancellative if, for all a, b, c, d ∈ M , the equality cad = cbd imposes a = b. An element b is called a factor of an element a if we can write a = cbd in M . We denote by Div(a) the set of factors of a. We denote by left divisibility in M (that is, for a, b ∈ M , we have a b when there exists c ∈ M such that b = ac). Right divisibility (defined similarly : b a when there exists c ∈ M such that b = ca) will rarely be used in this paper, so divisibility in M will simply mean left divisibility. When a is a left divisor of b in M , we say that b is a right multiple of a. An element a is said to be balanced if its sets of right-divisors and of left-divisors are equal, which in this case have to be equal to Div(a).

We say that M is atomic if there exists a mapping ν : M → N, called a norm, satisfying ν(a) > 0 for a = 1 and ν(ab) ≥ ν(a) + ν(b) for all a, b ∈ M . Note that the existence of such a mapping implies that the relations and are partial orders on M . When ν(ab) = ν(a) + ν(b) for all a, b ∈ M , we say that ν is additive. An atom in a monoid is an element a ∈ M satisfying: a = bc ⇒ b = 1 or c = 1 for all b, c ∈ M . We denote by S(M ) the set of atoms of M . Note that in an atomic monoid M , the set S(M ) has to be a generating set, and that any generating set of M contains S(M ). In particular, M is finitely generated if and only if S(M ) is finite. In an atomic monoid M , if ν(a) = ν(b) for all a, b ∈ S(M ), we say that ν is homogeneous. And it is said to be a preGarside monoid if it is both left preGarside and right preGarside monoid.

A monoid M is said to be a left preGarside monoid if (a L ) it is atomic and left cancellative (ca = cb ⇒ a = b); (b L ) for all a, b ∈ M , if the set {c ∈ M | a c
A Garside element of a preGarside monoid is a balanced element whose set of factors generates the whole monoid. When such an element exists, we say that the monoid is a Garside monoid.

Given a non empty finite set S, a Coxeter matrix is a symmetric matrix (m ab ) a,b∈S with entries in {1, 2, • • • , ∞}, such that m aa = 1 and m ab ≥ 2, for a = b. A Coxeter system associated to a Coxeter matrix (m ab ) a,b∈S is a pair (W, S), where W is the group with presentation

W = S | (ab) m ab = 1; m ab = ∞ .
The corresponding Artin monoid M is the monoid with presentation

M = S | [a, b m ab = [b, a m ab ; m ab = ∞ + .
where [a, b m denotes the alternating product aba • • • containing m terms. Since the defined Artin relations are homogeneous, M has a natural length function ℓ S compatible with the product. In [START_REF] Brieskorn | [END_REF], it is shown that every finite subset of M has a greatest common left divisor (gcd), and a greatest common right divisor (gcrd). It is also shown that a finite subset of S has a least common right multiple (lcm) if and only if it has a common right multiple, and that in that case, the least common right multiple and least common left multiple are equal. For a subset T ⊆ S, we denote its lcm by ∆(T ) when it exists.

The submonoids of fixed points and of periodic points

In general, a fixed points submonoid or any submonoid of a finitely generated monoid is not necessarily finitely generated. Clearly, a n ba n ∈ Fix(ϕ) for every n ∈ N. And one can show that a n ba n is not decomposable in Fix(ϕ), which means that Fix(ϕ) is not finitely generated.

In this section, we will show that for a finitely generated left preGarside monoid M , and an endomorphism ϕ ∈ End(M ), the submonoids Fix(ϕ) and Per(ϕ) are also finitely generated left preGarside monoids under some condition (Theorems 2.3 and 2.13). And that, in the particular case of Artin monoids, Fix(ϕ) and Per(ϕ) are not only finitely generated, but even finitely presented (Proposition 2.15).

For a monoid M , finitely generated by S, and an endomorphism ϕ ∈ End(M ), we define

n ϕ = max{k ∈ N * | ∃s ∈ S such that ϕ k (s) = 1 and ϕ k-1 (s) = 1} if 1 ∈ ϕ(S) 1 if 1 / ∈ ϕ(S) .
Given a non empty set X, we denote by X * the set of all finite words x 1 • • • x n over the elements of X, that we call letters, and by ε the empty word in X * . Assume M is a monoid generated by a set S, and let T ⊆ S. Let us denote by π * T : S * → T * the forgetting morphism of monoids defined by π * T (t) = t for t ∈ T , and by π * T (t) = ε for t ∈ S \ T . In the sequel, when it is well-defined, we denote by π T : M → M the morphism of monoids induced by π * T .

Example 2.2. For M = s, t | sts = tst + , with S = {s, t} and T = {t}, we have π * T (sts) = t and π * T (tst) = tt. Or in M , one has t = t 2 , then π T is not well-defined. But for M = s, t | stst = tsts + with similar S and T , the morphism π T is well-defined, since π T (stst) = π T (tsts) = t 2 .

The submonoid of fixed points

For this subsection, let M be a finitely generated left preGarside monoid, equiped with an additive and homogeneous norm ν, and ϕ be in End(M ). Set S = S(M ), S 0 = S ∩ Per(ϕ), S 1 = S ∩ (ϕ nϕ ) -1 {1}, S 2 = S \ S 1 , p = |S|!, and π := π S 2 when it is well-defined, which is always the case when 1 / ∈ ϕ(S).

Theorem 2.3. Let M be a finitely generated left preGarside monoid, with an additive and homogeneous norm ν, ϕ be in End(M ), and π as above. If the morphism π is well-defined, then Fix(ϕ) is a finitely generated left preGarside monoid.

The proof of this theorem is in the spirit of [START_REF] Rodaro | Fixed points of endomorphisms of trace monoids[END_REF]Theorem 3.1], where the particular case of trace monoids was considered. We will start though by proving some lemmas. Note that in the proof of the next lemma, where ϕ| S is a permutation (i.e. ϕ is an automorphism), we do not need the additivity or the homogeneity of the norm ν. In the case of preGarside monoids, this result is shown in [START_REF] Bessis | Springer theory in braids groups and the Birman-Ko-Lee monoid[END_REF]Proposition 2.26].

Lemma 2.4. If the restriction of ϕ to S is a permutation, then Fix(ϕ) is a finitely generated left preGarside monoid.

Proof. Since Fix(ϕ) ⊆ M , it is clear that property (a L ) holds for Fix(ϕ). Let a, b ∈ Fix(ϕ) be such that the set {c ∈ Fix(ϕ) | a c and b c} is nonempty, and set δ = a ∨ b their (left) lcm in M . We have ϕ(a) = a ϕ(δ) and ϕ(b) = b ϕ(δ), then δ ϕ(δ). Thus δ ϕ(δ) • • • ϕ p (δ), and so ν(δ) ≤ ν(ϕ(δ)) ≤ • • • ≤ ν(ϕ p (δ)).
Or ϕ| S is a permutation, and p = |S|!, then ϕ p = Id M . Therefore ν(δ) = ν(ϕ(δ)), with δ ϕ(δ). Thus, we have δ = ϕ(δ), and so δ ∈ Fix(ϕ). Let c ∈ Fix(ϕ) such that a c and b c. Write c = δδ ′ with δ ′ ∈ M . We have c, δ ∈ Fix(ϕ), then, by cancellativity, δ ′ ∈ Fix(ϕ). Hence property (b L ) holds, and so Fix(ϕ) is a left preGarside monoid.

