C R Tellier
email: ctellier@ens2m.fr

Etch Pits on (hk0) and (hhℓ) Silicon Surfaces. Experimental Shapes and Simulations

Keywords: . 07.05.Tp Computer modelling and simulation -81.05.Cy Elemental semiconductors-81.65.Cf Surface cleaning, etching, patterning

The present work is concerned with the development of etch pits on (hk0) and (hhl) silicon surfaces immersed in various etchants. Final shapes of small etch pits are found to be bounded by {111} facets solely. The orientation dependence of etching shapes is explained in terms of a kinematic and tensorial model involved in the simulator TENSOSIM. An agreement between experimental and theoretical etching shapes is observed showing that the simulator can generate quite accurate 3D etching shapes for pits blocked by limiting facets. After a fair adjustment of the database the simulator derives also etching shapes close to experimental shapes of large pits.

Introduction

Silicon is commonly used to fabricate by wet etching [START_REF] Korvink | MEMS: A Practical Guide of Design, Analysis and Application[END_REF][START_REF]Etching of Crystals[END_REF][START_REF] Petersen | Proc. IEEE[END_REF][START_REF] Seidel | [END_REF][5][6] micromechanical devices for sensors and actuators applications. KOH solutions are currently utilized [START_REF]Etching of Crystals[END_REF][START_REF] Petersen | Proc. IEEE[END_REF][START_REF] Seidel | [END_REF][5][6][7] for the micromachining of silicon structures but other etchants [8][9][10][11][START_REF] Hodebourg | Active Passive Electron[END_REF][START_REF] Tellier | Active Passive Electron[END_REF] may be adopted. In the last decade TMAH based solutions [8][9][10][11] were found to be fully compatible with MOS circuit technology. Furthermore some recent works [START_REF] Hodebourg | Active Passive Electron[END_REF][START_REF] Tellier | Active Passive Electron[END_REF] were devoted to the chemical etching of silicon plates in NaOH etchants.

A chemically inert masking film is lithography opened to fabricate mesas and membranes on silicon surfaces. During the opening process some mask pitting may be produced. As a result this pitting gives rise to the formation of small etched holes during the subsequent micromachining. In the following these holes are called etch pits even if crystal imperfections and especially emergent dislocations [START_REF]Etching of Crystals[END_REF][START_REF] Heimann | Silicon Chemical Etching[END_REF] are not at the origin of etch pit formation.

Up to now simulations of etch pits formation are Monte Carlo simulations [START_REF] Cuppen | [END_REF][16][START_REF] Van Veenendal | From an atomistic to a continuum description of crystal growth[END_REF] that consider the atomic structure of the starting surface and assume that the probability of removing a certain atom of the surface is connected to the number of nearest neighbour atomic bonds. Most of these works [START_REF] Cuppen | [END_REF][START_REF] Van Veenendal | From an atomistic to a continuum description of crystal growth[END_REF] are concerned with Si(100) surfaces. Among these works some [16,[START_REF] Van Veenendal | From an atomistic to a continuum description of crystal growth[END_REF] focus on the Monte Carlo simulation of pyramidal etch hillocks that develop when silicate particles are deposited on the (100) surface during the etching process. Moreover to our knowledge no simulations treat the development of etch pits on (hk0) and (hhℓ) surfaces. In the present work we propose to derive theoretical shapes of etch pits with a simulation tool TENSOSIM [START_REF] Tellier | [END_REF][START_REF] Tellier | Active Passive Electron[END_REF][START_REF] Tellier | Proceedings of the IEEE Sensors Conference[END_REF] that is based on an analytical kinematic model.

Experimental shapes of etch pits produced on differently oriented silicon surfaces immersed in various anisotropic etchants are presented in the first part of this paper. The major conclusion of this first part is that the final shape of a small etch pit is solely orientation dependent. The nature of etchant has no influence on this final shape and consequently theoretical shapes can be derived by choosing a convenient etchant. So a database for the chemical etching of silicon in a KOH 35% solution is constructed in the second part. In the third part this database is introduced in the simulation tool TENSOSIM to furnish the orientation dependent shape of etch pits.

Experimental results

Experimental details

Thin (hhℓ) and (hk0) plates were cut from a p-type silicon ingot with an electrical resistivity in the range 1-30 Ωcm. Orientations as defined by the IEEE standard on Piezoelectricity [START_REF] Meeker | IEEE Standard on Piezoelectricity[END_REF] are listed in Table 1. With this standard the reference cut is the (010) surface. A (hk0) surface (or x' 1 x 3 plane) is obtained by a single rotation of ϕ degrees about the x 3 axis (or [001] direction). Starting with the [110] surface (ϕ = -45°) a second rotation of θ degrees about the rotated x' 1 axis ( ] 0 1 1 [ direction) generates a (hhℓ) surface (or x' 1 x' 3 plane). The thickness of plates was approximately 500 µm. A two layer (silicon dioxide-silicon nitride) coating was applied on the upper face of plates. Combinations of circular masks were then patterned on this coating using conventional photolithography. Structures (membranes and mesa) were micromachined in various etchants (Table 2) at a constant temperature of 60°C. The etching duration was the same for all cuts immersed in a given solution. Moreover depending on orientation the etch depth was in the range 70 µm-120 µm. Etched surfaces were examined by Scanning Electron Microscopy. On SEM images of micromachined plates some etch pits due to mask pitting were depicted on the upper surface of mesas. In most cases the lateral sizes of these etch pits did not exceed 30 µm.

Experimental etch pits

Etch pits develop when the inert mask contains opened defects with concave contours. Let us first consider the case of a 35% KOH solution. The chemical etching of silicon in this KOH solution is governed by anisotropy of type 1 [5]. Both {100}, {110} and {111} planes are associated with protuberances of the dissolution slowness surface of equation L(ϕ,θ). The anisotropy ratio k= L {111} /L {100} takes a value larger than 50 and {110} planes dissolve about twice rapidly than {100} planes. The {111} protuberances are very peaky and in the vicinity of {111} orientations the trajectories of surface elements diverge markedly at concave intersections. Consequently {111} facets limit in majority etched holes and etch pits of small size that are only bounded by {111} planes exhibit a top contour composed of successive linear segments LS i . So etch pits on (010) surfaces are pyramidal with four {111} limiting facets. Figure 1 shows etch pits on (010) surfaces etched in NaOH and tetra-methyl ammonium hydroxide (TMAH) solutions. Here again the etch pits are limited by {111} facets and exhibit the characteristic pyramidal shape. We thus infer that the shape of etch pits is only determined by the orientation of the silicon surface provided the lateral size of mask defects remains small as compared with etch depth. In this condition the nature of etchant is irrelevant to shape of etch pits. This feature is conveniently illustrated by Figures 2 and3 that show etch pits related to various (hk0) and (hhℓ) surfaces. Let us turn attention to SEM images 2a and 2A (Figure 2) related to (110) surfaces. Care must be taken that in the case of TMAH solution (image 2a) on the one hand the bottom (110) surface is still visible and that on the second hand an additional facet (f) participates to the concave part of etched pit. However the four {111} facets that make an angle of 35° with the [001] direction as well as the two {111} facets that intersect the (110) surface along the horizontal <110> direction can be easily identified.

As under the above specified conditions the nature of etchant does not affect the final shape of {hk0} and {hhl} etch pits we have collected in Figures 4 and5 the SEM images of etch pits as provided by various etchants. All etch pits shown in these figures are bounded by {111} planes only. Turning attention to (hk0) planes we observe i) The progressive evolution of shape characterized by the destruction of the pyramidal shape of (010) pits.

ii) A symmetrical development of {111} facets with respect to the x' 1 axis that constitutes the intersecting line of the (001) plane with the reference surface. Let us effectively recall that the (001) plane acts as a mirror plane.

Etch pits on (11ℓ) (Figure 5) deviate also more and more from the pyramidal etch pit when the angle of cut θ decreases from 80° to 58° (see Table 1). The mirror symmetry associated with the ) 0 1 1 ( plane gives now shapes with a symmetry with respect to the rotated x' 3 axis if we start with a defect having an appropriate contour (circular contour or rectangular contour aligned along x' 1 axis). Departure from the mirror symmetry as observed in Figure 5d may be attributed to more complex shape of the contour defect. We can also observe that limiting {111} facets are sometimes terraced (see Figures 5b, 5c and5d for example). But other works [START_REF] Hodebourg | Active Passive Electron[END_REF]22] report also the presence of macrosteps on {111} surfaces. Etch pits that develop on (hh1) surfaces are collected in Figure 6. These etch pits exhibit moderate changes in shape with orientation in accord with the assumption that etch pits consist only of {111} facets. A simultaneous examination of Figures 5 and6 reveals clearly that when a doubly rotated surface falls over from the (001) surface (θ=90°) to the (110) surface (θ=0°) the more significant change in shape occurs when we pass through the (111) surface. Effectively the two outward {111} facets (called f O3 and f O3 * in Figure 5) that contribute to (11ℓ) etch pits do not remain outward facets on the SEM images of (hh1) pits (Figure 6). Consider now bordering f O1 and f O2 facets that intersect (11ℓ) and (hh1) surfaces along the ] 0 1 1 [ direction (Figures 5 and6). The f O1 facet corresponds to the (111) facet whereas the second outward facet f O2 is identified as facet for (hh1) and (11ℓ) pits respectively. It is thus easy by simple geometrical considerations to calculate the lateral extent LE {111} of these two limiting facets. Let be α f 1 O and α f 2 O the inclinations of these facets with respect to the reference surface. Obviously the ratio of lateral extents

K LE = LE 1 O f ) /LE is: 2 O f K LE = TAN α f 2 O /TAN α f 1 O (1)
For the (11ℓ) and (hh1) etch pits theoretical values of lateral extent ratio K LE are collected in Table 3. Also given in Table 3 ) lies perpendicular to the reference surface. This feature is verified experimentally (see Figure 5).

As a conclusion of the experimental study we naturally argue that the formation of etch pits on (hhℓ) and (hk0) surface is independent of the nature of the etchant as long as etch pits are bordered solely by {111} facets. The final shape of etch pits can thus be constructed numerically using a database specific to one of etchants listed in Table 2. For this purpose in the following section:

i) The theoretical framework of the simulation is briefly introduced.

ii) The database related to 35% wt. KOH solution is adjusted.

3 Theoretical framework and database for 35% KOH

Theoretical framework

The numerical simulation is based on an analytical model of the anisotropic dissolution presented several years ago by Tellier [23,24]. This model called KT model (for kinematic and tensorial model) constitutes a 3D extension of the kinematic model originally proposed by Frank [25,26]. In the 3D model to take in account the anisotropy we attach a dissolution slowness vector L(ϕ,θ) to a surface element dS(ϕ,θ) of orientation (ϕ,θ). The dissolution slowness vector is defined in such a way that: i) Its direction coincides with that of the unit inward normal n(ϕ,θ) to the surface element.

ii) Its magnitude L(ϕ,θ) is equal to the reciprocal of the etch rate related to the surface element.

When the orientation of the surface element varies the dissolution slowness vector describes in space a representative surface called the dissolution slowness surface. The originality of the KT model is to adopt a tensorial formulation to express the analytical equation for the dissolution slowness surface. This formulation involves Cartesian components of n(ϕ,θ) and dissolution constants. These dissolution constants are components of dissolution tensors and consequently classical tensor transformation rules are used to reduce the number of constants. As a result the equation for the dissolution slowness surface reflects naturally the symmetry elements of the point group related to the crystal into consideration.

To construct numerically the final shape of a micromachined structure it is necessary to follow the motion of starting surface elements within the crystal during the dissolution. Kinematic models [23,25,26] show that surface elements propagate along straight lines if the dissolution is governed solely by orientation. We thus need expression for the displacement vector P(ϕ,θ) of a surface element. Fortunately the displacement P can be calculated [23] from the analytical equation for the dissolution slowness surface. Calculation of displacements P of all surface elements potentially present at the starting surface allows us to derive the final shape of structure provided some specific algorithms [START_REF] Tellier | Proceedings of the 8 th European Frequency and Time Forum[END_REF] were carried out in the simulation tool. The accord between simulated and experimental etching shapes is dependent on values of the dissolution constants as determined from experiments. Effectively the non zero dissolution constants compose the database of the simulator. A database for the 35% wt. KOH etchant and silicon crystal was proposed in a previous work [5]. Some departures between theory and experiment were observed showing that is necessary to perform additional adjustments. These adjustments are presented in the following section.

