Philippe Marie

Jerome Rousseau
email: jerome.rousseau@univ-brest.fr

Recurrence for random dynamical systems

Keywords: . 2000 Mathematics Subject Classification. Primary: 37H, 37C45, 37B20; Secondary: 37A25 Random dynamical system, Poincaré recurrence, dimension theory, decay of correlations

This paper is a first step in the study of the recurrence behavior in random dynamical systems and randomly perturbed dynamical systems. In particular we define a concept of quenched and annealed return times for systems generated by the composition of random maps. We moreover prove that for super-polynomially mixing systems, the random recurrence rate is equal to the local dimension of the stationary measure.

Introduction

Random dynamical systems (e.g. [START_REF] Kifer | Ergodic theory of random transformations[END_REF][START_REF] Kifer | Random dynamics, Handbook of dynamical systems[END_REF][START_REF] Liu | Dynamics of random transformations: smooth ergodic theory[END_REF][START_REF] Ohno | Asymptotic behaviors of dynamical systems with random parameters[END_REF]) and quantitative description of the recurrence in deterministic dynamics (e.g. [START_REF] Boshernitzan | Quantitative recurrence results[END_REF][START_REF] Barreira | Hausdorff dimension of measures via Poincaré recurrence[END_REF][START_REF] Saussol | Recurrence rate in rapidly mixing dynamical systems[END_REF]) have been deeply studied in the last years. But despite of the success of these both surveys, we did not find in the literature any result about the recurrence of random dynamical systems. The purpose of this paper is to define the main needed objects to start this study and to give first results to describe the recurrence behavior in the random case.

The evolution of a random dynamical system generated by i.i.d. random transformations will be as described in what follows: we consider an indexed family of transformations {T λ } λ∈Λ defined on a compact Riemannian manifold X and a probability measure P on a metric space Λ. Let (λ n ) n≥0 be an i.i.d. stochastic process with common distribution P, a random evolution of an initial state x ∈ X will be generated by a realisation of this process, say λ := (λ 0 , λ 1 , . . . , λ n , . . .) ∈ Λ N , such as for every n ≥ 0:

x λ n := T λ n-1 • . . . T λ 0 x
The set {x λ n } ∞ n=0 ⊂ X will be the random orbit of x associated to the realisation λ ∈ Λ N . We actually do not require for the process (λ n ) n≥0 to be independent. We consider a weaker assumption that we explain below which allow us to get results available for more general random dynamical systems. In fact, we focus on systems for which there exists a stationary measure; in the i.i.d. case a stationary measure is a probability measure µ on X such that :

µ(A) = Λ µ(T -1 λ A)dP(λ)
for any Borelian A of X. The probability measure µ over the phase space X will play the role of the invariant one in the random setting. It is natural to wonder if the classical results of deterministic recurrence can be applied to the random orbits evolution. There are two ways of studying the long-term behavior of random orbits; we can describe them with respect to the stationary measure, firstly, fixing a realisation of the random component and secondly, taking into account the noise globally. The first case is a statistical study only over the space X, while the second one is a statistical study over both the space X and the noise space Ω. The first point of view is usually called quenched and the second one annealed. The annealed case provides a better global understanding of random dynamical systems as ergodic theory provides a great scope of deterministic orbits description. First we will introduce some new definitions that are the counterpart of usual recurrence objects in the random framework. More precisely, for random dynamical systems generated by the composition of random transformations we define a quenched and an annealed version of return times. The annealed one is defined in the i.i.d. case by

T B (x) := Λ N inf{n > 0 : T λ n-1 • . . . • T λ 0 x ∈ B}dP N (λ).
As we said above, this quantity appears to us as the best quantity for a relevant quantitative analysis of the recurrence in random dynamical systems. We will justify the choice of this definition applying classical results of deterministic recurrence (Poincaré's theorem, Kac's lemma) to a usual representation of random maps, namely the skew product transformation, and a good Borelian subset. We will moreover see that the annealed return time appears naturally in this study.

Then, in section 3, using a recent result from Rousseau and Saussol [START_REF] Rousseau | Poincaré recurrence for observations[END_REF] about the recurrence for observations applied to the skew product representation, we establish a link between random return times and local dimension of stationary measures in both cases: quenched and annealed. Moreover, in our main theorem (Theorem 18),we prove that, for systems whose decay of correlations hold super-polynomially fast for Lipschitz observables, the random recurrence rates are equal to the local dimension of the stationary measure. This is a first step in the description of the recurrence for non-deterministic dynamical systems.

In the fourth section we will give two examples of random toral automorphisms for which our results hold. The first one is a non-i.i.d. random dynamical system where the random maps are themselves chosen according to a Markov process. The second one deals with i.i.d. hyperbolic toral automorphisms.

We will then consider the particular case where random dynamical systems are used as a small stochastic perturbation of an initial dynamics. We will apply our results to two examples studied by Baladi and Young [START_REF] Baladi | On the spectra of randomly perturbed expanding maps[END_REF]: expanding maps of the circle and piecewise expanding maps of the interval.

The last part is devoted to the proofs of our main theorems.