Let Σ be the set of all ϕ-orbits B in S that have a right common multiple (and therefore a least right common multiple ∆(B) ([6, Lemma 2.1])). We claim that

Fix(ϕ) = ∆(B), B ∈ Σ + . (1)
Let B ∈ Σ, and b ∈ B. Since B is a ϕ-orbit, then there exists a ∈ B such that b = ϕ(a). But a ∆(B), so b = ϕ(a) ϕ(∆(B)). Thus b ϕ(∆(B)) for every b ∈ B, and then ∆(B) ϕ(∆(B)). Therefore, as above, we have

∆(B) = ϕ(∆(B)). Hence ∆(B), B ∈ Σ + ⊆ Fix(ϕ).
Conversely, let u ∈ Fix(ϕ) such that u = 1, and let s ∈ S such that s u; then u is left divisible by all the elements in the ϕ-orbit B 1 of s, so is left divisible by their lcm

u 1 := ∆(B 1 ) (which exists). Let v ∈ M such that u = u 1 v. Since u, u 1 ∈ Fix(ϕ), we have by cancellativity, v ∈ Fix(ϕ). By induction on ν(u), we get that v ∈ ∆(B) | B ∈ Σ + . Thus u ∈ ∆(B) | B ∈ Σ + and so (1) holds. The set S is finite, then so is Σ. Therefore Fix(ϕ) is finitely generated. Example 2.5. i. Let M = s, t, u | st = ts, sus = usu, tut = utu + , and ϕ ∈ End(M ) such that ϕ(u) = u, ϕ(t) = s and ϕ(s) = t.
The monoid M is an Artin monoid, so it satisfies all properties of preGarside monoids by [START_REF] Brieskorn | [END_REF]. Then we have Σ = {{u}, {s, t}}, and

Fix(ϕ) = u, st + . ii. Let M = a 1 , b 1 , a 2 , b 2 | a 1 b 1 a 1 = b 2 1 , a 2 b 2 a 2 = b 2 2 , a 1 a 2 = a 2 a 1 , b 1 b 2 = b 2 b 1 , a 1 b 2 = b 2 a 1 , b 1 a 2 = a 2 b 1 + , and ϕ ∈ End(M ) such that ϕ(a 1 ) = a 2 , ϕ(a 2 ) = a 1 , ϕ(b 1 ) = b 2 and ϕ(b 2 ) = b 1 . The monoid M is a preGarside monoid, as a direct product M = a 1 , b 1 | a 1 b 1 a 1 = b 2 1 + × a 2 , b 2 | a 2 b 2 a 2 = b 2 2
+ of two preGarside monoids (see [START_REF] Dehornoy | Gaussian groups and Garside groups, two generalisations of Artin groups[END_REF]). Then we have

Σ = {{a 1 , a 2 }, {b 1 , b 2 }}, and Fix(ϕ) = a 1 a 2 , b 1 b 2 + . Lemma 2.6. If 1 / ∈ ϕ(S), then S 0 = S ∩ Fix(ϕ p ) = S ∩ ϕ p (S), (2)
and ϕ(S 0 ) = S 0 .

(3)

Proof. Recall that S 0 = S ∩ Per(ϕ). Let a ∈ S 0 and m = min{k ∈ N * /ϕ k (a) = a}. Since 1 / ∈ ϕ(S) and ν is additive and homogeneous, we have ν(ϕ(u)) ≥ ν(u) for every u ∈ M . Hence ϕ n (a) ∈ S for every n ∈ N. Assume that there exist Let a ∈ S ∩ ϕ p (S) and b ∈ S such that ϕ p (b) = a. Since 1 / ∈ ϕ(S), using that ν is additive and homogeneous, and

0 < i < j ≤ m such that ϕ i (a) = ϕ j (a). So ϕ m-j • ϕ i (a) = ϕ m (a). Then ϕ m-(j-i) (a) = a and m -(j -i) < m, which contradicts the definition of m. Thus m = #{a, ϕ(a), • • • , ϕ m-1 (a)} ≤ |S|,
ϕ p (b) = a, we get {b, ϕ(b), • • • , ϕ p (b)} ⊆ S. The inequality |S| < p + 1 yields ϕ i (b) = ϕ j (b)
for some 0 ≤ i < j ≤ p. By composing with ϕ p-i , we get a = ϕ j-i (a). Then a ∈ Per(ϕ) ∩ S = S 0 , and therefore (2) holds.

Let a ∈ S 0 . As before, we have {a, ϕ(a), . . . , ϕ p-1 (a)} ⊆ S 0 = S ∩ Fix(ϕ p ). On the one hand, we have ϕ(a) ∈ S 0 , then ϕ(S 0 ) ⊆ S 0 . On the other hand, we have a = ϕ(ϕ p-1 (a)) and ϕ p-1 (a) ∈ S 0 , then ϕ(S 0 ) ⊇ S 0 , and so (3) holds.

Let M 0 , M 1 and M 2 , be the submonoids of M generated by S 0 , S 1 and S 2 respectively. By definition of S 1 and atomicity of M , note that ϕ nϕ (M 1 ) = {1} and ϕ(M 1 ) ⊆ M 1 , which we will be using more than once.

Lemma 2.7. If 1 / ∈ ϕ(S), then M 0 = Fix(ϕ p )
, it is a finitely generated left preGarside monoid, and the restriction of ν to M 0 is additive and homogeneous.

Proof. The order of every periodic point of S divides p = |S|!, then M 0 ⊆ Fix(ϕ p ). Let s 1 , . . . , s n ∈ S such that ϕ p (s

1 • • • s n ) = s 1 • • • s n .
In view of the homogeneity and additivity of ν, and the fact that 1 / ∈ ϕ(S), we have ϕ(s i ) ∈ S for all i. By (2), S 0 = S ∩ ϕ p (S). Then ϕ(s

1 • • • s n ) ∈ S 0 + , which means that s 1 • • • s n ∈ M 0 . Thus we have the equality M 0 = Fix(ϕ p ).
The submonoid M 0 is finitely generated by definition, and it is atomic and left cancellative because M 0 ⊆ M . The restriction ν| M 0 of ν is additive, and since S(M 0 ) = S 0 , it is also homogeneous. Let a, b lie in M 0 such that the set Γ 0 = {c ∈ M 0 | a c and b c} is nonempty, and set δ = a ∨ b their lcm in M . Set ψ = ϕ p . Since S 0 = S ∩ Fix(ψ), we have ψ| M 0 = Id M 0 . Let c lie in Γ 0 and a ′ , a ′′ belong to M such that c = aa ′ and δ = aa ′′ . One has a, c ∈ Fix(ψ), then a ′ = ψ(a ′ ) by cancellativity. We have δ c, then a ′′ a ′ , and so ψ n (a ′′ ) a ′ for all n ∈ N. The sequence of integers (ν(ψ n (a ′′ ))) n∈N is increasing because of the additivity and homogeneity of ν and the fact that 1 / ∈ ϕ(S). On the other hand, it is bounded by ν(a ′ ) because ψ n (a ′′ ) a ′ for all n.

