Minos Axenides
email: axenides@inp.demokritos.gr

Emmanuel Floratos
email: mflorato@phys.uoa.gr

Stam Nicolis
email: stam.nicolis@lmpt.univ-tours.fr

Arnol'd cat map lattices

We construct Arnol'd cat map lattice field theories in phase space and configuration space. In phase space we impose that the evolution operator of the linearly coupled maps be an element of the symplectic group, in direct generalization of the case of one map. To this end we exploit the correspondence between the cat map and the Fibonacci sequence. The chaotic properties of these systems can be also understood from the equations of motion in configuration space. These describe inverted harmonic oscillators, where the runaway behavior of the potential compete with the toroidal compactification of the phase space. We highlight the spatio-temporal chaotic properties of these systems using standard benchmarks for probing deterministic chaos of dynamical systems, namely the complete dense set of unstable periodic orbits, which, for long periods, lead to ergodicity and mixing. The spectrum of the periods exhibits a strong dependence on the strength and the range of the interaction.

Introduction and overview

Low dimensional systems with few degrees of freedom have provided a fertile ground for the development of the concepts and methods of deterministic chaos with their characteristic disordered behaviour at the classical and quantum level [START_REF] Arnold | Mathematical aspects of classical and celestial mechanics[END_REF][START_REF] Cvitanovic | Chaos: classical and quantum[END_REF][START_REF] Zaslavsky | Hamiltonian chaos and fractional dynamics[END_REF][START_REF] Gutzwiller | Chaos in classical and quantum mechanics[END_REF]. While the original focus of interest centered around dissipative and Hamiltonian low dimensional systems, progress was quickly followed by efforts to understand the complex dynamics of high dimensional systems consisting of many coupled chaotic degrees of freedom [START_REF] Crutchfield | Phenomenology of spatio-temporal chaos[END_REF][START_REF] Kaneko | Complex Systems: Chaos and Beyond: Chaos and Beyond: A Constructive Approach With Applications in Life Sciences[END_REF][START_REF] Vulpiani | Chaos: from simple models to complex systems[END_REF][START_REF] Badii | Complexity hierarchical structures and scaling in physics[END_REF]. They are spatially extended systems, which can be driven away from equilibrium and exhibit spatio-temporal chaos (STC). They give rise to diverse pattern formation [START_REF] Cross | Spatiotemporal chaos[END_REF][START_REF] Cross | Pattern formation outside of equilibrium[END_REF] as the result of their highly complex dynamical behavior. STC models possess either continuous or discrete time dynamics (maps). The spatial degrees of freedom are either discrete or continuous giving rise, respectively, to lattice dynamics or an effective hydrodynamic description in terms of continuous fields. Such systems described are described by their equations of motion: Partial differential equations, systems of coupled ordinary differential equations; or as coupled map lattices (CMLs) with continuous state spaces. Another way is using cellular automata with discrete state spaces [START_REF] Kaneko | Coupled maps[END_REF].

Indicators for chaos for spatiotemporal systems have been proposed-namely the finite amplitude Lyapunov exponents, covariant Lyapunov vector exponents [START_REF] Pikovsky | Lyapunov exponents: a tool to explore complex dynamics[END_REF], as well as benchmark dynamical entropies, like the Kolmogorov-Sinai entropy [START_REF] Sinai | Topics in Ergodic Theory[END_REF][START_REF] Cornfeld | Ergodic theory[END_REF].

While the approach to the problem of describing spatio-temporal chaos in coupled map lattices is mostly numerical and a comprehensive understanding from the analytical side is, still, lacking, there has been some activity recently in an effort to acquire an analytical understanding of the dynamics of linear CMLs [START_REF] Cvitanović | Chaotic field theory: a sketch[END_REF][START_REF] Gutkin | Classical foundations of many-particle quantum chaos[END_REF][START_REF] Gutkin | Linear encoding of the spatiotemporal cat[END_REF][START_REF] Cvitanovic | Spatiotemporal cat: a chaotic field theory[END_REF]. One aim of this program of research is to define chaotic field theories made up of chaotic oscillator constituents, in an effort to provide a local description of some of the coherent structures that emerge from the dynamics of continuous fluid systems in the régime of weak turbulent flows [START_REF] Holmes | Turbulence, coherent structures, dynamical systems and symmetry[END_REF]. However it isn't clear, whether the complexity of these structures is due to the known complex behavior of their constituents, the result of the way they are coupled, or both. The reason is that the typical way for establishing such a relation, namely the study of symmetries, has proven to be very difficult to follow for these systems.

There are, however, cases where this approach is possible. Our present work focuses on the systematic construction of a special class of CMLs, the lattice field theories of Arnol'd cat maps in various dimensions, taking into account their symmetries in phase space-namely covariance under symplectic transformations-and how these are related to the symmetries in configuration space.

In the present paper we provide the classical framework for describing the dynamics of chaotic oscillators via the dynamics of n, linearly coupled, Arnol'd cat maps (CACML), subject to periodic boundary conditions. Their phase space is the torus T 2n [Z]. and their dynamics is represented by elements M of the symplectic group, Sp 2n [Z]. This is the generalization for n cat maps, of the symmetry properties of one cat map. Within this framework we can vary the dimensionality of the lattice, the number of oscillators, the strength of their interactions as well 1 as the range of the non-locality thereof.

We can therefore study in detail the classical chaotic properties of this system, since it's possible to obtain explicit expressions, that can be reliably evaluated. We focus, as an example, on the classical spatio-temporal chaotic properties of CACML's in one dimension, using a benchmark of chaos of any chaotic system, namely, the set of all of its unstable periodic orbits [START_REF] Cvitanovic | Chaos: classical and quantum[END_REF][START_REF] Auerbach | Exploring chaotic motion through periodic orbits[END_REF].

The periodic orbits of the CACMLs are classified by initial conditions, which have rational coordinates in the toroidal phase space, with common denominator N. Upon varying N = 3, 5, 7, . . . , over the primes and, more, generally, the odd integers (even integers have subtle issues, particularly in the quantum case [START_REF] Floratos | Unitary evolution on a discrete phase space[END_REF][START_REF] Kaiblinger | Metaplectic operators for finite abelian groups and R d[END_REF], so require special study) we obtain all the periodic orbits (which are all unstable) [START_REF] Sinai | Topics in Ergodic Theory[END_REF]. For large N and for fixed size of the toroidal phase space we can approach a scaling limit. For long periods the periodic trajectories lead to ergodicity and strong mixing [START_REF] Sinai | Topics in Ergodic Theory[END_REF][START_REF] Cornfeld | Ergodic theory[END_REF][START_REF] Frigg | The Ergodic Hierarchy[END_REF].

This limit can be subtle, already for one map [START_REF] Axenides | The arithmetic geometry of AdS 2 and its continuum limit[END_REF].

In the case of translational invariant couplings we find explicitly: a) all the periodic orbits, b) the Lyapunov spectra and c)the Kolmogorov-Sinai entropy of the CACMLs as a function of the strength and the range of interactions. Armed with these analytic results we find that the maximum Lyapunov exponent of these systems is an increasing function of both the coupling constant and of the range of the interaction.

We provide also a method for determining the periods of the orbits based on the properties of matrix Fibonacci polynomials. These periods are random functions of N and they have stronger dependence on the coupling and the range of the interaction than in the non interacting case, i.e. for the single cat map. We present several numerical examples in support of this observation. The dependence of the periods on N provides information about the quantum spectra of these systems, which deserve a study in their own right.

For the case of the single cat map, which corresponds to n = 1, a detailed study of the periods, of their relation to the energy spectrum and its asymptotic properties for the quantum system, can be found in ref. [START_REF] Percival | A linear code for the sawtooth and cat maps[END_REF][START_REF] Percival | Arithmetical properties of strongly chaotic motions[END_REF][START_REF] Bird | Periodic orbits of the sawtooth maps[END_REF][START_REF] Keating | Asymptotic properties of the periodic orbits of the cat maps[END_REF]. Now we would like to discuss our particular motivation for this study. This derives from the realization that the physics of quantum black holes is a prime example of a chaotic many-body system, when the microstates can be resolved. Therefore it has become of topical interest to construct models for both the probes and for the near horizon geometry, that is defined by the microstates. This is why a consistent description of chaotic field theories has become a fascinating bridge that establishes novel relations between the subjects of interest to the high energy community and the community of classical and quantum chaos. This can be summarized as follows:

Black Holes (BH) are at present understood to be physical systems of finite entropy which, for an observer at infinity, is described by the dynamics of the microstates of the black hole, that live in the near horizon geometry. The chaotic dynamics of these microstates has new features, such as fast scrambling and non-locality. Specifically it has been conjectured that black holes are the fastest information scramblers in Nature [START_REF] Hayden | Black holes as mirrors: Quantum information in random subsystems[END_REF][START_REF] Sekino | Fast Scramblers[END_REF][START_REF] Shenker | Black holes and the butterfly effect[END_REF][START_REF] Maldacena | A bound on chaos[END_REF][START_REF] Bousso | Snowmass White Paper: Quantum Aspects of Black Holes and the Emergence of Spacetime[END_REF], that exhibit unitary quantum evolution. This, in turn, motivated the search for models that can capture these features. One class of such models builds upon the relation between the near horizon shock wave geometries and of the so-called gravitational memory effects. In these models it seems, indeed, possible-in principle-that the near horizon region of a black hole could form a chaotic memory, i.e. a basin of purely geometrical data of all of its past and recent history, through the 't Hooft mechanism of permanent space-time displacements caused by high energy scattering events of infalling wave packets [START_REF] Hooft | Diagonalizing the Black Hole Information Retrieval Process[END_REF][START_REF] Barrabes | Stochastically fluctuating black hole geometry, Hawking radiation and the transPlanckian problem[END_REF][START_REF] Papadodimas | An Infalling Observer in AdS/CFT[END_REF][START_REF] Banerjee | A toy model of black hole complementarity[END_REF][START_REF] Stanford | Complexity and Shock Wave Geometries[END_REF][START_REF] Shenker | Multiple Shocks[END_REF][START_REF] Mezei | On entanglement spreading in chaotic systems[END_REF][START_REF] Polchinski | Chaos in the black hole S-matrix[END_REF]. In the language of refs. [START_REF] Strominger | Gravitational Memory, BMS Supertranslations and Soft Theorems[END_REF][START_REF] Ellis | W ∞ Algebras, Hawking Radiation and Information Retention by Stringy Black Holes[END_REF][START_REF] Hawking | Soft Hair on Black Holes[END_REF] such data can be identified with the soft hair of the BH, whose origin is the infinite number of conservation laws, described by the BMS group. Proposals for a chaotic dynamics, within a discretized spacetime, for the microscopic degrees of freedom of the stretched horizon have been discussed for quite some time in the literature [START_REF] Hooft | How quantization of gravity leads to a discrete space-time[END_REF][START_REF] Brown | Quantum Complexity and Negative Curvature[END_REF][START_REF] Banks | Schwarzschild black holes from matrix theory[END_REF][START_REF] Iizuka | Matrix Models for the Black Hole Information Paradox[END_REF][START_REF] Avery | Qubit Models of Black Hole Evaporation[END_REF][START_REF] Magan | Black holes as random particles: entanglement dynamics in infinite range and matrix models[END_REF].