Random dynamical systems and return time

Let us first recall some generalities about random dynamical systems. Let Ω be a metric space with B(Ω) its Borelian σ-algebra and (Ω, B(Ω), P) a probability space which represent the space of the randomness. Let ϑ be a P-preserving map which represents the time evolution of the randomness. Ω indexes a family of maps {T ω } ω∈Ω from a compact Riemannian manifold X into itself. A random dynamical system T on X over (Ω, B(Ω), P, ϑ) is generated by mappings T ω so that the map (ω, x) → T ω x is measurable and so that the following cocycle property holds:

T n+m ω = T n ϑ m ω • T m ω
Each realisation ω ∈ Ω of the random component of the system defines a random evolution law, under the action of which, the evolution of an initial state x ∈ X will be given by its random orbit, defined by the set {x ω n }, where for any n ∈ N:

x ω n := T ϑ n-1 ω • . . . • T ω x := T n ω x.
One useful representation of this system is given by the following skew product transformation:

S : Ω × X -→ Ω × X (ω, x) -→ (ϑω, T ω x).
It is straightforward to see that for a fixed (ω, x) ∈ Ω × X the iteration of this map will generate on the second component the random orbit of x associated to the realisation ω since S n (ω, x) = (ϑ n ω, T n ω x). This drives to consider invariant measures of the skew product map to study the behavior of the random dynamical system. Let us suppose that ν is an invariant probability measure for S and define the canonical projection over Ω:

π Ω : Ω × X → Ω (ω, x) → ω
Then, denoting by F * µ the pushforward measure of µ by a measurable map F 1 , S * ν = ν implies ϑ * ((π Ω ) * ν) = (π Ω ) * ν and thus the marginal measure of ν over (Ω, B(Ω)) must be invariant for the dynamics ϑ. Since the random component of the system is considered to be given a priori [START_REF] Arnold | Random Dynamical Systems[END_REF], we will focus on invariant measures satisfying (π Ω ) * ν = P. Definition 1. A probability measure ν is invariant for the random dynamical system T if:

(i) S * ν = ν (ii) (π Ω ) * ν = P.
From now on, we deal with random dynamical systems for which the skew product invariant measure is a product measure ν = P ⊗ µ, this case is in fact quite general since it includes the important case we considered in introduction where the maps are chosen independently with the same distribution P. We recover this situation setting P = P N , ϑ = σ, the left shift defined on Ω = Λ N and T ω = T λ 0 for ω = (λ 0 , λ 1 , ..., λ n , ...) in the skew product definition. Definition 2. We say that µ is a stationary measure for the random dynamical system if the measure ν = µ ⊗ P is invariant for the skew product. This vocabulary comes from the fact that i.i.d. random maps generate Markov processes, namely, a random orbit is a Markov process whose transition probabilities are given by:

P (x, B) = P{ω ∈ Ω : T ω x ∈ B} (1)
A huge literature is devoted to the study of this special kind of Markov processes, see for example [START_REF] Bhattacharya | Asymptotics of a class of Markov processes which are not in general irreductible[END_REF][START_REF] Bhattacharya | Random dynamical systems: theory and applications[END_REF][START_REF] Diaconis | Iterated random functions[END_REF]. This property of random orbits provides a great interest to this study from the probabilistic point of view. We remark that for i.i.d. random dynamical systems under weak assumptions such measures always exist (for example if T ω is continuous for all ω ∈ Ω [START_REF] Kifer | Ergodic theory of random transformations[END_REF]).

Proposition 3. If T = (M, {T λ } λ∈Λ , P, σ) is an i.i.d. random dynamical system, then a probability measure µ is stationary if and only if for all Borelian B it verifies (see [START_REF] Arnold | Random Dynamical Systems[END_REF] for instance):

Λ (T λ ) * µ(B)dP(λ) = µ(B) 1 i.e. for all measurable subset A, F * µ(A) = µ(F -1 A)
That is to say that a stationary measure is a measure which is invariant in average over the random component.

Therefore, the study of the deterministic dynamical system (Ω×X, S, P⊗µ) will provide lots of informations about the behavior of random orbits.

We now introduce a return time concept for random orbits. Since (Ω × X, S, P ⊗µ) is a deterministic dynamical system, we can define the first return time into a Borelian subset

A × B ⊂ Ω × X as usual for (ω, x) ∈ A × B: τ S A×B (ω, x) = inf{k > 0 : S k (ω, x) ∈ A × B}. Choosing A = Ω we remark that S n (ω, x) ∈ Ω × B if and only if T n ω x ∈ B.
This naturally drives to the following definition: Definition 4. For a fixed ω ∈ Ω the first quenched random return time in a measurable subset B ⊂ X of the random orbit starting from a point x ∈ B is:

τ ω B (x) = τ S Ω×B (ω, x) = inf{k > 0 : T ϑ k-1 ω • • • • • T ω x ∈ B}.
Remark that the Poincaré recurrence theorem applied to the skew product ensures that this quantity is almost everywhere finite.