Thus (ν(ψ n (a ′′ ))) n∈N is stationary from some rank m a ∈ N * . Write ψ ma (a ′′ ) = s 1 • • • s r with s 1 , . . . , s r ∈ S. Since ν(ψ ma (a ′′ )) = ν(ψ ma+1 (a ′′ )), then ψ(s i ) ∈ S for all i. So {s i , ϕ(s i ), . . . , ϕ p (s i )} ⊆ S, and therefore ψ(s i ) ∈ S 0 for all i. Thus ψ ma+1 (a ′′ ) ∈ M 0 . Similarly, for b ′ , b ′′ ∈ M such that c = bb ′ and δ = bb ′′ , we have some rank m b ∈ N * such that ψ m b +1 (b ′′ ) is in M 0 . The inclusion M 0 ⊆ Fix(ψ) yields ψ ma+1 (a ′′ ) = ψ ma+m b (a ′′ ) and ψ m b +1 (b ′′ ) = ψ m b +ma (b ′′ ). We have ψ ma+m b (δ) = aψ ma+1 (a ′′ ) = bψ m b +1 (b ′′ ), then ψ ma+m b (δ) ∈ Γ 0 . If c ∈ Γ 0 , then δ c, and so ψ ma+m b (δ) ψ ma+m b (c) = c. Write c = ψ ma+m b (δ)c ′ with c ′ ∈ M . Both c and ψ ma+m b (δ) are in Fix(ψ), then by cancellativity, ψ(c ′ ) = c ′ . Thus, c ′ ∈ M 0 and ψ ma+m b (δ) is the least element of Γ 0 . Whence property (b L ) holds, so M 0 is a left preGarside monoid.
Lemma 2.8. If π is well-defined, then M 2 is a finitely generated left preGarside monoid, and the restriction of ν to M 2 is additive and homogeneous.

Proof. As in the previous proof, the submonoid M 2 is atomic, left cancellative and finitely generated, and the restriction ν| M 2 is additive and homogeneous. Let a, b lie in M 2 such that the set Γ 2 = {c ∈ M 2 | a c and b c} is nonempty, and set δ = a ∨ b their lcm in M . Let c ∈ Γ 2 and a ′ , a ′′ ∈ M such that c = aa ′ and δ = aa ′′ . We have δ c, then a ′′ a ′ . So write a ′ = a ′′ â with â ∈ M . By the homogeneity and additivity of ν, for all u ∈ M we have ν(π(u)) ≤ ν(u), and ν(π

(u)) = ν(u) ⇔ π(u) = u ⇔ u ∈ M 2 . We have ν(c) = ν(a) + ν(a ′′ ) + ν(â) and ν(π(c)) = ν(π(a)) + ν(π(a ′′ )) + ν(π(â)). But a and c belong to M 2 , then ν(a ′′ ) + ν(â) = ν(π(a ′′ )) + ν(π(â)). So ν(π(a ′′ )) = ν(a ′′ ) because ν(π(u)) ≤ ν(u) for all u ∈ M . Then a ′′ ∈ M 2 .
Similarly, there is b ′′ in M 2 such that δ = bb ′′ . Thus δ is the least element of Γ 2 , whence property (b L ). Therefore M 2 is a left preGarside monoid. Lemma 2.9. Let N 1 be a left preGarside monoid, and N 2 a monoid. Assume there exists a morphism f :

N 1 → N 2 that is a retraction. Then N 2 is left preGarside.
Proof. The morphism f is a retraction, then we have a section g :

N 2 → N 1 such that f • g = Id N 2 . Thus N 2 embeds in N 1 . So N 2 is
δ = g(a)a ′ = g(b)b ′ with a ′ , b ′ ∈ N 1 . Thus f (δ) = af (a ′ ) = bf (b ′ ). Let c ∈ Λ and c ′ ∈ N 1 such that g(c) = δc ′ . Then c = f (δ)f (c ′ )
, and so f (δ) divides c in N 2 . Hence, f (δ) is the least element of Λ, and therefore N 2 is left preGarside.

Lemma 2.10. For every d ∈ N * , we have

(π • ϕ) d = π • ϕ d . Proof. If 1 / ∈ ϕ(S)
∈ S 1 , then ϕ d (a) ∈ M 1 = S 1 + because ϕ(M 1 ) ⊆ M 1 . Thus, (π • ϕ) d (a) = π • ϕ d (a) = 1. If a ∈ S 2 , write ϕ(a) = u 0 a 1 u 1 • • • a k u k ,
with a 1 , . . . , a k ∈ S 2 and u 0 , . . . , u k ∈ M 1 . On the one hand, we have

π • ϕ d (a) = π • ϕ d-1 (u 0 a 1 u 1 • • • a k u k ) = π • ϕ d-1 (a 1 • • • a k )
in view of ϕ d-1 (u i ) ∈ M 1 for every i. On the other hand, and by the induction hypothesis,

(π • ϕ) d (a) = (π • ϕ) d-1 • π • ϕ(a) = π • ϕ d-1 • π(u 0 a 1 u 1 • • • a k u k ) = π • ϕ d-1 (a 1 • • • a k ). Thus (π • ϕ) d (a) = π • ϕ d (a) for every d ∈ N * .
In view of the previous lemmas, we may now prove Theorem 2.3 in two parts, depending on whether 1 lies in ϕ(S) or not.

Proof of Theorem 2.3. Case I: 1 / ∈ ϕ(S). By the equality (3) in Lemma 2.6, the morphism ϕ restricts to an endomorphism ϕ 0 of M 0 = S 0 + . We show that

Fix(ϕ) = Fix(ϕ 0 ). (4)
It is immediate that Fix(ϕ 0 ) = Fix(ϕ) ∩ M 0 , so it suffices to show that Fix(ϕ)

⊆ M 0 . Let u = a 1 • • • a k belong to Fix(ϕ), with a 1 , . . . , a k ∈ S. Then a 1 • • • a k = u = ϕ p (u) = ϕ p (a 1 ) • • • ϕ p (a k ). Since 1 / ∈ ϕ(S)
, and ν is additive and homogeneous, we have ν(ϕ p (a i )) = ν(a i ) and so ϕ p (a i ) ∈ S for all i. By Lemma 2.6, we have S 0 = S ∩ ϕ p (S), so ϕ p (a i ) ∈ S 0 for all i, and then u = ϕ p (a 1 ) • • • ϕ p (a k ) ∈ M 0 . Therefore, Fix(ϕ) ⊆ M 0 , and so Fix(ϕ) = Fix(ϕ 0 ). Now ϕ 0 | S 0 is a permutation, and by Lemma 2.7, M 0 is a finitely generated left preGarside monoid. Then by Lemma 2.4 (where in the case of a permutation, the norm of M 0 does not have to be additive or homogeneous), Fix(ϕ 0 ), and therefore Fix(ϕ), is a finitely generated left preGarside monoid.

Case II: 1 ∈ ϕ(S). Denote n = n ϕ , and recall that π = π S 2 . Consider the morphism ϕ 2 = (π • ϕ)| M 2 that is clearly in End(M 2 ). We have 1 / ∈ ϕ 2 (S 2 ). Indeed, if ϕ 2 (s) = π(ϕ(s)) = 1 for some s ∈ S 2 , then ϕ(s) ∈ M 1 , which means that ϕ n (ϕ(s)) = ϕ n+1 (s) = 1. But since n = max{k ∈ N * | ∃s ∈ S such that ϕ k (s) = 1 and ϕ k-1 (s) = 1}, then ϕ n (s) = 1, which contradicts the fact that s ∈ S 2 . Thus, by Lemma 2.8 and Case I, Fix(ϕ 2 ) is a finitely generated left preGarside monoid.

