Kamal Ghoumid
email: kghoumid@ensa.univ-oujda.ac.ma

Senior Member, IEEE Badr-Eddine Benkelfat
email: badr-eddine.benkelfat@int-edu.eu

Richard Ferriere
email: richard.ferriere@univ-fcomte.fr

Gwenn Ulliac
email: gwenn.ulliac@univ-fcomte.fr

Tijani Gharbi
email: tijani.gharbi@univ-fcomte.fr

R Ferriére

Wavelength-Selective Ti:LiNbO Multiple Y-Branch Coupler Based on Focused Ion Beam Milled Bragg Reflectors

Keywords: Bragg reflectors, drop channels, wavelength filtering devices, Y-branch coupler

This paper described a wavelength-selective couplers dedicated to drop functionality (spectral-selective couplers dedicated to wavelength demultiplexing). It is consist of two Y junctions: an input two drop channels and an output. The Y-branches distributors of optical power were carried out by a specific geometry of Ti:LiNbO waveguides. Each Y junction is composed by two identical bends jointed at one with a weak inclination angle to decrease losses. On each arm of the Y-branches, a Bragg grating (BG) is etched by Focused Ion Beam (FIB) in order to extract a specific wavelength band. A two-channel configuration is adopted which demonstrate a drop functionality of two spectral bands centered at 1123 nm and 1527 nm. The complementary spectrum is avalaible at the coupler output.

I. INTRODUCTION

D ENSE and Coarse Wavelength Division Multiplexing (WDM) technologies, for metro and access networks, are some applications based on the use of specific optical devices. In the last decade, these optical networks are rapidly evolved from static to dynamic architectures. The critical component in a reconfigurable network is the Reconfigurable Optical Add/Drop (ROADM) which require flexibility in the optical wavelength domain, based on Wavelength Selective Switching (WSS).

A wavelength selective switch is an optical device that routes different spectral components from an input port to the desired output ports without optical-to-electrical-to-optical conversion.

In this context, several approaches have been proposed for implementation of the WSS subsystem. Most of them have mainly focused on Liquid Crystal on Silicon (LCoS) and Micro-Electro-Mechanical-System (MEMS) technologies [START_REF] Sakurai | LCOS-based 4 4 wavelength cross-connect switch for flexible channel management in ROADMs[END_REF], [START_REF] Strasser | Wavelength-selective switches for ROADM applications[END_REF].

In a conventional WSS, the optical performance (e.g., channel spacing, filter shape, etc.) is determined by the physical design of the optical structure through which the light passes. More recently, an add/drop filter based on coupled vertical gratings on silicon has been proposed to implement 1 by 4 WDM functionality [START_REF] Tan | Wide bandwidth, low loss 1 by 4 wavelength division multiplexer on silicon for optical interconnects[END_REF]. The realization of such applications in integrated optics opens several applications.

Integrated optical components using Y-branch structures are used in such applications, are commonly employed in recent years like, for example, to realized Mach-Zehnder modulators [START_REF] Zhou | Performance limitations of a Y-branch directional coupler-based polymeric high-speed electro-optical modulator[END_REF], [START_REF] An | Polymeric electro-optic modulator based on 1A2 Y-fed directional coupler[END_REF], or 1 2 digital optical switch known as "Y-branch digital optical switch" (DOS) [START_REF] Chang | Switching characteristics of Ni:LiNbO directional couplers with two types of waveguide structures[END_REF]- [START_REF] Erdogan | Optical add-drop multiplexer based on an asymmetric Bragg coupler[END_REF]. Specific function, like variable power ratio division or TE/TM mode conversion, can be realized by using the electro-optic properties of lithium niobate (LN, LiNbO ) [START_REF] Hida | Polymer waveguide thermooptics switch with low electric power consumption at 1.3 m[END_REF]. Wavelength filtering can also be obtained by surface acoustic wave (SAW) on LN. Furthermore, with an asymmetrical Y-junction structure fabricated by diffusing different materials into Y-cut LN, the incident TE and TM modes can be split by two output arms due to their different preferences of polarizations [START_REF] Chang | A novel Y-branch waveguide for power dividing and mode splitting[END_REF]. More recently the propagation in two-dimensional photonic crystal slabs allows an amplitude division of a guided beam [START_REF] Sugimoto | Light-propagation characteristics of Y-branch defect waveguides in AlGaAs-based air-bridge-type two-dimensional photonic crystal slabs[END_REF].

The lithium niobate crystal stays a popular material for integrated optics, mainly due to its electro-optical, acousto-optical and non-linear properties that are exploited on various commercial devices [START_REF] Pant | Optimal control theory for optical waveguide design: Application to Y-branch structures[END_REF]- [START_REF] Schiek | Nonlinear directional coupler in periodically poled lithium niobate[END_REF]. It needs relatively simple techniques to obtain good performances compared with the sophisticated technologies used for semiconductor materials. The use of such a material is recommended in several integrated components.

The micro/nano-structuring of LiNbO for the manufacture of BGs represents however a challenging task, due to the well known resistance of the material to standard machining methods [START_REF] Kim | Narrowband Bragg reflectors in Ti:LiNbO optical waveguides[END_REF]- [START_REF] Takeshima | Fabrication of a periodic structure wit a high refractive-index difference by femtosecond laser pulses[END_REF]. Different approaches have been conducted for achieving nanostructures are employed [START_REF] Ghoumid | Optical performance of Bragg gratings fabricated in Ti:LiNbO waveguides by focused ion beam milling[END_REF]- [START_REF] Ferrière | Bragg miroir inscription on LiNbO waveguides by index microstructuration[END_REF]. The possibility of realizing miniature LiNbO components arises with these fabrication methods.

In this article, FIB method is implemented to etch BGs on Ti:LiNbO waveguides [START_REF] Chaganti | Sub-micron grating fabrication on hafnium oxide thin-film waveguides with focused ion beam milling[END_REF], [START_REF] Yin | Fabrication of high-aspect-ratio submicron-to-nanometer range microstructures in LiNbO for the next generation of integrated optoelectronic devices by focused ion beams (FIB)[END_REF] whose geometrical structure is composed by Y-branch junctions, in order to extract spectrum bands in awideband optical source. We chose this technique because we have already shown its effectiveness in achieving the proper Bragg reflectors with reflection coefficients that reach 97%, associated with bandwidth midheight width of about 100 nm [START_REF] Ghoumid | Optical performance of Bragg gratings fabricated in Ti:LiNbO waveguides by focused ion beam milling[END_REF], [START_REF] Ghoumid | Effect of depth etching on Bragg reflectors realized by focused ion beam in Ti:LiNbO waveguide[END_REF]. This method has already been employed to etch sub-micrometric one-dimensional structures in LiNbO or 2D-photonic crystals. In comparison with more standard methods, FIB milling the advantage concerne the direct etching without any additional technological steps. Indeed FIB is a high capacity tool providing direct, local and controlled waveguide property modifications [START_REF] Chaganti | Sub-micron grating fabrication on hafnium oxide thin-film waveguides with focused ion beam milling[END_REF], [START_REF] Yin | Fabrication of high-aspect-ratio submicron-to-nanometer range microstructures in LiNbO for the next generation of integrated optoelectronic devices by focused ion beams (FIB)[END_REF].

Here we show the feasibility and the realization of a specific coupler using as a wavelength filter which consist of two Y-branches and two BGs in order to extract spectrum bands in a wideband optical source. Our idea was based on the properties of BGs that are used generally to achieve very high reflection around a given central wavelength. Thus, at the first two outputs, two spectral bands around 1123 nm and 1527 nm are extracted by the mean of BGs. At the output, the complementary spectrum of the input source is available. This component could have a significant interest in convenient and reproducible fabrication technologies, 'The benefit of the proposed structure is that can be used as the switching element for wavelength routing in a wavelength selective switch.

We will consider as the future prospects will be to benefit from eletro-optical properties of LN for realizing tunable devices, that would be very useful and convenient for the switching element for wavelength routing in a Wavelength Selective Switch.

II. PRINCIPLE OF WAVELENGTH DEMULTIPLEXING BY MULTIPLE Y-BRANCH COUPLER

We present a wavelength-selective coupler; the common property desired in this type of device is the extraction of peculiar wavelengths from a continuum. Its basic scheme is given in Fig. 1; it consists in an input E and three outputs and . Two Bragg gratings noted and were etched in each arm of the Y junctions, they reflect the wavelengths around and respectively. The other wavelengths are transmitted and not affected by the BG.

If a continuous spectrum is injected at E, a wavelength bands reflected around and are recovered at and due to and respectively. It is obtained by the mean of a "white" fiber laser. At output, the entire spectrum is transmitted, except for the reflected bands. It should be noted that the wavelengths reflected around and recovered at have a power less than those recovered at , this is due to the alignment losses. We also note that the output , the presence a small portion of the spectrum reflected wavelengths around due to . We recall that the wavelengths reflected obey the Bragg condition given by following equation [START_REF] Erdogan | Fiber grating spectra[END_REF]- [START_REF] Matuschek | Exact coupled-mode theorys for multilayer interference coatings with arbitrary strong index modulations[END_REF]: [START_REF] Sakurai | LCOS-based 4 4 wavelength cross-connect switch for flexible channel management in ROADMs[END_REF] where is the effective index of the propagating mode in the waveguide, and are respectively the Bragg order and period.

is the Bragg wavelength.

III. FABRICATION PROCESS

The preliminary experiments were carried out on a LiNbO chip with a thickness of 0.5 mm and X-cut Z-propagation is used as substrate. In this configuration the two polarizations TE/TM of incident wave are affected by the same index nm during its propagation. A Titanium layer (90 nm thickness) is deposited on the crystal surface by means of an Alliance Concept Eva 450 sputtering. The evaporation speed in the chamber is controlled by an Inficon quartz oscillator which assures a homogenous deposition of the metal. The geometric pattern of waveguides is realized by classical photolithographic techniques and chemical etching of the titanium layer. The optical waveguides are obtained by standard thermal indiffusion (1020 C during 8 h) of titanium ribbons (90 nm-thick and 7 m wide). The thermal diffusion of the titanium layer causes a variation of the refractive index of LN proportional to the concentration of Ti. Therefore, we obtain an increase in ordinary and extraordinary indexes of LiNbO . Thus, the resulting layer is described as an index which is greater relatively to those of adjacent layers. It allows the confinement of light, and hence the realization of the waveguide Ti:LiNbO . The guiding layer is of an index gradient, it is designed to reduce scattering losses due to interface roughness, and also to minimized the modal and polarization dispersion of the optical mode traveling in the waveguide.