From now on we assume that X is a metric space with metric d. For ω ∈ Ω, we are interested in the behavior as r → 0 of the quenched random return time of a point x ∈ X into the open ball B(x, r) defined by

τ ω r (x) := inf {k > 0 : T ϑ k-1 ω • ... • T ω x ∈ B(x, r)} = inf {k > 0 : d(T ϑ k-1 ω • ... • T ω x, x) < r} .
Definition 5. The random dynamical system T on X over (Ω, B(Ω), P, ϑ) with a stationary measure µ is called random-aperiodic if

P ⊗ µ{(ω, x) ∈ Ω × X : ∃n ∈ N, T ϑ n-1 ω • . . . • T ω x = x} = 0.
Let T 0 = R α an irrational rotation of the circle for an irrational number α and T 1 the identity map of the circle. The i.i.d. random dynamical system constructed with this two maps chosen with the same probability P (0) = P (1) = 1 2 is not random-aperiodic and we have P{ω ∈ Ω : τ ω r (x) = 1} = P (1) = 1 2 for all r > 0. This means that after one iteration half of the points did not rotate and 1 2 n of them after n iterations. Anyway, almost every point will eventually rotate and their dynamics will be quite interesting providing that we wait for an enough long time. To avoid this kind of problem with the first return time for non-random-aperiodic system we need to introduce the non-instantaneous return times (more details are presented on the Section 2.3 of [START_REF] Rousseau | Poincaré recurrence for observations[END_REF]). Definition 6. Let r > 0. For x ∈ X, ω ∈ Ω and p ∈ N we define the p-non-instantaneous quenched random return time: Finally we recall that the lower and upper pointwise or local dimension of a Borel probability measure µ on X at a point x ∈ X are defined by

τ ω r,p (x) := inf {k > p : d(T ϑ k-1 ω • ... • T ω x,
d µ (x) = lim inf r→0 log µ (B (x, r)) log r and d µ (x) = lim sup r→0 log µ (B (x, r)) log r .
These dimensions are linked with the Hausdorff dimension of the measure (see e.g. [START_REF] Falconer | Techniques in fractal geometry[END_REF] for more details on dimensions):

Remark. If X ⊂ R n for some n ∈ N * , for a finite Borel measure µ on X dim H µ = ess-sup d µ .
3. Quenched recurrence rate, annealed recurrence rate and pointwise dimension

We now state our results about the recurrence behavior for the random maps. We first focus on the quenched case (random orbits) in section 3.1 and then on the annealed one (random recurrence) in section 3.2.

3.1.

Recurrence for random orbits. Theorem 7. Let T be a random dynamical system on X over (Ω, B(Ω), P, ϑ) with a stationary measure µ.

For P ⊗ µ-almost every (ω, x) ∈ Ω × X R ω (x) ≤ d µ (x) and R ω (x) ≤ d µ (x).
Proof. This theorem is proved using Theorem 2 of [START_REF] Rousseau | Poincaré recurrence for observations[END_REF] applied to (Ω×X, B(Ω×X), P⊗µ, S) with the observation f defined by

f : Ω × X -→ X (ω, x) -→ x.
With this observation, for all (ω, x) ∈ Ω × X, for all r > 0 and for all p ∈ N we identify the return time for the observation

τ f r,p (ω, x) = τ ω r,p (x), the pushforward measure f * (P ⊗ µ) = µ,
and the pointwise dimensions for the observation

d f P⊗µ (x) = d µ (x) and d f P⊗µ (x) = d µ (x).
This theorem is also satisfied for the first return time:

Corollary 8. Let T be a random dynamical system on X over (Ω, B(Ω), P, ϑ) with a stationary measure µ.

For P ⊗ µ-almost every (ω, x) ∈ Ω × X lim inf r→0 log τ ω r (x) -log r ≤ d µ (x) and lim sup r→0 log τ ω r (x) -log r (x) ≤ d µ (x).
Proof. Since for all (ω, x) ∈ Ω × X, for all r > 0 and for all p ∈ N we have

τ ω r (x) ≤ τ ω r,p (x) then lim inf r→0 log τ ω r (x) -log r ≤ R ω (x) and lim sup r→0 log τ ω r (x) -log r (x) ≤ R ω (x)
and the corollary is proved by Theorem 7.

Even if the inequalities in Theorem 7 can be strict, with more assumptions on the random dynamical system one can prove that the equalities hold. This drives us to introduce the decay of correlations for a random dynamical system: Definition 9. A random dynamical system with a stationary measure µ has a superpolynomial decay of correlations if for all n ∈ N * and ψ , ϕ Lipschitz observables from X to R:

| X Ω ψ(T ϑ n-1 ω • . . . • T ω x)ϕ(x)dP(ω)dµ(x) - X ψdµ X ϕdµ| ≤ ψ ϕ θ n
with lim n→∞ θ n n p = 0 for any p > 0 and where . is the Lipschitz norm.

Theorem 10. Let T be a random dynamical system on X over (Ω, B(Ω), P, ϑ) with a stationary measure µ. If the random dynamical system has a super-polynomial decay of correlations then

R ω (x) = d µ (x) and R ω (x) = d µ (x) for P ⊗ µ-almost every (ω, x) ∈ Ω × X such that d µ (x) > 0.
Proof. As previously, this theorem is proved using Theorem 5 of [START_REF] Rousseau | Poincaré recurrence for observations[END_REF] applied to (Ω × X, B(Ω × X), P ⊗ µ, S) with the observation f defined by

f : Ω × X -→ X (ω, x) -→ x.
We emphasize that in Theorem 5 of [START_REF] Rousseau | Poincaré recurrence for observations[END_REF] the condition of the super-polynomial decay of correlations is applied to the skew product, in fact in our case this is not necessary. A weaker assumption, the super-polynomial decay of correlations for the random dynamical system, allows us to prove Lemma 15 of [START_REF] Rousseau | Poincaré recurrence for observations[END_REF] for the observation f defined above. The proof of this lemma is identical since in our case the decay of correlations is used in a function which does not depend on ω.