Our contribution to this quest started with the study of single particle probes, sent by observers at infinity, in order to learn about the near horizon AdS 2 geometries of black holes, taken as discrete and nonlocal dynamical systems [START_REF] Axenides | Modular discretization of the AdS 2 /CFT 1 holography[END_REF][START_REF] Axenides | Chaotic Information Processing by Extremal Black Holes[END_REF][START_REF] Axenides | The quantum cat map on the modular discretization of extremal black hole horizons[END_REF].

More specifically we have shown how the so-called Arnol'd cat maps, acting in a AdS 2 discrete near horizon geometry, can capture the properties of its single particle probes. We constructed explicitly an exact discrete version of AdS 2 /CFT 1 correspondence with chaotic and mixing dynamics for Gaussian single-particle wave packets, that is shown to provide an example of the so-called "Eigenstate Thermalization Hypothesis" [START_REF] Srednicki | Chaos and quantum thermalization[END_REF]. Finally, we have demonstrated that the model for their discrete and chaotic [START_REF] Arnol | 'd, Dynamics, statistics and projective geometry of Galois fields[END_REF], near horizon geometry admits a continuum limit [START_REF] Axenides | The arithmetic geometry of AdS 2 and its continuum limit[END_REF], where the smooth classical geometry is recovered.

The long term objective of our recent work is to provide models of non-local chaotic quantum dynamics of the tuneable rate of mixing(and its quantum avatar, scrambling) for the degrees of the horizon itself by n-particle systems. Our conjecture is that this can be achieved through the construction of the quantum CMLs of Arnol'd cat maps [START_REF] Mantica | Many-Body Systems and Quantum Chaos: The Multiparticle Quantum Arnol'd Cat[END_REF].

Therefore, while our previous work focused on the properties of single particle probes of the near horizon geometry, in the present work, we construct many-body systems, that possess the necessary features expected of the interacting black hole microstates themselves-namely, non-locality, chaos and strong mixing (scrambling). Therefore these many-body systems can be considered as effective models of the dynamics of the near horizon geometry itself.

Below we present the plan and summarize the results of the paper.

In section 2 we present the general setting of the dynamics of systems of n particles with evolution maps that are integral toral automorphisms of the 2n dimensional phase space,T 2n i.e elements of the symplectic group Sp 2n [Z], acting on points of the torus T 2n of radius R ≡ 1 mod 1.

The completely chaotic and mixing dynamics is described by the maximally hyperbolic elements of this group i.e. whose eigenvalues are pairs of positive real numbers,(λ > 1, 1/λ < 1), thus decomposing the phase space into symplectic planes with hyperbolic motion [START_REF] Zaslavsky | Hamiltonian chaos and fractional dynamics[END_REF][START_REF] Sinai | Topics in Ergodic Theory[END_REF][START_REF] Mcduff | Introduction to symplectic topology[END_REF].

In section 3 we discuss how to obtain elements of Sp 2n [Z],which describe n coupled Arnold cat maps that are maximally hyperbolic. Starting from the most general way for linearly coupling Fibonacci integer sequences, we construct a family of coupled Arnold cat maps lattices (CACML), in any dimension d = 1, 2, .. imposing symplectic interactions of tuneable nonlocality and we show that they are all maximally hyperbolic toral automorphisms.

In section 4 for the case of translation invariant couplings, in d = 1, we determine explicitly all the orbits of the CACML (periodic and non periodic).

In section 5 we discuss further measures of STC, namely the Lyapunov spectra and the Kolmogorov-Sinai entropy and find that the latter scales as the volume of the system. This property is a significant check of the consistency of our calculations and shows that the CACML defined in this way does have sensible thermodynamics. We observe also that the Kolmogorov-Sinai entropy is a good proxy for the mixing time(scrambling) of the dynamical system: The bigger the K-S entropy the faster the mixing time, so tuning the K-S entropy with the parameters of the system we can tune its mixing time. The corresponding property for the quantum system pertains to the entanglement entropy of subsystems and its time evolution (cf. also [START_REF] Bianchi | Linear growth of the entanglement entropy and the Kolmogorov-Sinai rate[END_REF][START_REF] Palma | Linear growth of the entanglement entropy for quadratic Hamiltonians and arbitrary initial states[END_REF]).

We discuss the dependence of the Lyapunov spectra on the dimensionality,the size of the system, n, the strength G, and the range, l, of the interactions.

In section 6 we determine the periods of the periodic orbits.

To do that we discretize the toroidal phase space by considering all the initial conditions, which are rational numbers with a common demoninator N. This new phase space we call T 2n [N ]. In this-discrete-phase space the toral automorphisms are elements of the group Sp 2n [Z N ]. The set of all periodic orbits of the corresponding dynamical systems on the continuous phase space T 2n [R], are given by the set of all different orbits of the CACML in T 2n [N ] mod N, by considering all possible values of N.

The spectra of the periods T [N ] of the CACML are the lengths of its orbits and they are random functions of N. They are determined by properties of the matrix Fibonacci polynomials mod N.

We study numerically the spectrum of the periods, for fixed values of the number n of coupled Arnol'd cat maps, the modular integer N, the strength and the range of the interactions. We observe, as might be expected, a random and stronger dependence on N for larger values of n,as well for increasing values of the strength and the range of interactions.

In section 7 we present our conclusions, possible applications, as well as open problems.

In appendix A we review, for completeness, some useful properties of the Fibonacci polynomials and their matrix generalizations and in, appendix B we determine and discuss possible conserved quantities of the ACML systems, which can be expressed as quadratic functions of the position and momenta of the system.The corresponding conservation laws restrict the volume of the toroidal phase space available to the trajectories of the system and lead to the vanishing of some of the Lyapunov exponents of the system (namely through eigenvectors of the evolution operator with eigenvalue one).

2 Dynamics of symplectic automorphisms of the toroidal phase space T 2n

In this section we provide a short review of the mathematical tools, that are necessary for studying the dynamics of n particles, in a toroidal phase space, T 2n = R 2n /(Z n × Z n ). The discrete time evolution will be described by discrete time maps, M, that are elements of the symplectic group, Sp 2n [Z], that act on the toroidal phase space as

x m+1 ≡ x m M mod 1 (2.1)
at the m-th time step (m = 0, 1, 2, . . .), along with the initial condition x m=0 = x 0 ∈ T 2n . We have taken the length of the sides of the torus equal to 1. In this notation, x m = (q m , p m ), where q m and p m are the positions and the momenta of the n particles at time step m.

By definition any element M ∈ Sp 2n [Z] preserves the (symplectic) inner product, x , x of any two vectors x and x ,

x , x = n I=1 (q I p I -q I p I ) (2.2)
This inner product can be rewritten as

x , x = x T Jx (2.3)
where J is the symplectic matrix

J = 0 -I n×n I n×n 0 (2.4)
We can decompose M in blocks of n × n (integer) matrices

M = A B C D (2.5)
The invariance of J under the action of any element M ∈ Sp 2n [Z],

J = M T JM (2.6) implies the constraints A T D -C T B = I n×n A T C = C T A B T D = D T B (2.7)
The second and third constraints express that A T C and B T D are symmetric, integer-valued, matrices.

To visualize the motion, in the toroidal phase space, T 2n , under the action of M, it is useful to decompose it into "simpler" actions. For it is possible to show that any matrix M ∈ Sp 2n [Z] can be decomposed into the product of three, simpler, symplectic matrices, that generate the symplectic group, namely

M = I n×n 0 S L I n×n U T 0 0 U -1 I n×n S R 0 I n×n (2.8)
where S R,L are integer symmetric matrices and U an invertible, integer matrix, which can be determined, given the matrices A, B, C, D, as follows:

U = A T S L = CA -1 S R = A -1 B
(2.9)

These relations hold, iff A is invertible, with integer entries. If this isn't the case, it is possible to redefine M so that A does have the desired properties. This decomposition will be useful, also, for the study of the quantum mechanics of this system, since the properties of the unitary evolution operator of the latter are, indeed, those of the metaplectic representation of the symplectic group [START_REF] Axenides | Modular discretization of the AdS 2 /CFT 1 holography[END_REF][START_REF] Athanasiu | Holomorphic quantization on the torus and finite quantum mechanics[END_REF].

An explicit example is that of the single Arnol'd cat map (n = 1), which can be written as

M = 1 1 1 2 = 1 0 1 1 1 2×2 1 1 0 1 (2.10)
and the exercise we shall solve in the following sections is to find the generalization for n such maps.

Let us now consider the initial condition, x 0 = (k 1 /N, k 2 /N, . . . , k n /N, l 1 /N, l 2 /N, . . . , l n /N ), with 0 ≤ k I ≤ N, 0 ≤ l I ≤ N and k I , l I , N integers. Since the length of each side of the hypercube that defines T 2n is taken equal to 1, the evolution equation (2.1) can be rewritten as

(k, l) m+1 = (k, l) m M mod N (2.11)
The set of vectors (k, l) mod N, defines the lattice Z n N × Z n N which can be identified with T 2n [N ]. Since the number of rational points of T 2n , with fixed denominator N , is equal to N 2n , the matrix M mod N , which belongs to the finite group Sp 2n [Z N ], has a finite period,

T [N ], where T [N ] is the smallest integer such that M T [N ] ≡ I n×n mod N .
This period does not depend, generically, on the initial condition and, thus, defines the length of the corresponding periodic orbit. It does depend "randomly" on N and, for large N , there exist "short" periodic orbits, for which T [N ] << N. It is known, for the case n = 1 [START_REF] Dyson | Period of a discrete cat mapping[END_REF] and M the Arnol'd cat map, that the smallest period corresponds to N a Fibonacci integer, f q . In this case,

T [f q ] = 2q.
What is significant about the period, T [N ], of the-hyperbolic-map, M, is that it controls the rate of "spreading", with time, of a localized distribution of initial conditions and, due to the compactness of the phase space, the rate of mixing [START_REF] Dyson | Period of a discrete cat mapping[END_REF]. Although, for finite N , there isn't any rigorous notion of ergodicity and of mixing, we can get important information, for reasonably large values of N , about the mixing time, by considering evolution times equal to half of the period. In the case of the Arnol'd cat map, for N a Fibonacci integer, the mixing time, t mixing scales as log N .