Adjustment of the database

To determine the database we follow a two step procedure described elsewhere [START_REF] Leblois | [END_REF]. In the first step we analyze experimental results related to:

i) The etch rate of differently oriented silicon surfaces. ii) 2D etching shapes and especially out-of-roundness profiles that are representations of changes in the contour shape of starting circular plates. However this analysis leads to deviations because: i) Etched surfaces are covered by dissolution figures [START_REF] Hodebourg | Active Passive Electron[END_REF][29][30][31] and consequently measured etch rates differs more and less from the etch rate of a perfect planar surface.

ii) Thorough and faithful analysis of (hk0) and (hhℓ) out-of-roundness profiles furnishes with very good agreement orientations for which polar plots of the dissolution slowness L(ϕ,θ) exhibit minima and maxima. But never the successive amplitudes of minima and maxima can be extracted with a good accuracy from experimental out-of-roundness profiles. So when the first step is terminated some uncertainties in the elaborated database remain. It is the reason why successive comparisons between experimental shapes and simulated 3D etching shapes as derived with the simulator TENSOSIM (see section 4.1.) are performed in the second step in order to elaborate an iterative adjustment of the database.

Several works have shown [5,10,[START_REF] Hodebourg | Active Passive Electron[END_REF] that the adjustment of the dissolution constants is easier when we concentrate our study on simple structures such as membranes and mesas micromachined with a starting circular mask. Figures 7a and7a* give examples of (hk0) "circular" membranes and mesas etched in a 35% wt. KOH solution. A previous database (Law 1, [5]) has been determined by applying only the first step of the procedure. Figures 7b and7b* collect the theoretical shapes obtained using the Law 1. A recent adjustment of the database (New Law) furnishes the simulated structures of Figures 7c and7c*. Note that taken into account the mirror symmetry only simulated half structures are drawn in Figure 7. Comparing theoretical shapes of membranes with experimental shapes we see at first sight roughly qualitative agreement for the two laws. However comparing theoretical mesas with the observed mesas several differences can be noticed:

(1) The top contour of theoretical mesas derived with the Law 1 are more rounded than contours of mesas simulated with the new database. This difference is particularly marked for the (230) mesa. Turning attention to the SEM image of this mesa (Figure 7a*) we observe that the upper contour is effectively composed of linear segments in agreement with the New Law simulation.

(2) Mesas in Figures 7a and7a* are partly limited by slightly disorientated shoulders (such as shoulders S1, S2 and S3) that joint more vertical facets to the etched reference surface. Theoretical shapes corresponding to Law 1 have not evolved as experimental shapes. A very important difference is the absence of shoulders S2 and S3 on simulated mesas. Fortunately with the New Law shoulders S2 and S3 are formed in regions similar to experimental regions.

(3) Regions with bunching (see regions B1 and B2 in Figures 7a and 7a*) can be observed on the SEM images of mesas. Moreover in practice the convergence of facets may form bunching regions composed of terraces as shown by regions B1 and B2. At this point it is important to realize that simulations furnish only wrapping curves. Terraces or dissolution figures on etched surfaces are not incorporated in the simulation tool. Consequently comparison of theory and experiments focus essentially on identification of regions where bunching occurs. Comparison reveals that the New Law predicts convergence of facets in regions similar to observed regions.

So points ( 1) and ( 2) call for an adoption of the New Law to improve the simulation of theoretical shapes especially for mesas shapes. Effectively it is less easy to reproduce the adequate shape of an etched mesa because we are concerned simultaneously with concave intersections (beneath the mask) and with a convex top contour. Accordingly the testing of possible databases with simulated mesas is required. Hence Figure 8 furnishes additional comparison. Here again the New Law appears to be more optimal than the Law 1.

In conclusion for simulation of 3D etching shapes it is best to work with the database related to the New Law. This database is composed of dissolution constants of rank greater than seven and special attention is paid on dissolution constants that generate very peaky {111} facets on a narrow angular sector.

Simulation of etch pits

All shapes of etch pits are calculated introducing the database called New Law in the self elaborated simulation tool TENSOSIM. To simulate etch pits it is assumed that the starting contour of the mask defect exhibit a simple shape. In the following circular and square contours are primarily selected.

The simulator tool TENSOSIM

The framework of the simulator is the analytical and continuum description of the anisotropic etching of crystals presented in section 3.1. The anisotropy is taken into account through the dissolution constants (database) and the main foundation of the simulator remains independent to the crystal and to the etchant into consideration. Starting data are the database, the wafer orientation, the contour of mask layout and the etching duration t. The flow chart of the simulation is given in Figure 9a. It is composed of six major steps. In step 1 we decompose the starting surface in successive surface elements and the procedure of introduction of these surface elements depends on the nature concave or convex of the successive intersections of the mask contour. These surface elements cut successive vertical planes P Vi along surface profiles composed of linear elements. So in step 2 we calculate the displacements P i of all profile elements potentially present at mask edges that lie in successive cross sectional planes P Vi . At this point the surface profile S i obtained in a plane P Vi shows crossings (Figure 9b). So tests are made in step 3 to eliminate these unphysical crossings. In step 4 we calculate the intersection points I ij of previous surface profiles S i in successive horizontal planes P Hj . We thus obtain constant level contours C j in which crossings can be observed. The elimination of these crossing by specific tests is performed in step 5. Finally with this procedure the final etching shape of a micromachined structure is furnished in the form of a constant level contour diagram.

Simulations

Firstly let us suppose that the mask defect has a circular contour. The simulator gives the theoretical predictions shown in Figure 10 for (hk0) planes and in Figure 11 for (hhℓ) planes. Turning attention to Figure 10 we observe that the top contour of (010) and (110) etch pits are composed of four and six linear segments respectively showing that planar facets limit pits. It is obvious that the four segments of (010) pits correspond to intersections of the four

) 1 1 1 ( , ) 1 1 1 ( , ) 1 1 1 ( and ) 1 1 1
( planes with the upper surface. In the case of the (110) pit the simulator generates (Figure 10d) effectively six {111} facets with four vertical facets in close accord with experimental observation (Figure 4e). Figure 10 illustrates conveniently how when the angle ϕ varies from 0° to -45° the starting four {111} facets are progressively transformed into six {111} facets whose inclination changes progressively. The agreement between experimental shapes and theoretical shapes is very good for (010), ( 140) and (110) surfaces. At first sight the (230) pit does not look as the experiment etch pit. However with a careful examination we recognize in Figure 10c the contour elements LS 1 and LS 2 and symmetrical ones present in Figure 4d.

In Figure 11 we follow the theoretical evolution of etch pit when the angle θ falls from 90° ((001) surface) to 0° ((110) surface). A decrease of 10° in starting angle θ (θ = 90°) induces a significant change in the etch pit shape. Effectively the final shape of (118) pit (Figure 10b) differs from the well known (001) pyramidal shape. Figure 11 gives also evidence for a transition in etch pit shape that occurs for the (111) surface (θ ≅35°). It is sufficient to compare Figures 11e with Figure 11f corresponding to (112) and (221) pits respectively. Moreover in the case of (hh1) etch pits (Figures 11f to 11i) the more marked changes in shape with decreasing θ appears to be the slow evolution in inclination of f O1 and f O2 limiting facets.

Figures 12 and13 show theoretical shapes of etch pits in the special case of a square mask defect. Etching times were adjusted for the various (hhℓ) surfaces in order to derive etch pits bordered by {111} facets only. As expected {111} facets develops and propagate at the four square corners of the mask. Let us recall that the ] 0 1 1 [ direction lies in (111) and f O2 ( )

) limiting planes. So these two limiting planes are rapidly formed at corners whereas the four other {111} limiting facets develops progressively. As a result the two (111) and f O2 limiting facets cut the upper surface along two linear segments LS 1 and LS 4 (see Figure 13f for the definition of segments LS i ) that are the most longer of the top contour. The difference between lengths of intersecting segments LS 1 and (LS 2 , LS 3 ) is more marked for (hh1) pits (Figure 13) than for (11ℓ) pits (Figure 12). In contrast in the case of a circular mask (Figure 11) the six {111} facets are tangent to the starting mask contour. This means less difference in the length of the six segments LS i that compose the contour of (hhℓ) pits. Comparison of Figures 11g,11h and 11i with Figures 13a,13b and 13c conveniently illustrates the influence of starting mask contour on the final contour of etch pit. In view of theoretical results fitted in Figures 11 and13 and provided that pits are only limited by {111} facets we can conclude that changes in the shape of mask contour affect only the length of intersecting segments LS i .

At this point it is interesting to investigate the influence of a misalignment of the square mask with respect to the standard ] 0 1 1 [ direction on the final shape of (hhl) etch pits. Figure 14 shows theoretical (112) and (441) etch pits as derived assuming misalignments of few degrees. Clearly the mirror symmetry is destroyed and the simulator generates pairs of segments (LS 2 ,LS 2 *) and (LS 3 ,LS 3 *) with different lengths. Simulated etch pits as shown in Figures 14a and14b are quite successful for the experimental etch pits of Figures 3A and6c respectively. Moreover let us pay attention to (112) etch pits of Figures 3a and3A. Taking into account above arguments it is reasonable to assume that pits of Figures 3a and3A are the result of clockwise and anticlockwise misalignments respectively. In conclusion it can be seen that a misalignment does not affect the nature of limiting facets but has a marked influence on the final top contour of etch pits.

As {111} planes dissolve very slowly for all etchants investigated in this study the final etching shape of small etch pits does not depend on the etchant. This property remains true until the duration of etching leads to the formation of {111} facets only. For shorter etching durations the independence of pit shape on etchant is no longer valid because facets associated to secondary maxima of the dissolution slowness can also bound pits. In particular {100} planes are known [5,[START_REF] Tellier | Active Passive Electron[END_REF]22,30] to act as limiting facets for KOH, TMAH and NaOH etchants. The early formation, the subsequent development and finally the disappearance of these {100} facets is determined by the anisotropy ratio k. This ratio takes a value of about 100 for the KOH 35% solution [5,22] and falls to about 20 for the TMAH solution [30]. In addition {110} planes correspond also to protuberances (maxima) of the dissolution slowness surfaces in the case of KOH and to valleys (minima) in the case of TMAH. These differences in etching behaviours justify at posteriori the adjustment of the database. With a careful adjustment it becomes possible to follow the development of etch pits with increasing etching durations and to show that for short durations secondary limiting facets participate to etching shapes. Figure 15 illustrates the evolution of etch pit shape with the etching duration for ( 113) and ( 230) surfaces immersed in a KOH 35% solution. For short etching time a facet (facet F* in Figures 15a and15A) other than a {111} facet bound etch pits. In Figures 15b and15B the etching duration is adjusted to prevent the appearance of this facet F* that is just blocked by {111} facets. As soon as {111} facets limit only etch pits etching shapes are not longer modified by prolonged etching (compare Figures 15b and15B with Figures 15c and15C respectively). Let us recall for the KOH etchant the dissolution slowness ratio L {110} /L {100} is close to .5 so that in terms of dissolution criteria [START_REF]Etching of Crystals[END_REF] etch pits are primarily bounded by {111} facets and secondarily by {100} facets. The observation of {110} facets is highly improbable. The F* facet in Figure 15a intersects the reference (113) surface along the ] 0 1 1 [ direction whereas the F* facet in Figure 15A cuts the (230) surface along the [001] direction. Hence it is reasonable to assume that F* facets are (001) and (010) facets for (113) and (230) etch pits respectively. Moreover careful examination of Figure 5c reveals the presence of a F* facet in agreement with theoretical prediction of Figure 15a. Figure 16 shows results for a "large" pit (more than 100 µm × 100 µm in size). The experimental (551) pit is obtained with a contour mask that deviates from the standard rectangular contour aligned along the ] 0 1 1 [ direction. Together with usual {111} facets the etched pit is bounded with two other facets identified as facets F1 and F1* in Figure 16a. One can easily depict in Figure 16a the bottom (551) surface covered with dissolution figures like grooves aligned along the ] 0 1 1 [ direction. The theoretical (551) pit of Figure 16b is composed of eight limiting facets and with a long rectangular mask the simulator yields elongated symmetrical F1 and F1* facets that correspond well with the two additional facets identified in Figure 16a. So compared with experimental etching shapes, the simulated shapes of Figures 15 and16 show the interest to perform an accurate adjustment of the dissolution constants.