The following proposition establishes that the non-instantaneous return time notions are not necessary for random-aperiodic systems.

Proposition 11. If the system is random-aperiodic, for P⊗µ-almost every (ω, x) ∈ Ω×X and for every p ∈ N, τ ω r,p (x) = τ ω r (x) for r small enough.

Proof. Since the system is random-aperiodic, for

P × µ-almost all (ω, x) ∈ Ω × X, for all k > 0, d(T ϑ k-1 ω • . . . • T ω x, x) > 0, then τ ω r (x) → r→0
+∞. Thus, for all p > 0, there exists r(p, x, ω) such that for every r < r(p, ω, x), we have τ ω r (x) > p. Therefore, for every r < r(p, ω, x), we have τ ω r (x) = τ ω r,p (x).

The previous theorem is proved for the instantaneous quenched random lower and upper recurrence rate: Corollary 12. Let T be a random dynamical system on X over (Ω, B(Ω), P, ϑ) with a stationary measure µ. If the random dynamical system is random-aperiodic and has a super-polynomial decay of correlations then

lim inf r→0 log τ ω r (x) -log r = d µ (x) and lim sup r→0 log τ ω r (x) -log r = d µ (x) for P ⊗ µ-almost every (ω, x) ∈ Ω × X such that d µ (x) > 0.
Proof. This is just a consequence of Theorem 10 and Proposition 11.

3.2. Random recurrence. For B a Borelian subset of X, Poincaré's recurrence theorem applied to S ensures that the first return time is finite for

P ⊗ µ-almost all (ω, x) ∈ Ω × B.
The other main result in deterministic recurrence is Kac's lemma. To study what happens in the random case, we first need to recall the concept of ergodicity in the random setting:

Definition 13. We say that the random dynamical system T on X over (Ω, B(Ω), P, ϑ) is ergodic with respect to a stationary measure µ if the deterministic system (Ω × X, S, P ⊗ µ) is ergodic.

Let us now suppose that the random dynamical system is ergodic then we can apply Kac's lemma and we get:

Ω×B τ S Ω×B (ω, x) dP ⊗ µ(ω, x) = Ω×B τ ω B (x) dP ⊗ µ(ω, x) = B Ω τ ω B (x) dP(ω)dµ(x) = 1.
(

) 2
Which is an argument to study the quantity:

Ω τ ω B (x) dP(ω).
Therefore, the latter naturally appears in the study of random recurrence and while the quenched version of return times will allow a description realisation by realisation, this one will provide a global description of the system, including the random part in its totality. This is coherent with the ergodic theory point of view and the classical statistical study of dynamical systems. Let us moreover remark that averaged quantities are also omnipresent in the study of i.i.d. random dynamical systems, see for example the Viana's course [START_REF] Viana | Stochastic dynamics of deterministic systems[END_REF], where the random evolution operator, respectively the transfert one, is the average of the evolution, respectively the transfert, operators associated to each random maps T ω over the random component. We have already remarked in the first section that stationary measures were defined by an average too.

Definition 14. The first annealed random return time for x ∈ B in the set B is:

T B (x) = Ω τ ω B (x) dP(ω).
With this definition and (2), we immediately get a random version of Kac's lemma:

Proposition 15 (Random Kac's lemma). Let T be an ergodic random dynamical system with respect to the stationary probability measure µ, then for any Borelian subset B of X such that µ(B) > 0, we have:

B T B (x)dµ(x) = 1.
As previously, we are interested in the behaviour of the return time of x ∈ X into B(x, r) when r → 0, which drives to the following definitions: The main results are the following two theorems (whose proofs can be found in Section 6): Theorem 17. Let T be a random dynamical system on X over (Ω, B(Ω), P, ϑ) with a stationary measure µ. For µ-almost every x ∈ X R(x) ≤ d µ (x) and R(x) ≤ d µ (x).

Theorem 18. Let T be a random dynamical system on X over (Ω, B(Ω), P, ϑ) with a stationary measure µ. If the random dynamical system has a super-polynomial decay of correlations then

R(x) = d µ (x) and R(x) = d µ (x)
for µ-almost every x such that d µ (x) > 0.

The first inequality is still satisfied using the first return time:

Corollary 19. Let T be a random dynamical system on X over (Ω, B(Ω), P, ϑ) with a stationary measure µ. For µ-almost every x ∈ X lim inf

r→0 log T r (x) log r ≤ d µ (x) and lim sup r→0 log T r (x) log r ≤ d µ (x).
Proof. Since for every x ∈ X and for every p ∈ N, T r (x) ≤ T r,p (x), the result follow from Theorem 17.

As in the previous section, for random-aperiodic dynamical systems we do not need the non-instantaneous return time (see Section 6 for the proof):

Proposition 20. Let T be a random dynamical system on X over (Ω, B(Ω), P, ϑ) with a stationary measure µ. If the random dynamical system is random-aperiodic and has a super-polynomial decay of correlations then

lim inf r→0 log T r (x) log r = d µ (x) and lim sup r→0 log T r (x) log r = d µ (x)
for µ-almost every x such that d µ (x) > 0.