It is this property that implies that the dynamics of the Arnol'd cat map is chaotic and that it is possible to acquire information about the chaotic behavior by studying orbits whose period takes "large" values. The reason is that it is known that chaotic orbits, whether for Hamiltonian or non-Hamiltonian systems, essentially can be identified as unstable periodic orbits of infinite period [START_REF] Cvitanovic | Chaos: classical and quantum[END_REF][START_REF] Gutkin | Linear encoding of the spatiotemporal cat[END_REF]. For chaotic systems, described by the dynamics of symplectic linear maps, i.e. elements of Sp 2n [Z] and phase space T 2n , these linear maps must be hyperbolic, i.e. their eigenvalues must be real and positive.

One consequence of the above is that these eigenvalues come in pairs, (λ, 1/λ), with λ > 1. These pairs define planes in the 2n-dimensional phase space, spanned by the corresponding eigenvectors, where the flow expands along the eigenvector, corresponding to the eigenvalue λ and contracts along the eigenvector, corresponding to the eigenvalue 1/λ.

Closing this section we recall, for completeness, properties of the finite group Sp

2n [Z N ].
The mod N reduction of the symplectic group, Sp 2n [Z], defines the finite symplectic group, Sp 2n [Z N ] as follows:

Sp

2n [Z N ] = {M mod N, ∀M ∈ Sp 2n [Z]} (2.12)
The mod N reduction is a homomorphism from Sp This allows to obtain the order of the group, Sp 2n [Z N ], since the order of each term of this decomposition is known, namely [START_REF]Order formulas for symplectic groups[END_REF],

ord Sp 2n [Z p k ] = p (2k-1)n 2 +(k-1)n n i=1 p 2i -1 (2.

16)

.

3 Interacting Arnol'd cat maps from symplectic couplings of n, k-Fibonacci sequences

In the previous section we defined evolution operators Sp 2n [Z] M : T 2n → T 2n , that act on the points of the 2n-torus. These evolution operators describe the dynamics in phase space of n oscillators, each defined on a lattice of n sites. However this construction doesn't show how the oscillators are actually coupled. So we must show how it is possible to obtain the evolution operator of n oscillators, in terms of the evolution operator of one oscillator, how the phase space of one is embedded in the phase space of all. This is the subject of the present section. We shall show how to couple n Arnol'd cat maps. We know that the evolution operator for each is an element of Sp 2 [Z], the set of rational points of a 2-torus, T 2 . The total phase space of the system will be T 2n , the 2n-dimensional torus and the proposed dynamics will be described by appropriate elements of the symplectic group, Sp 2n [Z].

The idea is to exploit the known correspondence between the Arnol'd cat map and the Fibonacci sequence and describe the coupling between the Arnol'd cat maps by the coupling between the sequences, so that the evolution operators of n coupled Arnol'd cat maps can be understood as iteration matrices of n coupled generalized Fibonacci sequences.

Coupled Fibonacci sequences have been considered in the literature, for instance in [START_REF] Atanassov | A new perspective to the generalization of the Fibonacci sequence[END_REF][START_REF] Singh | Coupled Fibonacci sequence of fifth order and some properties[END_REF][START_REF] Rathore | Generalized Coupled Fibonacci Sequences[END_REF]. However, in these papers the possible applications to Hamiltonian dynamics were not the topic of interest and, moreover, the corresponding maps were not symplectic.

In the literature it has been a matter of debate, how to couple together Arnol'd cat maps. What we propose in this paper is to define the coupling between Arnol'd cat maps, through the coupling between generalized Fibonacci sequences, by requiring that the resulting map be symplectic.

To understand how this works, let us show how this works for two maps. We shall define the coupling between two n = 2 cat maps using the coupling between n = 2 generalized Fibonacci sequences, {f m } and {g m } (but we write the expressions in a way that generalizes immediately to arbitrary n), as follows:

f m+1 = a 1 f m + b 1 f m-1 + c 1 g m + d 1 g m-1 g m+1 = a 2 g m + b 2 g m-1 + c 2 f m + d 2 f m-1 (3.1)
where the a i , b i , c i , d i are integers, f 0 = 0 = g 0 and f 1 = 1 = g 1 are the initial conditions and m = 1, 2, 3, . . .. A byproduct of our analysis will be how to define the coupling between k-Fibonacci sequences (in particular the case k = 1, which corresponds to the case of two Arnol'd cat maps).

Based on the properties of symplectic matrices, that we reviewed in the previous section, we write these equations in the following matrix form

X m+1 ≡     f m g m f m+1 g m+1     =     0 0 1 0 0 0 0 1 b 1 d 1 a 1 c 1 d 2 b 2 c 2 a 2         f m-1 g m-1 f m g m     Xm (3.2)
In this expression we now focus on the 2×2 matrices

D ≡ b 1 d 1 d 2 b 2 C ≡ a 1 c 1 c 2 a 2 (3.3)
in terms of which the one-time-step evolution equation (3.2) can be written in block form as

X m+1 = 0 n×n I n×n D C X m (3.4)
In analogy with the case of a single Fibonacci sequence and its relation with the Arnol'd cat map, we impose the constraint (cf. (A.3))

0 n×n I n×n D C T J 0 n×n I n×n D C = -J (3.5)
This condition implies that

D = I n×n C = C T (3.6) Therefore a 1 = k 1 , a 2 = k 2 , c 1 = c 2 = c.
In terms of these parameters, the recursion relations take the form

f m+1 = k 1 f m + f m-1 + cg m g m+1 = k 2 g m + g m-1 + cf m (3.7)
and can be identified as describing a particular coupling between a k 1 -and a k 2 -Fibonacci sequence (cf. appendix A). This particular coupling is determined by the condition that the square of the evolution matrix is an element of Sp 4 [Z]:

A = 0 1 1 C ⇒ M = A 2 = 1 C C 1 + C 2 (3.8)
The role of the coupling is played by the integer c. In components:

    0 0 1 0 0 0 0 1 1 0 k 1 c 0 1 c k 2     2 =     1 0 k 1 c 0 1 c k 2 k 1 c c 2 + k 2 1 + 1 c(k 1 + k 2 ) c k 2 c(k 1 + k 2 ) c 2 + k 2 2 + 1     (3.9)
and represents the discrete time evolution matrix for the coupling of two "generalized" Arnol'd cat maps. We remark that, if c = 0 and k 1 = k 2 = 1, we recover two, decoupled, Arnol'd cat maps; if c = 0, and k 1 = k 2 = 1 we can, thereby, identify two "coupled" Arnol'd cat maps, while, if k 1 = k 2 = k, the system decouples into two, independent, (k + c), resp. (k -c) cat maps, for f m ± g m .

The generalization to n cat maps proceeds as follows: We choose two diagonal matrices, of positive integers, K IJ = K I δ IJ and G IJ = G I δ IJ , with I, J = 1, 2, . . . , n. If K I = 1, the cat maps are Arnol'd cat maps, with their coupling defined by the vector G I .

One way to write the coupling between the maps is, once more, to work with the (generalized) Fibonacci sequences. To this end, we define the translation operator, P and, for n > 2, we can distinguish between a "closed" and an "open" chain, of maps, by defining P I,J = δ I-1,J mod n for the former (and setting P 1,n = 0 = P n,1 for the latter). The periodicity is expressed by the fact that P n = I n×n . Moreover, P is orthogonal, since PP T = I n×n . Now we can define the coupling matrix for n sequences as

C = K + PG + GP T (3.10)
The corresponding 2n × 2n evolution matrix, A is given by

A = 0 n×n I n×n I n×n C (3.11)
and satisfies the relation A T JA = -J. Its square,

M = A 2 = I n×n C C I n×n + C 2 (3.12)
This satisfies the relation

M T JM = J, showing that M ∈ Sp 2n [Z].
Since A is symmetric, (from the property that C = C T ), M is positive definite and its eigenvalues come in pairs, (λ, 1/λ), with λ > 1. This property implies that, for all matrices K and G this system of coupled maps is hyperbolic.

It is possible to decompose the classical evolution matrix M in terms of the generators of the symplectic group (2.8)

M = I n×n 0 n×n C I n×n I n×n C 0 n×n I n×n (3.13)
Moreover each factor generates, for any symmetric, integer, matrix C, an abelian subgroup of Sp 2n [Z]. These factors are called "left" (resp. "right") translations.

Eq. (3.12) is, indeed, a key result of our paper, since it shows how the evolution operator of n cat maps is defined from the evolution operator of the individual maps, in a way consistent with symplectic covariance.

If C = I n×n , we have n decoupled Arnol'd cat maps, while the off-diagonal elements of C describe their interaction. If C is diagonal, we have decoupled cat maps and the band structure of C encodes the (non-)locality of the interactions. (This will become clearer when we shall discuss the dynamics in configuration space.)

An important special case arises if we impose translation invariance along the chain of maps, i.e. K I = K and G I = G for all I = 1, 2, . . . , n.

As an example of the translation invariant closed chain, we present below the matrix C, for n = 3:

C =   K G G G K G G G K   (3.14)
The corresponding evolution map, A, that describes three, coupled, K-Arnol'd cat maps, is a 6 × 6 matrix, given by the expression

A =         0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 0 0 K G G 0 1 0 G K G 0 0 1 G G K         (3.15)
and the symplectic map, M = A 2 , of three, coupled K-Arnol'd cat maps, is given by the expression

M =         1 0 0 K G G 0 1 0 G K G 0 0 1 G G K K G G 2G 2 + K 2 + 1 G 2 + 2GK G 2 + 2GK G K G G 2 + 2GK 2G 2 + K 2 + 1 G 2 + 2GK G G K G 2 + 2GK G 2 + 2GK 2G 2 + K 2 + 1         (3.16)
It is interesting that the coupling G appears, not only, in the off-diagonal 3 × 3 blocks, but, also, in the diagonal elements of the lower block. On the other hand, setting G = 0 we recover the case of three, decoupled "k-Arnol'd" cat maps.

Let us now consider the case of the open chain. The only change involves the operator P, which, now, must be defined as P IJ = δ I-1,J , for I, J = 1, 2, . . . , n. Due to the absence of the mod n operation, the "far non-diagonal" (upper right and lower left) elements are, now, zero. This express the property that the n-th Fibonacci is not coupled to the first one (and vice versa).

For both, closed or open, chains, we observe certain algebraic properties of the evolution matrix, A.