With these experimental parameters the refractive indexes of the waveguide and the adjacent layers are: nm nm nm nm . In addition, it is worth noting that the waveguide is single mode obtained in the range of wavelengths [1050-1600] nm (with cutoff wavelength nm). The profile observed on a monitor via an infrared camera is shown in Fig. 2. These dimensions measured at relatively to its maximum are m and m, and its maximum is measured at m. A top views of Ti:LiNbO waveguide and the two Y-branches are represented on the Fig. 3. It is noted that the two Y junctions are designed with an inclination angle in order to avoid significant propagation losses in the waveguide splitter [START_REF] Lee | Enhacing alignment tolerance of silicon waveguide by using a wide grating coupler[END_REF]- [START_REF] Papakonstantinou | Insertion loss and misalignment tolerance in multimode taperedwaveguide bends[END_REF]. In our conception, the angle of inclination of the arms is even lower than this value, it is 0.6 .

Once the waveguides were manufactured, the following step deals with the etching of and . The FIB method was adopted to realize a micro-structuration of the waveguides. FIB technique produces a localized sputtering of the material. Consequently it induces significant transformations of the guide structure and a strong modification of the propagation properties. In these experimentations a dual beam Orsay Physics Canion 31 LEO 4400 FIB was used, the command of the beam deflection was realized by a Raith Elphy Quantum 4.0 driver. Starting with the sputtering of a Chromium Cr layer whose main role is to prevent the accumulation of Ga ions, with a thickness of 12 nm on the substrate, which was then inserted into the FIB chamber and exposed to a Ga liquid metal ion source (LMIS) with ion acceleration energy of 30 keV.

For the design of both BGs, we used (1), with the Bragg order and nm. As a result, we obtain the following parameters: the first has a period m, a period number and an etching depth m, whereas the second (with and nm) has a period m, a period number and an etching depth m. The BG obtained by FIB etching is illustrated in Fig. 4.

It should be noted the etching depth of the FIB process ( m) doesn't match the modal depth of the Ti diffused waveguides (about 2.2 m evaluated at of maximum mode intensities [START_REF] Ghoumid | Optical performance of Bragg gratings fabricated in Ti:LiNbO waveguides by focused ion beam milling[END_REF]). This means for sure reduced grating effectiveness and additional out of plane losses. At this etching depth m, we expect an improvement of losses of in order of about 2.2 dB compared to the same BG the same quality but etched up to a depth of m.

IV. RESULTS AND DISCUSSIONS

To assess the spectral responses in each output, a continuous spectrum source with a wavelength range of nm is launched at the coupler input. This continuum laser uses a strong nonlinear effect induced by a picosecond pulse (1064 nm) injected in a photonic crystal fiber specially designed [START_REF] Dudley | Supercontinuum generation in photonic crystal fiber[END_REF]. It allows to obtain radiation with a flat broadband spectrum from nm. The measurement setup included: The measurement setup included: elements (V-grooves :

) for the injection and recovery of transmitted and reflected optical signals, waveguide + selectif coupler, as well as three optical spectrum analysers and to measure reflectivities and the transmittances respectively (Fig. 5).

Figs. 6 and7, illustrate the reflectivities curves versus wavelengths measured at the outputs and respectively, they were centered around nm and nm. The coefficients of reflection of each curve were measured and found about 95%. The bandwidth at half maximum are respectively nm and nm. Fig. 8 shows the measured transmission spectrum at the output versus wavelengths of the signal of our device, and the whole transmitted spectrum was obtained except for the reflected wavelengths. It should be noted that the reflectance PSD and transmittance PSD are not the exact reciprocal of each other (because of the loss, absorption ).

This study of experimental measurements implies the announcement of the existence of various types of losses which is related to: the injection and alignment process, material absorption, Y-junctions (inclination) and Bragg gratings corrugations. In the absence of corrugations, the three first types of front losses mentioned, can be estimated from Fig. 8 to be around 6 dB (this value is found to be the mean difference between curves (a) and (b)). The losses due to etching can be estimated between 2 and 2.5 dB (the mean difference between curves (b) and (c)) [START_REF] Ghoumid | Optical performance of Bragg gratings fabricated in Ti:LiNbO waveguides by focused ion beam milling[END_REF].

A good results should be pointed out, a nice and non trivial feature of the presented experimental results is the absence of parasitic coupling to cladding modes, that usually induce unwanted losses at wavelengths shorter than the nominal one.

A significant remark concerning the existence of the parasitic peaks in the spectra of the reflectivities or of the transmittance measured at the exits and , which are due to the roughness of the holes obtained after etching by FIB [START_REF] Ghoumid | Optical performance of Bragg gratings fabricated in Ti:LiNbO waveguides by focused ion beam milling[END_REF]. Indeed, by the engraving of a material by the mean of the FIB, the later is insulated by Ga ions, these induced modifications by the ionic insulation seems to be insignificant and the reason of the imperfections in the transmitted spectra and the Optical propagations is certainly due to the conical shape of the engraved BGs [START_REF] Burr | The impact of finite-depth cylindrical and conical holes in lithium niobate photonic crystals[END_REF] and does not seem to be a modification of the substrate from the insulation. V. CONCLUSION A concrete application for extraction of wavelength bands by using Ti:LiNbO waveguides with two Y junctions, comprising an input and a several outputs had been clearly demonstrated. The waveguides were manufactured by thermal diffusion process, and the Bragg gratings were etched by FIB in each Y-branch. Thus, a successful extraction from the bands around 1123 nm and 1527 nm was achieved. The principal advantages of this directional coupler, is that it can be used in both directions. Moreover, it can contain several Y junctions and several BGs, in order to extract other wavelengths. Nevertheless, we noted that the milling process of BGs was very long (eight hours).

Our perspective is to design, manufacture and characterize this kind of components suited for applications like DWDM. We will seek to remove wavelengths below 1 nm by changing the BGs parameters used for obtaining highly selective filters. Another objective will be to benefit from eletro-optical properties of LN for realizing tunable devices that would be very useful and convenient in practical applications. Also, we note that this structure can be used in an application in the field of optical networks. He presently holds a position of Professor at Institut TELECOM-TELECOM Sud-Paris, Evry, France. His currently research interest include optoelectronic systems for optical signal processing, optical devices for high-speed WDM optical fiber communications. In 1978, he joined the Centre National de la Recherche Scientifique (CNRS) as a Physicist, where he was involved with optical processing in polychromatic light and holographic optical elements. He current research deals with telemetry, femto-second laser milling, electroptic signal processing and integrated optics components in the field of telecommunications Gwenn Ulliac received the M.Sc. and Ph.D. degrees in microwave and microtechnology from Lille University, France, in 2001 and 2005, respectively.

Richard

In 2006, he joined the FEMTO-ST Institute, where he is currently a research engineer in the clean room MIMENTO. His research interests include the fabrication and the technological developments of opto-electronic and acousto-optical components on lithium niobate material (waveguides, bragg gratings, photonic crystals, phononic crystals, acoustic resonator. Tijani Gharbi received the D.Sc. degree in state molecular physics from the University of Franche-Comté, Besancon, France, in 1982.

He has extensive experience in research, training and valorization from the medical sector. He served for 30 years in the biomedical field. He is responsible for the team "Microsystems for Life Sciences" in the Optics Laboratory of Besancon, then head of the team "Biophotonics" of the FEMTO-ST UMR CNRS 6174. He assured the management of 20 Ph.D. theses. On the education side, it was the precursor of the rapprochement of the engineering and the medical technology in the region of Franche-Comté. As such, he is one of the founding members of the Higher Institute of Engineers of Franche-Comté (ISIFC). He is the author of numerous scientific papers published in major international journals. He is also an inventor of some 30 patents related to biomedical applications.

Fig. 1 .

 1 Fig. 1. Wavelength selectif coupler composed by two Y junctions cascading: an input , two drop channels , and output , including two Bragg gratings and on each arm.

Fig. 2 .

 2 Fig. 2. The profile of the mode propagated in the plan of Ti:LiNbO waveguide observed on a monitor TV via an infrared camera. It is single-mode at the wavelength 1550 nm.

Fig. 3 .

 3 Fig. 3. (a) First junction Y, (b) second junction Y, (c) Ti:LiNbO waveguide obtained by standard diffusion (90 nm thick and 7 m wide titanium layers at 1020 C during 8 h).

Fig. 4 .

 4 Fig. 4. (a) m, m) etched in a Ti:LiNbO waveguide by FIB, (b) zoom of and Ti:LiNbO waveguide.

Fig. 5 .Fig. 6 .

 56 Fig. 5. Experimental setup : continuous spectrum [850-1750] nm, V-grooves , optical fibers (O.F.), waveguide + selectif coupler and and .

Fig. 7 .

 7 Fig. 7. Reflectivity versus wavelengths measured at the output : peak centered around nm with nm.

Fig. 8 .

 8 Fig. 8. Transmission versus wavelengths measured at the output . The peak at 1064 nm corresponds to the pump of the 'white' laser employed to provide the wide spectrum at E input. (a) Source 'white' laser, (b) Ti:LiNbO waveguide, (c) measured spectrum at the output .

Kamal

  Ghoumid received the Ph.D. degree from the Institut TELECOM, TELECOM Sud-Paris, Evry, France, and the University of Franche-Comté, Besanon, France), in 2008. He has worked as postdoctoral researcher at Institut Jean Lamour of University of Henri Poincaré, Nancy, France, during 2009. His area of research work was to study Pirani Vacuum SAW sensor. He has taught at the ENSAO School, Oujda, Morocco. Since 2010, he has been working in LOPMD Laboratory of Institut FEMTO-ST, France. He has also the experience in research areas of electro-optic modulator, optical devices for high-speed WDM, signal processing and integrated optic components in the field of telecommunications. Badr-Eddine Benkelfat (SM'03) was born on August 18, 1958. He received the Ph.D. degree from the University of Franche-Comté, Besanon, France, in 1983.

  Ferriere was born in 1947. He received the Doctorat de 3eme cycle in 1975 and the Doctorat d'Etat degree from the University of Franche-Comté, Besancon, France, in 1982.

ACKNOWLEDGMENT

The authors wish to acknowledge R. Salut, B. Guichardaz, D. Bitschene, J.-Y. Rauch, V. Petrini for technical assistance, and all personnel of the MIMENTO technology centre of the Institut FEMTO-ST, Besançon, France.