In addition values of the lateral extent K LE as estimated from the theoretical diagrams of Figures 11,12 and 13 are also collected in Table 3. This table reveals that the agreement between theoretical values calculated from geometrical considerations and values furnished by simulations is very good for (11ℓ) pits with deviations less than 3 %. More marked departures are observed for (hh1) etch pits. Effectively the departure reaches about 10 % for most of (hh1) pits (the (221) pit excepted). The theoretical formation of a slightly curved (111) facet instead of a perfectly planar facet (see Figure 13 for example) explains the observed discrepancies.

Conclusions

The formation of etch pits on various singly and doubly rotated silicon surfaces is studied. For long durations of etching small etch pits are exclusively bordered by {111} facets making final 3D etching shapes independent on the nature of etchants. This independence is not still valid for short durations or large etch pits.

The orientation dependence of final 3D etching shapes of etch pits is investigated using the kinematic and tensorial model of the anisotropic dissolution. The simulator TENSOSIM that is based on the KT model generates theoretical shapes close to experimental shapes for small (hk0) and (hhℓ) etch pits. The simulator gives also a fair description of the influence of mask misalignment and of etching duration on etching shapes. It is also found that for a given etchant and for large etch pits the simulator furnishes good etching shapes provided the database specific to the etchant was accurately adjusted. Thus it is concluded that the KT model and correlated simulations satisfactory account for micromachining of (hk0) and (hhℓ) pits even when only blocking facets are active. Let us recall that micromachined mechanical structures may be partly composed of regions with blocking facets. So the simulator TENSOSIM appears to be a convenient tool for the study of such blocked regions. a

Captions to figures

Step 1: Mask Pattern decomposition in surface elements and then in profile elements lying in vertical planes Crystal, Reference surface, Etchant, Etching time

Step 2: Calculation of displacement P i of profile elements

Step 3: Elimination of crossings in vertical planes

Step 4: Determination of etching contours lying in horizontal planes

Step 5: Elimination of crossings in horizontal planes

Figure 1 :Figure 2 :

 12 Figure 1: Etch pits formed on (100) surfaces. Images a, b are for NaOH 35% and TMAH 25% etchants respectively

Figure 3 :

 3 Figure 3: Resemblance of shapes for etch pits formed on (hhℓ) surfaces. Images a, A are for (112) surfaces and KOH 35% and NaOH 35% etchants respectively. Images b, B are for (661) surfaces and KOH 35% and NaOH 35% etchants respectively

Figure 4 :

 4 Figure 4: Evolution of shapes with orientation for etch pits formed on (hk0) surfaces. Images a, b, c, d and e are for (010), (160), (140), (230) and (110) surfaces respectively. Etch pits in images b, c, d and e are obtained with a KOH 35 % etchant. Etch pit in image a is for a TMAH solution. Arrows indicate the rotated x' 1 direction.

Figure 5 :

 5 Figure 5: Evolution of shapes with orientation for etch pits formed on (11ℓ) surfaces. Images a, b, c and d are for (118), (114), (113) and (112) surfaces respectively. Etch pits in images c and d are obtained with a KOH 35 % etchant. Etch pits in images a and b are for TMAH and NaOH solutions. Arrows indicate the rotated <110> direction (x' 1 direction).

Figure 6 :

 6 Figure 6: Evolution of shapes with orientation for etch pits formed on (hh1) surfaces. Images a, b, c and d are for (221), (331), (441) and (661) surfaces respectively. Etch pits in images a, b and C are obtained with a NaOH 35 % etchant. Etch pits in images d are for a KOH 35 % Solution. Arrows indicate the rotated <110> direction (x' 1 direction).

Figure 7 :

 7 Figure 7: Micromachining of membranes and mesas on (hk0) surfaces. SEM images a and a* are for (370) and (230) surfaces respectively. Diagrams b and b* were derived in a previous work using the Law 1. Diagrams c and c* are obtained with the new database (New Law).

Figure 8 :

 8 Figure 8: Micromachining of a mesa on a (140) surface. SEM image (a) and theoretical shapes as derived with the previous database (b) and with the new database (c).

Figure 9 .

 9 Figure 9. The flow chart of the simulation (a) and cross sectional etch path with crossings (b).

Figure 10 :

 10 Figure 10: Simulated shapes for etch pits formed with a circular mask. Diagrams a, b, c and d are for (010), (140), (230) and (110) surfaces respectively.

Figure 11 :

 11 Figure 11: Simulated shapes for etch pits formed with a circular mask. Diagrams a,b, c, d, e, f, g, h, and i are for (118), (115), (114), (113), (112), (221), (331), (441) and (661) surfaces respectively.

Figure 12 :

 12 Figure 12: Simulated shapes for etch pits formed on (11ℓ) surfaces with a starting square mask. Diagrams a, b, c, d and e are for (118), (115), (114), (113) and (112) surfaces respectively.

Figure 13 :

 13 Figure 13: Simulated shapes for etch pits formed on (hh1) surfaces with a starting square mask. Diagrams a, b, c, d and e are for (331), (441), (551), (661) and (110) surfaces respectively.

Figure 14 :

 14 Figure 14: Influence of misalignment ψ. Diagrams a and b are for (112) surface with ψ = 3° and (441) surface with ψ=5° respectively. The linear segment LS i are identified in schematic diagram c.

Figure 15 :

 15 Figure 15: Influence of the etching duration t (in arbitrary unit) on (113) and (230) surfaces.

Figure 16 :

 16 Figure 16: SEM image (a) of a "large" (551) pit etched in a KOH 35% solution and simulation (b) as derived starting with a rectangular mask.

Figure 1 :Figure 2 :Figure 3 :Figure 4 :Figure 5 :

 12345 Figure 1: Etch pits formed on (100) surfaces. Images a, b are for NaOH 35% and TMAH 25% etchants respectively

Figure 6 :

 6 Figure 6: Evolution of shapes with orientation for etch pits formed on (hh1) surfaces. Images a, b, c and d are for (221), (331), (441) and (661) surfaces respectively. Etch pits in images a, b and C are obtained with a NaOH 35 % etchant. Etch pits in images d are for a KOH 35 % Solution. Arrows indicate the rotated <110> direction (x' 1 direction).

Figure 7 :

 7 Figure 7: Micromachining of membranes and mesas on (hk0) surfaces. SEM images a and a* are for (370) and (230) surfaces respectively. Diagrams b and b* were derived in a previous work using the Law 1. Diagrams c and c* are obtained with the new database (New Law).

Figure 8 :

 8 Figure 8: Micromachining of a mesa on a (140) surface. SEM image (a) and theoretical shapes as derived with the previous database (b) and with the new database (c).

Step 6 :Figure 9 .Figure 10 :Figure 11 :Figure 12 : 1 Figure 13 :Figure 14 :Figure 16 :

 691011121131416 Figure 9. The flow chart of the simulation (a) and cross sectional etch path with crossings (b).

Table 1 :

 1 Orientation of the various silicon surfaces and corresponding values of angles of cut ϕ and θ.

		(hk0) surfaces
	Crystal surface	Angle ϕ (°)	Angle θ (°)
	(010) (160) (140) (230) (110)	0 -9.46 -14.04 -33.69 -45	0 0 0 0 0
		(hhℓ) surfaces
	Crystal surface (118) (114) (113) (112) (221) (331) (441) (551) (661)	Angle ϕ (°) -45 -45 -45 -45 -45 -45 -45 -45 -45	Angle θ (°) 79.97 70.53 64.76 58.17 19.47 13.26 10.0 8.05 6.72

Table 2 :

 2 The various etchants. TMAH is for tetra-methyl ammonium hydroxide.

	KOH-water	KOH-water	TMAH-water	NaOH-water
	35% in weight	54% in weight	25% in weight	35% in weight

Table 3 :

 3 Experimental and theoretical values of the lateral extent ratio K LE . Values of K LE as estimated from simulated pits are also collected in this table.

	Surface	Experiment	Theory	Simulation
	(118)	2.14	2.10	2.11
	(115)	-	3.5	3.54
	(114)	4.97	4.98	5.05
	(113)	9.82	10	9.75
	(221)	4.83	5	7.5
	(331)	2.69	2.8	3.1
	(441)	2.09	2.14	2.35
	(551)	1.92	1.83	1.97
	(661)	1.65	1.65	1.74


Ël D Micka

Jean Chekroun

Roux

HOMEOMORPHISMS GROUP OF NORMED VECTOR SPACE: CONJUGACY PROBLEMS AND THE KOOPMAN OPERATOR

Keywords: . 1991 Mathematics Subject Classification. 37C15, 20E45, 39B72, 57S05 Conjugacy problems, Homeomorphisms group, Koopman operator, functional equations and inequalities. 1 where L f,g : ψ → f, ψ, g -

This article is concerned with conjugacy problems arising in homeomorphisms group, Hom(F ), of non-compact subsets F of normed vector spaces E. Given two homeomorphisms f and g in Hom(F ), it is shown how the existence of a conjugacy may be related to the existence of a common generalized eigenfunction of the associated Koopman operators. This common eigenfunction serves to build a topology on Hom(F ), where the conjugacy is obtained as limit of a sequence generated by the conjugacy operator, when this limit exists. The main conjugacy theorem is presented in a class of generalized Lipeomorphisms.

Introduction

In this article we consider the conjugacy problem in the homeormophsims group of a finite dimensional normed vector space E. It is out of the scope of the present work to review the problem of conjugacy in general, and the reader may consult for instance [START_REF] Chaperon | Géométrie Différentielle et Singularités de Systèmes Dynamiques[END_REF][START_REF] Cong | Topological Dynamics of Random Dynamical Systems[END_REF][START_REF] Katok | Introduction to the Modern Theory of Dynamical Systems, With a supplement by Anatole Katok and Leonardo Mendoza[END_REF][START_REF] Kotus | Global structural stability of flows on open surfaces[END_REF][START_REF] Irwin | Smooth Dynamcal Systems[END_REF][START_REF] Palis | A global perspective for non-conservative dynamics[END_REF][START_REF] Smale | Differentiable dynamical systems[END_REF] and references therein, to get a partial survey of the question from a dynamical point of view. The present work raises the problem of conjugacy in the group Hom(F ) constituted by homeomorphisms of a non-compact subset F of E and is intended to demonstrate how the conjugacy problem, in such a case, may be related to spectral properties of the associated Koopman operators. In this sense, this paper provides new insights on the relationships between the spectral theory of dynamical systems and the geometrical problem of conjugacy.

More specifically, given two homeomorphisms f and g of F , we show here that the conjugacy problem in Hom(F ) is subject to the existence of a common generalized eigenfunction for the associated Koopman operators U f and U g (cf. Definition 2.2), i.e. a function Φ satisfying, (1.1) U f (Φ) ≥ λΦ, U g (Φ) ≥ µΦ, for some λ, µ > 0, where Φ lives within some cone K of the set of continuous real-valued functions on F . The elements of this cone present the particularity to exhibit a behavior at infinity prescribed by a subadditive function R; see Section 2. More precisely, when such a Φ exists, it is shown how Φ can be used to build a topology such that the sequence of iterates {L n f,g (h 0 )} n∈N , of the conjugacy operator 1 initiated to some h 0 ∈ Hom(F ) close enough to L f,g (h 0 ) in that topology, converges to the conjugacy h satisfying f • h = h • g, provided that {L n f,g (h 0 )} is bounded on every compact of F ; cf. Theorem 4.3. The topology built from Φ, relies on a premetric on Hom(F ) where Φ serves to weight the distance to the identity of any homeomorphism of F , with respect to the composition law.

The plan of this article is as follows. Section 2 sets up the functional framework used in this article, where in particular the main properties of the topology built from any member Φ ∈ K are derived with a particular attention to closure properties and convergence in that topology of sequences in Hom(F ); cf Propositions 2.7 and 2.10. Section 3 establishes a fixed point theorem, Theorem 3.1, for mappings acting on Hom(F ), when this last one is endowed with the topology discussed in Section 2. In section 4 the main theorem of conjugacy, Theorem 4.3, is proved based on Theorem 3.1 applied to the conjugacy operator, where the contraction property is shown to be conditioned to a generalized eigenvalue problem of type (1.1). This related generalized eigenvalue problem for the Koopman operators associated with the conjugacy problem is then discussed in Section 4.3 where in particular connections with relatively recent results about Schröder equations and Abel equation are established. The results obtained here were motivated in part by [START_REF] Chekroun | Averaging of time-periodic systems without a small parameter[END_REF] where some of the results derived in that work where conditioned to a conjugacy problem on non necessarily compact manifold.