The k-Fibonacci sequence has the important property that the elements of the matrix A(k) m are arranged in columns of consecutive pairs of the sequence. We shall show that this property can be generalized for n interacting k-Fibonacci sequences as follows:

Theorem 1. The m-th power of the evolution matrix, A (cf. eq. (3.11)) can be written as

A m = C m-1 C m C m C m+1 (3.17)
where C 0 = 0 n×n , C 1 = I n×n and C m+1 = CC m + C m-1 , with m = 1, 2, 3, . . .. This matrix recursion relation generalizes to matrices the k-Fibonacci sequence for numbers, holds for any matrix, C and, in particular for the (symmetric, integer) matrix C, defined by eq. (3.10).

Proof. The proof is by induction. For m = 1 it is true, by definition. If we assume it holds for m > 1, then, by the relation A m+1 = A • A m , we immediately establish that it holds for m + 1.

It is straightforward to generalize this construction, to take into account interactions between next-to-nearest neighbors, and so on. For the case of the closed chain, the most general construction is encoded in the matrix C. Its definition (3.10) can be written as

C = KI n×n + [n/2]-1 l=1 P l G l + G l P T l (3.18)
The label l refers to the neighborhood: l = 1 labels the nearest neighbors, l = 2 the next-tonearest neighbors and so on. The farthest neighbors, on the closed chain, are [n/2] -1 sites apart.

The matrices G l are all diagonal, with integer entries and represent the couplings between the maps, at different sites of the lattice. By construction, the matrix C is symmetric, therefore the evolution matrix M ∈ Sp 2n [Z].

Imposing, once more, translation invariance, the matrices G l = G l I n×n , therefore, C becomes

C = KI n×n + [n/2]-1 l=1 G l P l + P T l (3.19)
The chaotic properties of the corresponding matrix M, depend strongly on how G l depends on l; namely, whether G l decreases, is independent of, or increases with l.

As an illustration, we show the matrix C for n = 7, that can describe couplings up to third nearest neighbors:

C =           K G 1 G 2 G 3 G 3 G 2 G 1 G 1 K G 1 G 2 G 3 G 3 G 2 G 2 G 1 K G 1 G 2 G 3 G 3 G 3 G 2 G 1 K G 1 G 2 G 3 G 3 G 3 G 2 G 1 K G 1 G 2 G 2 G 3 G 3 G 2 G 1 K G 1 G 1 G 2 G 3 G 3 G 2 G 1 K           (3.20)
Here we have assumed translation invariance, K = KI n×n and we remark that all pairs of symmetric diagonals contain identical elements, G 1 , G 2 , . . . , G [n/2]-1 = G 3 (for n = 7).

Another direction involves considering higher dimensional lattices of coupled Arnol'd cat maps.

For example the case of the square lattice (with periodic boundary conditions) can be described in the following way:

Let f (I,J) m be the family of sequences, at timestep m, where I, J = 0, 1, 2, . . . , n-1. Therefore we have n 2 Fibonacci sequences. The neighbors of the site (I, J) are taken as (I ± 1, J) and (I, J ± 1). The translation operators, that connect any site with its neighbors, are P ⊗ I n×n , P T ⊗ I n×n , I n×n ⊗ P and I n×n ⊗ P T . It is easy to convince oneself that these translation operators determine the order of the n 2 Fibonacci sequences along a vector of length n 2 . More explicitly, the ordering is the "lexicographic" ordering. The second index, J, of f (I,J) n is the "fast" index, while the index I is the "slow" index. In row form the ordering is the following: (f

(0,0) m , f (0,1) m , . . . , f (0,n-1) m , f (1,0) m , f (1,1) m , . . . , f (1,n-1) m , . . . , f (n-1,0) m , f (n-1,1) m , . . . , f (n-1,n-1) m
). The corresponding matrix C, which encodes the couplings between nearest neighbors, contains two, diagonal, n 2 × n 2 , matrices, one of which is K, just as for the case of the chain, along with another matrix G, which contains all the nearest-neighbor couplings.

Schematically,

C = K + (P ⊗ I + I ⊗ P ) G + G P T ⊗ I + I ⊗ P T (3.21)
In analogy with the one-dimensional case, interactions involving larger neighborhoods can be described by replacing P, respectively P T , by P l , respectively [P T ] l . Higher dimensional (hypercubic) lattices of Arnol'd cat maps can be described in the same way.

In summary we have constructed the evolution operator for n Arnol'd cat maps in a way that is consistent with its action as an element of Sp 2n [Z] on the torus T 2n and have shown how it is built up from the evolution operator of the individual maps.

If the coordinates of the initial condition are rational, then, as we have explained, the mod 1 operation, which expresses the fact that the action takes place on the torus, is replaced by the mod N operation, where N is the least common multiple of the denominators of the coordinates. These symplectic maps are all elements of Sp 2n [Z], since the matrices K and G l are all integer-valued. Applying the restriction of the mod N operation, these maps belong to the group Sp 2n [Z N ] and act on the toroidal lattice T 2n [N ]. As noted before, all the orbits will be periodic, with period T [N ] of the corresponding map M (3.12).

The next step in the study of the dynamics of the map, M, entails computing its spectrumfrom which we can deduce the Lyapunov exponents and hence the Kolmogorov-Sinai entropyand its eigenvectors. This calculation is facilitated by studying the equations of motion in configuration space, where, indeed, locality makes more sense than in phase space.

In the next section, therefore, we shall construct, explicitly, starting from Hamilton's equations, the corresponding Newton's equations, which describe the discrete time evolution of the position variables, as well as their solutions. For the case of translation invariant couplings, i.e.

K and G constant (first, for nearest-neighbor interactions, and, subsequently, for any range 1 < L ≤ [n/2] -1), they take into account the degree of locality of the interactions through their dependence on L, the number of interacting neighbors. L = 1 means nearest-neighbor interactions and so on. A particularly striking property of the equations in configuration space is that these n coupled maps, describe a system of n coupled inverted harmonic oscillators that don't exhibit runaway behavior, since this is "cured" by the compactness of the phase space.

From Hamilton's to Newton's discrete time equations

The classical equations of motion

x m+1 = x m M (4.1)
where m is the iteration timestep of the map, can be written in terms of positions, q m and momenta, p m as

(q m+1 , p m+1 ) = (q m , p m ) 1 C C 1 + C 2 q m+1 = q m + p m C p m+1 = q m C + p m 1 + C 2 (4.2)
Since C is symmetric, we can unclutter notation considerably by omitting the "transpose" symbol. Henceforth the row vectors are written without it:

We may solve the first of the last two equations for p m ,

p m = q m+1 C -1 -q m C -1 ⇔ p m+1 = q m+2 C -1 -q m+1 C -1 (4.3)
and insert the result in the second, in order to obtain a recursion relation for q m only-i.e. Newton's equations of motion:

q m+1 -2q m + q m-1 = q m C 2 (4.4)
This equation describes the discrete time evolution of n coupled Arnol'd cat maps, using only the coordinates in position space. It highlights that the locality properties of the system are encoded in the band structure of the symmetric matrix C. This procedure assumes that C is invertible; which fails to hold when C has a zeromode, that corresponds to a conserved quantity. In this case, the evolution matrix, M, has an eigenvalue equal to 1 (for each zeromode). This, in turn, implies that, if we choose as initial conditions the zeromode itself, this will not evolve in time.

However we do not need to assume that C is invertible, in order to obtain eqs. (4.4).

In the following, we shall consider the case of the non-zero modes and discuss the zeromodes separately.

In the subspace of the non-zero modes, the matrix C 2 is, by construction, positive definite and eq. ( 4.4) describes coupled inverted harmonic oscillators-the coupling is repulsive. The interest for this system stems from the fact that the phase space of each of these particles is a two-dimensional torus, so the motion is strongly chaotic and mixing (cf. also [START_REF] Palma | Linear growth of the entanglement entropy for quadratic Hamiltonians and arbitrary initial states[END_REF]).

We now decouple the modes of Newton's equations and diagonalize C by (finite) Fourier transform, F † CF ≡ D where

F IJ = e 2πiIJ/n / √ n ≡ ω IJ n / √ n.
We define the mode variable r m by q m ≡ r m F. The mode variable r m satisfies Newton's equation of motion in the form:

r m+1 -2r m + r m-1 = r m D 2 (4.5)
where D IJ = δ IJ D J , with

D J = K + 2G cos 2πJ n (4.6)
for the case of nearest-neighbor interactions. We note here that, if n is even, then the mode J 0 = n/2 has zero eigenvalue, when K = 2G. If n is odd, on the other hand, a zeromode cannot exist, because K and G are positive integers. It is possible to include the case of couplings beyond nearest-neighbors, i.e. G l , with 1 < l ≤ (n -1)/2, as follows:

D J = K + 2 n-1 2 l=1 G l cos 2πlJ n (4.7)
We shall now determine the discrete time evolution of the normal modes of the chain, by setting (r m ) I = r I,m ≡ δ IJ ρ m I a J (where I, J = 1, 2, . . . , n). We duly find a quadratic equation for ρ I :

ρ 2 I -(2 + D 2 I )ρ I + 1 = 0 ⇔ ρ ±,I = 2 + D 2 I 2 ± |D I | 2 D 2 I + 4 (4.8)
Thus the general solution, for r m can be written as

r m = a + ρ m + + a -ρ m - (4.9)
and the solution, in terms of q m , is

q m = r m F (4.10)
The initial conditions are (q 0 , p 0 ) = (q 0 , q 1 C -1 -q 0 C -1 ) or, equivalently, (q 0 , q 1 ). We can express the coefficients, a ± , in terms of the modes, r 0 and r 1 (where ρ ± are the diagonal matrices, with elements ρ I,± )

r 0 = a + + a - r 1 = a + ρ + + a -ρ - ⇔ a + = (r 1 -r 0 ρ -) (ρ + -ρ -) -1 a -= (r 0 ρ + -r 1 ) (ρ + -ρ -) -1 (4.11)
These equations, of course, only hold for the non-zero mode sector, since they degenerate for the zeromodes.

For the zeromodes (r m ) I = (r 0 ) I and, similarly, for the corresponding zeromodes of the momenta.

By defining ν ± ≡ F † ρ ± F, we obtain the exact solution of eq. (4.4), for the discrete time evolution of the positions, q m in the form

q m = (q 0 ν + -q 1 ) ν m + + (q 1 -q 0 ν + ) ν m -(ν + -ν -) (4.12)
We notice here that, for initial conditions, (q 0 , q 1 ) integer vectors, q m will be integer, also, for all times. This is obvious, since the coupling matrix, C, appearing in eq. (4.4) is integer-valued.

Studying the periodic trajectories, we restricted the initial conditions to be integer vectorsmod N. In order to find, at any time step, m, the position q m inside the torus T 2n [Z N ] we have to realize mod N reduction in eq. (4.12). The modes are coupled, in position space, through the matrices ν ± . In components, these read

(ν ± ) IJ = F † IM (ρ ± ) M K F KJ = 1 n ω (J-I)K n (ρ ± ) K (4.13)
since ρ ± are diagonal. We remark that ν ± are real, symmetric and positive definite matrices since (ρ ± ) K = (ρ ± ) n-K , for K = 0, 1, 2, . . . , n -1 and their product ν + ν -= I n×n , since ρ + ρ -= I n×n .