David Burguet
email: david.burguet@upmc.fr

Ruxi Shi
email: rshi@impan.pl

ZERO-DIMENSIONAL AND SYMBOLIC EXTENSIONS OF TOPOLOGICAL FLOWS

A zero-dimensional (resp. symbolic) flow is a suspension flow over a zero-dimensional system (resp. a subshift). We show that any topological flow admits a principal extension by a zero-dimensional flow. Following [Bur19] we deduce that any topological flow admits an extension by a symbolic flow if and only if its time-t map admits an extension by a subshift for any t = 0. Moreover the existence of such an extension is preserved under orbit equivalence for regular topological flows, but this property does not hold more true for singular flows. Finally we investigate symbolic extensions for singular suspension flows. In particular, the suspension flow over the full shift on {0, 1} Z with a roof function f vanishing at the zero sequence 0 ∞ admits a principal symbolic extension or not depending on the smoothness of f at 0 ∞ .

Introduction

A symbolic extension of a discrete topological system (X, T ) is a topological extension π : (Y, S) → (X, T ) by a subshift (Y, S). Existence of symbolic extensions and their entropy are related with weak expansive entropy properties of the system [START_REF] Boyle | The entropy theory of symbolic extensions[END_REF]. Building on [START_REF] Boyle | The entropy theory of symbolic extensions[END_REF] the first author and T.Downarowicz also investigate uniform generators, which are symbolic extensions with a Borel embedding ψ : (X, T ) → (Y, S) with π • ψ = Id X .

To study symbolic extensions for discrete systems, M. Boyle and T. Downarowicz [START_REF] Boyle | The entropy theory of symbolic extensions[END_REF] have developed a new entropy theory. The first step in their construction of symbolic extensions consists in building a zero-dimensional principal extension (see Section 2.2 for precise definitions). In [START_REF] Boyle | The entropy theory of symbolic extensions[END_REF] this is done by using the small boundary property for finite topological entropy systems with a minimal factor [START_REF] Lindenstrauss | Mean dimension, small entropy factors and an embedding theorem[END_REF]. In this way one even get a strongly isomorphic zero-dimensional extension. Later T. Downarowicz and D. Huczek gave a constructive proof of a zero-dimensional principal extension for any topological system (with finite topological entropy or not).

More recently the first author developed in [START_REF] Burguet | Symbolic extensions and uniform generators for topological regular flows[END_REF] a theory of uniform generators for topological regular (i.e. without fixed points) flows. In this context a symbolic extension (resp. uniform generator) is a topological extension by a regular suspension flow over a subshift (resp. with an embedding). To investigate existence of uniform generators he considered topological flows satisfying the so-called small flow boundary property 1 . Under this extra assumption, the flow admits a strongly isomorphic zero-dimensional extension. Here we investigate only symbolic extensions for topological flows and not uniform generators for these, so that to be reduced to the case of zero-dimensional suspension flows, we only need to build a principal zero-dimensional extension (this is achieved in Section 2). In this way we may also consider the symbolic extensions of a topological singular flow 2 . In Section 3 we build on [START_REF] Burguet | Symbolic extensions and uniform generators for topological regular flows[END_REF] to study symbolic extension for general topological flows. In particular we show that two regular topological flows which are orbit equivalent either both admit a (resp. principal) symbolic extension or not. This result was only proved in [START_REF] Burguet | Symbolic extensions and uniform generators for topological regular flows[END_REF] for regular flows with the small flow boundary property. We also give counterexamples for singular flows.

In the rest part (Section 4) we define singular suspension flows as suspension flows over a discrete system (X, T ) with a roof function vanishing only at a fixed point of T . We investigate their symbolic extensions. In the case (X, T ) is expansive, an associated singular suspension flow 1 The small flow boundary property (for topological flows) is an analogy to the small boundary property (for discrete topological systems). This notion is introduced by the first author in [START_REF] Burguet | Symbolic extensions and uniform generators for topological regular flows[END_REF].

2 An isomorphic extension of a singular flow is necessarily singular (so that if one wants to develop a theory of uniform generators one should consider singular suspension flows over a subshift).

1 with finite topological entropy does not admit in general a principal symbolic extension. This depends on the behaviour of the roof function near the singularity. For (X, T ) being the two full shift we illustrate this phenomenon by computing the symbolic extension entropy function for a large class of roof functions. In fact we explicitly build a symbolic extension of the corresponding singular suspension flow.

Zero-dimensional principal extension

By a (discrete) topological system (X, T ), we mean that X is a compact metrizable space and T : X → X is continuous. Moreover, the system is said to be invertible if T is a homeomorphism. A pair (X, Φ) is called a topological semi-flow (resp. topological flow) if X is a metrizable compact space and Φ = (φ t ) t≥0 (resp. Φ = (φ t ) t∈R ) are continuous maps from X to itself satisfying that φ 0 (x) = x and φ t (φ s (x)) = φ t+s (x) for all t, s ≥ 0 (resp. t, s ∈ R). A point x ∈ X is a fixed point of Φ when φ t (x) = x for all t ≥ 0. A flow without fixed point is called regular, otherwise the flow is said singular. We let M T (X) or M Φ (X) (resp. M e T (X) or M e Φ (X)) be the compact set of Borel probability measures invariant (resp. ergodic) by the topological system or flow. We recall that the measure-theoretic entropy h Φ (µ) of µ ∈ M Φ (X) is defined as the the entropy of its time-1 map, i.e. h Φ (µ) = h φ1 (µ). Through this paper, we use the following notations: R ≥0 = [0, ∞) and R >0 = (0, ∞).

2.1. Suspension semi-flows. Let (X, d) be a compact metric space and T : X → X a continuous map. Let f : X → R >0 be a continuous map. Let X f be the compact subset of X × R ≥0 defined by

X f := {(x, t) : 0 ≤ t ≤ f (x), x ∈ X}. (2•1)
We consider the equivalence relation ∼ on X f with (x, f (x)) ∼ (T x, 0) for all x ∈ X and we denote by X f the quotient space X f / ∼ endowed with the quotient topology. By abuse of notations we also write (x, t) to denote the equivalence class of (x, t) ∈ X f . The suspension semi-flow over T under the roof function f , written by (X f , T f ), is the semi-flow (T f t ) t∈R on the space X f induced by the time translation T t on X × R ≥0 defined by T t (x, s) = (x, t + s). If T is a homeomorphism, then (X f , T f ) defines a flow. Such (semi-)flows are regular.

Bowen and Walters define a metric d f compatible with the quotient topology on X f as follows [BW72, Section 4]. For a point (x, t) ∈ X f we let u (x,t) = t/f (x). A pair of points A = (x A , t A ) and B = (x B , t B ) in X f is said to be

• horizontal if u A = u B , then its length is |AB| := (1 -u A )d(x A , x B ) + u A d(T x A , T x B ). • vertical if x A = x B , x B = T x A or x A = T x B , then its length is |AB| := |u A -u B |, 1 - u B + u A or 1 -u A + u B respectively. A sequence A 1 , • • • , A n of n points in X f is said admissible when (A i , A i+1 ) is either vertical or horizontal for i = 1, • • • , n -1
and the length of this sequence is defined as the sum of the length of its corresponding pairs. Then the distance d f between two points A and B is defined as the infimum of the length of all admissible sequences

A 1 , • • • , A n with A 1 = A and A n = B.
Denote by L the Lebesgue measure on R. Let (X, T ) be a discrete topological system. Let f : X → R >0 be a continuous function and (X f , T f ) be the associated suspension semi-flow. For µ ∈ M T (X) the product measure µ × L induces a finite T f -invariant measure on X f , which defines a homeomorphism Θ between M T f (X f ) and M T (X):

Θ : M T (X) → M T f (X f ) µ → (µ × L) |X f f dµ .
Due to Abramov [START_REF] Abramov | On the entropy of a flow[END_REF], the entropy of µ and Θ(µ) are related by the following formula

(2•2) h T f (Θ(µ)) = h T (µ) f dµ , ∀µ ∈ M T (X).
2.2. Extensions. A suspension flow over a zero-dimensional invertible dynamical system will be called a zero-dimensional suspension flow and a topological extension by a zero-dimensional suspension flow is said to be a zero-dimensional extension. Similarly a suspension flow over a symbolic discrete topological dynamical system (a.k.a. Z-subshift) will be called a symbolic suspension flow and a topological extension by a symbolic suspension flow is said to be a symbolic extension.

Let (X, Φ) and (Y, Ψ) be two topological semi-flows. Suppose that π : Y → X is a topological extension from (X, Φ) to (Y, Ψ). The topological extension is said to be

• principal when it preserves the entropy of invariant measures, i.e. h(µ) = h(πµ) for all Ψ-invariant measure µ, • with an embedding when there is a Borel embedding ψ : (X, Φ) → (Y, Ψ) with π • ψ = Id X , • isomorphic when the map induced by π on the sets of invariant Borel probability measures is bijective and π : (Y, Ψ, µ) → (X, Φ, πµ) is a measurable isomorphism for any Ψ-invariant measure µ, • strongly isomorphic when there is a set E ⊂ X with µ(E) = 1 for all µ ∈ M Φ (X) such that the restriction of π to π -1 E is one-to-one. Clearly, we have the following implication: strongly isomorphic + 3 isomorphic with an embedding principal.

Construction.

In this section we build a zero-dimensional principal extension for any topological semi-flow.

Theorem 2.1. Every topological semi-flow has a zero-dimensional principal extension. Moreover the roof function of this extension may be chosen constant equal to 1.

To prove this theorem, we first show that every suspension semi-flow has a zero-dimensional principal extension (Proposition 2.3). Then, for a general topological flow we build a principal extension by a suspension flow. 2.3.1. Zero-dimensional principal extension of suspension flow. Let (Z, T ) be a discrete topological system. Let f : Z → R >0 be a continuous function. In this section, we construct a zero-dimensional principal extension of the suspension semi-flow (Z f , T f ). Due to T. Downarowicz and D. Huczek [START_REF] Downarowicz | Zero-dimensional principal extensions[END_REF], there exists an invertible dynamical system (X, S) which is a zero-dimensional principal extension of (Z, T ). Denote by ρ : X → Z the factor map. We see that the map g :

= f • ρ : X → R >0 is continuous. Define ρ : X g → Z f by ρ(x, t) → (ρ(x), t) for all (x, t) ∈ X g . We have ρ(x, f (x)) = (ρ(x), f • ρ(x)
) for all x ∈ X, so that ρ induces a continuous map ρ : X g → Z f . Moreover ρ commutes with the translation on the second coordinate, therefore ρ is a continuous factor map.