A functional framework on the homeomorphisms group

In this section we introduce family of subgroups of homeomorphisms for the composition law. These subgroups associated with the framework from which they are derived, will be used in the analysis of the conjugacy problem in the homeomorphism group itself. The topology with which they are endowed is also introduced here and the main properties are derived. The extension of these topologies to the whole group of homeomorphisms is also presented and the related closure properties and convergence of sequences in the homeomorphisms group are discussed.

2.1. Notations and preliminaries. In this article E denotes a d-dimensional normed vector space (d ∈ N * ), endowed with a norm denoted by • and F denotes a noncompact subset of E. The following class of functions serves to specify some behavior at infinity of homeomorphisms and to build topologies that will be central in our approach; cf Proposition 2.5.

Definition 2.1. The space E R F . Let R : R + → R + -{0}, be a continuous function, which is subadditive, i.e., R(u + v) ≤ R(u) + R(v), ∀ u, v ∈ R + . We denote by E R F the set of continuous functions Φ : F → R + , satisfying: (G 1 ) ∃ m > 0, ∀ x ∈ F, m ≤ Φ(x), (G 2 ) Coercivity condition: Φ(x) -→ +∞, as x ∈ F and x → +∞ , (G 3 ) Cone condition: There exist β and γ, such that β > γ > 0, and, (2.1) ∀ x ∈ F, γR( x ) ≤ Φ(x) ≤ βR( x ).

For obvious reasons, R will be called sometimes a growth function.

Remark 2.1.

(a) It is interesting to note that the closure K := E R F , is a closed cone with non-empty interior in the Banach space X = C(F, R) of continuous functions Ψ : F → R, endowed with the compact-open topology [START_REF] Hirsch | Differential Topology[END_REF]

, i.e. K + K ⊂ K, tK ⊂ K for every t ≥ 0, K ∩ (-K) = {0 X } and Int K = ∅.
(b) Note that the results obtained in this article could be derived with weaker assumptions than in (G 3 ), such as relaxing (2.1) for x ≥ ν for some ν > 0, and assuming measurability on R and Φ (with respect to the Borel σ-algebras of R + and F respectively) instead of continuity. However, further properties have to be derived in order to extend appropriately the approach developed in this paper. For instance assuming only measurability of R, it can be proved, since R is assumed to be subadditive, that R is bounded on compact subsets of R + , see e.g. [START_REF] Gosselin | A maximal theorem for subadditive functions[END_REF], lemma 1, p. 167; a property that would appear to be important for extending the results of this article in such a context. We leave for the interested reader these possible extensions of the results presented hereafter. (c) Other generalization about R could be also considered, such as

R(u + v) ≤ C(R(u) + R(v)), ∀ u, v ∈ R + ,
for some C > 0, allowing the fact that any positive power of a subadditive function is subadditive in that sense; but this condition would add complications in the proof of Theorem 3.1 for instance. We do not enter in all these generalities to make the expository less technical.

We need also to consider a function r : R + → R + , verifying the following assumptions.

Assumptions on the function r. We assume that r(x) = 0 if and only if x = 0, r is continuous at 0, r is nondecreasing, subadditive and for some statements we will assume further that, (A r ) r is lower semi-continuous for the simple convergence on F , i.e., r( lim

n→+∞ inf f n (x) ) ≤ lim n→+∞ inf r( f n (x) ),
for any sequence {f n } n∈N of self-mappings of F .

Cross condition. Finally, we will consider the following cross condition between the growth function R and the function r,

(C r,R ) ∃ a > 0, ∃ b > 0, ∀ u ∈ R + , R(u) ≤ ar(u) + b.
As simple example of functions Φ, R, and r satisfying the above conditions (including

(A r )), we can cite r(u) = u, Φ(x) = R( x ) =
x + 1, that will be used to illustrate the main theorem of this article later on; see subsection 4.2.

Hereafter in this subsection, condition (A r ) is not required. We introduce now the following functional on Hom(F ) with possible infinite values,

(2.2) | • | Φ,r :    Hom(F ) → R + f → |f | Φ,r := sup x∈F r( f (x)-x ) Φ(x) . Note that, |f | Φ,r = 0 if and only if f = Id F (separation condition),
where Id F denotes the identity map of F .

Definition 2.2. The Koopman operator on E R F associated to f ∈ Hom(F ) is defined as the operator U f given by:

(2.3) U f : E R F → E R F Φ → U f (Φ), where U f (Φ)(x) = Φ(f (x)), ∀ x ∈ F.
Remark 2.2. Classically, the Koopman operator is given with other domain such as L p (F ) [START_REF] Ding | The point spectrum of Perron-Frobenius and Koopman operators[END_REF] (generally p = 2) and arises naturally with the Frobenius-Perron operator in the study of ergodicity and mixing properties of measure-preserving transformations [START_REF] Cornfeld | Ergodic Theory[END_REF]. The Koopman operator addresses the evolution of phase space functions (observables), such as Φ above, described by the linear operator U f rather than addressing a direct study of the dynamics generated by f . This idea has been introduced by Koopman and von Neumann in the early 30's [START_REF] Koopman | Dynamical systems of continuous spectra[END_REF], and has paved the road of what is called today the spectral analysis of dynamical systems [START_REF] Cornfeld | Ergodic Theory[END_REF]. We propose here more specifically to link this spectral analysis with the geometrical problem of conjugacy.

We can now state the following proposition.

Proposition 2.3. Consider R given as in Definition 2.1, and Φ ∈ E R F . Let r satisfy the above assumptions except (A r ), and such that (C r,R ) is satisfied. Introduce the following subset of Hom(F ),

(2.4) H Φ,r := {f ∈ Hom(F ) : |f | Φ,r < ∞, and |f -1 | Φ,r < ∞}.
Then (H Φ,r , •) is a subgroup of (Hom(F ),•) and, for any f ∈ H Φ,r , the Koopman operator associated to f satisfies,

(2.5) ∀ Φ ∈ E R F , U f (Φ) ≤ Λ(f )Φ, with, Λ(f ) := aβ|f | Φ,r + bm -1 β + βγ -1
, and where the constants appearing here are as introduced above.

Proof. We first prove the subgroup property. Let x be arbitrary in F , and f , g in H Φ,r . Then,

(2.6) r( f • g -1 (x) -x ) Φ(x) ≤ r( f • g -1 (x) -g -1 (x) ) Φ(g -1 (x)) • Φ(g -1 (x)) Φ(x) + r( g -1 (x) -x ) Φ(x) .
From (G 3 ) and the subadditivity of R,

Φ(g -1 (x)) ≤ β(R( g -1 (x) -x ) + R( x )),
and since R( x ) ≤ γ -1 Φ(x), we get by using (C r,R ) and (G 1 ),

Φ(g -1 (x)) Φ(x) ≤β a r( g -1 (x) -x) ) Φ(x) + b Φ(x) + βγ -1 ≤C := aβ|g -1 | Φ,r + bm -1 β + βγ -1 , (2.7)
with C finite since |g -1 | Φ,r exists by definition of H Φ,r .

Going back to (2.6) we deduce that,

(2.8) r( f • g -1 (x) -x ) Φ(x) ≤ C|f | Φ,r + |g -1 | Φ,r < ∞, which concludes that f • g -1 ∈ H Φ,r
, and H Φ,r is a subgroup of Hom(F ). The proof of (2.5) consists then just in a reinterpretation of (2.7).

Remark 2.4. Fairly general homeomorphisms are encompassed by the groups, H Φ,r , introduced above. For instance, in the special case Φ(x) = R( x ) := x + 1, and r(x) = x, denoting by H 0 , the group H Φ,r , and | • | Φ,r by | • | 0 for that particular choice of Φ, and r, the following two classes of homeomorphisms belong to H 0 and exhibit non-trivial dynamics.

(a) Mapping f of R d which are perturbation of linear mapping in the following sense:

(2.9)

f (x) = T x + ϕ(x),
with T a linear automorphism of R d and ϕ a C 1 map which is globally Lipschitz with Lipschitz constant, Lip(ϕ), satisfying Lip(ϕ

)< ||T -1 || -1 L(R d )
-that ensures f to be an homeomorphism of R d from the Lipschitz inverse mapping theorem (cf. e.g. [START_REF] Irwin | Smooth Dynamcal Systems[END_REF] p. 244) -and ϕ(x) ≤ C( x + 1) (that ensures |f | 0 < ∞) for some positive constant; and such that the inverse of the differential of f has a uniform upper bound M > 0 in the operator norm, i.e., [Df (u)] -1 L(R d ) ≤ M for every u ∈ R d , which ensures |f -1 | 0 < ∞ by the mean value theorem. For instance, f (x) = x + 1 2 log(1 + x 2 ) provides such an homeomorphism of R. Note that every ϕ that is a C 1 map of R d with compact support and with appropriate control on its differential leads to an homeomorphism of type (2.9) that belongs to H 0 . (b) Extensions of the preceding examples to the class of non-smooth (non C 1 ) perturbations of linear automorphisms can be considered; exhibiting non-trivial homeomorphisms of H 0 . For instance, the two-parameter family of homeomorphisms, {L a,b , b ∈ R\{0}, a ∈ R}, known as the Lozi maps family [START_REF] Lozi | Un attracteur étrange (?) du type attracteur de Hénon[END_REF]:

(2.10) L a,b : R 2 -→ R 2 (x, y) → (1 -a|x| + y, bx) ,
where | • | denotes the absolute value here; constitutes a family of elements of H 0 . Indeed, it is not difficult to show that L a,b and its inverse,

for b = 0, M a,b : (u, v) → ( 1 b v, -1 + u + a |b| |v|) have finite | • | 0 -values.
This family shares similar properties with the Hénon maps family. For instance there exists an open set in the parameter space for which generalized hyperbolic attractors exists [START_REF] Misiurewicz | Strange attractors for the Lozi mappings[END_REF].

We introduce now the following functional on H Φ,r × H Φ,r ,

(2.11) ρ Φ,r (f, g) := max(|f • g -1 | Φ,r , |f -1 • g| Φ,r ),
which is well-defined by Proposition 2.3 and non-symmetric. Since obviously ρ Φ,r (f, g) ≥ 0 whatever f, and g, and ρ Φ,r (f, g) = 0 if and only if f = g, then ρ Φ,r is in fact a premetric on H Φ,r . Note that hereafter, we will simply denotes f • g by f g. Due to the non-symmetric property, two natural type of "balls" can be defined with respect to the premetric ρ Φ,r . More precisely, Definition 2.

3. An open ρ Φ,r -ball of center f to the right (resp. left) and radius α > 0 is the subset of H Φ,r defined by

B + ρ Φ,r (f, α) := {g ∈ H Φ,r : ρ Φ,r (g, f ) < α} (resp. B - ρ Φ,r (f, α) := {g ∈ H Φ,r : ρ Φ,r (f, g) < α}).
Proposition 2.5. Consider R given as in Definition 2.1, and Φ ∈ E R F . Let r satisfy the above assumptions except (A r ), and such that (C r,R ) is satisfied. Then, the premetric as defined in (2.11) satisfies the following properties.

(i) For every f, g, h, in H Φ,r , the following relaxed triangle inequality holds,

(2.12) ρ Φ,r (f, g) ≤ aβρ Φ,r (f, h)ρ Φ,r (h, g) + (bm -1 β + βγ -1 )ρ Φ,r (f, h) + ρ Φ,r (h, g).
(ii) The following families of subsets of H Φ,r ,

T + (ρ Φ,r ) := {H ⊂ H Φ,r : ∀ f ∈ H, ∃ α > 0, B + ρ Φ,r (f, α) ⊂ H} and, T -(ρ Φ,r ) := {H ⊂ H Φ,r : ∀ f ∈ H, ∃ α > 0, B - ρ Φ,r (f, α) ⊂ H} are two topologies on H Φ,r . (iii) For all f ∈ H Φ,r
, for all α * > 0, and for all g ∈ B - ρ Φ,r (f, α * ), the following property holds:

ρ Φ,r (f, g) < α * bm -1 β + βγ -1 ) ⇒ (∃ α > 0, B - ρ Φ,r (g, α) ⊂ B - ρ Φ,r (f, α * )),
and thus for all f ∈ H Φ,r , α>0 B - ρ Φ,r (f, α) is a fundamental system of neighborhoods of f , which renders T -(ρ Φ,r ) first-countable. Analogue statement holds with "+" instead of "-".

(iv) Let H Φ,r (-) denote the closure of H Φ,r for the topology T -(ρ Φ,r ), then

H Φ,r (-) Hom(F ) ⊂ H Φ,r .
Remark 2.6. Proof of (iii) below shows that an arbitrary open ρ Φ,r -ball (centered to the right or left) is not necessarily open in the sense of not being an element H of T +/-(ρ Φ,r ), since b > 0 and γ < β.