Having determined explicitly all the periodic orbits of the system, we shall now study the spectrum of the Lyapunov exponents.

Tuneable non-locality, Lyapunov spectra and K-S entropy

In this section we shall obtain analytic expressions for the Lyapunov spectra and the Kolmogorov-Sinai entropy, based on the calculations of the previous sections and we shall discuss their significance for the mixing (scrambling) properties of the ACML chaotic systems.

Let us start with the spectrum of the Lyapunov exponents, λ I , I = 0, 1, 2, . . . , n -1, which characterize the spatio-temporal chaotic properties of the chain.

We shall consider two cases:

1. The case of nearest neighbor (nn) interactions, viz. when G l = G for l = 1 and 0 for l > 1.

2. The case of longer range interactions, viz. when G l = 0, for l > L,L = 2, 3, ..,

[(n -1)/2].
In both cases, the Lyapunov exponents are defined by cf. (4.8),

λ ±,I = log ρ ±,I = log 2 + D 2 I 2 ± |D I | 2 D 2 I + 4 (5.1)
In general, the D J are given by the expression

D J = K + 2 [ n-1 2 ] l=1 G l cos 2πlJ n (5.2)
In the first case, typical density plots for the spectra are shown in fig. 1.

In the second case, we shall consider uniform couplings, viz. G l = G, for l = 1 to l = L < (n -1)/2 and G l = 0 for l > L.

This particular choice is interesting for two reasons: First, we can compute the sum explicitly, viz.

D (L) J = K + 2G sin(JLπ/n) sin(Jπ/n) cos(π(L + 1)J/n) (5.3)
for J = 1, 2, . . . , n -1 (and 1 ≤ L ≤ (n -1)/2). The case J = 0 must be treated separately, since D (L) 0

= K + 2GL. A typical example for the density plot of the spectrum of the Lyapunov exponents in this case, is shown in fig. 2.

An important consistency check of our calculations is that the sum of the positive Lyapunov exponents, for large values of n, is a linear function of n. Indeed this sum can be identified with the rate of entropy production, which is known as the Kolmogorov-Sinai entropy [START_REF] Pesin | Characteristic Lyapunov exponents and smooth ergodic theory[END_REF][START_REF] Sinai | Dynamical systems II: Ergodic theory with applications to dynamical systems and statistical mechanics[END_REF]. We plot it, for the same values of K and G as above, for nearest-neighbor interactions, L = 1, as a function of the length of the chain, from n = 3 to n = 100, in fig. 3. We remark that, for small sizes, there are deviations from linear behavior, that takes over at large sizes. We observe that the slope of the K-S entropy, is an increasing function of K.

S KS = n-1 I=0 λ +,I (5.4)
For longer-range, uniform interactions, L = 5, the corresponding K-S entropy is displayed in fig. 4. Let us discuss now the importance of the above typical behavior of the Lyapunov spectra and the K-S entropy. In the examples we considered above we notice the following four properties of the ACML systems.

First the increase of the average magnitude of the Lyapunov exponents as functions of the size of the system ,n,the coupling constant G and the constant K which plays a role for the mass of the coupled chaotic units, i.e.the Arnold cat maps.

Second the shape(curvature) of the Lyapunov spectra is tuneable, i.e we may control if we have many large or many small Lyapunov exponents or a flat region in the middle.

Third the increase of the slope of the K-S entropy as a function of K of the system, for fixed G and nearest-neighbor interactions.

Fourth-somewhat surprisingly-the long range interactions, generically, do not increase the average magnitude of the Lyapunov exponents and consequently the K-S entropy-this can be understood as a result of the oscillations in the spectra of the Lyapunov exponents.

All these four properties are important for the tuneability of the mixing properties of the system.

This system of n-coupled chaotic units (ACM) can be proven by general theorems that it is mixing, because it is ergodic and it has a compact-bounded phase space, the 2n dimensional torus. The mixing time of the system is defined as the logarithm of the deviation from the uniform distribution in time T for an initially chosen probability distribution in the phase space,divided by the time T, in the large T limit.

The problem of the calculation of the mixing time is an interesting exercise, whose solution, for the present system, will be described in detail in a future publication [START_REF] Axenides | The mixing time for Arnol'd cat map lattices[END_REF]. For the case of n = 1 it is equal to the 1/logarithm of the golden ratio [START_REF] Arnold | Ergodic problems of classical mechanics[END_REF]. For n > 1 it is expected to be proportional to 1/S K-S .

In general we would expect that the mixing time is faster, the greater the K-S entropy, but there are known counterexamples depending on the choice of the initial probability distribution [START_REF] Crawford | Decay of correlations in a chaotic measure-preserving transformation[END_REF].

For quantum systems there is the conjecture, as we discussed in the introduction, that the black holes are the fastest scramblers of the universe and their scrambling time is proportional to the logarithm of their entropy. However, whether this scrambling time can be identified with the mixing time of the quantum dynamical system is an open question and different scenaria have been proposed [START_REF] Latora | Kolmogorov-Sinai entropy rate versus physical entropy[END_REF].

The spectra of periods of ACML systems

In this section we study the problem of finding the spectrum of the periods of the evolution operator M of n coupled Arnol'd cat maps mod N . They are expected to be a random function of N ; however, for special values of N, a thorough along the lines of Falk and Dyson [START_REF] Dyson | Period of a discrete cat mapping[END_REF], which has been done only for single cat map-n = 1-can only lead to bounds and, only in some cases, to exact expressions.

In this section we shall present an algorithm for finding the period of the evolution operator of n cat maps, using properties of the matrix Fibonacci polynomials.

Since the dynamics is that of a system of coupled, "inverted" harmonic oscillators-that are, however, constrained to a (compact) toroidal phase space, T 2n [N ]-we expect that this system describes maximally chaotic and mixing motion.

To be concrete consider the action of the evolution operator M (cf. eq. (3.12) ), for the system of n coupled Arnol'd cat maps, on the discrete phase space T 2n [N ]. According to Theorem 3.17 the m-th power of M is given by the expression

M m = A 2m = C 2m-1 C 2m C 2m C 2m+1 (6.1)
where

C 0 = 0, C 1 = I n×n and C m+1 = CC m + C m-1 , with m = 1, 2, 3, . . .. Since M ∈ Sp 2n [Z], its mod N reduction belongs to Sp 2n [Z N ].
The order of the latter-finitegroup can be determined using the relations (2.15) and (2.16). These imply that the order of the evolution matrix M mod N , T (N ), must be a divisor of the order of Sp 2n [Z N ].

In order to determine T (N ) we make use of Theorem 3.17 as follows: Since M T (N ) = I 2n×2n mod N , it is obvious that C 2T (N )-1 ≡ I n×n mod N and C 2T (N ) ≡ 0 mod N, which reduces the problem of finding the period to finding the least value of m = T (N ), for which these two relations hold simultaneously.

The period of M mod N is the smallest integer, T (N ), such that

M T (N ) ≡ I 2n×2n mod N (6.2)
This implies that

M T (N ) = C 2T (N )-1 C 2T (N ) C 2T (N ) C 2T (N )+1 mod N ≡ I n×n 0 n×n 0 n×n I n×n mod N (6.3) Therefore C 2T (N ) ≡ 0 mod N , C 2T (N )-1 ≡ I n×n mod N and C 2T (N )+1 = C • C 2T (N ) + C 2T (N )-1 ≡ I n×n mod N .
From these relations it is easy to show that 2T (N ) is the period of the sequence of matrices {C m mod N }:

Proof. The starting point is the property that

C 2T (N )-1 ≡ I n×n mod N C 2T (N ) ≡ 0 n×n mod N (6.4)
This implies that

C 2T (N )+1 = C • C 2T (N ) + C 2T (N )-1 ≡ I n×n = C 1 mod N C 2T (N )+2 = C • I n×n ≡ C = C 2 mod N C 2T (N )+3 = C • C 2T (N )+2 + C 2T (N )+1 ≡ I n×n + C 2 = C 3 mod N (6.5)
Now the key observation is that C m is a polynomial in the matrix C, with positive integer coefficients and its degree is equal to m -1, which is even for m odd and odd for m even.

In fact these polynomials turn out to be nothing else but the so-called Fibonacci polynomialsnow defined over the space of integer matrices mod N.

The Fibonacci polynomials are defined by the recursion relation [START_REF] Philippou | Fibonacci polynomials[END_REF]

F m+1 (x) = xF m (x) + F m-1 (x) (6.6)
with x a formal variable and initial conditions F 0 (x) = 0 and F 1 (x) = 1. The Fibonacci polynomials have been extensively studied, for x ∈ R; what we note here is that they can be defined for x = C, i.e. matrices; and many of their remarkable properties carry over to this case. In order, therefore, to find the periods mod N we must find, for a given evolution matrix C, the values of m, for which the matrix Fibonacci polynomial F 2T (N ) (x = C), vanishes and, simultaneously, F 2T (N )-1 (x = C) = I n×n (mod N ).

Using that C m = F m (C) and their explicit formula we readily find that

F m (C) = [ m-1 2 ] j=0 -j + m -1 j C -2j+m-1 , (6.7)
where [•] denotes the integer part of the argument.

The reason these polynomials are particularly useful here is that the evolution in phase space is described by the recursion relation

(q m , p m ) = (q m-1 , p m-1 ) • M = (q 0 , p 0 ) • M m = (q 0 , p 0 ) F 2m-1 (C) F 2m (C) F 2m (C) F 2m+1 (C) (6.8)
and we realize that the coefficients in the last expression are the Fibonacci polynomials for matrices-which allow us to write the evolution in phase space in closed form:

q m = q 0 F 2m-1 (C) + p 0 F 2m (C) p m = q 0 F 2m (C) + p 0 F 2m+1 (C) (6.9)
These equations highlight that at step m = T (N ), which is defined by eqs. (6.4), q T (N ) = q 0 and p T (N ) = p 0 , consistent with T (N ) being the period of motion.

Our framework provides a lot of freedom for choosing the dynamics:

1. The number of maps, n.

2. The positive integer, K, from the K-Fibonacci sequence.

3. The range of the non-locality, l; l = 1 is for nearest neighbor interactions, l = 2 for next-to-nearest neighbor interactions, and so on; l ≤ l max = integer part(n -1)/2.