We claim that Fix(ϕ) = ϕ n (Fix(ϕ 2 )).

(

) 5
As seen before, we have ϕ n (M 1 ) = {1} and ϕ(M 1 ) ⊆ M 1 . Let u ∈ Fix(ϕ). We may factor u =

u 0 a 1 u 1 • • • a k u k , with a 1 , . . . , a k ∈ S 2 and u 0 , . . . , u k ∈ M 1 . It follows that u = ϕ n (u) = ϕ n (a 1 a 2 • • • a k ).
For every i, we have π

• ϕ(u i ) = 1, in view of ϕ(M 1 ) ⊆ M 1 . Now a 1 a 2 • • • a k ∈ M 2 ,
and

ϕ 2 (a 1 a 2 • • • a k ) = π • ϕ(a 1 a 2 • • • a k ) = π • ϕ(u 0 a 1 u 1 • • • a k u k ) = π(u) = a 1 a 2 • • • a k . Hence a 1 a 2 • • • a k ∈ Fix(ϕ 2 ), and so u = ϕ n (a 1 a 2 • • • a k ) ∈ ϕ n (Fix(ϕ 2 )). Thus Fix(ϕ) ⊆ ϕ n (Fix(ϕ 2 )). Conversely, let v = a 1 a 2 • • • a k ∈ Fix(ϕ 2 )
, with a 1 , . . . , a k ∈ S 2 . Clearly,

ϕ n • π = ϕ n . (6)
Hence v = ϕ 2 (v) = π • ϕ(v) yields ϕ(ϕ n (v)) = ϕ n • ϕ(v) = ϕ n • π • ϕ(v) = ϕ n (π • ϕ(v)) = ϕ n (v) and so ϕ n (v) ∈ Fix(ϕ). Thus ϕ n (Fix(ϕ 2 )) ⊆ Fix(ϕ) and so Fix(ϕ) = ϕ n (Fix(ϕ 2 )).
In view of (5), we have a morphism f := ϕ n | Fix(ϕ 2 ) : Fix(ϕ 2 ) → Fix(ϕ). Let u ∈ Fix(ϕ). By Lemma 2.10, we get

ϕ 2 (π(u)) = π •ϕ•π(u) = π •ϕ•π •ϕ(u) = π •ϕ 2 (u) = π(u). Then π(Fix(ϕ)) ⊆ Fix(ϕ 2 )
, and so we have another morphism g := π| Fix(ϕ) : Fix(ϕ) → Fix(ϕ 2 ). In view of ( 6), one has ϕ n • π(u) = ϕ n (u) = u for every u ∈ Fix(ϕ), and then f • g = Id| Fix(ϕ) . So the morphism f is a retraction with section g. We established that Fix(ϕ 2 ) is a left preGarside monoid. So by Lemma 2.9, Fix(ϕ) is also a left preGarside monoid. The submonoid Fix(ϕ 2 ) is finitely generated, then so is Fix(ϕ), in view of (5).

We will see in the proof of Proposition 2.15 that the equality (5) induces an isomorphism between Fix(ϕ) and Fix(ϕ 2 ).

The submonoid of periodic points

As in the previous subsection, we consider a finitely generated left preGarside monoid M , equiped with an additive and homogeneous norm ν, and we fix ϕ ∈ End(M ). We also set n = n ϕ , S = S(M ), S 0 = S ∩ Per(ϕ), S 1 = S ∩ (ϕ n ) -1 {1}, S 2 = S \ S 1 , p = |S|!, and π := π S 2 when it is well-defined. Proposition 2.12. If the morphism π is well-defined, then we have

Per(ϕ) = Fix(ϕ pn ).
The proof of this proposition is also in the spirit of [START_REF] Rodaro | Fixed points of endomorphisms of trace monoids[END_REF]Theorem 3.2], where the particular case of trace monoids was considered.

Proof. Case I: 1 / ∈ ϕ(S). By definition, n = 1 in this case. We will use induction on |S|. The case |S| = 0 being trivial, assume that |S| > 0 and the result holds for smaller sets.

We may assume S 0 S, otherwise ϕ| S would be a permutation, and since the order of ϕ| S must divide the order of the symmetric group on S, which is p, we would get (ϕ| S ) p = Id S and therefore ϕ p = Id M , yielding Fix(ϕ p ) = M = Per(ϕ).

For every r ∈ N * , if we replace ϕ by ϕ r , then S 0 remains the same in view of Per(ϕ) = Per(ϕ r ), and so does M 0 . On the other hand, by (3), we restrict ϕ to ϕ 0 = ϕ| M 0 , and we have ϕ

r | M 0 = (ϕ| M 0 ) r = ϕ r 0 . Hence Fix(ϕ r ) = Fix(ϕ r 0 ) (7)
by applying (4) to ϕ r . By the induction hypothesis and Lemma 2.7, we have Per(ϕ 0 ) = Fix(ϕ

|S 0 |! 0 ). Since |S 0 |! divides p, we get Per(ϕ 0 ) = Fix(ϕ |S 0 |! 0 ) ⊆ Fix(ϕ p 0 ) ⊆ Per(ϕ 0
) and so Per(ϕ 0 ) = Fix(ϕ p 0 ). Together with [START_REF] Kummetz | The topology of mazurkiewicz traces[END_REF], this yields

Per(ϕ) = ∪ r≥1 Fix(ϕ r ) = ∪ r≥1 Fix(ϕ r 0 ) = Per(ϕ 0 ) = Fix(ϕ p 0 ) = Fix(ϕ p )
as required.

Case II: 1 ∈ ϕ(S). By definition, we have Per(ϕ) ⊇ Fix(ϕ pn ). Conversely, let u ∈ Per(ϕ), say u ∈ Fix(ϕ r ). We may factor u

= u 0 a 1 u 1 • • • a k u k , with a 1 , . . . , a k ∈ S 2 and u 0 , . . . , u k ∈ M 1 . It follows that u = ϕ rn (u) = ϕ rn (a 1 a 2 • • • a k ). Now a 1 a 2 • • • a k ∈ M 2 , and Lemma 2.10 yields a 1 a 2 • • • a k = π • ϕ rn (a 1 a 2 • • • a k ) = (π•ϕ) rn (a 1 a 2 • • • a k ). Consequently a 1 a 2 • • • a k belongs to Fix(ϕ rn 2 ) ⊆ Per(ϕ 2 ).
As in the proof of Theorem 2.3, we have 1 / ∈ ϕ 2 (S 2 ). Thus, by Lemma 2.8 and Case I, we have Per(ϕ 2 ) = Fix(ϕ

|S 2 |! 2 ). We get a 1 a 2 • • • a k ∈ Fix(ϕ |S 2 |! 2 ) ⊆ Fix(ϕ pn
2 ), and so

a 1 a 2 • • • a k = π • ϕ pn (a 1 a 2 • • • a k ) in view of Lemma 2.10. Hence ϕ pn (u) = ϕ pn (a 1 a 2 • • • a k ) = v 0 a 1 v 1 • • • a k v k for some v 0 , v 1 , . . . , v k in M 1 . Thus ϕ 2pn (u) = ϕ pn • π • ϕ pn (u) = ϕ pn • π(v 0 a 1 v 1 • • • a k v k ) = ϕ pn • π(u 0 a 1 u 1 • • • a k u k ) = ϕ pn (u). Since ϕ r (u) = u, this yields to u = ϕ r (u) = ϕ 2r (u) = • • • = ϕ pnr (u) = ϕ pn(r-1) (u) = • • • = ϕ pn (u).
Therefore Per(ϕ) = Fix(ϕ pn ). Theorem 2.13. If the morphism π is well-defined, then Per(ϕ) is also a finitely generated left preGarside monoid.