Proof. We first prove (i). Using the triangle inequality for • and subadditivity of r, it is easy to note that for all x ∈ F ,

(2.13) r( f g -1 (x) -x ) Φ(x) ≤ r( f h -1 hg -1 (x) -hg -1 (x) ) Φ(x) + |hg -1 | Φ,r .
From the following trivial equality,

r( f h -1 hg -1 (x) -hg -1 (x) ) = r( f h -1 hg -1 (x) -hg -1 (x) ) Φ(hg -1 (x)) Φ(hg -1 (x)) ,
we deduce from Proposition 2.3, that,

r( f g -1 (x) -x ) Φ(x) ≤ Λ(hg -1 )|f h -1 | Φ,r , which reported in (2.13) gives, (2.14) sup x∈F r( f g -1 (x) -x ) Φ(x) ≤ Λ(hg -1 )|f h -1 | Φ,r + |hg -1 | Φ,r ,
leading to (2.12), by re-writing appropriately (2.14) and repeating the computations with the substitutions f ← f -1 , g -1 ← g and h -1 ← h for the estimation of |f -1 g| Φ,r . The proof of (ii) is just a classical game with the axioms of a topology and is left to the reader.

We prove now (iii), only for T -(ρ Φ,r ); the proof for T + (ρ Φ,r ) being a repetition. Let f ∈ H Φ,r and α * > 0. Let g ∈ B - ρ Φ,r (f, α * ), then from (2.12) we get for all h ∈ H Φ,r , (2.15)

ρ Φ,r (f, h) ≤ aβρ Φ,r (f, g)ρ Φ,r (g, h) + (bm -1 β + βγ -1 )ρ Φ,r (f, g) + ρ Φ,r (g, h).
We seek now the existence of α > 0 such that

B - ρ Φ,r (g, α) ⊂ B - ρ Φ,r (f, α * ). Denoting ρ Φ,r (f, g) by α
, such a problem of existence is then reduced from Eq. (2.15) to the existence of a solution α > 0 of, (2.16) aβαα

+ (bm -1 β + βγ -1 )α + α < α * .
A necessary condition of existence is,

(2.17) α < α * * := α * bm -1 β + βγ -1
that turns out to be sufficient since any α > 0 satisfying,

(2.18) α < α * -(bm -1 β + βγ -1 )α 1 + aβα ,
is a solution because the RHS is positive.

The second part of (iii) is a reinterpretation of the result just obtained. Indeed, we have proved that for all f ∈ H Φ,r , and for all α * > 0 there exists 0 < α * * < α * (since γ < β and b > 0 by definition), such that

B - ρ Φ,r (f, α * * ) ∈ T -(ρ Φ,r ). Now if we introduce B(f ) := α * >0 B - ρ Φ,r (f, α * ), then the family F(f ) of subsets of H Φ,r defined by, F(f ) := {V ∈ 2 H Φ,r : ∃ B ∈ B(f ), s. t. V ⊃ B}, is a family of neighborhoods of f , since every V ∈ F(f ) contains by definition a subset B - ρ Φ,r (f, α * ) (that is not necessarily open)
, and therefore a subset of type B - ρ Φ,r (f, α * * ) which is open from what precedes. The first-countability holds then naturally and the proof of (iii) is complete.

We prove now (iv

). Let f ∈ H Φ,r (-)
Hom(F ), then by the property (iii), there exists a sequence

{f n } n∈N ∈ H N Φ,r , such that ρ Φ,r (f, f n ) → n→∞ 0.
Then by definition of ρ Φ,r , we get in

particular that |f f -1 n | Φ,r and |f -1 f n | Φ,r exist from which we deduce that |f | Φ,r and |f -1 | Φ,r exist, since H Φ,r is a group. 2.2.
Closure properties for extended premetric on Hom(F ). In this section, we extend the closure property (iv) of Proposition 2.5 to Hom(F ) itself (cf. Proposition 2.7) and prove a cornerstone proposition (Proposition 2.10) concerning the convergence in T -(ρ Φ,r ) of sequences taking values in Hom(F ), where ρ Φ,r denotes the extension of the premetric ρ Φ,r to Hom(F )×Hom(F ).

The result described in Proposition 2.7 will allow us to make precise conditions for which the solution of the fixed point theorem proved in the next section, lives in H Φ,r . This specific result is not fundamental for the proof of the main theorem of conjugacy of this article, Theorem 4.3; whereas Proposition 2.10 will play an essential role in the proof of the fixed point theorem, Theorem 3.1, and by the way in the proof of Theorem 4.3. Important related concepts such as the one of incrementally bounded sequence are also introduced in this subsection.

We define ρ Φ,r as the extension of the premetric ρ Φ,r to Hom(F )×Hom(F ), by classically allowing ρ Φ,r to take values in the extended real line R := R ∪ {∞} instead of R; with the usual extensions of the arithmetic operations2 .

According to this basic extension procedure, it can be shown that T +/-(ρ Φ,r ) is a topology on Hom(F ) and that Proposition 2.5 can be reformulated for ρ Φ,r with the appropriate modifications. Note that in particular the relaxed triangle inequality (2.12) holds for ρ Φ,r and any f, g and h in Hom(F ). Indeed, either f g -1 and f -1 g belong to H Φ,r and (2.12) obviously holds; either f g -1 or f -1 g does not belong to H Φ,r and then -because of the group structure of H Φ,r -for all h ∈ Hom(F ), we get that necessarily at least one element of the following list {f h -1 , f -1 h, hg -1 , h -1 g} does not belong to H Φ,r ; which still leads in all the cases to a validation of analogues to (2.12) for ρ Φ,r .

We are now in position to introduce contingent conditions to our framework that are required to obtain closure type results in Hom(F ). These conditions present the particularity to hold in T + (ρ Φ,r ) for any sequences involved in the closure problem related to the topology T -(ρ Φ,r ). More precisely we introduce the following concepts. Definition 2.4. We say that a sequence {f n } of elements of Hom(F ) is incrementally bounded in T + (ρ Φ,r ) with respect to q, for some q ∈ N, if and only if:

∃ C + q , ∀ p ∈ N, (p ≥ q) ⇒ (ρ Φ,r (f p , f q ) ≤ C + q ).
and |g| Φ,r = ∞ do not necessarily imply that ρ Φ,r (f, g) = ∞. This for instance the case for f

(x) = g(x) = Ax, with A ∈ Gl d (R), A = I d , r(x) = x, and Φ(x) = p x + 1.
When no precision on q is given, we just say that the sequence is incrementally bounded. Furthermore, the sequence is said to be uniformly bounded in T + (ρ Φ,r ) if and only if:

∃ C + , ∀ p, q ∈ N, (p ≥ q) ⇒ (ρ Φ,r (f p , f q ) ≤ C + q ).
The "(-)-statements" consist of changing the role of p and q in the above statements. We denote by IB + the set of incrementally bounded sequences in T + (ρ Φ,r ) and by IB + u its subset constituted only by uniformly incrementally bounded sequences.

-(ρ Φ,r ) of sequences {f n } ∈ Hom(F ), such that, ∃ n 0 ∈ N : f n 0 ∈ H Φ,r , for which {f n } is incrementally bounded in T + (ρ Φ,r ) with respect to n 0 .
We have then the important proposition that completely characterizes the limits in T -(ρ Φ,r ) (that leaves Hom(F ) stable) of sequences of Hom(F ) which are incrementally bounded in T + (ρ Φ,r ) with respect to some n 0 for which f n 0 belongs to H Φ,r . Proposition 2.7. Let Hom(F ) (-),b + be as introduced in Definition 2.5, then,

(2.19) Hom(F ) (-),b + Hom(F ) ⊂ H Φ,r . Proof. Let f ∈ Hom(F ) (-),b +
Hom(F ), we want to show that |f | Φ,r and |f -1 | Φ,r exist. By assumptions, there exists

{f n } ∈ IB + , such that ρ Φ,r (f, f n ) -→ n→∞ 0. Consider n 0 such that f n 0 ∈ H Φ,r resulting from the definition of Hom(F ) (-),b + . From (2.12), ρ Φ,r (f p , Id F ) ≤ aβρ Φ,r (f p , f n 0 )ρ Φ,r (f n 0 , Id F )+ (bm -1 β + βγ -1 )ρ Φ,r (f p , f n 0 ) + ρ Φ,r (f n 0 , Id F ). (2.20)
Since {f n } is incrementally bounded in T + (ρ Φ,r ) with respect to n 0 , then from (2.20), we deduce that the real-valued sequence {|f n | Φ,r } n≥n 0 is bounded.

Besides, for any x ∈ F , and any n ≥ n 0 ,

r( f (x) -x ) ≤ |f f -1 n | Φ,r Φ(f n (x)) + r( f n (x) -x )
, and therefore by using the estimate (2.5) in Proposition 2.3,

(2.21) r( f (x) -x ) Φ(x) ≤ |f f -1 n | Φ,r • Λ(|f n | Φ,r ) + |f n | Φ,r , since Λ(|f n | Φ,r ) is well defined for n ≥ n 0 because |f n | Φ,r exist for such n. We get then triv- ially that {Λ(|f n | Φ,r )} n≥n 0 is bounded because {|f n | Φ,r } n≥n 0 is. Now since |f f -1 n | Φ,r -→ n→∞ 0
by assumption, we then deduce that |f | Φ,r exist by taking n sufficiently large in (2.21). To conclude it suffices to note that (2.20) shows thanks to the incrementally bounded property, that {|f -1 n | Φ,r } n≥n 0 is bounded as well, leading to the boundedness of {Λ(|f -1 n | Φ,r )} n≥n 0 which by repeating similar estimates leads to the existence of |f -1 | Φ,r by using the assumption:

|f -1 f n | Φ,r -→ n→∞ 0.
In the sequel we will need sometimes to use the following property verified by the function r. Let r satisfy the condition of the preceding subsection and let G denote a continuous function G : F → E, then for any K compact of F , there exists

x K ∈ K, such that r sup x∈K G(x) = r( G(x K ) ) = sup x∈K r( G(x)
), since r is increasing. Since we will need this property of r later, we make it precise as the condition, (S) : for all compact set

K ⊂ F, r sup x∈K G(x) = sup x∈K r( G(x) ),
for every continuous function G : F → E, which holds therefore for r as defined above.

In what follows, ρ Φ,r will refer for both ρ Φ,r when applied to elements of H Φ,r , and to ρ Φ,r when applied to elements of Hom(F ), without any sort of confusion.

Let us now introduce the following concept of Cauchy sequence in H Φ,r or more generally in Hom(F ), which is adapted to our framework.

Definition 2.6. A sequence {f n } in H Φ,r or in Hom(F ), is called ρ + Φ,r -Cauchy (resp. ρ - Φ,r - Cauchy), if the following condition holds, ∀ > 0, ∃ N ∈ N, (p ≥ q ≥ N ) ⇒ (ρ Φ,r (f p , f q ) ≤ ). (resp. ∀ > 0, ∃ N ∈ N, (q ≥ p ≥ N ) ⇒ (ρ Φ,r (f p , f q ) ≤ ).
Remark 2.8. Note that, since ρ Φ,r is not symmetric, the role of p and q are not symmetric as well, to the contrary of the classical definition of a Cauchy sequence in a metric space.

Remark 2.9. By definition, every sequence {f n } in H Φ,r which is ρ + Φ,r -Cauchy (resp. ρ - Φ,r -Cauchy) belongs to IB + u (resp. IB - u ). However, a sequence {f n } in Hom(F ) which is ρ + Φ,r -Cauchy is not an element of IB + u in general but is an element of IB + . We are now in position to prove the following cornerstone proposition concerning the convergence in T -(ρ Φ,r ) of ρ + Φ,r -Cauchy sequences in Hom(F ). Proposition 2.10. Assume that F is a non-compact subset of E which is locally connected, σ-compact and locally compact 3 . Consider R given as in Definition 2.1, and Φ ∈ E R F . Let r satisfy the above assumptions including (A r ), and such that (C r,R ) is satisfied. Let {f n } be a sequence in Hom(F ). If the following conditions hold:

(C 1 ) For every compact K ⊂ F , the sequence of the restriction of f n to K, {f n | K }, is bounded, as well as the sequence of the restrictions of the inverses,

{f -1 n | K }, (C 2 ) {f n } is ρ + Φ,r -Cauchy, then {f n } converges in T -(ρ Φ,r
) towards an element of Hom(F ).