4. The couplings, G l , for l = 1, 2, . . . , l max , that are, also, positive integers.

To simplify matters, we shall provide numerical examples for the periods, T [N ], by choosing all the couplings, G l ≡ G and for a given range of non-locality. Moreover we shall choose N = p a prime and n = 2 -5 (so up to five maps). (The case n = 1, of the one cat map, has been thoroughly studied in the past [START_REF] Keating | Asymptotic properties of the periodic orbits of the cat maps[END_REF][START_REF] Axenides | The quantum cat map on the modular discretization of extremal black hole horizons[END_REF].) For these choices, the expression for the order of the group simplifies considerably:

ord (Sp 2n [Z p ]) = p n 2 n i=1 p 2i -1 (6.10)
To conclude this section we report on the results of the numerical investigations for T (N ) for selected values of N and K = G = 1 (cf. fig. 5)

Conclusions and outlook

The study of classical and quantum chaotic field theories has received considerable attention recently, inspired by work in turbulence, the problem of statististical n-body thermalization (the so-called Eigenstate Thermalization Hypothesis), as well as from the motivation to describe the quantum dynamics of black holes. To this end we started by setting forth the description of the classical n-body lattice field theories, whose fundamental constituents are Arnol'd cat maps. The underlying quantum dynamics will be the subject of future work.

More specifically, in this work, we have presented the consistent Hamiltonian dynamics of coupled map lattices of n classical chaotic oscillators-Arnol'd cat maps (ACM)-in phase space and in cofinguration space-with guiding principle the symplectic invariance of the phase space of the n-body system.

In configuration space, each ACM is located on a single site of the lattice and acts on a two-dimensional toroidal phase space in a way that is hyperbolic and exhibits maximal mixing.

Our method is based on the representation of the Arnol'd cat map in terms of Fibonacci sequences with the n-body coupled ACM generalization being realized through the coupling of n k-Fibonacci sequences.

The n-body system is thus defined on a 2n-dimensional toroidal phase space, by the action of elements of the symplectic group Sp 2n [Z], which is maximally hyperbolic and mixing, thanks to the periodic boundary conditions.

The corresponding equations of motion in configuration space are those of n, linearly coupled, inverted harmonic oscillators. It is interesting to stress that it is the boundary conditions in the phase space that ensure that the system doesn't have runaway behavior. This is an example of how mechanical systems, with unbounded potentials, can be understood as chaotic systems, upon imposing periodic boundary conditions-in phase space.

The chaotic properties of the system as a whole are quite intricate, despite the simplicity of the couplings. They depend, indeed, on the possibility of tuning their locality and strength and can be understood through their periods, Lyapunov spectra and Kolmogorov-Sinai entropies. An interesting point, which will be the subject of future work is the discrete conformal symmetry, that emerges, when the range of the couplings becomes maximal.

A further topic is the construction of quantum Arnol'd cat map lattices, along with their continuum (scaling) limit as field theories. It is here that the issues of closed subsystem thermalization dynamics, the Eigenstate Thermalization Hypothesis, as well as the saturation of the fast scrambling bound, for many-body systems, can be framed and consistently treated. A useful diagnostic for this is the set of so-called "out of time-order correlation functions" [START_REF] Foini | Eigenstate thermalization hypothesis and out of time order correlators[END_REF][START_REF] Foini | Eigenstate Thermalization and Rotational Invariance in Ergodic Quantum Systems[END_REF].

Acknowledgements: This research was funded by the CNRS IEA (International Emerging Actions) program "Chaotic behavior of closed quantum systems" under contract 318687.

These imply that f pq-1 (x) f pq (x) f pq (x) f pq+1 (x) = f p-1 (x) f p (x) f p (x) f p+1 (x) q = f q-1 (x) f q (x) f q (x) f q+1 (x) p (A.8)

For future reference, we call A 2 , ACM 1 , to indicate that it describes the motion of a single particle. Below we shall study the dynamics of n particles.

Since the matrix A(x) doesn't depend on m, we can solve the recursion relation in closed form, by setting f m ≡ Cρ(x) m and find the equation, satisfied by ρ(x) ρ(x) m+1 = xρ(x) m + ρ(x) m-1 ⇔ ρ(x) 2 -xρ(x) -1 = 0 ⇔ ρ(x) ≡ ρ(x) ± =

x ± √ x 2 + 4 2

Therefore, we may express f m (x) as a linear combination of ρ m + (x) and ρ m -(x) = (-) m ρ -m + (x):

f m (x) = A + ρ m + (x) + A -ρ m -(x) ⇔ f 0 = A + + A -= 0 f 1 = A + ρ(x) + + A -ρ(x) -= 1 (A.9)
whence we find that Applying such a transformation to the equation of motion, we find that the condition

A + = -A -= 1 ρ + (x) -ρ -(x) = 1 √ x 2 + 4 therefore,
R T CR = C (B.6)
guarantees that these rotations are symmetries of the equations of motion.

All such transformations define a subgroup of O n [Z], that is, therefore the invariance group of the ACML. In this case, the existence of a zeromode a leads to the existence of additional zeromodes, given by aR. This defines a linear subspace of zeromodes, labeled by all such matrices R.

That this group isn't empty follows from the particular form of the matrix C, which commutes with the matrix P. The above transformations describe discrete spatial translations as well as rotations in the target space; however, there exists a further, important, symmetry, of the equations of motion (B.1), namely that of discrete translations in time, m → m + 1.

An immediate consequence of this symmetry is that, from the block form for M,

M = A B C D (B.8)
we find that the quadratic form, Q(x), given by the expression Q(x) = q T B T q + q T D T -A pp T C T p (B.9) is conserved in time:

Q(x m+1 ) = Q(x m ) (B. 10
)
where x m = (q m , p m ).

For the particular case of the CACML, A = I n×n , B = C and D = 1 + C 2 , we find the expression Q(x) = q T Cq + q T C 2 pp T Cp (B.11) since C is symmetric.

Of course what is significant is that the global sign is arbitrary, therefore, we can write it in a more "conventional" form as Q(x) = p T Cpq T Cqq T C 2 p (B.12)

where the mod 1 operation is implicit. This, particular, solution will play a significant role in the construction of the quantum, unitary, evolution operator U (M) for the corresponding quantum mechanical system of n coupled Arnol'd cat maps.

  2n [Z] to Sp 2n [Z N ], with kernel the principal congruent subgroup, Γ sp [N ] = {M ∈ Sp 2n [Z] |M ≡ I 2n×2n mod N } (2.13) We can describe the decomposition of the finite group Sp 2n [Z N ] into the direct product of finite groups of the form Sp 2n [Z p k ], where p is a prime and k is a positive integer, corresponding to the decomposition in prime factors of N N = L l=1 p k l l (2.14) Indeed, using the Chinese Remainder Theorem [62], we can show that Sp 2n [Z N ] = L l=1 Sp 2n Z p k l l (2.15)

Figure 1 :

 1 Figure 1: Histogram of the sorted Lyapunov spectra, of λ (K) +,I vs. I = 0, 1, 2, . . . , n -1, for n = 100, G = 1, nearest-neighbor interactions-L = 1-and K = 1, 2, 3.

Figure 2 :

 2 Figure 2: Histogram of the sorted Lyapunov spectra, λ (L) +,I , vs. I = 0, 1, 2, . . . , n -1, for uniform couplings, namely, K = 3, G l = G = 1, n = 101 and L = 1, 2, 3.

Figure 3 :

 3 Figure 3: The Kolmogorov-Sinai entropy, S KS , of eq. (5.4) as a function of the chain length, n = 5 -101, for K = 1, 2, 3 and G = 1. We remark deviations from linear behavior at small chain lengths, that become negligible at larger lengths.

Figure 4 :

 4 Figure 4: Kolmogorov-Sinai entropy, for n = 5 -101 Arnol'd cat maps, K = 3, G = 1 with ranges L = 1 and L = 5.

Figure 5 :

 5 Figure 5: Period T (N ) for N = p 11 = 31 (the eleventh prime) to p 31 = 113, (the thirty-first prime) for l = 1, n = 1 and 2. We remark the dramatic change from n = 1, one map, to n = 2, two coupled maps. This reflects the dramatic increase in size of the order of the group.

  f m (x) = ρ(x) m + -(-) m ρ(x) quite fascinating that the LHS of this expression is a polynomial in x that, moreover, takes integer values for integer values of x! Since the target space is a torus, there exists a group of transformations, beyond the translations mod 1, namely the orthogonal group over the integers, O n[Z]. By definition, an element R ∈ O n [Z] satisfies the condition RR T = I n×n .

  The matrix P belongs to O n [Z] and represents, by a shift along the lattice, a rotation in the target space! This particular symmetry is the reason for the degeneracy in the spectrum, D I , I = 1, 2, . . . , n namely D I = D n-I (B.7) So, finally, degeneracies of the spectrum of C, are explained by the existence of rotations R ∈ O n [Z], which commute with P.

A The Fibonacci polynomials and their basic properties

The Fibonacci sequence is one of the integer sequences, which has been studied, for a long time and there are journals dedicated to its properties and their applications. One generalization is provided by the sequence of polynomials, f m (x), defined by

which can be written in matrix form

The matrix A(x) is not a symplectic matrix, but it satisfies

where J is the symplectic matrix (2.4), for n = 1. The integer Fibonacci sequence, f m = f m (x = 1) and the integer k-Fibonacci sequence corresponds to f m (x = k) with k = 2, 3, . . . These sequences are related to the "golden" and "silver ratios" by

for x = 1 and 2 respectively. The Fibonacci polynomials are given explicitly by the relation

where [•] denotes the integer part of the argument. The Fibonacci polynomials are generated by powers of the matrix A(x) viz.

This relation is the origin, in fact, of the properties of the Fibonacci polynomials:

B The conservation laws of ACML systems

In this section we study the conservation laws of the discrete evolution equations,

where C is the symmetric, integer-valued matrix, given by the expression (3.19), i.e.

In this equation q ∈ T n , whose compactness ensures mixing and implies that in eq. (B.1) there is an implicit "mod 1", to ensure that q m ∈ T n for all timesteps.

A conservation law is related to, either, to the existence of an eigenvalue equal to 1 of the evolution operator, M, or to a degeneracy of eigenvalues of M. These, in turn, can be recast in terms of properties of the matrix C.The first case corresponds to a zero eigenvalue of C. The corresponding eigenvector, a ∈ T n , allows us to identify the symmetry as the translation q m → q m + a. In this case, there exists a solution of Newton's equations, which is linear in time, viz.

This means that the corresponding Lyapunov exponent vanishes.

For the matrix C defined by eq. (B.2), when G l = G for l = 1 and G l = 0 for l > 1, (i.e. nearest-neighbor interactions) we can verify that such an eigenvector always exists, when K = 2G, and n is even and is given by the expression

where I = 1, 2, . . . , n.