Random toral automorphisms

We now give two examples of random dynamical systems for which our theorems hold. We emphasize that the first example is a non-i.i.d. random dynamical system where there exists a stationary measure.

4.1. Non-i.i.d. random linear maps. Let X = T 1 be the one-dimensional torus. Consider the two linear maps which preserve Lebesgue measure Leb on X

T 1 : X -→ X and T 2 : X -→ X x -→ 2x x -→ 3x.
The random orbit is constructed by choosing one of these two maps following a Markov process with the stochastic matrix

A = 1/2 1/2 1/3 2/3 .
In fact, this random dynamical system is represented by the following skew product

S : Ω × X -→ Ω × X (ω, x) -→ (ϑ(ω), T ω x) with Ω = [0, 1], T ω = T 1 if ω ∈ [0, 2/5), T ω = T 2 if ω ∈ [2/5, 1
] and where ϑ is the following piecewise linear map

ϑ(ω) =        2ω if ω ∈ [0, 1/5) 3ω -1/5 if ω ∈ [1/5, 2/5) 2ω -4/5 if ω ∈ [2/5, 3/5) 3ω/2 -1/2 if ω ∈ [3/5, 1].
Since ϑ preserves Lebesgue measure P = Leb, the skew product S is Leb ⊗ Leb-invariant and so Leb is a stationary measure for this random dynamical system. One can easily see that this system is random-aperiodic. Since S has an exponential decay of correlations [START_REF] Baladi | Positive transfer operators and decay of correlations[END_REF], Corollary 12 and Proposition 20 hold, so for Leb ⊗ Leb-almost every (ω, x)

∈ [0, 1] × T 1 lim r→0 log τ ω r (x) -log r = 1
and for Leb-almost

every x ∈ T 1 lim r→0 log [T r (x)] -log r = 1.
4.2. Random hyperbolic toral automorphisms. Let X = T 2 , we recall that a hyperbolic toral automorphism is a map A : T 2 → T 2 acting through the matrix x → Ax (mod 1), such that the matrix A has integer entries, eigenvalues with modulus different from 1 and det A = ±1. We will restrict our example to the case where the matrix has positive entries, it is possible to consider more general automorphisms under an invariant cone assumption, see [START_REF] Ayyer | Exponential decay of correlations for randomly chosen hyperbolic toral automorphisms[END_REF]. Let Ω = {0, 1} N and ϑ = σ be the left shift on Ω. Let A 0 , A 1 two hyperbolic automorphisms with positive entries. Let A 0 be chosen with a probability q and A 1 with a probability 1q, i.e. P = P N with P (0) = q and P (1) = 1q. Then the Lebesgue measure is stationary and the decay of correlations is exponentially fast for Lipschitz observables (indeed for strong Hölder observables (see [START_REF] Ayyer | Exponential decay of correlations for randomly chosen hyperbolic toral automorphisms[END_REF])) f and g which satisfy

T 2 f (x)dx = T 2 g(x)dx = 0.
Proposition 21. This system is random-aperiodic.

Proof. For any fixed ω ∈ Ω and n ∈ N * , the matrix M corresponding to A n ω = A ωn . . . A ω 2 A ω 1 (i.e, A n ω x = M x mod 1) has positive integers entries. By the Perron-Frobenius Theorem, M has a simple largest eigenvalue λ > 0 and since | det M | = 1, λ > 1. Therefore, A n ω is also a hyperbolic toral automorphism. By [START_REF] Pollicott | Dynamical systems and ergodic theory[END_REF], we have for all n ∈ N * Card{x ∈ T 2 : A n ω x = x} < +∞ and so

Leb({x ∈ T 2 : ∃n ∈ N * , A n ω x = x}) = 0. Then, this system satisfies

P ⊗ Leb({(ω, x) ∈ {0, 1} N × T 2 : ∃n ∈ N * , A n ω x = x}) = 0.
Proposition 21 and Corollary 12 give for P ⊗ Leb-almost every (ω, x) ∈ {0,

1} N × T 2 lim r→0 log τ ω r (x) -log r = 2
and Proposition 20 gives for Leb-almost

every x ∈ T 2 lim r→0 log [T r (x)] -log r = 2.

Small random perturbations

A particular case of random dynamical systems is the one where all the maps are chosen arbitraly close to a fixed initial map, namely the random dynamical system is a small random perturbation of a map T . More precisely we consider a measurable map T : X → X which plays the role of an initial dynamics, and we add a small amount of noise during its evolution, modelled by a random dynamical system. More precisely, for any small ε > 0 (which is the noise level) we consider a probability space (Λ ε , B(Λ ε ), P ε ) where Λ ε is a metric space and B(Λ ε ) the Borelian σ-algebra. Then let us consider a parametrized family of maps {T λ } λ∈Λε which are ε-close to T in some C 0 sense for small ε. The framework is the following Ω = Λ N ε , P = P N ε , ϑ = σ the left shift on Ω and T ω = T λ 1 for all ω = (λ 1 , λ 2 , . . . ) ∈ Ω. Let us consider the following map:

Φ ε : Λ ε × X -→ X (λ, x) -→ T λ (x)
such that for all x ∈ X, Φ ε

x := Φ ε (., x) : Λ ε → X is measurable and there is a λ * ∈ Λ ε which satisfies Φ ε (λ * , x) = T (x) for all x ∈ X. We consider the probability measure (Φ ε x ) * P ε i.e. the probability defined by:

(Φ ε x ) * P ε (A) := P ε {λ ∈ Λ ε : T λ x ∈ A}
Note that it is exactly the family of transition probabilities defining the associated Markov process of the random maps (see 1). We set the following classical assumptions:

(RT1) For all x ∈ X, (Φ ε x ) * P ε is absolutely continuous with respect to the Lebesgue measure for any small ε > 0. (RT2) For all x ∈ X, Φ ε

x (Λ ε ) ⊂ B ε (T x) for any small ε > 0.