If furthermore, either {f n } is a sequence of homeomorphisms living in H Φ,r , or more generally

{f n } is such that f n 0 ∈ H Φ,r for some n 0 , then {f n } converges in T -(ρ Φ,r ) towards an element of H Φ,r .
Proof. The proof is divided in several steps.

Step 1. Let {f n } be a sequence of homeomorphisms of F fulfilling the conditions of the proposition. Let > 0 be fixed. Then from (C 2 ), there exists an integer N such that p ≥ q ≥ N implies,

∀ x ∈ F, r( f p (x) -f q (x) ) =r( f p f -1 q f q (x) -f q (x) ) ≤ρ Φ,r (f p , f q )Φ(f q (x)) ≤ Φ(f q (x)), (2.22) which for every compact K ⊂ F , leads to, (2.23) ∃ M K > 0, ∀ x ∈ K, r( f p (x) -f q (x) ) ≤ M K ,
by assumption (C 1 ) and the continuity of Φ. Similarly, p ≥ q ≥ N , implies,

∀ x ∈ F, r( f -1 p (x) -f -1 q (x) ) =r( f -1 p f q f -1 q (x) -f -1 q (x) ) ≤ρ Φ,r (f p , f q )Φ(f -1 q (x)) ≤ Φ(f -1 q (x)), (2.24)
which, for every compact K ⊂ F , can be summarize with (2.23), as,

(2.25) ∃ M K > 0, r sup x∈K f p (x) -f q (x) ≤ M K , and r sup x∈K f -1 p (x) -f -1 q (x) ≤ M K ,
by using (S) and labeling still by M K the greater constant. Now from the continuity of r at 0 and r(0) = 0 if and only if x = 0, we deduce that the sequences {f n } and {f -1 n } converge uniformly on each compact towards respectively a map f : F → F and a map g : F → F . Note that by choosing appropriately a family of compact subsets of F , covering F , we get furthermore that f and g can be chosen continuous on F .

Step 2. Our main objective here is to show that f = g -1 , i.e. f ∈ Hom(F ). Since F is assumed to be σ-compact and locally compact, there exists an exhaustive sequence of compacts sets {K k } k∈N of F ; see e.g. corollary 2.77 in [START_REF] Aliprantis | Infinite Dimensional Analysis: A Hitchhiker's Guide[END_REF]. From step 1, (2.25) is valid for any p ≥ q ≥ N, with N which does not depend on the compact K, and therefore we get that {f n } is Cauchy for the metric ∆ on Hom(F ), given by,

∆(φ, ψ) = δ(φ, ψ) + δ(φ -1 , ψ -1 ), where δ(φ, ψ) := ∞ k=0 2 -k φ -ψ k (1 + φ -ψ k ) -1 , and φ -ψ k := max x∈K k φ(x) -ψ(x) for
any compact K k , and any homeomorphisms of F , φ and ψ. Indeed, it suffices to note that for a given > 0, from (2.25) there exists l, and N such that,

∞ k=l+1 2 -k ≤ 4
, and

l k=0 2 -k f p -f q k ≤ 4 ,
which leads to δ(f p , f q ) ≤ /2, and similarly to δ(f -1 p , f -1 q ) ≤ /2, for any p ≥ q ≥ N . Now, since F is locally compact and locally connected from a famous result of Arens (see theorem 4 of [START_REF] Arens | Topologies for homeomorphism groups[END_REF]), Hom(F ) is complete for ∆ which is a metric compatible with the compact-open topology [START_REF] Hirsch | Differential Topology[END_REF], making Hom(F ) a Polish group 4 . Therefore {f n } converges in the compact-open topology towards an element h ∈ Hom(F ). By recalling that the compact-open topology is here equivalent to the topology of compact convergence [START_REF] Ren | Topologies on homeomorphism spaces of certain metric spaces[END_REF], we obtain by uniqueness of the limit that f = h ∈ Hom(F ); f being the limit of {f n } in the topology of compact convergence from step 1.

Step 3. Let us summarize what has been proved. We have shown under the assumptions (C 1 ) and (C 2 ), that we can "exhibit" from any sequence {f n } which is ρ + Φ,r -Cauchy, an element f ∈ Hom(F), such that {f n } converges uniformly to it on each compact subset of F , and {f -1 n } does the same towards f -1 . In fact we can say more with respect to the topology of convergence.

Indeed, going back to (2.22), we have that,

∀ > 0, ∃N ∈ N, (p ≥ q ≥ N ) ⇒ ∀x ∈ F, r( f p (x) -f q (x) ) ≤ • Φ(f q (x))
i.e. by making p → +∞ and using (A r ),

∀ > 0, ∃N ∈ N, (q ≥ N ) ⇒ ∀x ∈ F, r( f (x) -f q (x) ) ≤ • Φ(f q (x)) , which leads to, ∀ > 0, ∃N ∈ N, (q ≥ N ) ⇒ (|f f -1 q | Φ,r ≤ ).
From similar estimates, we get,

∀ > 0, ∃N ∈ N, (q ≥ N ) ⇒ (|f -1 f q | Φ,r ≤ ).
We have thus shown the convergence of

{f n } in T -(ρ Φ,r ), that is, ρ Φ,r (f, f q ) -→ q→+∞ 0.
At this stage, we have proved that {f n } converges in T -(ρ Φ,r ) towards an element of Hom(F ).

Step 4. This last step is devoted to the proof of the last statement of the theorem concerning the membership of the limit of {f n } to H Φ,r . This fact is simply a consequence of Proposition 2.5-(iv) in the case {f n } ∈ H N Φ,r , and a consequence of Remark 2.9, and Proposition 2.7 in the case {f n } ∈ (Hom(F)) N , which gives in all the cases that the limit in T -(ρ Φ,r ) of {f n } lives in H Φ,r . The proof is therefore complete.

Lastly, it is worth to note that it is only in step 4 of the preceding proof that was needed assumption (A r ), but since Proposition 2.10 will be used in the proof of Theorems 3.1 and 4.3, we will make a systematic use of this assumption in the sequel.

A fixed point theorem in the homeomorphisms group

In this section we state and prove a new fixed point theorem valid for self-mappings acting on Hom(F ), which holds within the functional framework developed in the preceding section. This fixed point theorem uses a contraction mapping argument that involves a Picard scheme that has to be controlled appropriately due to the relaxed inequality (2.12). In that section ρ Φ,r will stand for the extended premetric introduced in the last subsection. Theorem 3.1. Consider R given as in Definition 2.1, and Φ ∈ E R F , with F as in Proposition 2.10. Let r satisfy the above assumptions including (A r ), and such that (C r,R ) is satisfied. Let Υ : Hom(F ) → Hom(F ) be an application. Let {f n } be a sequence in Hom(F ). We assume that there exists h 0 ∈Hom(F ) such that the following conditions hold:

(i) δ := ρ Φ,r (Υ(h 0 ), h 0 ) < min(1, A -1 ), where A = max(aβ, bm -1 β + βγ -1 ), (ii) {Υ n (h 0 )} n∈Z is bounded on every compact of F . Assume furthermore that there exists a constant 0 < C < 1, such that,

(3.1) ∀ (f, g) ∈ Hom(F ) × Hom(F ), ρ Φ,r (Υ(f ), Υ(g)) < Cρ Φ,r (f, g),
then there exists a unique h ∈ Hom(F ) such that Υ(h) = h, which is obtained as a limit in

T -(ρ Φ,r ) of {Υ n (h 0 )} n∈N . Furthermore, if there exists n 0 ∈ N such that Υ n 0 (h 0 ) ∈ H Φ,r , then h ∈ H Φ,r .
Proof. From Proposition 2.10, it suffices to show that {Υ n (h 0 )} n∈N is ρ + Φ,r -Cauchy for any h 0 ∈ Hom(F ) satisfying (i). For simplifying the notations we denote by ρ m n the quantity ρ Φ,r (Υ m (h 0 ), Υ n (h 0 )), for m ≥ n. Note that since δ is finite, by recurrence and using the contraction property (3.1) we can show that all the quantities ρ m n are finite as well.

By using the relaxed inequality (2.12), the contraction property (3.1) and the definition of A in condition (i), it's easy to get for any integers n, m ≥ n + 1, and k ≥ 1,

ρ Φ,r (Υ m+k (h 0 ), Υ n (h 0 )) ≤ AδC m+k-1 ρ Φ,r (Υ m+k-1 (h 0 ), Υ n (h 0 ))+ AδC m+k-1 + ρ Φ,r (Υ m+k-1 (h 0 ), Υ n (h 0 )), (3.2)
which leads to,

(3.3) ρ m+k n < C m+k-1 ρ m+k-1 n + C m+k-1 + ρ m+k-1 n
, by using Aδ < 1 from assumption (i) and the notations specified above. Let > 0 be fixed. For any m and n, we introduce now the two-parameters sequence {F k (m, n)} k∈N defined by recurrence through,

(3.4) F k (m, n) = C m+k-1 F k-1 (m, n) + C m+k-1 + F k-1 (m, n), ∀ k ≥ 1, F 0 (m, n) = .
When no ambiguity is possible, F k will simply stand for F k (m, n). The role of m and n will be apparent in a moment. Moreover, from (3.3), it's easy to show that for any n and m ≥ n + 1,

(ρ m n ≤ F 0 (m, n)) ⇒ (∀ k ≥ 1, ρ m+k n ≤ F k (m, n)).
As we will see, it suffices to consider m = n + 1 to prove the theorem, a choice that we make in what follows. Since C < 1, obviously,

∃ N 1 : ∀ n ≥ N 1 , ρ n+1 n < C n δ < = F 0 (n + 1, n),
and therefore we get that

ρ n+1+k n ≤ F k (n + 1, n), from what precedes.
The key idea is now to note that if for all k ≥ 0, F k (n + 1, n) ≤ 2 , for n sufficiently big, then the sequence {Υ n (h 0 )} n∈N is ρ + Φ,r -Cauchy for h 0 ∈ Hom(F ) fulfilling condition (i). In the sequel we prove that it is indeed the case.

To do so, an easy recurrence shows that, ∀ k ∈ N, F k > 0, and {F k } is strictly increasing.

In particular, ∀ k ∈ N, F k ≥ , and therefore for any k ≥ 1 and m,

(3.5) F k F k-1 = C m+k-1 + 1 + C m+k-1 F k-1 ≤ C m+k-1 1 + 1 + 1.
Thus by using (3.4) and iterating (3.5), we obtain for any k ≥ 1,

(3.6) v k := F k -F k-1 < C m+k-1 (F k-1 + 1) ≤ C m+k-1 k l=2 C m+l-2 (1 + 1 ) + 1 • + 1 ,
with the convention k=1 l=2 C m+l-2 1 + 1 + 1 ≡ 1, making valid (3.6) for k = 1. Since C < 1, then for any l ∈ {2, ..., k}, and k ≥ 2, C m+l-2 ≤ C m , which leads from (3.6) to,

(3.7) v k ≤ C m C m+1 1 + -1 + C k-1 • + C k-1 ,
which is also valid for k = 1, by simply computing F 1 -F 0 .

Besides,

(3.8) ∃ N 2 : ∀ m ≥ N 2 , a m := C m+1 (1 + -1 ) + C < 1,
which shows for m ≥ N 2 , (3.9)

j=k j=1 v j ≤ C m 1 -a m + 1 1 -C ,
since C < 1 and a m < 1. Now it can be shown,

(3.10) ∃ N 3 : ∀ m ≥ N 3 , C m 1 -a m + 1 1 -C ≤ .
Fixing m = n + 1, we conclude from (3.9) and the trivial identity

F k = j=k j=1 v j + F 0 that, (3.11) ∀ n ∈ N, ∀k ∈ N, (n ≥ max(N 1 , N 2 , N 3 )) ⇒ (F k (n + 1, n)) ≤ 2 ),
which shows in particular that, (3.12)

∀ n ∈ N, ∀k ∈ N, (n ≥ max(N 1 , N 2 , N 3 )) ⇒ (ρ n+k+1 n ≤ 2 ).
We have thus proved that {Υ n (h 0 )} n∈N is ρ + Φ,r -Cauchy for any h 0 ∈ Hom(F ) fulfilling conditions (i) and (ii), and thus by Proposition 2.10,

h := lim n→∞ Υ n (h 0 ) exists in T -(ρ Φ,r ).
It can be shown furthermore that, for every h 0 ∈ Hom(F ), {Υ n (h 0 )} n∈N is incrementally bounded with respect to any n 0 ∈ N, due to the contraction property and the fact that {Υ n (h 0 )} n∈N is ρ + Φ,r -Cauchy. In consequence, if there exists n 0 such that Υ n 0 (h 0 ) ∈ H Φ,r then by applying Proposition 2.10 again (last part) we obtain h ∈ H Φ,r .