Proof. In Proposition 2.12, we showed that Per(ϕ) = Fix(ψ), where ψ = ϕ pn . Denote

S 1 (ϕ) = S 1 , S 1 (ψ) = S ∩ (ψ n ψ ) -1 {1}, S 2 (ϕ) = S 2 , S 2 (ψ) = S \ S 1 (ψ), π(ϕ) = π and π(ψ) = π S 2 (ψ) . We have S 1 (ϕ) = {s ∈ S | ∃k ∈ N such that ϕ k (s) = 1} = {s ∈ S | ∃k ∈ N such that ψ k (s) = 1} = S 1 (ψ). Then S 2 (ψ) = S 2 (ϕ)
, and so π(ψ) = π(ϕ) = π. Thus we can apply Theorem 2.3 to ψ, which means that Fix(ψ), and therefore Per(ϕ), is a finitely generated left preGarside monoid.

The case of Artin monoids

A symmetry of an Artin group A generated by S, is an endomorphism ϕ of A such that ϕ |S is a permutation. In [4, Lemma 10] and [START_REF] Michel | A note on braid monoids[END_REF]Corollary 4.4], it is shown that, given a group G of symmetries of an Artin group A, the submonoid of elements fixed by G, is isomorphic to another Artin monoid. In particular, given an Artin monoid M generated by S, and ϕ ∈ End(M ) such that ϕ |S is a permutation (i.e. ϕ ∈ Aut(M )), the submonoid Fix(ϕ) is also an Artin monoid. Below, we will show that this is also the case for Per(ϕ), and for every ϕ ∈ End(M ).

Let M = S | [a, b m ab = [
b, a m ab ; m ab = ∞ + be an Artin monoid, and ϕ be in End(M ). By [START_REF] Brieskorn | [END_REF], Artin monoids satisfy all properties of preGarside monoids. The set of atoms S(M ) of M is S, and the length ℓ S is an additive and homogeneous norm over M . Thus, we can apply the results from the previous subsections. As before, set n = n ϕ , S 0 = S ∩ Per(ϕ), S 1 = S ∩ (ϕ n ) -1 {1}, S 2 = S \ S 1 , p = |S|!, and π := π S 2 when it is well-defined. It is known that the submonoids M 0 = S 0 + , M 1 = S 1 + and M 2 = S 2 + are Artin monoids too. Proposition 2.15. Let M be an Artin monoid, and ϕ be in End(M ). Then the submonoids Fix(ϕ) and Per(ϕ) are also Artin monoids.

Proof. Assume first 1 / ∈ ϕ(S). In the proof of Theorem 2.3, we showed that Fix(ϕ) = Fix(ϕ 0 ), with ϕ 0 ∈ End(M 0 ) and ϕ| S 0 is a permutation. Then by [4, Lemma 10], Fix(ϕ 0 ), and therefore Fix(ϕ), is an Artin monoid. In this case, n = 1, and by Proposition 2.12, we have Per(ϕ) = Fix(ϕ p ). Since 1 / ∈ ϕ(S), then ℓ S (ϕ(u)) ≥ ℓ S (u) for all u ∈ M , and so 1 / ∈ ϕ p (S). Thus, Fix(ϕ p ), and therefore Per(ϕ), is again an Artin monoid. Assume now 1 ∈ ϕ(S). In the proof of Theorem 2.3, we showed that Fix(ϕ) = ϕ n (Fix(ϕ 2 )), with ϕ 2 ∈ End(M 2 ) and 1 / ∈ ϕ 2 (S 2 ). Let u, v ∈ Fix(ϕ 2 ) such that ϕ n (u) = ϕ n (v). Then π • ϕ n (u) = π • ϕ n (v), and so, by Lemma 2.10,

(π • ϕ) n (u) = (π • ϕ) n (v). Thus u = ϕ n 2 (u) = ϕ n 2 (v) = v.
Hence, the morphism ϕ n | Fix(ϕ 2 ) : Fix(ϕ 2 ) → Fix(ϕ) is not only surjective, but also injective. Therefore, Fix(ϕ) is isomorphic to Fix(ϕ 2 ). By Case I, Fix(ϕ 2 ) is an Artin monoid, then so is Fix(ϕ). By Proposition 2.12, we have Per(ϕ) = Fix(ϕ pn ). Since 1 ∈ ϕ(S), one has 1 ∈ ϕ pn (S). Thus Fix(ϕ pn ), and so Per(ϕ), is an Artin monoid.

Inequalities and some equivalences between metrics

The purpose of this section is to define three metrics d 1 , d 2 and d 3 in finitely generated preGarside monoids, to compare them in general, and in the particular cases of trace monoids and Garside monoids.

Metrics and normal forms

In order to define our three metrics, we start by introducing the following general framework. Recall that given a non empty set X, we denote by X * the set of all finite words over X. Henceforth, these words will be denoted as tuples, to avoid any confusion with the monoids elements. Let M be a monoid, X be a non empty set, and ι : M ֒→ X * be an injective map. For u, v ∈ M with ι(u) = (u 1 , . . . , u n ) and ι(v) = (v 1 , . . . , v m ), we define

r(u, v) = max{k ≥ 0 | u 1 = v 1 , . . . , u k = v k } if u = v ∞ if u = v .
The metric d over M , associated to ι, is defined, for all u, v ∈ M , by v) .

d(u, v) = 2 -r(u,
When ι(u) = (u 1 , . . . , u n ) for some u ∈ M , then for all k ≤ n, we denote ι [k] (u) = (u 1 , . . . , u k ) ∈ X * .

Let M be a finitely generated preGarside monoid. For each metric d i over M , we will define X i , ι i and r i as above. The set X 1 for the first distance d 1 is defined in [START_REF] Bessis | Springer theory in braids groups and the Birman-Ko-Lee monoid[END_REF], where it is denoted by P ; the subset of M with a preGarside structure. It contains the finite set of atoms S = S(M ), and whenever it contains an element, it also contains all its left and right divisors ([1, Proposition 2.4]). We will denote it by M red , since in the case of an Artin monoid, it is just the set of reduced elements, that we will recall bellow. The properties of M red shown in [START_REF] Michel | A note on braid monoids[END_REF] for Artin monoids, hold in finitely generated preGarside monoids with the same proofs, as stated in [START_REF] Bessis | Springer theory in braids groups and the Birman-Ko-Lee monoid[END_REF]. Namely ([1, Proposition 2.12]), there is a unique function α : M → M red which induces the identity on M red , and satisfies There is also a length function on W (see [START_REF] Michel | A note on braid monoids[END_REF]), which we denote also by ℓ S . It is known that two minimal expressions of an element of W are equivalent by using Artin relations only. The length of an element is defined by the length of any of its minimal expressions as products of elements of S. This implies that the induced quotient map from M to W has a canonical section (as a map of sets), whose image M red consists of those elements of M which have the same length as their image in W .