Remark. Our assumption (RT2) implies a C 0 closeness between T and every T λ .

Again, we do not need non-instantaneous return time:

Proposition 22. Under the hypothesis (RT1), the system is random-aperiodic.

Proof. We will prove that for all x ∈ X

P N ε ({ω = (λ 1 , λ 2 , ...) ∈ Λ N ε : ∃n ∈ N * , T n ω x = x}) = 0 where T n ω = T λn • • • • • T λ 2 • T λ 1 . Let x ∈ X. Let A ⊂ X with Leb(A) = 0. Since the measure (Φ ε x ) * P ε is absolutely continuous by (RT1) we have (Φ ε x ) * P ε (A) = P ε ({λ 1 ∈ Λ ε : T λ 1 x ∈ A}) = 0. Moreover, we remark that (Φ ε x ) 2 * P 2 ε (A) := P 2 ε ({(λ 1 , λ 2 ) ∈ Λ 2 ε : T λ 2 T λ 1 x ∈ A}) = 0. Indeed (Φ ε x ) 2 * P 2 ε (A) = Λε (Φ ε T λ 1 x ) * P ε (A)dP ε (λ 1 )
but since T λ 1 x ∈ X for all λ 1 ∈ Λ ε , (RT1) gives (Φ ε T λ 1 x ) * P ε (A) = 0 for all λ 1 ∈ Λ ε . Using this idea, with an easy induction argument, we can prove that for every n ∈ N *

(Φ ε x ) n * P n ε (A) := P n ε ({(λ 1 , . . . , λ n ) ∈ Λ n ε : T λn . . . T λ 1 x ∈ A}) = 0. (3)
Finally, we get

P N ε ({ω ∈ Λ N ε : ∃n ∈ N * , T n ω x = x}) = P N ε ( n∈N * {ω ∈ Λ N ε : T n ω x = x}) ≤ n∈N * P N ε ({ω ∈ Λ N ε : T n ω x = x}) ≤ n∈N * P n ε ({(λ 1 , . . . , λ n ) ∈ Λ n ε : T λn . . . T λ 1 x = x}) ≤ n∈N * P n ε ({(λ 1 , . . . , λ n ) ∈ Λ n ε : T λn . . . T λ 1 x ∈ {x}}) =0 by (3)
≤ 0.

5.1. Small random perturbations of expanding maps of the circle. Let X = S 1 and T be an expanding C r (2 ≤ r < ∞) transformation of S 1 . We put an additive noise to this system, namely T λ x = T x + λ where λ is a random variable distributed according to a density supported on (-ε, + ε). Then it is well known (see for example [START_REF] Baladi | On the spectra of randomly perturbed expanding maps[END_REF]) that the random dynamical system admits an absolutely continuous stationary measure µ whose density is C r-1 and is exponentially mixing for C r-1 observables. Even if the decay of correlations is exponential for not Lipschitz observables, it is possible to go from C 5.2. Small random perturbations of piecewise expanding maps of the interval. Let X = [0, 1] and T be a C 2 -piecewise expanding map without periodic turning point (see [START_REF] Baladi | On the spectra of randomly perturbed expanding maps[END_REF]) 2 , then for the same additive perturbation than in the previous example, Baladi and Young [START_REF] Baladi | On the spectra of randomly perturbed expanding maps[END_REF] have proved the existence (and the stability) of an absolutely continuous stationary measure µ and the exponential decay of correlations for observables that are of bounded variations, and therefore for Lipschitz ones. We obtain by Corollary 12 and Proposition 22 that for P ⊗ µ-almost every (ω,

x) ∈ (-ε, ε) N × [0, 1] lim r→0 log τ ω r (x) -log r = 1
and we obtain by Proposition 20 that for µ-almost every

x ∈ [0, 1] lim r→0 log [T r (x)] -log r = 1.
2 Remark that similar results can be obtained for maps whose derivative is uniformly larger than 2, see [START_REF] Kifer | Random dynamics, Handbook of dynamical systems[END_REF].

Since τ ω 4r,p (y, x) is bounded by the p th return time of (y, ω) in the set B(x, 4r) × Ω, Kac's lemma provides the following inequality: B(x,4r)×Ω τ ω 4r,p (y, x) dµ(y)dP(ω) ≤ p.