A conjugacy theorem and the generalized spectrum of the Koopman operator

4.1. The conjugacy theorem. We prove in this section the main result of this article, i.e the conjugacy Theorem 4.3. To do so, we need further preliminary tools and notations that we describe hereafter. In that section we assume the previous assumptions on r (see Section 2) including the condition (A r ). As in Section 3 the premetric ρ Φ,r will stand for the extended premetric introduced in subsection 2.2. In what follows, we endow again Hom(F ) with a topology T -(ρ Φ,r ) where r and R satisfy the condition (C r,R ) of Section 2, except that here Φ belonging to E R F is not arbitrary and has to solve a generalized eigenvalue problem to handle the conjugacy problem; cf Theorem 4.3.

For any self-mapping f of F , we introduce the following r-Lipschitz constant,

(4.1) λ r (f ) := sup r( f (x) -f (y) ) r( x -y ) , x, y ∈ F, x = y ,
which can takes infinite values. This quantity is clearly the direct extension assessed in the map r( • ), of the Lipschitz constant Lip(f ) classically assessed in • .

Definition 4.1. We denote by L r (F ) the set of homeomorphisms of F such that λ r (f ), and λ r (f -1 ) exist. Such an homeomorphism is called an r-Lipeomorphim of F .

We will also need the following concept of generalized eigenvalue of the Koopman operator, outlined in the introduction.

E R F . A generalized eigenvalue of U f is any λ ∈ R such that, ∃ Φ ∈ E R F : U f (Φ) ≥ λΦ,
where in case of existence, Φ is the corresponding generalized eigenfunction.

Lastly, given f and g in Hom(F ), we introduce the following classical conjugacy operator,

(4.2) L f,g : Hom(F) → Hom(F) h → L f,g (h) := f • h • g -1 .
We are now in position to state and prove the main theorem of this article, a conjugacy theorem which is conditioned to a generalized eigenvalue problem of the related Koopman operators.

Theorem 4.3. Given f and g in Hom(F ) where F is as in Proposition 2.10, assume that there exist a growth function R given as in Definition 2.1 and a function r satisfying the above assumptions including (A r ), such that (C r,R ) is satisfied; and such that the following conditions are fulfilled, (a) f, g ∈ L r (F ), (b) there exists α > 1 and a common generalized eigenfunction Φ ∈ E R F of the Koopman operators U f and U g , which solves the following generalized eigenvalue problem, (

P α : U f (Φ) ≥ αλ r (f )Φ, U g (Φ) ≥ αλ r (g)Φ. 4.3)
Under these conditions, for any Φ solving (4.3), assume further that there exists an homeomorphism h 0 of F satisfying the following properties:

(i) δ := ρ Φ,r (L f,g (h 0 ), h 0 ) < min(1, A -1 )
, where A = max(aβ, bm -1 β + βγ -1 ), (ii) {L n f,g (h 0 )} n∈Z is bounded on every compact of F . Then f and g are conjugated by a unique element h of Hom(F ), which is the limit in T -(ρ Φ,r ) of {L n f,g (h 0 )} n∈N . Furthermore, if there exists n 0 ∈ N such that L n 0 f,g (h 0 ) ∈ H Φ,r , then h ∈ H Φ,r . Proof. Let f and g be two homeomorphisms of F . Let the function r and the growth function R be such that the conditions (a) and (b) of Theorem 4.3 are satisfied. Let Φ ∈ E R F be a solution of (4.3). We endow then Hom(F ) with the topology T -(ρ Φ,r ) for such a Φ.

Since the existence of a solution in Hom(F ) to the conjugacy problem is equivalent to the existence of a fixed point in Hom(F ) of L f,g , it suffices from Theorem 3.1 to examine if L f,g is a contraction in the premetric ρ Φ,r . To do so, we need to estimate |L f,g (h 1 )

• (L f,g (h 2 )) -1 | Φ,r as well as |(L f,g (h 1 )) -1 • L f,g (h 2 )| Φ,r for all h 1 , h 2 ∈ Hom(F ). Simple computations show that for all x ∈ F , (4.4) r( f • h 1 • h -1 2 • f -1 (x) -x ) Φ(x) ≤ λ r (f ) r( h 1 • h -1 2 • f -1 (x) -f -1 (x) ) Φ(x) , which leads to, (4.5) |L f,g (h 1 ) • (L f,g (h 2 )) -1 | Φ,r ≤ λ r (f ) • sup u∈F Φ(u) Φ(f (u)) • |h 1 • h -1 2 | Φ,r . Similar computations show, (4.6) |(L f,g (h 1 )) -1 • L f,g (h 2 )| Φ,r ≤ λ r (g) • sup u∈F Φ(u) Φ(g(u)) • |h -1 1 • h 2 | Φ,r ,
and since Φ solves the generalized eigenvalue problem P α , we get for all u ∈ F , (

) λ r (f ) Φ(u) Φ(f (u)) ≤ 1 α < 1, and λ r (g) Φ(u) Φ(g(u)) ≤ 1 α < 1, 4.7
which allows us to conclude that,

(4.8) ρ Φ,r (L f,g (h 1 ), L f,g (h 2 )) ≤ 1 α • ρ Φ,r (h 1 , h 2 ),
for all h 1 , and h 2 in Hom(F ), i.e. the conjugacy operator L f,g is a contraction for the premetric ρ Φ,r . The rest of the assumptions (i) and (ii) of Theorem 4.3 are just a translation of the ones used in Theorem 3.1, and thus by using this last theorem the proof of the present one is easily achieved.

Remark 4.4. This theorem presents the particularity to provide conditions under which, when it exists, the conjugacy h, lives in H Φ,r . This means in such a case that h satisfies some behavior at infinity prescribed by Φ, which has in turn to solve P α , a spectral problem related to the Koopman operators U f , and U g which involves structural constants of f and g: λ r (f ) and λ r (g) as introduced above. This aspect could be of interest in control theory.

Remark 4.5. Condition of type (i) in Theorem 4.3 is often met in a stronger form when dealing with (local) conjugacy problems that arise around a fixed point. Indeed it is often required that the convergence of the sequence {f n g -n } holds in C 0 provided that f and g are tangent to sufficiently high order (cf. e.g. [START_REF] Chaperon | Géométrie Différentielle et Singularités de Systèmes Dynamiques[END_REF], Lemma 3 p. 95).

4.2.

An illustrative example. To simplify we set E = R and we consider F = [0, +∞) which fulfills the conditions of Theorem 4.3. We consider furthermore r(x) = x and Φ(x) = R(x) = √ x + 1 (for x ∈ F ) that are subadditive function on F . Note that such an R satisfies the condition of Section 2, and that Φ ∈ E R F with β = 2 and γ = 1/2 for instance. Note also that r satisfies (A r ) (and (S)) and fulfill all the other standing assumptions. The cross condition (C r,R ) of Section 2 is fulfilled since,

1 + √ u ≤ u + 5 4 , ∀ u ∈ R + .
We consider the following dynamics on F : f = ηx with 0 < η < 1 and g(x) = ηx + ϕ(x), where ϕ ∈ C 1 (R + , R + ) with compact support is such that g is an homeomorphism of F ; ϕ will be further characterized in a moment. Other conditions on ϕ will be imposed below. Lastly note that λ r (f ) = Lip(f ) for r(x) = x. Since η < 1, there exists > 0 such that, (4.9)

√ η η > 1 + . Now since η(1 + ) ≤ √ η < 1, we get for all x ∈ F, αLip(f )Φ(x) = η(1 + )( √ x + 1) ≤ √ ηx + 1 = Φ(f (x)), with α = 1 + , which shows that Φ is a generalized eigenfunction in E R F of U f with eigenvalue λ = (1 + )Lip(f ) in that particular context.
From (4.9), we get, (4.10) ∃ 2 > 0, :

√ η η + 2
> (1 + ), and 2 < η.

Besides, for such an 2 , there exists a function ϕ ∈ C 1 (R + , R + ) with compact support such that, (4.11) Lip(g) = max

x∈F |η + ϕ (x)| = η + 2 ,
and such that g is still an homeomorphism of F (since 2 < η). From (4.11) and (4.10) we get now, (4.12)

(1 + )Lip(g)Φ(x) = (1 + )(η + 2 )( √ x + 1) ≤ √ ηx + 1 ≤ ηx + ϕ(x) + 1 = Φ(g(x)),
which shows that Φ is a generalized eigenfunction in E R F of U g with λ = (1 + )Lip(g). We are then left with a common eigenfunction of U f and U g satisfying P α with α = 1 + . From our assumptions, it's easy to check furthermore that f and g belong to L r (F ), and therefore conditions (a) and (b) of Theorem 4.3 are satisfied in that particular setting. Recall from the proof of Theorem 4.3 that these conditions ensure the contraction of the conjugacy operator L f,g for the premetric ρ Φ,r .

To apply Theorem 4.3 we have now to check the remaining conditions (i) and (ii). We check first condition (i) for h 0 = g. Note that there exists ψ ∈ C 1 (R + , R -) such that g -1 (x) = x/η + ψ(x), and ψ is with compact support. Furthermore in that particular setting, A = 9/4 for A as defined in condition (i) of Theorem 4.3. We have to estimate

|L f,g (h 0 ) • h -1 0 | Φ,r = |f g -1 | Φ,r and |(L f,g (h 0 )) -1 • h 0 | Φ,r = |f -1 g| Φ,r since h 0 = g.
Note that we can find ϕ and thus ψ such that,

ν := max max x∈F |ϕ|, max x∈F |ψ| < ηA -1 ,
without violating (4.11) and thus having Φ still satisfying P α . For such a choice, ην < η 2 A -1 < A -1 since η < 1, and in particular, which allows us to conclude that condition (i) is checked with h 0 = g. Since ϕ and ψ are with compact support then it can be shown that for h 0 = g, the sequence {L n f,g (h 0 )} n∈Z is bounded on every compact of F . We leave the details to the reader. We can thus apply Theorem 4.3 to conclude that f and g are conjugated which was of course obvious for Lip(ϕ) sufficiently small, from a trivial application of the global Hartman-Grobman theorem, cf. for instance [START_REF] Irwin | Smooth Dynamcal Systems[END_REF].

This modest example is just intended to illustrate some mechanisms of the approach developed in this article. Of course, further investigations are needed with respect to the existence of solutions of the generalized eigenvalue problem P α in spaces of type E R F , for more general homeomorphisms. We postpone this difficult task for a future work, discussing in the next subsection some related issues. 4.3. Generalized eigenvalues of the Koopman operator. We describe here two possible approaches to examine the generalized eigenvalue problem P α , the first one is based on Schröder equations and the second one is based on cohomological equations. The point of view retained is based on functional equations techniques coming from different part of that literature where we emphasize the overlapping. In both cases, by shortly reviewing the existing results, we provide hereafter conditions under which the generalized eigenvalue problem P α may possess continuous solutions, without being able to specify -in a general setting -if Φ can live in E R F . We make precise these aspects below, in the perspective to gather some results that were connected to our problem, and that we have found to be sparsely distributed in the literature. Note that in the sequel, we will focus more precisely on the generalized eigenvalue problem for U f and not P α itself, in order to exhibit already the main issues for the associated single existence problem of a generalized eigenfunction. 4.3.1. Approach based on Schröder equations. We recall first some background concerning Schröder equation. Here E denotes a real or complex normed vector space, and H denotes an arbitrary space of self mapping of E. Let f be such a map, then the Schröder's equation in H is the equation of unknown Ψ:

(4.15) Ψ • f = λ • Ψ,
for some λ. It is the equation related to the spectrum of the Koopman operator5 U f : Ψ → Ψ•f , in the space H. The properties of this spectrum are closely related to the function f as well the space H. The Schröder's equation has a long history and has been extended and studied in various settings. In the early 1870s, Ernst Schröder [START_REF] Schröder | Ueber unendlich viele Algorithmen zur Auflösung der Gleichungen[END_REF] studied this type of functional equation in the complex plane for the composition operator, for Ψ(z) = z 2 and f (z) = z + 1. The functional equation named after him is Ψ • f = λΨ where f is a given complex function, and the problem consists of finding Ψ and λ to satisfy the equation, i.e. an eigenvalue problem for U f . An important part of the results in the literature are devoted to contexts where f is a function mapping the unit disc in the complex plane onto itself initiated by the seminal work of Gabriel Koenigs in 1884 [START_REF] Koenigs | Recherches sur les intégrales de certaines équations fonctionnelles[END_REF]. The reader may consult [START_REF] Shapiro | Composition Operators and Schroder's Functional Equation[END_REF] or [START_REF] Clahane | Spectra of compact composition operators over bounded symmetric domains[END_REF] with references therein, for a recent account about this part of the literature. This functional problem has also been considered historically for map of the half-line [START_REF] Kuczma | Functional equations in a single variable[END_REF] or more general Banach spaces in the past decades [START_REF] Walorski | On the continuous smooth solutions of the Schröder equation in normed spaces[END_REF]. We mention lastly, that the Schröder equation is sometimes encountered under the form of the Poincaré functional equation [START_REF] Kuczma | Functional equations in a single variable[END_REF][START_REF] Julia | Sur une classe d'équations fonctionnelles[END_REF] and arises in various applications such that iterated function theory [START_REF] Denjoy | Sur l'itération de fonctions analytiques[END_REF][START_REF] Kuczma | Iterative Functional Equations[END_REF], branching process [START_REF] Seneta | Functional equations and the Galton-Watson process[END_REF] or dynamical systems theory (see for instance [START_REF] Belitskii | Functional equations in real-analytic functions[END_REF] or [START_REF] Yoccoz | Théorème de Siegel, nombre de Bruno et polynômes quadratiques[END_REF]). Naturally, the generalized eigenvalue problem P α can be related to Schröder equations. Indeed, if the following Schröder equation,

(4.16) Ψ • f = αλ r (f ) • Ψ,
has a solution Ψ : F → F for α > 1, then the generalized eigenvalue problem for U f has an obvious solution, provided that Φ(•) = Ψ(•) ∈ E R F and f ∈ L r (F ). It is known that such an equation can be solved for particular domain F and particular space of function over F such as Hardy spaces; see [START_REF] Cowen | Composition Operators on Spaces of Analytic Functions[END_REF]. It is interesting to note that most of the results typically demand some compacity assumptions of the Koopman operator U f which involve that f possesses at least a fixed point in F ; cf. [START_REF] Caugran | Spectra of compact composition operators[END_REF], cf. also Theorem 5.1 in [START_REF] Clahane | Spectra of compact composition operators over bounded symmetric domains[END_REF] for extensions of results of [START_REF] Caugran | Spectra of compact composition operators[END_REF].

There exist other techniques coming from functional analysis rather than complex analysis, to deal with P α from the point of view of Schröder equations. It consists of considering a more general type of Schröder equations where the unknown is a map Ψ : F → E aiming to satisfy, (4.17)

U f (Ψ) = A • Ψ,
where A is a linear map of E. If we assume that A is invertible, and that there exist a C 0functional N : E → R with a constant m > 0 satisfying,

(4.18) ∀ x ∈ F, N (f (x)) ≥ N (x) + m,
then Theorem 2.1 of [START_REF] Walorski | On the continuous smooth solutions of the Schröder equation in normed spaces[END_REF] permits to conclude to the existence of a continuous nonzero solution of (4.17). Note that if F ⊂ E\{0} and f (x) ≤ κ x on F , with κ ∈ (0, 1) then the functional inequality (4.18) is satisfied on F by simply taking N (x) := -log x and m := -log(κ). Now by assuming A invertible,

∀ ξ ∈ E, ξ ≤ A -1 Aξ , i.e., A -1 -1 ≤ inf ξ∈E-{0} Aξ ξ ,
and thus if,

(4.19) A -1 -1 ≥ αλ r (f ),
and there exists a couple (m, N ) satisfying (4.18), we obtain that Φ(•) := Ψ(•) with Ψ a solution of (4.17), is a solution of U f (Φ) ≥ αλ r (f ) • Φ. Thus in order to have a solution of the generalized eigenvalue problem for U f , we only need to know about the asymptotic behavior of Φ.

However, the examination of the growth of an eigenfunction Ψ solution of (4.17) is a difficult task in general (see e.g. [START_REF] Bourdon | Mean growth of Koenigs eigenfunctions[END_REF] or [START_REF] Banas | On Existence and asymptotic behavior of solutions of some functional equations[END_REF] for particular cases more related to situations like the "basic" Schröder equation (4.16)), which renders the generalized eigenvalue problem for U f and thus the functional problem P α introduced here non-trivial to solve in general.

To conclude, we emphasize that the Schröder equation is related to Abel's functional equation [START_REF] Abel | Détermination d'une fonction au moyen d'une équation qui ne contient qu'une seule variable[END_REF], which is a well known functional equation often presented into the form ϕ(f (x)) = ϕ(x) + 1, where ϕ : X → C is an unknown function and f : X → X is a given continuous mapping of a topological space X [START_REF] Kuczma | Functional equations in a single variable[END_REF]. 6 With respect to our purpose, if the Abel equation possesses a continuous solution, then it provides a continuous solution to (4.18) that attributes to Abel equation a central role in our approach. In fact, the relationships between both equations with respect to our functional problem are deeper as illustrates the following subsection.

Approach based on cohomology equations.

Even if, to the best of the knowledge of the authors, Theorem 4.3 exhibits new relationships between the existence of a conjugacy between two homeomorphisms and the spectrum of the related Koopman operators, relationships between conjugacy problems and functional equations are far to be new. They arise classically under the form of the Livshitz cohomology equation [START_REF] Livshitz | Homology properties of Y -systems[END_REF][START_REF] Livshitz | Cohomology of dynamical systems[END_REF], φ = Φ • f -Φ, where f : M → M is a dynamical system of some manifold M , φ : M → R, a given function, and Φ maps M into R or a multidimensional space; see for instance [START_REF] Banyaga | Cohomology equations and commutators of germs of contact diffeomorphisms[END_REF][START_REF] Banyaga | Cohomology equations near hyperbolic points and geometric versions of Sternberg linearization theorem[END_REF][START_REF] Katok | Introduction to the Modern Theory of Dynamical Systems, With a supplement by Anatole Katok and Leonardo Mendoza[END_REF][START_REF] De La Llave | Canonical perturbation theory of Anosov systems and regularity results for the Livsic cohomology equation[END_REF]. As pointed by Livshitz [START_REF] Livshitz | Homology properties of Y -systems[END_REF][START_REF] Livshitz | Cohomology of dynamical systems[END_REF] the existence of a continuous solution Φ strictly depends on the dynamics generated by f and the topological as well as geometrical properties of M . For instance if we consider the particular case of the Abel equation, Φ(f (x)) = Φ(x) + 1, there is no continuous solution if M is compact, since if such solution exists, Φ(f n (x)) = Φ(x) + n, which is impossible.

In the case of non-compact topological manifold, Belitskii and Lyubich have proved the following theorem (see [START_REF] Belitskii | The Abel equation and total solvability of linear functional equations[END_REF]; Corollary 1.6), that we present in a slightly less general setting, adapting their statements with respect to our purpose: Theorem 4.6. (From [START_REF] Belitskii | The Abel equation and total solvability of linear functional equations[END_REF]) Assume that M is locally compact and countable at infinity. If f : M → M is continuous and injective then the following statements are equivalent, (a) There exists a continuous solution ϕ : M → C of the Abel equation, ϕ(f (x)) = ϕ(x) + 1. 6 The link between the both is trivial when X ≡ C where every solution ϕ of the Abel equation leads to a solution Ψ : x → exp(log(λ)ϕ(x)) for every λ > 0 of the Schröder equation. In such a case the spectrum of U f contains (0, +∞). (c) Every compact subset of M is wandering for f .

In the above theorem, a compact set K ⊂ M , is qualified to be wandering if there exists an integer ν ≥ 1 such that

f n (K) ∩ f m (K) = ∅ (n -m ≥ ν),
in particular such a dynamical system f is fixed-point free and periodic-point free, which might be consistent with dynamical restriction imposed by any solution to the functional problem P α in the case λ r (f ) ≥ 1. 7 This powerful theorem provides however an incomplete solution to the problem P α . For instance, Theorem 4.6 shows that for f satisfying condition (c) above, there exits a solution of the equation ϕ(f (x)) = λϕ(x) with |λ| ≥ αλ r (f ), and therefore a solution of the generalized eigenvalue problem for U f ; obtained by taking its module, Φ(•) := |ϕ(•)|. The missing step for having a full solution of that problem is still the knowledge of the asymptotic behavior of Φ, which is a difficult task also for cohomology equations; see e.g. [START_REF] Banyaga | Cohomology equations near hyperbolic points and geometric versions of Sternberg linearization theorem[END_REF].

Remark 4.7. It is worth mentioning that in the framework developed in this article, the subset F plays a central role in the existence of solutions of functional problem of type P α . The reader may consult for instance [START_REF] Belitskii | Functional equations in real-analytic functions[END_REF], §5.2, where it is shown that the Abel equation associated to a contracting mapping of the cut plane R 2 \(-∞, 0], has a real-analytic solution, whereas the same map considered on the whole plane leads to an Abel equation without any continuous solution, since such a map possesses obviously a fixed point which is excluded by Theorem 4.6.

Whatever F and the approach retained, we can conclude that the main issue concerns therefore the asymptotic behavior of a possible eigenfunction of the generalized eigenvalue problem P α . The regularity of the common eigenfunction is also an important aspect of the problem, in that respect, the continuity assumption on R and thus on Φ could be relaxed using Remark 2.1. Dynamical properties such as condition (c) of Theorem 4.6 might play also a role in the existence of such eigenfunctions. Note also, that since the closure of E R F in the compact-open topology [START_REF] Hirsch | Differential Topology[END_REF] is a closed cone with non empty interior in C(F, R) (cf. Remark 2.1-(a)), it would be interesting to study the spectral properties of U f and U g within an approach of type Krein-Rutman theorem [START_REF] Schaefer | Topological Vector Spaces[END_REF]. However, the task is more difficult than usually since F is non-compact, σ-compact and locally compact, and therefore C(F, R) has a Fréchet structure [START_REF] Aliprantis | Infinite Dimensional Analysis: A Hitchhiker's Guide[END_REF](and not a Banach one), which renders non straightforward an extension of the classical Krein-Rutman theorem in that context to analyze the existence of a principal eigenfunction in E R F . Finally, we have intentionally not considered important dynamical properties such as hyperbolicity [START_REF] Katok | Introduction to the Modern Theory of Dynamical Systems, With a supplement by Anatole Katok and Leonardo Mendoza[END_REF] that could lead to other spectral problem than P α ; the purpose of the present work was indeed more to set up, in a general context, a framework that makes apparent certain relationships between the spectral theory of dynamical systems and the geometrical problem of conjugacy. Likewise, the idea of using some observable -here a common eigenfunction of the Koopman operators -to build a specific topology to deal with the conjugacy problem does not seem to be limited to the case of non-compact phase-space. We leave open all these interesting fields of investigations.

7 Indeed, if there exists a generalized eigenfunction Φ of U f and a periodic orbit of f of period p emanating from some x * , then by repeating p-times the change of variable x ← f (x) in Φ(f (x)) ≥ αλ r (f )Φ(x), we deduce necessarily that (αλr(f )) p ≤ 1 (since Φ > 0), which imposes that f cannot possess such a periodic orbit in the case λ r (f ) ≥ 1.

Definition 2 . 5 .

 25 We define Hom(F ) (-),b + to be the set constituted by the adherence values in T

Definition 4 . 2 .

 42 Let f ∈ Hom(F ), and U f its Koopman operator with domain

(4. 13 )

 13 |f g -1 | Φ,r = sup x∈F

( b )

 b For every continuous functions p : M → C\{0} and γ : M → C there exits a continuous solution ϕ : M → C of (4.20) ϕ(f (x)) = p(x)ϕ(x) + γ(x).

Note that similarly, | • | Φ,r may be extended in such a way, but it is important to have in mind that |f | Φ,r = ∞

Note that, since E is a finite dimensional normed vector space, which is locally compact Haussdorf space, ifF is an open or closed subset of E, it is locally compact; cf. e.g 3.18.4, p. 66 in[START_REF] Dieudonn | Éléments d'Analyse[END_REF]

Note that this reasoning is independent from the choice of the metric rendering Hom(F ) complete, since it is known that Hom(F ) as a unique Polish group structure (up to isomorphism); see[START_REF] Kallman | Uniqueness results for homeomorphism groups[END_REF]

The Koopman operator is also known as the composition operator in other fields[START_REF] Cowen | Composition Operators on Spaces of Analytic Functions[END_REF].