Lemma 2.2. ρ is principal.

Proof. Let µ ∈ M S (X). Since ρ is principal, we see that h(ρµ) = h(µ). It is clear that

ρΘ(µ) = ρ(µ × L) |X g g dµ = (ρµ × L) |Z f f d(ρµ) = Θ(ρµ).
Then by (2•2), we obtain that

h( ρΘ(µ)) = h(Θ(ρµ)) = h(ρµ) f d(ρµ) = h(µ) g dµ = h(Θ(µ)).
To sum up, we obtain the following proposition.

Proposition 2.3. Every suspension semi-flow has a zero-dimensional principal extension.

2.3.2. General case. We present the proof of Theorem 2.1 in the general case.

Proof of Theorem 2.1. Let (X, Φ) be a topological semi-flow. Let us denote by 1 the constant function on X equal to 1. Then the suspension semi-flow (X 1 , (φ 1 ) 1 ) over the time-1 map φ 1 under 1 defines an extension of (X, Φ) via the factor map π : (x, t) → φ t (x) for x ∈ X and t ≥ 0.

Notice that a (φ 1 ) 1 -invariant measure on X 1 has the form µ × L [0,1] where µ is a φ 1 -invariant measure and L [0,1] is the Lebesgue measure on [0, 1]. Pick arbitrary (φ 1 ) 1 -invariant measure µ × L [0,1] . It follows from Fubini's theorem that for all Borel subset B of X 1 :

π(µ × L [0,1] )(B) = µ × L [0,1] (π -1 ((B)), = µ × L [0,1] {(x, t) ∈ X 1 , φ t (x) ∈ B} , = 1 0 µ(φ -1 t (B)) dt, = 1 0 φ t µ(B) dt.
For all t ≥ 0, we observe that h φ1 (φ t µ) ≥ h φ1 (µ). Indeed, for s ≥ 0 with t + s ∈ N we have

h φ1 (φ t µ, φ -1 s P ) = h φ1 (µ, φ -1 s+t P ) = h φ1 (µ, φ - (t+s) 1
P ) = h φ1 (µ, P ) for any Borel finite partition P of X. Since the entropy function h φ1 is harmonic, we get then

h Φ (π(µ × L [0,1] )) = h φ1 (π(µ × L [0,1] )), = 1 0 h φ1 (φ t µ) dt, ≥ h φ1 (µ), ≥ h (φ1) 1 (µ × L [0,1] ).
Therefore, we conclude that (X 1 , (φ 1 ) 1 ) is a principal extension of (X, Φ). By Proposition 2.3, the suspension flow (X 1 , (φ 1 ) 1 ) has a zero-dimensional principal extension. By composition we get a principal zero-dimensional extension of (X, Φ). This completes the proof.

Applications to symbolic extensions

By the previous construction of a zero-dimensional principal extension, the results related to symbolic extensions obtained in [START_REF] Burguet | Symbolic extensions and uniform generators for topological regular flows[END_REF] for flows with the small flow boundary property may be straightforwardly extended to general topological flows. We first recall the framework of the entropy theory of M. Boyle and T. Downarowicz.

3.1. Superenvelope of entropy structures. Entropy structures are particular sequences of nonnegative real functions H = (h k ) k defined on the set X of invariant probability measures, X = M T (X) or = M Φ (X) depending on the context, which are converging pointwisely to the entropy function h. For a zero-dimensional system, the entropy with respect to any sequence (P k ) k of clopen partitions with diam(P k ) k -→ 0 defines an entropy structure. For a precise definition and examples we refer to [START_REF] Downarowicz | Entropy in dynamical systems[END_REF] for discrete systems and to [START_REF] Burguet | Symbolic extensions and uniform generators for topological regular flows[END_REF] for topological flows.

For a function f : X → R we let f ˜be the upper semi-continuous envelope of f , i.e.

∀µ ∈ X, f ˜(µ) = lim sup ν→µ f (ν).

A superenvelope E : X → R is an affine function such that for some (any) entropy structure

H = (h k ) k lim k (E -h k ) ˜= E -h.
3.2. Characterization of symbolic extensions. For a symbolic extension π : (Y, S) → (X, T ) of a topological discrete system (X, T ), we let h π be the associate entropy function defined as

∀µ ∈ M T (X), h π (µ) = sup ν∈M S (Y ), πν=µ h(ν).
We may define in the same way the entropy function associated to the symbolic extension of a topological flow. Then the fundamental theorem in the theory of symbolic extension is the following characterization of the entropy function h π in terms of superenvelopes.

Theorem 3.1 ([BD04], Theorem 5.5). For a discrete topological system (X, T ), any function h π is an affine superenvelope and conversely for any affine superenvelope E there is a symbolic extension π with h π = E.

The first author proved the corresponding statement for topological flows with the small flow boundary property (Theorem 3.6 in [START_REF] Burguet | Symbolic extensions and uniform generators for topological regular flows[END_REF]). The existence of a zero dimensional extension given by Theorem 2.1 implies the general following version : Theorem 3.2. For a topological flow (X, Φ), any function h π is an affine superenvelope and conversely for any affine superenvelope E there is a symbolic extension π with h π = E.

Similarly, we relate the symbolic extensions of a general topological flow with the symbolic extensions of the discrete systems given by its time-t maps (see Lemma 3.19 in [START_REF] Burguet | Symbolic extensions and uniform generators for topological regular flows[END_REF]).

Theorem 3.3. A topological flow admits a symbolic extension (resp. principal) if and only if φ t admits a symbolic extension (resp. principal) for some (any) t = 0.

We recall some terminology of the theory of symbolic extensions, which will be used in the next sections. Firstly the (topological) symbolic extension entropy h sex (T ) (resp. h sex (Φ)) of a topological system (resp. flow) is the infimum of the topological entropy over all its symbolic extensions. The corresponding measure theoretic quantity is the real function h sex defined on the set of invariant measures as the infimum of the functions h π over all the symbolic extensions π.

3.3.

Invariance under orbit equivalence. Two topological flows (X, Φ) and (Z, Ψ) are said to be orbit equivalent when there is a homeomorphism Γ from X onto Z mapping Φ-orbits to Ψ-orbits, preserving their orientation. In other words, the topological flows (X, Φ) and (X, Φ) with Φ = ( φ t ) t = (Γ -1 • φ t • Γ) t has the same orbits with the same direction, i.e.

{φ t (x) : t ∈ R ≥0 } = { φ t (x) : t ∈ R ≥0 }, ∀x ∈ X.
Assume that (X, Φ) is regular. Then we can define the continuous map θ : X × R ≥0 → R ≥0 with the following properties (see [START_REF] Beboutoff | Sur la mesure invariante dans les systemes dynamiques qui ne different que par le temps[END_REF]):

(i) φ t (x) = φ θ(x,t) (x), (ii) θ(x, t + s) = θ(x, s) + θ(φ s (x), t), (iii) θ(x, 0) = 0 and θ(x, t) is strictly increasing in t.
Zero and infinite topological entropy are invariants for orbit equivalent regular flows [START_REF] Ohno | A weak equivalence and topological entropy[END_REF]. In [START_REF] Burguet | Symbolic extensions and uniform generators for topological regular flows[END_REF] the first author showed existence of symbolic extensions is preserved under orbit equivalence for topological regular flows with the small boundary property. Here by using the zerodimensional principal extension built in Theorem 2.1 we remove this extra assumption. The proof differs completely from the one given in [START_REF] Burguet | Symbolic extensions and uniform generators for topological regular flows[END_REF] for flows with the small flow boundary property; it is not a direct consequence of the existence of principal zero dimensional extension as in the above Theorem 3.2 and Theorem 3.3.

Theorem 3.4. The existence of symbolic extensions (resp. principal) is preserved by orbit equivalence for regular flows.

Proof. Let Φ and Φ be two regular topological flows on a compact metric space X with the same orbits and let θ be defined as above. According to Theorem 2.1 the suspension flow (X 1 , (φ 1 ) 1 ) is a principal extension of (X, Φ) via the factor map π : (x, t) → φ t x. On the other hand, the function g : x → θ(x, 1) is continuous and positive on X (due to (iii)). Then by the fact that φ θ(x,1) (x) = φ 1 (x), the suspension flow (X g , (φ 1 ) g ) is the extension of (X, Φ) via the factor map π : (x, t) → φ t x :

(X 1 , (φ 1 ) 1 ) π (X g , (φ 1 ) g ) π (X, Φ) (X, Φ)
We claim that (X g , (φ 1 ) g ) is a principal extension of (X, Φ). To simplify the notations we denote by Ψ = (ψ t ) t the suspension flow (φ 1 ) g on X g . By Ledrappier-Walters formula

3 [LW77], it is enough to check h top (ψ 1 , π -1 z) = 0 for all z ∈ X. Fix z ∈ X. Since the map π : X g → X is given by π(x, t) = φ t x for x ∈ X and 0 ≤ t < g(x), the set π -1 z is contained in the set {( φ -t z, t) : 0 ≤ t ≤ max g}.
We recall that d g denotes the Bowen-Walters metric on X g . Let min g > > 0 and N ∈ N * . We will show the h top ( π -1 z, c ) ≤ log 2 N min g with c = 1 + 3 max g (min g) 2 , where h top ( π -1 z, c ) is the Bowen entropy of π -1 z for d g at the scale c with respect to ψ 1 , i.e.

h top ( π -1 z, c ) = lim sup n 1 n log max{ E n , E n is (c , n) -separated for d g w.r.t. ψ 1 }.
This will conclude the proof as and N are chosen arbitrarily and

h top ( π -1 z) = lim →0 h top ( π -1 z, c ).
By uniform continuity of g and the flow map, there are η, δ > 0 which can be assumed less than

2 such that |g(x) -g(y)| < 4N ∀x, y with d(x, y) < η and d (φ t z, φ s z) < η ∀t, s ∈ R with |t -s| < δ.
We let S n g be the Birkhof sum

S n g(x) = n-1 k=0 g • φ k (x).
Observe that

S n+m g = S n g + S m g • φ n .
We denote by |J| the diameter of a subset J of R. From the choice of δ the set |S n g(φ I z)| := {S n g(φ t z), t ∈ I} satisfies |S n g(φ I z)| < 4 for any subset I with |I| < δ and any integer 0 ≤ n ≤ N .