Combining ( 7) with ( 9), we have:

µ(D r,x ) ≤ pr 2ε µ (B (x, 4r)) (10)
and thus:

µ(A ε (r)) ≤ x∈C µ(D r,x
) by ( 4) and ( 6)

≤ p r 2ε x∈C µ (B(x, 4r
)) by ( 10)

≤ p r ε x∈C µ (B(x, r)) since µ is wdr ≤ p r ε by definition of C. Finally: n,e -n <δ µ(A ε (e -n )) = n>-log δ µ(A ε (e -n )) ≤ p n e -εn < ∞.
We can thus apply the Borel-Cantelli lemma to get for large n and for µ-almost every

x ∈ X δ T 6e -n ,p (x)µ B(x, 2e -n ) ≤ e 2εn i.e. log T 6e -n ,p (x) n ≤ 2ε + log µ(B(x, 2e -n )) -n . (11)
One can easily prove that for all a > 0 we have:

d µ (x) = lim inf n→∞ log µ (B (x, ae -n )) -n and d µ (x) = lim sup n→∞ log µ (B (x, ae -n )) -n R(x) = lim p→∞ lim inf n→∞ log T ae -n ,p (x) n and R(x) = lim p→∞ lim sup n→∞ log T ae -n ,p (x)
n Since ε can be chosen arbitrarily small we get the result considering the inferior (resp. the superior) limit when n → +∞ and the limit when p → +∞ in [START_REF] Keller | Stability of the spectrum for transfer operators[END_REF].

Proof of Theorem 18. Let x ∈ X be such that R ω (x) ≥ d µ (x) > 0 for P-almost every ω ∈ Ω, the existence of such a x is ensured by Theorem 10. Let 0 < ε < 1, by the proof of Theorem 5 in [START_REF] Rousseau | Poincaré recurrence for observations[END_REF], we know that for P-almost every ω ∈ Ω it exists N ∈ N such that for every p > N , lim inf r→0 µ(B(x, r)) 1-ε τ ω r,p (x) = +∞, that is to say that: Finally we have that for all p > N 1 and M > 0, it exists R > 0 such that for all r < R, µ(B(x, r)) 1-ε T r,p (x) ≥ M and then:

P( ω ∈ Ω : ∃N ∈ N, ∀p > N, lim inf
By Theorem 10, the equation ( 14) is satisfied for µ-almost every x such that d µ (x) > 0 and then the theorem is proved using Theorem 17.

As in the previous proof, the principal idea of the proof of Proposition 20 is to use Fatou's lemma:

Since ( 16) is satisfied for µ-almost every x such that d µ (x) > 0 by Corollary 12, we get the result by Corollary 19.

  x) < r} . Then we define the non-instantaneous quenched random lower and upper recurrence rates: R ω (x) := lim p→∞ lim inf r→0 log τ ω r,p (x) log r and R ω (x) := lim p→∞ lim sup r→0 log τ ω r,p (x) log r .

Definition 16 .

 16 Let r > 0 and p ∈ N. We define the p-non-instantaneous annealed random return time T r,p (x) := Ω τ ω r,p (x)dP(ω). When p = 0, we denote this return time T r (x). Then we define the non-instantaneous annealed random lower and upper recurrence rates R(x) := lim p→∞ lim inf r→0 log T r,p (x) log r R(x) := lim p→∞ lim sup r→0 log T r,p (x) log r .

r- 1

 1 observables to Lipschitz observables with some simple approximation arguments. Then by Corollary 12 and Proposition 22 for P ⊗ µ-almost every (ω, x) ∈ (-ε, ε) N × S 1 we have lim r→0 log τ ω r (x) log r = 1 and by Proposition 20 we get for µ-almost every x ∈ S 1 lim r→0 log [T r (x)] log r = 1.

  r→0 µ(B(x, r)) 1-ε τ ω r,p (x) = +∞ ) = P(Ω) = 1. (12)Let us denote byΩ(N ) := ω ∈ Ω : ∀p > N, lim inf r→0 µ(B(x, r)) 1-ε τ ω r,p (x) = +∞ . It exists N 1 ∈ N such that P( Ω(N 1 )) > 0, otherwise P(∪ N ∈N Ω(N )) ≤ N ∈N P( Ω(N )) = 0and it would contradict the fact that:1 = P( N ∈N Ω(N )) = P( ω ∈ Ω : ∃N ∈ N, ∀p > N, lim inf r→0 µ(B(x, r)) 1-ε τ ω r,p (x) = +∞ ).Let p > N 1 , we therefore have:Ω lim inf r→0 µ(B(x, r)) 1-ε τ ω r,p (x)dP(ω) = +∞. x, r)) 1-ε τ ω r,p (x)dP(ω) ≤ Ω lim inf r→0 µ(B(x, r)) 1-ε τ ω r,p (x)dP(ω) ≤ lim inf r→0 Ω µ(B(x, r)) 1-ε τ ω r,p (x)dP(ω) = lim inf r→0 µ(B(x, r)) 1-ε T r,p(x).We thus get that: lim inf r→0 µ(B(x, r)) 1-ε T r,p (x) = lim sup r→0 µ(B(x, r)) 1-ε T r,p (x) = +∞.

( 1

 1 ε) log µ(B(x, r)) log r ≤ M log r -log T r,p (x) log r which drives to: R(x) ≥ (1ε)d µ (x) and R(x) ≥ (1ε)d µ (x).Since these inequalities hold for every 0 < ε < 1, we get: R(x) ≥ d µ (x) and R(x) ≥ d µ (x).