α(uv) = α(uα(v)), (8)
Let M be a finitely generated preGarside monoid. To every element of M , can be associated a (left) normal form (n.f ), that is called the (left) greedy normal form, and defined as follows. To 1 M , we associate the empty sequence. And for u ∈ M \ {1} and u 1 , . . . , u n ∈ M red , we say that u = u 1 • • • u n is in normal form (n.f), if and only if no u i is equal to 1 and for any i we have u i = α(u i • • • u n ). In view of (8), the normality of a form can be seen locally ([1, Proposition 2.21]): u 1 • • • u k is a normal form if and only if u i u i+1 is for all i. This implies that any segment u i • • • u j of a normal form is normal. For u = u 1 • • • u n (n.f), we define ι 1 (u) = (u 1 , . . . , u n ), and denote n = |u| 1 . Let u, v ∈ M with ι 1 (u) = (u 1 , . . . , u n ) and ι 1 (v) = (v 1 , . . . , v m ). We define r 1 (u, v) exactly as r(u, v) above. Using the convention 2 -∞ = 0, the metric d 1 is defined by

d 1 (u, v) = 2 -r 1 (u,v) .
Another important normal form ι 2 over M , that we call the Foata normal form, is defined as follows. Let X 2 = {u ∈ M | ∃T ⊆ S, u = ∆(T )}, where S = S(M ) and ∆(T ) is the least right common multiple of the elements of T , which exists if and only if there is a right common multiple ([6, Lemma 2.1]). For u ∈ M \ {1}, there exists a unique ι

2 (u) = (u 1 , . . . , u n ) ∈ X * 2 such that u = u 1 • • • u n and u i = ∆({s ∈ S | s u i • • • u n }). When ι 2 (u) = (u 1 , . . . , u n ), we denote n = |u| 2 .
And similarly, the metric d 2 associated to ι 2 , is known as the FNF metric, and defined in [START_REF] Bonizzoni | About infinite traces[END_REF], for all u, v ∈ M , by

d 2 (u, v) = 2 -r 2 (u,v) .
When the monoid M is equiped with an additive and homogeneous norm ν, we can assume that ν(s) = 1 for all s in S, call this norm the length over S, and denote it by ℓ S . In this case, and in addition to d 1 and d 2 , there is a third and useful metric, decribed in [START_REF] Rodaro | Fixed points of endomorphisms of trace monoids[END_REF] for the particular case of trace monoids, that we will denote by d 3 . Given u, v ∈ M , we say that v is a prefix of u, when v left-divides u. For every n ∈ N, denote by Pref n (u) the set of all prefixes of u of length n. Let X 3 = P(M ) be the set of all parts of M . For u ∈ M \ {1}, set ι 3 (u) = (u 1 , . . . , u n ) with n = ℓ S (u) and u i = Pref i (u) for all i. And set ι 3 (1) = ({1}). Then the metric d 3 , known as the prefix metric, is defined in [START_REF] Kwiatkowska | A metric for traces[END_REF] as above, for all u, v ∈ M , by

d 3 (u, v) = 2 -r 3 (u,v) .
Proposition 3.4. Let M be a finitely generated preGarside monoid. Then we have

d 2 ≤ d 1 . Proof. Let u, v be in M and distinct. Set ι 1 (u) = (u 1 , . . . , u n ), ι 1 (v) = (v 1 , . . . , v n ′ ), ι 2 (u) = (u ′ 1 , . . . , u ′ m ) and ι 2 (v) = (v ′ 1 , . . . , v ′ m ′ )
. By Lemma 3.3, n ≤ m and n ′ ≤ m ′ . If m = r 2 (u, v) or m ′ = r 2 (u, v), then r 1 (u, v) ≤ min(n, n ′ ) ≤ min(m, m ′ ) = r 2 (u, v). Therefore d 2 (u, v) ≤ d 1 (u, v). So assume m < r 2 (u, v) and m ′ < r 2 (u, v) and set k = r 2 (u, v). By assumption u ′ k+1 = v ′ k+1 . We can therefore assume without restriction that u ′ k+1 does not left divide v ′ k+1 . By Lemma 3.3 i. and ii., ι If the identity mappings between (X, d) and (X, d ′ ) are uniformly continuous, we say that the metrics d and d ′ are uniformly equivalent. It is immediate that two equivalent metrics are consequently uniformly equivalent Below, we will show that in Garside monoids, d 1 is uniformly equivalent to d 3 , and in trace monoids, d 2 and d 3 are uniformly equivalent. However, these metrics are not equivalent, nor uniformly equivalent in general. Here are some examples to illustrate that. But in our example, r 3 = 2n and r 2 = n, so there is no C ′ in R such that r 3 ≤ r 2 + C ′ . Therefore, d 2 and d 3 cannot be equivalent.

iii. In a non-abelian Artin monoid M , the metric d 1 is not equivalent to d 2 , nor to d 3 . Indeed, let a, b ∈ S such that m ab ≥ 3. Consider u n = (abba) n and v n = (abba) n+1 . Then r 1 (u n , v n ) = 2n and r 2 (u n , v n ) = r 3 (u n , v n ) = 4n. Or, as in ii., the existence of some C > 0 such that d 1 ≤ Cd 2 and d 1 ≤ Cd 3 , means there is a C ′ in R such that r 3 ≤ r 1 + C ′ and r 2 ≤ r 1 + C ′ , which is impossible for our example.

  and b c} is nonempty, then it has a least element, denoted by a ∨ L b or a ∨ b. It is said to be a right preGarside monoid if (a R ) it is atomic and right cancellative (ac = bc ⇒ a = b); (b R ) for all a, b ∈ M , if the set {c ∈ M | c a and c b} is nonempty, then it has a least element, denoted by a ∨ R b.

Example 2 . 1 .

 21 Consider the cancellative monoid M = a, e, b | ae = ea, ebe = b + and the endomorphism ϕ such that ϕ(a) = ae, ϕ(e) = e and ϕ(b) = b.

  and then m divides p. Hence ϕ p (a) = a. So S 0 ⊆ S ∩ Fix(ϕ p ), and we clearly have S ∩ Fix(ϕ p ) ⊆ S ∩ ϕ p (S).

  atomic and left cancellative. Let a, b ∈ N 2 such that the set Λ = {c ∈ N 2 | a c and b c} is nonempty. Set δ = g(a) ∨ g(b) the least common right multiple of g(a) and g(b) in N 1 , and write

, then S 1 =

 1 ∅ and π = Id M . Assume 1 ∈ ϕ(S). We show the result by induction on d. The case d = 1 being trivial, assume d > 1 and the result holds for smaller integers. It suffices to check the equality on the generators. By definition of S 1 , if a

Example 2 . 11 .

 211 Let M = a, b, c | abab = baba, ac = ca + , and ϕ ∈ End(M ) such that ϕ(a) = b, ϕ(b) = a and ϕ(c) = 1, with ν additive and ν(a) = ν(b) = ν(c) = 1. By using the notations above, we have n = 1, S 1 = {c}, S 2 = {a, b}, M 2 = a, b | abab = baba + and ϕ 2 ∈ End(M 2 ), such that ϕ 2 (a) = b and ϕ 2 (b) = a. Then one has Fix(ϕ 2 ) = abab + , and Fix(ϕ) = ϕ(Fix(ϕ 2 )) = Fix(ϕ 2 ) = abab + .