Claim 1. For any subset I of R with |I| < δ, we can find a cover

F k = F k (I) of I with F k ≤ 2 k satisfying |S n g(φ J z)| < 2 for all k ∈ N, all n ≤ kN and J ∈ F k .
Proof of Claim 1. Fix such an interval I with |I| < δ. We will argue by induction on k. For k = 1 we let F 1 = {I}. Assume F k already built and take J ∈ F k . Then we may cover J by two subsets J 1 and J 2 in such a way the diameter of S kN g(φ J i z) is less than 4 for i = 1, 2. Since the diameter of J i + kN is less than δ, we have also |S j g(φ J i +kN z)| < 4 for every 0 ≤ j ≤ N and therefore

|S kN +j g(φ J i z)| ≤ |S kN g(φ J i z)| + |S j g(φ J i +kN z)| < 2 , ∀0 ≤ j ≤ N.
We conclude by letting We go back to the proof of h top ( π -1 z, c ) ≤ log 2 N min g . The flow Φ and Φ being orbit equivalent, there are finite families I and K of intervals with length respectively less than δ and /2 such that {( φ -t z, t) : 0 ≤ t < max g} is contained in I∈I, K∈K φ I z × K.

F k+1 = {J 1 , J 2 : J ∈ F k }.
Let m = [kN min g] and let E m be a maximal (m, c )-separated set inside π -1 z for the time-one map ψ 1 of the suspension flow on (X g , (φ 1 ) g ).

Claim 2. Any set of the form φ I z × K with |I| < δ and |K| < /2 contains at most 2 k points of E m . Suppose Claim 2 holds. Then we have

E m ≤ I K • 2 k .
We conclude that

h top ( π -1 z, c ) = lim sup m 1 m log E m ≤ lim sup k k log 2 + log I K kN min g = log 2 N min g .
It remains to prove Claim 2.

Proof of Claim 2. Let |I| < δ and |K| < /2. Let x = (φ t z, s) and x = (φ t z, s ) in φ J z × K for some J ∈ F k (I). We will show that x and x are (c , n)-closed. This would imply that there is at most one point of E m in φ J z × K and Claim 2 then follows. By Claim 1, we have |S n g(φ J z)| < 2 for all n ≤ kN . For l ≤ m the point x l = (φ t z, s+l) of X ×R has for representative in the quotient space X g the point (φ n+t z, s l ) where n is the largest integer with s l := s + l -S n g(φ t z) ≥ 0. As l ≤ kN min g we have n ≤ kN . Similarly we may define similarly the integer n and the real number s l associated to x l := (φ t z, s + l). It follows from |S n g(φ J z)| < 2 , |K| < 2 and < min g

that we have |n -n | ≤ 1. Let u l = s l g(φn+tz)
and u l = s l g(φ n +t z) in [0, 1]. We consider the point x l in X g defined as x l = (φ n+t z, u l g(φ n+t z). The pairs (x l , x l ) and (x l , x l ) are respectively horizontal and vertical. As t, t both lie in I we have |t -t | < δ so that

|x l x l | = u l d(φ n+t (z), φ n+t (z)) + (1 -u l )d(φ n+1+t (z), φ n+1+t (z)) ≤ .
Then the length |x l x l | may be bounded from above depending on |n -n | :

• either n = n . Then |g(φ n+t z) -g(φ n+t z)| < and |s l -s l | ≤ |s -s | + |S n g(φ t z) -S n g(φ t z)|, ≤ 2 + 2 = .
Therefore we get

|x l x l | = |u l -u l |, ≤ |s l -s l |g(φ n+t z) + s l |g(φ n+t z) -g(φ n+t z)| g(φ n+t z)g(φ n+t z) , ≤ 2 max g (min g) 2 ,
• or |n -n | = 1, say n = n + 1. Then s l < and |s l + g(φ n +t z) -s l | < so that with the previous notations we get

|x l x l | = |u l -u l + 1|, = s l g(φ n+t z) - s l -g(φ n +t z) g(φ n +t z) , ≤ s l g(φ n+t z) + s l -g(φ n +t (z)) g(φ n +t z) , ≤ min g + 2 min g ≤ 3 max g (min g) 2 .
Consequently we get in any case

d g (x l , x l ) ≤ |x l x l | + |x l x l |, ≤ + 3 max g (min g) 2 = c .
Assume that (X, Φ) admits a symbolic extension. Then so does (X, Φ 1 ) by Theorem 3.3. It follows from that (X g , (φ 1 ) g ) also has symbolic extension (resp. principal) by Lemma 3.18 in [START_REF] Burguet | Symbolic extensions and uniform generators for topological regular flows[END_REF]. Since (X g , (φ 1 ) g ) is a principal extension of (X, Φ), the flow (X, Φ) admits a symbolic extension (resp. principal).

Singular suspension flows

In Section 2.1, we have defined the suspension over a discrete invertible topological system (X, T ) with a positive continuous roof function f : X → R >0 . Assume now (X, T ) has a fixed point * and f : X → R ≥0 only vanishes on the fixed point * . Then we may define again the topological quotient space X f by (2•1). We assume that k∈N f (T k x) and k∈N f (T -k x) goes uniformly to infinity out of any neighborhood of * , i.e.

∀U open with

* ∈ U, ∀M > 0, ∃K ∈ N, s.t. ∀x ∈ X \ U K k=0 f (T k x), K k=0 f (T -k x) > M. (4•1)
Lemma 4.1. Under the hypothesis (4•1) the time translation still induces a topological flow T f on X f . Moreover the Bowen-Walters metric d f 4 is compatible with the quotient topology on X f .

The flow (X f , T f ) is called a singular suspension flow. When (X, T ) is a subshift we speak of singular symbolic flow. In the following we write * f = ( * , 0) ∈ X f the singularity of the flow.

Proof. The properties k∈N f (T k x) = +∞ and k∈N f (T -k x) = +∞ clearly ensures the orbit of the induced flow T f are defined on R at any (x, t)

∈ X f with x = * . Moreover T f is continuous on this open set X f \ { * f }. We let T f t ( * f ) = * f for all t ∈ R.
Let us show that for a fixed s ∈ R, the time-s map T f s is continuous at * f . Without loss of generality we may assume s > 0. We argue then by contradiction : assume there is a neighborhood V of * f in X f such that T f s (A) does not lie in V for A ∈ X f arbitrarily close to * f . For any x ∈ X \{ * }, we let k(x) be the largest integer k with

k l=0 f (T -l x) < s. Let U be an open neighborhood of * with {(x, t) ∈ X f , x ∈ U } ⊂ V and we write T f s (A) = (x s A , t s A ). Then x s A ∈ X \U for A arbitrarily close to * f . It follows from (4•1) that k(x s A ) is bounded for A arbitrarily close to * f by some K > 0. Then T -k(x s A ) x s A does not belong to the open neighborhood K k=0 T -k U of * . But A = (T -k(x s A ) x s A , t s A + s - k(x s A ) l=0 f (T -l x s A )
) is close to * f = ( * , 0). This is a contradiction.

To prove the Bowen-Walters metric is compatible with the topology on X f , in comparison with the regular case, one only needs to check that for two points A = (x A , t A ) and B = (x B , t B ) lying on the same orbit of the flow (in particular x B = T k (x A ) for some integer k) with d f (A, B) small, the minimum d(T x A , x B ) for ∈ {0, -1, 1} is also small. Indeed if A is close to * f it could happen that the length of a sequence

A = A 1 , • • • , A n = B with A i A i+1 vertical for i = 1, • • • , n -1 is
small, but k so large that x B = T k (x A ) and x A are far from each other, in particular x B would lie far form the fixed point * . But (4•1) prevent this case as it implies k is bounded.

We let M * T (X) (resp. M * T f (X f )) denote the convex subset (not closed) of M T (X) (resp. M T f (X f ))
given by measures distinct from δ * (resp. with * f zero measure). We also let

M † T (X) = M T (X) \ M * T (X) (resp. M † T f (X f ) = M T f (X f ) \ M * T f (X f )). Notice that if µ ∈ M † T f (X f ) then µ( * f ) > 0. Then the map Θ : M * T (X) → M * T f (X f ) µ → (µ × L) |X f f dµ .
is a homeomorphism (not affine in general). Sometimes, we write

ν µ = Θ(µ) for µ ∈ M * T (X). Moreover Θ -1 extends continuously to M T f (X f ) in such a way that Θ -1 (ξ) = δ * for any ξ ∈ M † T f (X f ).
It follows from Abramov formula and the variational principle for the entropy that

h top (T f ) = sup µ∈M * T (X) h(Θ(µ)) = sup µ∈M * T (X)
h(µ) f dµ .

4.1. Entropy structure of zero dimensional singular suspension flow. For a partition P of X we let A * = A * P be the atom of P containing * , then X * = X \ A * and P * the partition of X * induced by P . Fix also δ = δ(P ) = 1/p with p ∈ N * such that f (x) > 1 p for all x ∈ X * . Lemma 4.2. For all µ ∈ M T (X) with µ(X * ) = 0

h(ν µ , φ δ , P δ ) δ ≥ µ(X * ) h(µ * , T * , P * ) f dµ , (4•2)
where P δ is the partition of X f given by P δ := {X f \ (X * × [0, δ[), B × [0, δ[ : B ∈ P * }, T * is the first return map in X * w.r.t. T and µ * ∈ M T * (X * ) the measure induced by µ on X * .

We follow the lines of the proof of Lemma 3.8 in [START_REF] Burguet | Symbolic extensions and uniform generators for topological regular flows[END_REF].

Proof. Let X * δ be the subset of X f given by X * δ = X * × [0, δ[/ ∼. For the partition P * of X * we denote by P * δ = {B × [0, δ[, B ∈ P * } the partition induced by P on X * δ . We also let R δ be the partition of X * δ with respect to the first return time in X * δ w.r.t. φ δ and we let φ * δ the corresponding first return map. By applying Lemma 3.7 in [START_REF] Burguet | Symbolic extensions and uniform generators for topological regular flows[END_REF] to ν µ = Θ(µ), φ δ and P * δ we get

h ν δ µ , φ * δ , P * δ ∨ R δ = h (ν µ , φ δ , P δ ) ν µ (B δ ) , = f dµ δµ(A δ ) h (ν µ , φ δ , P δ ) . (4•3) But the partition n-1 k=0 (φ * δ ) -k P * δ of X * δ is just the partition n-1 k=0 T -k * P * × [0, δ[. Therefore we get, with H µ (Q) = C∈Q -µ(C) log µ(C) : h ν δ µ , φ * δ , P * δ = lim n 1 n H ν δ µ n-1 k=0 T -k * P * × [0, δ[ , = lim n 1 n C∈ n-1 k=0 T -k * P * ×[0,δ[ -(ν δ µ (C) log(ν δ µ (C),
where ν δ µ denotes the probability measure induced by ν µ on X * δ . For any

B ∈ n-1 k=0 T -k * P * and C = B × [0, δ[ we have ν δ µ (C) = νµ(C) νµ(X * δ ) = µ(B).
Therefore we obtain finally : h ν δ µ , φ * δ , P * δ = h(µ * , T * , P * ), which together (4•3) implies the required inequality (4•2).