When couplings beyond those between nearest-neighbors are non-zero, n must, still, be even and the condition on the couplings, for the existence of a zeromode, becomes

Let us now consider the case, when the spectrum of the matrix C shows degeneracies, which, in turn, correspond to degeneracies in the spectrum of the Lyapunov exponents. We observe that translation invariance in the target space, q → q + a, corresponds to a symmetry that is an inhomogeneous transformation, which is the hallmark of zeromodes, while degeneracies correspond to symmetries given by homogeneous transformations.

To look for such transformations we work as follows:


Description of the work

  David Rodriguez (Icam Toulouse, CERTOP) Supervisors : Yann FERGUSON (Icam Toulouse, CERTOP), Jean-Pierre POULAIN (CERTOP) Objectives of the thesis • Map initiatives on human-centered robotics (HCR) • Identify key points to fulfil gaps between ethics reflexions and practices in robotics • Qualify work transformations linked to robotization • Analyze robots' adoption and acceptance in food industry and hotel restaurant sector SOROBOT : SOcial Representations and acceptability Of human-centred roBOTics Future work • Interviews and observations in workplaces • Analyze statistics related to robotization in food industry and hotel restaurant field • Extend map on HCR mobilization to include political, economical or civil society actors [1] Xu, W., Dainoff, M. J., Ge, L., & Gao, Z. (2023). Transitioning to human interaction with AI systems: New challenges and opportunities for HCI professionals to enable human-centered AI. International Journal of Human-Computer Interaction, 39(3), 494-518.


Keywords: additive manufacturing, titanium alloy, martensitic transformation, strainhardening

The Ti-22Zr-9Nb-2Sn (at%) metastable -titanium alloy was manufactured by in situ laser powder bed fusion process from elemental powders. Mechanical characterization via cyclic tensile tests revealed an exceptionally high strain-hardening. Such a strainhardenability is induced by a massive stress-induced martensitic transformation from the  to the '' phase.

Outstanding strain-hardening of a new metastable -titanium alloy elaborated by in situ additive manufacturing L-PBF process

Hugo Schaal

Introduction

Metastable -titanium alloys are considered of great potential for biomedical applications, thanks to their low Young's modulus, preventing stress-shielding effect, and their excellent biocompatibility due to the use of non-toxic alloying elements for the stabilization of the  phase, such as Ta, Nb, Zr… [START_REF] Laheurte | Mechanical properties of low modulus  titanium alloys designed from the electronic approach[END_REF][START_REF] Geetha | Ti based biomaterials, the ultimate choice for orthopaedic implants -A review[END_REF][START_REF] Niinomi | Development of new metallic alloys for biomedical applications[END_REF][START_REF] Miyazaki | Development and characterization of Ni-free-Tibase shape memory and superelastic alloys[END_REF]. These titanium alloys can also exhibit stress-induced martensitic (SIM) transformation from the bcc  phase to the orthorhombic '' phase, which can lead to superelastic behaviour and shape memory effect when the transformation is reversible [START_REF] Castany | Design of strain-transformable titanium alloys[END_REF][START_REF] Bertrand | Deformation twinning in the full-α″ martensitic Ti-25Ta-20Nb shape memory alloy[END_REF][START_REF] Gordin | How a new strain transformable titanium-based biomedical alloy can be designed for balloon expendable stents[END_REF], or TRIP (TRansformation Induced Plasticity) effect, that can be combined with TWIP (TWinning Induced Plasticity) effect, leading to alloys with large deformation and high strain-hardening capability [START_REF] Sun | Investigation of early-stage deformation mechanisms in a metastable β titanium alloy showing combined twinning-induced plasticity and transformation-induced plasticity effects[END_REF]. etastable -titanium alloys obtained by usual melting techniques (cold crucible levitation melting, arc melting, etc…) have been widely studied and discussed since the last three decades, with a focus of interest on the mechanical properties induced by the SIM transformation. Additive manufacturing, and especially the laser powder bed fusion (L-PBF) technique, is of great interest for the medical fields, because of the capability of the process to manufacture very complex parts, and therefore patient specific biomedical devices. However, elaboration of metastable -titanium alloys by L-PBF process is challenging due to the poor diversity of pre-alloyed powders available for sale, which are mainly commercial alloys (Ti-6Al-4V, CoCr alloys, stainless steels, superalloys…). Some studies have investigated the manufacturing capability of the in situ technique, which promotes the manufacturing of parts with a blend of elemental powders instead of pre-alloyed J o u r n a l P r e -p r o o f powders, achieving alloying operation during the scanning of the laser beam [START_REF] Fischer | In situ elaboration of a binary Ti-26Nb alloy by selective laser melting of elemental titanium and niobium mixed powders[END_REF][START_REF] Vrancken | Microstructure and mechanical properties of a novel β titanium metallic composite by selective laser melting[END_REF]. This technique seems more suited when it comes to the prospection of chemical compositions with desired mechanical properties, such as alloys that exhibit SIM transformation. However, this method can lead to the formation of unmolten alloying elements in the metallic matrix, such as -stabilizing elements with high melting point like Ta, Nb and Mo [START_REF] Sing | Selective laser melting of titanium alloy with 50 wt% tantalum: Microstructure and mechanical properties[END_REF][START_REF] Wang | Microstructural homogeneity and mechanical behavior of a selective laser melted Ti-35Nb alloy produced from an elemental powder mixture[END_REF]. These inclusions can be detrimental to the local chemical homogeneity, and therefore shift the mechanical properties of the fabricated alloys. Consequently, design of experiments (DOE) must be used to optimize the manufacturing parameters, achieving parts with both low porosity and low unmolten particles for a limited number of tries. This optimization relies on the variation of the main parameters that drives the melt pool, which are laser power, speed of laser scanning and hatch distance between two laser tracks. Even though the in situ elaboration can lead to dense and homogeneous parts, the use of this technique to obtain biocompatible metastable -titanium alloys that exhibit SIM transformation has been rarely investigated.

Materials and methods

In this study, the Ti-22Zr-9Nb-2Sn (%at) metastable -titanium alloy was obtained by the in situ technique. An SLM 125 HL (SLM solutions Group AG, Germany) was used to carry out the fabrication of the different specimens. Elemental additive manufacturing grade powders of titanium, zirconium, niobium, and tin were used to achieve our powder mix. The blend of powders was performed using a T2F Turbula device, at a speed of 34 rpm, for 2h of blending. The average powder size of the blend of elemental powders used was between 15 and 45 µm. Optimized building parameters presented in Table 1 were selected through a preliminary study by iterative DOE in order to obtain the highest density by using a 67° hatch strategy [13]. Those building parameters provided parts with a density of about 99.5%, with unmolten niobium fraction of less than 0.05% of the parts volume. Those two different quantities were characterized and measured by X-ray µ-Tomography. Regarding the parts manufactured for this study, two different types of specimens were elaborated: for microstructural characterization, 10x10x10 mm 3 cubes were used, and 50x5x2 mm 3 walls were built to extract by electrical discharge machining flat tensile specimens with a gauge section of 3x1 mm² and a gauge length of 15 mm. The tensile direction was taken orthogonal to the building direction. Additive manufacturing process are known to induce important residual stress [START_REF] Ahmad | Residual stress evaluation in selective-laser-melting additively manufactured titanium (Ti-6Al-4V) and Inconel 718 using the contour method and numerical simulation[END_REF][START_REF] Liu | A study on the residual stress during selective laser melting (SLM) of metallic powder[END_REF]. As thermal treatments are an effective way to relieve this residual stress, the as-fabricated samples were stress-relieved at 700°C for 30 min, and then water quenched to maintain the fully- microstructure. Cyclic tensile tests were performed to characterize the mechanical behaviour of the alloy, with a strain rate of 10 -4 s -1 . Those cyclic tests consist in strain increments of 0.5% followed by stress unloading, up to the rupture of J o u r n a l P r e -p r o o f the specimen. The unloading parts of the curve enable the identification of potential superelastic behaviour. An extensometer was used to measure the deformation of the sample.

10 different specimens were tested and the mechanical performances are presented via the mean value measured along its standard deviation. The microstructure was characterized by X-ray diffraction (XRD), electron backscattered diffraction (EBSD) and transmission electron microscopy (TEM). EBSD analyses were carried out on a JEOL JSM 7100F scanning electron microscope with a step of 0.1 µm and an accelerating voltage of 20 kV. EBSD specimens were prepared by mechanical polishing using SiC papers with decreasing grid size, followed with colloidal silica suspension polishing mixed with H2O2 to get a "mirror finished" state. Optical microscopy specimens were prepared the same way, and then chemically etched with a 5% HF, 5% HNO3 and 90% H2O (vol) solution. TEM specimens were thinned by a three steps ion milling process with 4 keV, 2 keV and 0.5 keV argon ions using a Gatan PIPS II machine. Texture was investigated through {200}  and {110}  poles figures acquisition on a Rigaku SmartLab X-ray diffractometer using Cu-K 1 radiation at 40 kV and 50 mA.

Results and discussion

Microstructural analyses of the Ti-22Zr-9Nb-2Sn alloy are presented in Fig. 1. A fully  microstructure is expected after water quenching for this alloy. Fig. 1a confirms this assumption, as it shows the XRD pattern of the specimen, indexed as  phase. Fig. 1b details the microstructure of the alloy through EBSD analyses. The IPF-EBSD map exhibits columnar and elongated grains along the building direction. The grains are about 170 µm long, for an average 60 µm width. This microstructure is typical of parts built by L-PBF technique [START_REF] Fischer | In situ elaboration of a binary Ti-26Nb alloy by selective laser melting of elemental titanium and niobium mixed powders[END_REF][START_REF] Thijs | A study of the microstructural evolution during selective laser melting of Ti-6Al-4V[END_REF], due to the thermal gradient and the layer-by-layer building, promoting epitaxial growth along the building direction (BD). The EBSD analysis corroborates the XRD analysis (Fig. 1a), as every grain of the microstructure is successfully indexed with  phase.