Lemma 2 .

 2 14. The morphism π is well-defined. Proof. If 1 / ∈ ϕ(S), then S 2 = S and π = Id M . Suppose 1 ∈ ϕ(S). It suffices to verify that π([a, b m ab ) = π([b, a m ab ) for all m ab = ∞. Let a, b ∈ S such that m ab = ∞. If m ab is even, or if a and b are both in S 1 or in S 2 , the equality holds trivially. Suppose we have m ab = 2k + 1 for some a ∈ S 1 , b ∈ S, and k > 0. Then ϕ n ([a, b m ab ) = (ϕ n (b)) k and ϕ n ([b, a m ab ) = (ϕ n (b)) k+1 . Thus, by cancellativity, ϕ n (b) = 1, so b ∈ S 1 and we are done as remarked above.

  for all u, v ∈ M . Further, α(u) is the unique maximal element (for ) in the set {v ∈ M red | v u}. Let M = S | [a, b m ab = [b, a m ab ; m ab = ∞ + be an Artin monoid, whose natural length function is denoted, as in the preliminaries, by ℓ S . And let W = S | (ab) m ab = 1; m ab = ∞ be the corresponding Coxeter group.

[k+1] 2 (v 1 •

 21 • • v k+1 ) = (v ′ 1 , . . . , v ′ k+1 ). Since ι [k+1] 2 (u ′ 1 • • • u ′ k+1 ) = (u ′ 1 , . . . , u ′ k+1 ) and u ′ k+1 does not left divide v ′ k+1 , it follows from Lemma 3.3 that u ′ 1 • • • u ′ k+1 does not left divide v 1 • • • v k+1 . But on the other hand, u ′ 1 • • • u ′ k+1 left divides u 1 • • • u k+1 by Lemma 3.3 i.. Thus, v 1 • • • v k+1 = u 1 • • • u k+1 and r 1 (u, v) ≤ k. Hence r 1 (u, v) ≤ r 2 (u, v) and d 2 (u, v) ≤ d 1 (u, v).Definition 3.5. A mapping ϕ : (X, d) → (X ′ , d ′ ) between metric spaces is uniformly continuous if ∀ε > 0, ∃δ > 0, ∀x, y ∈ X 1 : (d(x, y) < δ ⇒ d ′ (ϕ(x), ϕ(y)) < ε).

Example 3. 6 .

 6 i. In an Artin monoid M with m ab = ∞ for some a, b ∈ S, the metric d 1 is not uniformly equivalent (and so not equivalent) to d 2 , nor to d 3 . Indeed, write u n = (ab) n and v n = (ab) n+1 . By definition of the metrics, we haver 2 (u n , v n ) = r 3 (u n , v n ) =2n, and since u n , v n ∈ M red , then r 1 (u n , v n ) = 0. Thus we have d 1 (u n , v n ) = 1 for all n, and lim n→∞ d 2 (u n , v n ) = lim n→∞ d 3 (u n , v n ) = lim n→∞ 2 -2n = 0. So d 1 cannot be uniformly equivalent to d 2 or d 3 . ii. The metrics d 2 and d 3 are not equivalent in general. Indeed, Let M be an Artin monoid, with a, b, c ∈ S such that m ab = 2 and m ac , m bc ≥ 3. Consider u n = (ab) n and v n = (ab) n c. Then r 2 (u n , v n ) = n and r 3 (u n , v n ) = 2n. We have ∃C > 0, d 2 ≤ Cd 3 ⇔ ∃C > 0, -r 2 ≤ log 2 (C)r 3 ⇔ ∃C ′ ∈ R, r 3 ≤ r 2 + C ′ .

Note that in this case, for u, v ∈ M , we have r

Relations between d 1 , d 2 and d 3

In this subsection, we will compare the first distance d 1 with the other two for a finitely generated pre-Garside monoid M , equiped with a length ℓ S . We start with d 1 and d 3 .

Lemma 3.1. Let u lie in M . Set ι 1 (u) = (u 1 , . . . , u n ). Then, for 1 ≤ k ≤ n, we have

Proposition 3.2. Let M be a finitely generated preGarside monoid, equiped with a length ℓ S . Then we have

Thus, r 3 (u, v) ≥ r 1 (u, v) and so

We turn now to d 1 and d 2 . Note that the existence of a length ℓ S is only necessary for d 3 , and we do not need it to compare d 1 and d 2 . The inclusion X 2 ⊆ M red , deduced from [1, Proposition 2.19], will be useful for us.

Proof.

i. The fact that n ≤ m is a consequence of [9, Proposition 4.8], also true for preGarside monoids, as stated in [START_REF] Bessis | Springer theory in braids groups and the Birman-Ko-Lee monoid[END_REF]. On the other hand, by using X 2 ⊆ M red and the same proof as [9, Proposition 4.10], we get u

. Hence, the property is true for j = 1. Assume j ≥ 2 plus the induction hypothesis. Write

2 (u ′ j v)). On the other hand, ι

The case of trace monoids

In this subsection, we focus on right angled Artin monoids (RAAM iii. if M is the free abelian monoid,

Indeed, point i. was already proved in [START_REF] Kummetz | The topology of mazurkiewicz traces[END_REF] by a topological argument. We provide an algebraic one. Let us start with the following remark :

For the remaining of the section, we fix a trace monoid M = S | ab = ba; m ab = ∞ + , and set p = |S|. For every u ∈ M , let ξ(u) denote the support of u, i.e. the set of atoms (elements of S) occurring in any expression of u.

Proof. Let w be in M and s be in S. Assume ι 2 (w) = (w 1 , . . . , w l ), and set ι 2 (ws) = (w ′ 1 , . . . , w ′ l ′ ). By [START_REF] Van Wyk | Graph groups are biautomatic[END_REF], ι 2 (ws) can be obtained in the following way. If s / ∈ ξ(w) and sw = ws, then ι 2 (ws) = (w 1 s, . . . , w l ) and |ws| 2 = l. If s ∈ ξ(w l ) or sw l = w l s, then ι 2 (ws) = (w 1 , . . . , w l , s) and

We have j 0 < l, ι 2 (ws) = (w 1 , . . . , w j 0 s, . . . , w l ) and |ws| 2 = l. In all cases, l ′ ≥ l and

s m in S, and apply the above argument to all the pairs

Lemma 3.10. We have

By Lemma 3.9, we have

Lemma 3.11. We have

Since we have m st ∈ {2, ∞} for all s, t ∈ S, then

We may now prove our proposition.