A subset of X is said to have a small boundary when its boundary has zero µ-measure for any µ ∈ M T (X). The partition P has a small boundary, when its atoms have small boundary. We assume now (X, T ) has the small boundary property, i.e. X has a topological base in which every element has a small boundary. Therefore there exists a sequence (P k ) k of small boundary partitions of X with diam(P k ) k -→ 0. Then the sequence of entropy functions with respect to (P k ) k is an entropy structure (see [START_REF] Downarowicz | Entropy structure[END_REF]). Let (δ k ) k = (1/p k ) k be the associated sequence of parameters (δ(P k )) k . Then

h(νµ,φ δ ,(P k ) δ ) δ = h(ν µ , φ 1 , Q k ) with Q k := p k -1 l=0 φ -l δ (P k ) δ k . The partitions Q = Q k have
also small boundary (w.r.t. the suspension flow) and for any µ ∈ M * T (X) we get with

H(t) = -t log t -(1 -t) log(1 -t) for all t ∈ [0, 1]: h(ν µ , φ 1 , Q) ≥ µ(X * ) h(µ * , T * , P * ) f dµ , ≥ h(µ, T, P ) -h(µ, T, {X * , A * }) f dµ , ≥ h(µ, T, P ) -H(µ(A * )) f dµ , (4•4) Q=Q k , k→+∞ ----------→ h(µ, T ) f dµ = h(ν µ , φ 1 ).
From Corollary 3.2 in [START_REF] Burguet | Symbolic extensions and uniform generators for topological regular flows[END_REF] it follows that (h(•, Q k )) k defines an entropy structure of the flow (X f , T f ). Following [START_REF] Downarowicz | Entropy structure[END_REF] we say for two sequences G = (g k ) k and H = (h k ) k of real functions defined on the same metric space X that H yields G, written H G, when for all > 0 and µ ∈ X there exists a neighborhood U of µ such that lim sup

k→∞ lim sup j→∞ sup µ∈U (g j -h k )(µ) ≤ . When X is compact this is equivalent to lim sup k→∞ lim sup j→∞ sup µ∈X (g j -h k )(µ) ≤ 0.
Two entropy structures H and G of a topological system or flow satisfy H G and G H (see [START_REF] Downarowicz | Entropy structure[END_REF][START_REF] Burguet | Symbolic extensions and uniform generators for topological regular flows[END_REF]).

Lemma 4.3. Let H T = (h k ) k be an entropy structure of T and G = (g k ) k be the sequence of real functions

g k defined on M * T f (X f ) by g k = h k • Θ -1 . Then the restriction to M * T f (X f ) of any entropy structure of (X f , T f ) yields G.
Proof. Without loss of generality we may assume H is the sequence (h(•, P k )) k with (P k ) k as above and consider the associated entropy structure h

(•, φ 1 , Q k ) of the suspension flow. Fix ν = Θ(µ) ∈ M * T f (X f ). For j large enough, µ(A * Pj ) is so small that H(ξ(A * P j
))

f dξ ≤ for all ξ in a neighborhood U of µ in M * T (X). Then it follows from (4•4) with ν µ = Θ(µ) that lim sup k→∞ lim sup j→∞ sup ν ξ ∈Θ(U ) h(ν ξ , φ 1 , Q j ) - h(ξ, T, P k ) f dξ ≥ lim sup j→∞ sup ξ∈U h(ν ξ , φ 1 , Q j ) - h(ξ, T, P j ) f dξ , ≥ -lim sup j sup ξ∈U H(ξ(A * )) f dξ ≥ -.
4.2. Singular suspension flows with small entropy at the singularity. Under some criterion on the entropy function at the singularity, we manage to build a symbolic extension of the suspension flow (X f , T f ) from a symbolic extension of (X, T ).

Proposition 4.4. Let (X f , T f ) be a singular suspension flow. Assume the associated discrete dynamical system (X, T ) admits a symbolic extension π with lim µ→δ * h π (µ)

f dµ = 0. Then the suspension flow admits a symbolic extension π with h π ≤ g har where g har denotes the harmonic extension of the function g :

M T f (X f ) → R ≥0 defined by g(ν) = h π (µ) f dµ for ν = Θ(µ) ∈ M * T f (X f
) and g(ν) = 0 for others ν. Proof. The function h π is upper semi-continuous on M T (X) and µ → f dµ is continuous and positive on M * T (X). Therefore G : µ → h π (µ) f dµ is upper semi-continuous on M * T (X) and thus so is

g = G • Θ -1 on M * T f (X f ). When ν belongs to M † T f (X f ), we have lim ξ→ν g(ξ) ≤ g(ν) = 0. Indeed we can assume ξ lies in M * T f (X f ) because g is zero on M † T f (X f ). Then lim ξ→ν g(ξ) ≤ lim ξ→ν G • Θ -1 (ξ), ≤ lim µ→δ * G(µ) = 0.
Arguing as in Lemma 3.13 in [START_REF] Burguet | Symbolic extensions and uniform generators for topological regular flows[END_REF] the function g is also affine on M

* T f (X f ). It is also affine on M † T f (X f ) as g is identically zero on this set. Then if ν ∈ M † T f (X f ) and ξ ∈ M * T f (X f ), the measure λν + (1 -λ)ξ belongs to M † T f (X f ) for all λ ∈]0, 1], therefore 0 = g(λν + (1 -λ)ξ) ≤ λg(ν) + (1 -λ)g(ξ).
In conclusion, the function g is upper semi-continuous and convex. Its harmonic extension g har is also upper semi-continuous by Fact A2.20 in [START_REF] Downarowicz | Entropy in dynamical systems[END_REF]. Observe also g har ≥ h T f . We show now g har is a (affine) superenvelope, which will conclude the proof of the proposition by Theorem 3.1. By Lemma 8.2.14 in [START_REF] Downarowicz | Entropy in dynamical systems[END_REF], it is enough to check for some entropy structure

H T f = (h T f k ) k with harmonic functions h T f k lim k (g har -h T f k ) ˜e = g har -h T f with f ˜e(ν) = lim sup ξ→ν ξ∈M e T f (X f )
f (ξ) for any real function defined on M T f (X f ). When ν = Θ(µ) belongs to M * T f (X f ) any ergodic measure ξ going to ν also lies in M * T f (X f ), so that by Lemma 4.3 for any entropy structure

H T = (h T k ) k we get lim k (g har -h T f k ) ˜e(ν) ≤ lim k (h π -h T k ) ˜e(µ) f dµ , ≤ h π -h T (µ) f dµ = (g har -h T f )(ν).
Finally let ν / ∈ M * T f (X f ) and ξ n be a sequence of ergodic measures going to ν. Then we may write

ξ n = Θ(µ n ) with µ n n -→ δ * so that lim k (g har -h T f k ) ˜e(ν) ≤ (g har ) ˜e(ν), ≤ lim sup µ→δ * h π (µ) f dµ , = 0, ≤ (g har -h T f )(ν).
Corollary 4.5. Assume (X, T ) admits a principal symbolic extension and lim µ→δ * h(µ) f dµ = 0, then the suspension flow (X f , T f ) also admits a principal symbolic extension. In particular, if (X, T ) has topological entropy zero, then (X f , T f ) admits a symbolic extension with zero topological entropy.

4.3.

Noninvariance under orbit equivalence. Contrarily to regular flows, two singular flows with finite topological entropy may be orbit equivalent but one admitting a symbolic extension and not the other.

Proposition 4.6. There are two orbit equivalent (singular) flows (X, Φ) and (X, Ψ) with h top (Φ) = h top (Ψ) < +∞ such that (X, Φ) admits a symbolic extension but not (X, Ψ).

Proof. For any n ∈ N there exists a topological system (X n , T n ) with h top (T n ) = 4 -n and h sex (T n ) = 3 -n (See Theorem D.1 in [START_REF] Boyle | Residual entropy, conditional entropy and subshift covers[END_REF]). We may let (X 0 , T 0 ) be a subshift. Let X = n∈N X n ∪ { * } be the one point compactification of the X n 's. We let T : X defined by T | Xn = T n and T * = * . The symbolic extension entropy h sex (T ) of (X, T ) satisfies h sex (T ) = sup n h sex (T n ) = 1. Then we consider the roof functions f and f with f

( * ) = f ( * ) = * and f | Xn = 1 2 n , f | Xn = 1
4 n for all integers n. The hypothesis (4•1) is easily checked in this case. The topological entropies of the associated singular suspended flow Φ f and Φ f satisfy

h top (T f ) = h top (Φ f ) = h top (T 0 ) = 1. Moreover we have h sex (T f ) ≥ h sex (T n ) f | Xn = (4/3) n
for all n ∈ N so that Φ f does not admit any symbolic extension. We check now h sex (T f ) = 1. For any n there is a symbolic extension of the time-1 map of (T f )| X f n with topological entropy less than (3/4) n . We let E n : M(X f n , (T f ) 1 ) → R be the associated superenvelope given by the entropy function of this extension. Each E n may be extended to an affine upper semi-continuous function

E n on M(X f , (T f ) 1 ) with E n (µ) = 0 for µ(X f n ) = 0.
As E n and thus E n is bounded from above by (3/4) n , the function E := n E n defines an affine upper semi-continuous function. Let H = (h k ) be an entropy strucutre of X f , (T f ) 1 . By using again Lemma 8.2.14 in [START_REF] Downarowicz | Entropy in dynamical systems[END_REF], to show E is superenvelope it is enough to check for all µ in the closure of ergodic measures and for all > 0, there exists k such that lim sup ν→µ, ν ergodic

(E -h k )(ν) ≤ (E -h)(µ) + .
Either µ is supported on some X n , then so does any ergodic measure ν close enough to µ and we may conclude in this case since E(µ) = E n (µ), E(ν) = E n (ν) and E n is a superenvelope of the time-1 map of the flow on (X n ) f . Or µ is the Dirac mass at * and lim sup ν→µ, ν ergodic (E-h k )(ν) ≤ lim sup n sup ξ E n (ξ) = 0 = E(δ * ). Therefore E is an affine superenvelope of the time-1 map of T f , so that by Theorem 2.1 the suspension flow T f admits a symbolic extension with topological entropy equal to sup µ E(µ) = 1.