Texture of the specimen is also investigated through EBSD, thanks to the inverse pole figures presented in Fig. 1b. The alloy exhibits a strong <100>  texture in the building direction, which is a consequence of the manufacturing conditions. This texture is characteristics of bcc materials elaborated by L-PBF process and was already reported, either by direct observations or by reconstitution from the room temperature phase [START_REF] Vasquez | Elaboration of oxide dispersion strengthened Fe-14Cr stainless steel by selective laser melting[END_REF][START_REF] Zhou | Anisotropic mechanical behavior of biomedical Ti-13Nb-13Zr alloy manufactured by selective laser melting[END_REF][START_REF] Simonelli | On the Texture Formation of Selective Laser Melted Ti-6Al-4V[END_REF]. However, the texture in the tensile direction (TD) does not exhibit preferential direction. As the EBSD analysis relies on a limited population of grains, the texture is also more accurately evaluated using XRD measurements. The pole figures (Fig. 1c) and inverse pole figures (Fig. 1e) obtained by XRD confirm the strong <100>  texture in the building direction, and the absence of preferential direction in the tensile direction. Fig. 2a shows the engineering stress-strain curve obtained on the Ti-22Zr-9Nb-2Sn sample after cyclic tensile test to rupture. The yield strength and the ultimate tensile strength reach J o u r n a l P r e -p r o o f 421±44 MPa and 1049±15 MPa, respectively. A total elongation of 10.4±0.3% is obtained, and the Young's modulus is evaluated at 45±0.4 GPa. It can be noticed that the unloading part of the cyclic tensile curve shows no reversibility, and therefore no superelastic effect. The specimen exhibits very high strain-hardening, as the difference between yield strength and ultimate tensile strength is about 630 MPa. The strain-hardening rate was obtained by the true stress-strain curve, which is shown in Fig. 2b. The strain-hardening curve is composed of three different stages, which are highlighted on the graph. During the early deformation domain (stage I), the strain-hardening rate decreases rapidly. Then, as the material undergoes plastic deformation, the strain-hardening rate increases until it reaches a peak (stage II). Finally, the strain-hardening rate decreases until the rupture of the material (stage III). In our case, the maximum strain-hardening rate is obtained at 753 MPa and its value is 14.6 GPa. A high strain-hardening is quite common in metastable -titanium alloys that exhibit TRIP and/or TWIP effects [START_REF] Sun | Investigation of early-stage deformation mechanisms in a metastable β titanium alloy showing combined twinning-induced plasticity and transformation-induced plasticity effects[END_REF][START_REF] Fu | Ultrahigh strain hardening in a transformation-induced plasticity and twinning-induced plasticity titanium alloy[END_REF]. However, the present strain-hardening rate value is exceptionally higher than those obtained with other metastable -titanium obtained by conventional melting process such as Ti-12Mo [START_REF] Sun | Investigation of early-stage deformation mechanisms in a metastable β titanium alloy showing combined twinning-induced plasticity and transformation-induced plasticity effects[END_REF], Ti-9Mo-6W [START_REF] Sun | A new titanium alloy with a combination of high strength, high strain hardening and improved ductility[END_REF], Ti-8.5Cr-1.5Sn [START_REF] Brozek | A -titanium alloy with extra high strain-hardening rate: Design and mechanical properties[END_REF] and Ti-10V-2Al-3Fe [START_REF] Danard | Microstructure design and in-situ investigation of TRIP/TWIP effects in a forged dual-phase Ti-10V-2Fe-3Al alloys[END_REF], where the maximum strain-hardening rate is about 2-2.5 GPa.

The origin of such an outstanding strain-hardening rate is investigated from microstructural characterizations after rupture. Fig. 2c displays the superposition of the XRD patterns of the specimens before deformation and after being strained until rupture. It shows a drastic change of microstructure in the deformed state, which is composed of both '' and  phases. The lattice parameters measured by XRD analysis are a  = 0.338 nm, and a '' = 0.309 nm, b '' = 0.518 nm and c '' = 0.479 nm. Despite the presence of some peaks of the original  phase, the relative intensity of these peaks indicates that the microstructure is mainly constituted of '' phase after tensile test. This observation implies that the alloy undergoes stress-induced martensitic (SIM) transformation, and that a large volume fraction of  phase transformed into '' phase. This SIM transformation is most probably the cause of the high strainhardening observed on Fig. 2a, as the nucleation of the '' martensitic phase hinders dislocation slip by the multiplication of interphase boundaries.

Post mortem microstructural characterizations were performed to identify the stress-induced martensite responsible for the high strain-hardening observed during mechanical characterization. Fig. 3 exhibits optical micrographs and TEM observations performed on specimens after rupture. The microstructure before deformation is shown in Fig. 3a, and is composed of columnar  grains. After deformation until rupture, numerous new black strips are clearly visible within the columnar  grains of the microstructure (Fig. 3b). It should be stated that the formation of the stress-induced martensitic '' phase is similar to a germination/growth process [START_REF] Bignon | Modelling martensitic transformation in titanium alloys: The influence of temperature and deformation[END_REF] which can be described as followed. First, the martensite forms in the initial  grain, followed by a radial development until it come across an obstacle such as grains boundaries. Then, the martensitic laths growth along their thicknesses until an J o u r n a l P r e -p r o o f energetic local equilibrium is reached. Fig. 3b displays thin and numerous laths of martensitic '' phase within the initial  grains, which implies that the thickening of those laths is limited, and that the formation of new '' laths is favoured in comparison to the thickening of previously formed '' laths.

TEM bright field image of the specimen microstructure after rupture corroborates this result (Fig. 3c), in which '' martensitic laths are clearly identified. The corresponding selected area electron diffraction (SAED) pattern is presented along the [113]  /[1 ̅ 12] '' zone axis (Fig. 3d).

In this diffraction pattern, the brightest diffraction spots belong to the martensitic '' phase and the weakest ones to  phase. The indexation of this diffraction pattern implies that the '' martensite observed was formed accordingly to the classical crystallographic relationship between the  and '' phase and shows no evidence of twinning during deformation.

Considering that the of the observed martensitic laths is only about 400 nm, it seems that the local equilibrium that drives the growth of '' laths is quickly achieved, in opposition with the formation of new '' phase, which seems more significant. The evolution of the microstructure through deformation can explain the evolution of the strain-hardening curve (Fig. 2b): The transition between stage I and stage II (Fig. 2b) corresponds to the triggering of the SIM transformation, in which '' laths begin to form in the initial  grains.

As the strain increases, more and more laths of martensite appear in the microstructure, generating interfaces, causing the stress to increase. The transition between stage II and stage III corresponds to a slow-down of the transformation rate of the martensitic phase, causing the strain-hardening rate to decrease until rupture as the number of new interfaces in the microstructure does not increase anymore. The formation of the martensitic laths can occur during stages II and III but does not contribute to the strain-hardening effect, as it does not hinder dislocation slip. The exceptional strain-hardening rate of the present Ti-22Zr-9Nb-2Sn alloy is then due to the SIM '' transformation without any contribution of twin formation, while both effects are mostly observed concomitantly in literature. Fig. 4 displays the strain-hardenability (ultimate tensile strength -yield strength) as a function of maximum elongation of titanium alloys elaborated by the L-PBF technique. This diagram exhibits the highest strain-hardenability for the present Ti-22Zr-9Nb-2Sn alloy compared to other alloys. Only the heat-treated Ti-10V-2Fe-3Al (900°C/30min followed by water quench) [START_REF] Mantri | Suppression and reactivation of transformation and twinning induced plasticity in laser powder bed fusion additively manufactured Ti-10V-2Fe-3Al[END_REF] shows similar high strain-hardenability due to a combination of TRIP/TWIP effect. The strain-hardening of this latter is however lower than the one obtained in the present study. This implies that twinning does not affect the strain-hardenability as much as the stressinduced martensite ''.

Conclusion

J o u r n a l P r e -p r o o f

As a conclusion, the Ti-22Zr-9Nb-2Sn (%at) alloy was successfully manufactured from a blend of elemental powders thanks to the in situ L-PBF technique. After a stressrelieved treatment at 700°C for 30 minutes followed by a water quench, microstructure is composed of fully- restored columnar grains, with a strong <100>  texture along the building direction. The mechanical characterization of the alloy was achieved through cyclic tensile tests. The stress-strain curve displays a very high strain-hardening, with an outstanding maximal strain-hardening rate of about 15 GPa, which is greatly higher than those commonly observed in literature for such alloys. Post mortem characterization by XRD and TEM shows that the alloy undergoes a non-reversible stress-induced martensitic transformation during tensile test. As the SIM transformation triggers, the formation of new '' laths is shown to be important, as their growth is limited. This important formation of new interphase boundaries hinders the dislocation slip and increase the strain-hardening rate. This Study Ti64+10Mo(%wt) [START_REF] Vrancken | Microstructure and mechanical properties of a novel β titanium metallic composite by selective laser melting[END_REF] Cp-Ti [START_REF] Sing | Selective laser melting of titanium alloy with 50 wt% tantalum: Microstructure and mechanical properties[END_REF] Ti50Ta(%wt) [START_REF] Sing | Selective laser melting of titanium alloy with 50 wt% tantalum: Microstructure and mechanical properties[END_REF] Ti-35Nb(%wt) [START_REF] Wang | Microstructural homogeneity and mechanical behavior of a selective laser melted Ti-35Nb alloy produced from an elemental powder mixture[END_REF] Ti1313(%wt) [START_REF] Zhou | Anisotropic mechanical behavior of biomedical Ti-13Nb-13Zr alloy manufactured by selective laser melting[END_REF] Ti10V2Fe3Al(%wt) [START_REF] Mantri | Suppression and reactivation of transformation and twinning induced plasticity in laser powder bed fusion additively manufactured Ti-10V-2Fe-3Al[END_REF] Ti64(%wt) [START_REF] Vrancken | Heat treatment of Ti6Al4V produced by Selective Laser Melting: Microstructure and mechanical properties[END_REF] Ti2448(%wt) [START_REF] Zhang | Manufacture by selective laser melting and mechanical behavior of a biomedical Ti-24Nb-4Zr-8Sn alloy[END_REF] Ti25332(%wt) [START_REF] Liu | Transformation-induced plasticity and high strength in beta titanium alloy manufactured by selective laser melting[END_REF] Ti-22Zr-9Nb-2Sn Fig. 4. Strain-hardenability of titanium alloys elaborated by L-PBF process as a function of maximum elongation.

Fig. 1 .Fig. 2 .Fig. 3 .

 123 Fig.1. XRD pattern of the Ti-22Zr-9Nb-2Sn alloy (a), IPF-EBSD map of the specimen in the building direction and its inverse pole figures (b). Pole figures (c) and inverse pole figures (d) of the specimen obtained by XRD measurements. BD and TD stands for Building and Tensile Directions.

1,2 , Philippe Castany 1 , Thierry Gloriant 1*

	1 Univ Rennes, INSA Rennes, CNRS, ISCR UMR 6226, 35000 Rennes, France

2 SLS France, 32 Boulevard de la Haie des Cognets, 35136 Saint-Jacques-de-la-Lande, France

Table and figures : Table 1 .

 and:1 Optimized elaboration parameters determined by DOE.

	Laser Power	Scanning speed	Hatching space	Layer thickness
	200 W	303 mm.s -1	0.06 mm	20 µm

=500 µm; Map4; Step=2 µm; Grid751x567

Acknowledgment

The authors the SCANMAT platforms of the University of Rennes for providing access to XRD, SEM and TEM facilities.