Proof of Proposition 3.7.

i. Gathering Lemmas 3.10 and 3.11, we get immediately that d 2 and d 3 are uniformly equivalent.

ii. Example 3.6 i.

iii. Assume M ≃ N p is the free abelian monoid, i.e. m ab = 2 for all a, b ∈ S. Let u, v be in M . Set ι 1 (u) = (u 1 , . . . , u n ) and ι 1 (v) = (v 1 , . . . , v m ). Since ab = ba for all a, b ∈ S, we have

Let q = min(T). We may suppose q = f j for some 1 ≤ j ≤ k. Thus, for all i ≤ q, we have u i = v i , s j ⊀ u q+1 and s j v q+1 . Therefore r 1 (u, v) = r 2 (u, v) = q. On the one hand, s q+1 j v and s q+1 j ⊀ u, therefore Pref q+1 (u) = Pref q+1 (v). On the other hand, let w ∈ M such that w u and ℓ S (w) = q. Then we can write

And if f i = g i , since q = min(T) and h i ≤ q, one has h i ≤ g i . So for all 1 ≤ i ≤ k, we have h i ≤ g i , which means that w v. Similarly, if w v with ℓ S (w) = q, then w u as well. Hence, Pref q (u) = Pref q (v), and therefore r 3 (u, v) = q. Thus,

The case of Garside monoids

In this subsection, we show that in a finitely generated Garside monoid, equiped with a length ℓ S , the metrics d 1 and d 3 are uniformly equivalent. Recall that a Garside monoid is a preGarside monoid containing a Garside element, i.e. a balanced element whose set of divisors generates the whole monoid. let M be a Garside monoid with a Garside element ∆. One of the Garside element important properties is that for all u ∈ M , α(u) is the greatest common (left) divisor of u and ∆, denoted by u ∧ ∆. In other words, we have u ∈ M red ⇔ u ∆. Proposition 3.12. Let M be a finitely generated Garside monoid, equiped with a length ℓ S . Set ℓ = ℓ S (∆). We have

Thus, by definition of the greedy normal form, u r 1 +1 v r 1 +1 and v r 1 +1 u r 1 +1 . So u r 1 +1 = v r 1 +1 , which contradicts the definition of r 1 (u, v). Therefore, one has either

Gathering propositions 3.2 and 3.12, we get : Theorem 3.13. In a finitely generated Garside monoid, equiped with a length ℓ S , the metrics d 1 and d 3 are uniformly equivalent.

Question : Are d 1 and d 2 uniformly equivalent in Garside monoids ?

Contractability of endomorphisms of Artin monoids

The aim of this section is to extend [START_REF] Rodaro | Fixed points of endomorphisms of trace monoids[END_REF]Theorem 4.1] to all Artin monoids. However, it is easy to verify that the assertions stated in [START_REF] Rodaro | Fixed points of endomorphisms of trace monoids[END_REF]Theorem 4.1] can be not equivalent in an Artin monoids (see Example 4.3 below). So [START_REF] Rodaro | Fixed points of endomorphisms of trace monoids[END_REF]Theorem 4.1] can not be directly extended. Actually, in the general case, the metric d 1 appears as more natural than d 2 , mainly because of Property [START_REF] Kwiatkowska | A metric for traces[END_REF]. Moreover, one can verify that Property (iii) of Lemma 4.1 is the exact translation of Property ( 14) in [START_REF] Rodaro | Fixed points of endomorphisms of trace monoids[END_REF]Theorem 4.1(iii)] when replacing d 2 by d 1 . So Theorem 4.2 provided a convenient generalisation of [START_REF] Rodaro | Fixed points of endomorphisms of trace monoids[END_REF]Theorem 4.1] to the context of Artin monoids.

Let M = S | [a, b m ab = [b, a m ab ; m ab = ∞ + be an Artin monoid, and ϕ be in End(M ). As shown in [START_REF] Rodaro | Fixed points of endomorphisms of trace monoids[END_REF], the metric space (M, d 1 ) admits a completion ( M , d 1 ) defined as follows. Let ∂M consist of all infinite sequences of the form u 1 u 2 • • • , such that u i ∈ M red for all i, and u 1 • • • u n is a (greedy) normal form for all n ∈ N. We have M = M ∪ ∂M .

The metric d 1 extends to M in the obvious way, and it is easy to check that ( M , d 1 ) is complete: given a Cauchy sequence (U n ) n with U n = u n1 u n2 • • • , it follows easily that each sequence (u nk ) k is stationary with limit, say, u k , and we get u

is indeed the completion of (M, d 1 ). We may refer to ∂M as the boundary of M .

Assume that ϕ is uniformly continuous with respect to d 1 . Since ( M , d 1 ) is the completion of (M, d 1 ), ϕ admits a unique continuous extension Φ to ( M , d 1 ). By continuity, we must have Φ(X) = lim n→∞ ϕ(u n ) whenever X ∈ ∂M and (u n ) n is a sequence on M satisfying X = lim n→∞ u n .

Lemma 4.1. The following properties are equivalent:

(iv) for all u ∈ M , α(ϕ(u)) = α(ϕ(α(u))).

Proof. We prove that (ii) ⇒ (iv)

). We also have α(uv 1 ) = α(uα(v)) = α(uv) = u and |v 1 | 1 = 1 < k. Then, by the case k = 1, we get α(ϕ(uv 1 )) = α(ϕ(u)). Thus α(ϕ(uv)) = α(ϕ(u)) and (ii) holds.

Recall that a mapping ϕ : (X, d) → (X, d) on a metric space is called a contraction with respect to d, if d(ϕ(u), ϕ(v)) ≤ d(u, v) for all u, v ∈ X. (ii) ϕ is a contraction with respect to d 1 ;

(iii) for all u, v ∈ M red , α(uv) = u ⇒ α(ϕ(uv)) = α(ϕ(u));

(iv) for all u ∈ M , α(ϕ(u)) = α(ϕ(α(u))).

Furthermore, in these cases

Proof. The equivalence (i) ⇔ (ii) is clear, and (iii) ⇔ (iv) follows from Lemma 4.1. Assume (ii). Let u, v be in M red such that α(uv) = u. Then d 1 (uv, u) = 1 2 , and by (ii), d 1 (ϕ(uv), ϕ(u)) ≤ 1 2 . Thus α(ϕ(uv)) = α(ϕ(u)). So (iii) holds. Conversely, assume (iii). Let u belong to M . Set ι 1 (u) = (u 1 , . . . , u n ) and ι 1 (ϕ(u)) = (U 1 , . . . , U N ). We prove by induction on k that for k ∈ {1, . . . , n}, one has ι 1 (ϕ(u 1 • • • u k )) = (U 1 , . . . , U k , . . . ). In particular n ≤ N . For k = 1, the result holds by (iv). So assume k ≥ 2 plus the induction hypothesis. By the induction hypothesis, we can write ϕ(u The following example illustrates that the equivalence between (ii) and (iii) in [START_REF] Rodaro | Fixed points of endomorphisms of trace monoids[END_REF]Theorem 4.1] is not true for all Artin monoïds with respect to d 2 , which is why we used d 1 to extend [START_REF] Rodaro | Fixed points of endomorphisms of trace monoids[END_REF]Theorem 4.1] into our Theorem 4.2.

Example 4.3. Let M = s, t | ststststs = tstststst + , and ϕ ∈ End(M ) such that ϕ(s) = sts and ϕ(t) = tst. Set ∆ = ststststs, and define α 2 (u) by ι [START_REF] Bessis | Springer theory in braids groups and the Birman-Ko-Lee monoid[END_REF] 2 (u) = (α 2 (u)) for u in M . We have X 2 = {s, t, ∆}, and {(u, v) ∈ X 2 2 | α 2 (uv) = u} = {(s, t), (t, s), (s, s), (t, t), (∆, s), (∆, t), (∆, ∆)}. Then, for all u, v ∈ X 2 , we have α 2 (uv) = u ⇒ α 2 (ϕ(uv)) = α 2 (ϕ(u)). However, the morphism ϕ is not a contraction with respect to d 2 , since d 2 (ϕ(s), ϕ(sts)) > d 2 (s, sts).