Remark 4.7. In the above example, the discrete system (X, T ) admits a symbolic extension, however the singular suspension flow (X f , T f ) has finite topological entropy, but no symbolic extensions.

Universal symbolic suspension flow.

In the following we consider the two full shift with the singularity given by the infinite zero sequence, i.e. (X, T ) = {0, 1} Z , σ and * = 0 ∞ . We let d be the usual distance on X given by

d((u n ) n , (v n ) n ) = 2 -min{|n|, un =vn} . In particular we have d((u n ) n , * ) = 2 -ku with k u = min{|n|, u n = 1}.
We investigate the entropy and symbolic extension of a singular symbolic suspension flow over (X, T ) associated to a roof function of the form f (x) = R(d(x, * )) for some continuous function R : [0, 1] → R ≥0 with R( * ) = 0 and R(x) > 0 for x = * . We may also write f as f (u) = g(k u ) for u = 0 and for a function g : N → R ≥0 with lim k→+∞ g(k) = 0 by letting g = R(2 -• ). With these notations the singular symbolic flow X f is well defined (i.e. (4•1) holds true) if and only if k∈N g(k) = +∞. Theorem 4.8. Assume the sequence (kg(k)) k∈N is converging to l ∈ R + ∪ {+∞} when k goes to infinity. Then

(1) if l = 0, the topological entropy of (X f , T f ) is infinite, (2) if 0 < l < +∞, then the symbolic extension entropy of (X f , T f ) satisfies

∀ν ∈ M T f (X f ), h sex (ν) = h ˜(ν) = h(ν) + 1 2l ν( * f ),
(3) if l = +∞ the flow (X f , T f ) admits a principal symbolic extension.

Question 4.9. Does any singular symbolic suspension flow with finite topological entropy admit a symbolic extension?

As a consequence of Theorem 4.8, we present the following examples. Let

g(k) = 1/k, g (k) = 1/ √ k and g (k) = 1/k log k for all k.
The associated suspension flows (X f , T f ), (X f , T f ) and (X f , T f ) are orbit equivalent. Since kg(k) → 1 and kg (k) → +∞ as k → +∞, it follows from Theorem 4.8 that (X f , T f ) admits a principal symbolic extension contrarily to (X f , T f ). Moreover they have both finite topological entropy, but h top (T f ) = +∞. There exist also orbit equivalent singular topological flows (X, Φ) and (X, Ψ) with h top (Φ) = 0 and h top (Ψ) = +∞ [START_REF] Sun | Zero entropy versus infinite entropy[END_REF].

Corollary 4.10. There are two orbit equivalent singular symbolic flows (X, Φ) and (X, Ψ) with h top (Φ) = h top (Ψ) < +∞ such that (X, Φ) admits a principal symbolic extension but not (X, Ψ). This clearly implies the first item. Moreover the symbolic extension entropy function h sex is upper semi-continuous and concave, therefore writing

ν ∈ M T f (X f ) as ν = ν( * f )δ * f + (1 -ν( * f ))ξ with ξ ∈ M * T f (X f ) we get a first inequality in the second item h sex (ν) ≥ (1 -ν( * f ))h sex (ξ) + ν( * f )h sex (δ * f ), ≥ (1 -ν( * f ))h(ξ) + ν( * f ) lim sup η→δ * f h(η), ≥ h(ν) + 1 2l ν( * f ).
To prove (4•5) we consider the Bernoulli measure µ λ ∈ M σ ({0, 1} Z ) with parameter λ = µ([1]) where [1] denotes the cylinder [1] := {(u n ) n ∈ {0, 1} Z : u 0 = 1}. We also denote by [0 2k+1 ] for k ∈ N the cylinder [0 2k+1 ] := {(u n ) n ∈ {0, 1} Z : u m = 0 for |m| ≤ k}. We compute for λ close to 0 :

f dµ λ = µ λ ([1]) + k∈N g(k)µ λ [0 2k+1 ] \ [0 2(k+1)+1 ] , = λ + l k∈N λ(2 -λ)(1 -λ) 2k+1 k + O(λ), = λ -lλ(1 -λ)(2 -λ) log(λ(2 -λ)) + O(λ), = -2lλ log λ + O(λ),
where lim sup λ→0 O(λ) λ < +∞. As the entropy of µ λ is equal to

-λ log λ -(1 -λ) log(1 -λ) = -λ log λ + O(λ), we get h(ν µ λ ) = h(µ λ ) f dµ λ = -λ log λ + O(λ) -2lλ log λ + O(λ) , λ→0 ---→ 1 2l .
Moreover the T f -invariant measure ν µ λ is going to the Dirac measure at the singularity when λ goes to zero. Indeed for all fixed k, we have for some polynomial P k with P k (0) = 0:

ν µ λ ([0 2k+1 ] × R/ ∼) = 1 - X\[0 2k+1 ] f dµ λ X f dµ λ , ≥ 1 - λP k (λ) + O(λ) X f dµ λ , λ→0 ---→ 1.
(3) Let us show now the last item assuming the second one. Assume kg(k) goes to infinity as k tends to infinity. For a > 0, let g a (k) = min(g(k), a k ) for all k. The associated roof function f a satisfies the hypothesis of the second item, that is, lim k→∞ kg a (k) = a. The symbolic extension entropy function is upper semi-continuous, therefore h ˜≤ h sex and we get for all a > 0 :

lim sup µ→δ * h(µ) f dµ ≤ lim sup µ→δ * h(µ) f a dµ , ≤ h ˜(δ * fa ), ≤ h sex (δ * fa ) = 1 2a
.

By letting a go to +∞, we get lim sup µ→δ * h(µ) f dµ = 0. By applying Corollary 4.5, the singular suspension flow (X f , T f ) admits a principal symbolic extension.

(2) We prove now the second item. For g with lim k kg(k) = l we build a symbolic extension of (X f , T f ) with h π (ν) ≤ h(ν) + ν( * f ) 2l for all ν ∈ M(X f , T f ). This will conclude the proof of Theorem 4.8.

Step 1 : Construction of a principal extension by a regular suspension flow π : (Y f , S f ) → (X f , T f ). We first build a principal extension of (X f , T f ) by a regular suspension flow (Y f , S f ). For x ∈ {0, 1} Z we let k + x = min{n ∈ N * , x n = 1} et k - x = min{n ∈ N, x -n = 1} and we denote by k x the pair (k - x , k + x ). We consider the partition of E := (N ∪ {+∞}) × (N ∪ {+∞}) \ {(+∞, +∞)} into the following subsets of points k = (k -, k + ) ∈ E:

• R 1 := {k -= 0}, • R 2 := {0 < k -≤ 1 3 k + }, • R 3 := {k + > k -> 1 3 k + > 0}, • R 4 := {0 < k + ≤ k -}.
We let L : E → N be the function satisfying for all k = (k -, k + ) ∈ E :

(1)

L(k) = 1 for k ∈ R 1 , (2) L(k) = k -for k ∈ R 2 , (3) L(k) = k + -(k -+k + ) 2 8k - for k ∈ R 3 , (4) L(k) = k + /2 for k ∈ R 4 .
Lemma 4.12. For all k = (k -, k + ) ∈ R 3 , we have with R(p, q) = q j=p 1 j for positive integers q > p:

(1) L(k) > k + -k - 2 ≥ 0, (2) L(k) ≤ k + 2 , (3) k + + k -- 8k -(k + -L(k)) ≤ 4, (4) R k -, k + +k - 2 + R k + -L(k), k + +k - 2 k -→+∞ ------→ log 2.
Proof. One checks easily the two first items. Let us just show the two last ones. From the definition of L on R 3 we have by using

√ 1 -x ≥ 1 -x for all x ∈ [0, 1] : 8k -(k -+ k + ) 2 8k - = 8k -(k + -L(k)) ≤ k + + k -, (k -+ k + ) 2 -8k -≤ 8k -(k + -L(k)) ≤ k + + k -, (k + + k -) 1 - 8k - (k -+ k + ) 2 ≤ 8k -(k + -L(k)) ≤ k + + k -, k + + k --4 ≤ 8k -(k + -L(k)) ≤ k + + k -.
The second line makes sense only if 8k -≤ (k + + k -) 2 . But 8k -> (k + + k -) 2 implies k -≤ 1 and k + ≤ 2. In this remaining case we have therefore again

k + + k -- 8k -(k + -L(k)) ≤ 4.
Finally we prove item (4). Observe that k + and k -are going simultaneously to infinity for k ∈ R 3 . Then as R(p, q) = log q p + o(p) p for large p where lim p→∞ o(p) p = 0, we get

R k -, k + + k - 2 + R k + -L(k), k + + k - 2 = log (k + + k -) 2 4k -(k + -L(k)) + o(k -) k -, k -→+∞ ------→ log 2.
Let S : {0, 1} Z be the topological system defined as Sx = σ L(kx) x for x = * and S * = * . The map S is continuous at any x = * because in this case L(k y ) is constant for y in some neighborhood of x. Now if x is close to the zero sequence then k +

x and k - x are large. Moreover we have always 0

≤ L(k x ) ≤ k + x 2
and therefore y = σ l(kx) x, which satisfies y n = 0 for n

= -k - x , • • • , k + x 2
, is also close to the zero sequence. For x ∈ X with x 0 = 1, we let p x = min{n ∈ N * , (S n x) 0 = 1}. Lemma 4.13. For all x ∈ X with x 0 = 1 and 3 ≤ k +

x < +∞ there exists a unique 0 < r < p x such that

• k S q x ∈ R 2 for 1 ≤ q < r, • k S r x ∈ R 3 , • k S q x ∈ R 4 for r < q ≤ p x -1.

Moreover

(1) for 1 ≤ q ≤ r we have k - S q x = 2 q-1 , (2) there exists a sequence r+1 , • • • , px-2 ∈ {0, 1} depending only on k +

x such that for r < q < p x we have k + S q x = 2 px-1-q + 0≤i<px-q-1 q+i 2 i .

(3) p x -2 -r ≤ px 2 -1. Proof. We only prove the two last items (2) and (3), as the other conclusion is easily checked. For

p x -1 > q > r we have k + S q+1 x = k + S q x - k + S q x 2
, therefore k + T q x = 2k + S q+1 x + q with q ∈ {0, 1} depending on the parity of k + S q x . Also k + S px-1 x = 1. Then we get by a direct induction the desired formula for k + T q x . Concerning the last item, we have k

S r x ∈ R 3 , so that k + S r y > k - S r y = 2 r-1 > k + S r y 3 . Then we have 2 px-r-2 ≤ k + S r+1 x ≤ k + S r y ≤ 3 • 2 r-1 , thus 2 px ≤ 2 2r+3 , i.e. r ≥ px-3 2 .
Remark 4.14. The integers p x , r and the sequence r+1 , • • • , px-2 ∈ {0, 1} only depend on k +

x for x ∈ X satisfying 3 ≤ k + x < +∞. Now we consider the roof function f on X given by ∀x

∈ X \ { * }, f (x) = L(kx)-1 k=0 f (σ k x) f ( * ) = l log 2. Lemma 4.15. f is continuous on X.
Proof. Clearly it is enough to check the continuity at * , i.e. when k -

x and k + x both go to infinity. We have

• f (x) ∼ lR(k - x , 2k - x ) when k x lies in R 2 , • f (x) ∼ lR k - x , k + x +k - x 2 + lR k + x -L(k x ), k + x +k - x 2 , when k x lies in R 3 , • f (x) ∼ lR( k + x 2 , k + x )
, when k x lies in R 4 . In all cases (see Lemma 4.12 (4) for the second case) we get f (x) x→ * ---→ l log 2.

Let Y = n∈N S n X. Any sequence x = (x n ) n ∈ X = {0, 1} Z with x n = 0 for n ≤ 0 or with x 0 = 1 belongs to Y . In particular for any x ∈ X there is k > 0 with σ -k x ∈ Y . We let (Y f , S f ) be the suspension flow over (Y, S| Y ) with roof function f . We also denote by Z the subset of X given by sequences with infinitely many 1's in the future and in the past. The set of recurrent points of (Y, S| Y ) is given by (

Y ∩ Z) ∪ { * }. Lemma 4.16. The map π : (Y f , S f ) → (X f , T f ), (x, t) → T f t (x, 0) is a principal topological extension.
Proof. To prove the extension property, it is enough to see π(x, f (x)) = π(Sx, 0), i.e. T f f (x) (x, 0) = (Sx, 0) for all x ∈ Y which follows from

T f f (x) (x, 0) = T f L(kx )-1 l=0 f (σ k x) (x, 0), = T f f (σ L(kx )-1 x) • • • • • T f f (x) (x, 0), = (σ • • • • • σ L(kx) times (x), 0), = (Sx, 0).
The factor map π is surjective. Indeed for any x ∈ {0, 1} Z there is k > 0 with σ

-k x = y ∈ Y . Therefore for (x, t) ∈ X f there is s ≥ 0 with π(y, s) = T f s (y, 0) = (x, t). The recurrent points different from * f in X f are contained in the subset Z ×R/ ∼ of X f . For x ∈ Z we let y = σ -k x ∈ Y where k is the smallest nonnegative integer with σ -k x ∈ Y . Then π -1 (x, t) = {(y, s)} for some 0 ≤ s ≤ f (y). Moreover π -1 * f is contained in the subset {0} × R/ ∼ of Y f
and the suspension flow T f restricted to this set is topologically conjugated to the translation flow on the circle. In any case we have h top (π -1 (x, t)) = 0 for all recurrent points (x, t) of (X f , T f ). By Ledrappier-Walters formula, π is a principal extension.

Step 2 : A symbolic extension χ of (Y, S). To conclude the proof of Theorem 4.8 we investigate the symbolic extensions of (Y, S). We will build a symbolic extension χ of (Y, S) (with an embedding) with entropy function h χ equal to µ → h(µ) + µ( * ) log 2 2 . Let us call a block any finite word of the form 10 l := 1 0 • • • 0 l times with l ∈ N. Any sequence in Y ∩ Z is an infinite concatenation of such blocks. A map ψ : Y ∩ Z → A Z for some alphabet A is said a block code map if for any x ∈ Z ∩ Y whose orbit is given by the concatenation of blocks (B n ) n∈Z the orbit of ψ(x) under the shift on A Z is the concatenation of (Ψ(B n )) n for some map Ψ from the set of all blocks to n∈N A n . For Ψ given, the map ψ will be completely defined by letting the zero coordinate of ψ(x) be equal to the zero-coordinate of Ψ(B 0 ) for any x ∈ Y ∩ Z with x 0 = 1 and by ensuring ψ • S = σ • ψ where σ denotes here the shift on A Z .

We define now the map Ψ. The alphabet A is given by A = {y z , y ∈ {1, 2, 3, 4} and z ∈ {1, 2, 3, 4, ×}}.

For y z ∈ A we will refer to y as the y-coordinate of y z . For any l ∈ N, we let Ψ(10 l ) = y z1 1 • • • y zp p be the word of length p = p x for any x ∈ Y with x 0 = x l+1 = 1 and x 1 = • • • = x l = 0 such that we have y q = i whenever k S q-1 x ∈ R i for any 1 ≤ q ≤ p. Finally we put with the notations of Lemma 4.13:

• z 1 = k + + k -- 8k -(k + -L(k)
) with (k -, k + ) = k T r x , then 0 ≤ z 1 ≤ 4 by Lemma 4.12 (3), • z p-2i = p-2-i for 0 ≤ i ≤ p -3 -r (note that p -2(p -3 -r) > 1 by Lemma 4.13 (3)).

• z i = × for others i.

Lemma 4.17. The block code map ψ defines a uniform generator of (Y, S), i.e. ψ is a Borel embedding of (Y ∩ Z, S) to (A Z , σ) such that χ := ψ -1 extends in a symbolic extension of (Y, S) on ψ(Y ∩ Z) ⊂ A Z .

Proof. We first check Ψ is injective on the set of blocks. Assume Ψ(10 l ) = y z1 1 • • • y zp p . Then we may recover l from the y i and z i as follows. For x ∈ Y with x 0 = x l+1 = 1 and x 1 = • • • = x l = 0 we have

• r = {i, y i = 2} + 1, • z p-2i = p-2-i for 0 ≤ i ≤ p -3 -r. • k - S r x = 2 r-1 , • k + S r+1 x = 2 p-r-2 + 0≤i<p-r-2 r+1+i 2 i , • l = k - S r x + k + S r x = z 1 + 8k - S r x k + S r+1
x . Then we may rebuild the decomposition of x ∈ Y ∩ Z into blocks which are delimited by the y-coordinates equal to 1 in the sequence ψ(x). Therefore, to prove the injectivity of ψ on Y ∩ Z, it only remains to show how to repair the position of x inside the blocks, that is k - x or k + x . With the above notations, assume (ψ(x)) 0 = y zq q . We have :

• if y q = 1 then k - x = 0 • if y q = 2 or 3, we get k - x = 2 q-2 , • if y q = 4, then k + x = 2 p-1-l + 0≤i<p-q-1 q+i 2 i . Let us check now that ψ -1 extends to a symbolic extension of (Y, S). Any u ∈ ψ(Y ∩ Z)\ψ(Y ∩ Z) may be written as the concatenation of words of the form Ψ(B) for blocks B and semi-infinite or bi-infinite words in the alphabet A without any y-coordinate equal to 1. In the last case, when there is no y-coordinate equal to 1 in u, we just let χ(u) be the zero sequence, i.e. χ(u) = * . Moreover a semi-infinite block without y-coordinates equal to 1 is sent to a semi-infinite sequence of 0's. Then the position of x = χ(u) inside such a semi-infinite sequence is defined as below :

• if the semi-infinite sequence lies in the future then k - x = 2 l-2 , • if it lies in the past k + x = 2 p-1-l + 0≤i<p-l-1 l+i 2 i where l is the position of u inside the semi-infinite block.

Defined in this way, the extension χ of ψ -1 on ψ(Y ∩ Z) is continuous. Observe finally that χ is surjective on Y because the image of χ contains Y ∩ Z which is dense in Y .

Remark 4.18. By letting ψ( * ) be the 1 × sequence (1 × ) ∞ , we get a Borel embedding of the set of recurrent points of (Y, S) to A Z . By Remark 1.4 in [START_REF] Burguet | Uniform generators, symbolic extensions with an embedding, and structure of periodic orbits[END_REF] we may then extend ψ to a Borel embedding of the whole system (Y, S) satisfying χ • ψ = Id Y .

We finish now the proof of the second item of Theorem 4.8. The only recurrent points in ψ(Z ∩ Y ) \ ψ(Z ∩ Y ) belong to the two subshifts of finite type generated 5 respectively by the 2-words 2 0 2 × , 2 1 2 × and 4 0 4 × , 4 1 4 × , which are contained in χ -1 * . Therefore h χ (µ) = h(µ) for any µ ∈ M S (Y ) with µ(Y ∩ Z) = 1. Moreover h χ (δ * ) = h top (χ -1 * ) = log 2 2 . The function h χ being affine we get h χ (µ) = h(µ) + µ( * ) log 2 2 for all µ ∈ M S (Y ).

Step 3 : Conclusion. Finally the symbolic extension π induced by χ of (Y f , S f ) by the suspension flow over (ψ(Y ∩ Z), σ) under the roof function f • χ satisfies h π (ν) = h(ν) for all ν ∈ M * S f (Y f ) and h π (δ * f ) = h χ (δ * ) f ( * ) = 1 2l . As the extension π : (Y f , S f ) → (X f , T f ) is principal and satisfies π -1 M * X f (X f ) = M * S f (Y f ), the extension Π = π • π also satisfies h Π (ν) = h(ν) for all ν ∈ M * T f (X f ) and h Π (δ * f ) = 1 2l . This concludes the proof of the second item of Theorem 4.8, because the function h Π is affine.

3

  Let π : (X, T ) → (Y, S) be an extension. Then for any S-invariant measure ν, we have that sup µ:πµ=ν hµ(T ) = hν (S) + Y htop(π -1 y)dν(y).

Corollary 4 .

 4 10 and Proposition 4.6 raise the following question : Question 4.11. Do there exist orbit equivalent singular topological flows with equal topological entropy one without any symbolic extension and the other with a principal one? Now we present the proof of Theorem 4.8. Proof of Theorem 4.8. (1) We first show that (