  Proof of Proposition 20. Let x ∈ X be such that lim inf r→0 log τ ω r (x) log r ≥ d µ (x) > 0 for P-almost every ω ∈ Ω; the existence of such a x is ensured by Corollary 12. Let 0 < ε < 1, for P-almost every ω ∈ Ω we have lim inf r→0 µ(B(x, r)) 1-ε τ ω r (x) = +∞ and thenΩ lim inf r→0 µ(B(x, r)) 1-ε τ ω r (x)dP(ω) = +∞. x, r)) 1-ε τ ω r (x)dP(ω) ≤ lim inf r→0 Ω µ(B(x, r)) 1-ε τ ω r (x)dP(ω) = lim inf r→0 µ(B(x, r)) 1-ε T r (x).So we obtainlim inf r→0 µ(B(x, r)) 1-ε T r (x) = lim sup r→0 µ(B(x, r)) 1-ε T r (x) = +∞.Then, it exists R > 0 such that for all r < R we have µ(B(x, r)) 1-ε T r (x) ≥ M and so:(1ε) log µ(B(x, r)) log r ≤ M log r -log T r (x) log r which gives us: lim inf r→0 log T r (x) log r ≥ (1ε)d µ (x) and lim sup r→0 log T r (x) log r ≥ (1ε)d µ (x).These inequalities are satisfied for ε arbitrarily small, so we get: lim inf r→0 log T r (x) log r ≥ d µ (x) and lim sup r→0 log T r (x) log r ≥ d µ (x).

Acknowledgement. The authors would like to thank B. Saussol for his useful help and comments.

Proofs

We will use the notion of weakly diametrically regular measures: Definition 23. A measure µ is weakly diametrically regular (wdr) on the set Z ⊂ X if for any η > 1, for µ-almost every x ∈ Z and every ε > 0, there exists δ > 0 such that if r < δ then µ (B (x, ηr)) ≤ µ (B (x, r)) r -ε .

Proof of Theorem 17. It is well known (see for instance [START_REF] Barreira | Hausdorff dimension of measures via Poincaré recurrence[END_REF]) that any probability measure is weakly diametrically regular on R d for all d ∈ N * and then so is the measure µ on X. Let us emphasize that in the previous definition the function δ(•, ε, η) can be made measurable for every fixed ε and η. Let us fix ε > 0 and η = 4. We choose δ > 0 small enough to get:

where X δ := {x ∈ X : δ(x, ε, η) > δ} . We now set the following lemma in order to use the Borel-Cantelli's one:

Proof.

Let p ∈ N, r > 0 and C ⊂ X δ be a maximal 2r-separated set for X δ .

For y ∈ B(x, 4r), we define:

4r,p (y, x)dP(ω).

If d(x, y) < 2r, for all ω ∈ Ω we have τ ω 4r,p (y, x) ≥ τ ω 6r,p (y) and then: T 4r,p (y, x)µ (B (x, 4r)) ≥ T 6r,p (y)µ (B (y, 2r)) .

(

It follows that for any x ∈ C:

where: D r,x := y ∈ B (x, 4r) : T 4r,p (y, x)µ (B (x, 4r)) ≥ r -2ε .

We then use the Markov's inequality and we get:

µ(D r,x ) ≤ r 2ε µ (B (x, 4r))

B(x,4r) T 4r,p (y, x) dµ(y)

= r 2ε µ (B (x, 4r))

B(x,4r)×Ω τ ω 4r,p (y, x) dµ(y)dP(ω).


Nahuel Palumbo
email: nahuel.palumbo@inria.fr

Guillermo Polito
email: guillermo.polito@inria.fr

Pablo Tesone
email: pablo.tesone@inria.fr

Stéphane Ducasse
email: stephane.ducasse@inria.fr

Ordering Optimisations in Meta-Compilation of Primitive Methods

Meta-compilation schemes help to automatically build Just-in-Time (JIT) compilers from interpreters by performing a meta-interpretation of the VM interpreter. Generated JIT compilers face the well-known problem of phase ordering: selecting a good optimisation sequence to apply to the compiled programs. Manual optimisation lists are hard to maintain and are one-size-fits-all solutions that assume that a single sequence is equally effective in all possible programs. Generating such a list automatically is still challenging nowadays.

In this paper, we explore the phase-ordering problem in the case of the meta-compilation of Pharo VM interpreter primitives. In addition to a manual strategy, we present three automatic strategies to find good-enough optimisation sequences: a search-based approach, a predictive approach based on code shape, and an automatically-found fixed list approach. We compare them altogether by measuring the relative compiled code size and their rate of convergence. We evaluate this work over 17 of Pharo's language interpreter primitives. On average, the predictive strategy gives its optimal result before the rest with 21% fewer optimisations, the search strategy finds better results in complex cases. This article shows that automatic approaches seem promising for primitive meta-interpretation.

Introduction

Meta-compilation schemes help to automatically build Just-in-Time (JIT) compilers from interpreters (Section 2) [START_REF] Rigo | PyPy's approach to virtual machine construction[END_REF][START_REF] Vergu | Specializing a meta-interpreter: Jit compilation of dynsem specifications on the graal vm[END_REF]. A meta-compiler performs a meta-interpretation of VM interpreter code to generate JIT compiler code. Such systems use meta-interpretation because they are implemented as an abstract interpreter interpreting a VM interpreter. The meta in meta-interpreter comes to the fact that they interpret an interpreter. The automatic generation of JIT compilers from an interpreter eases programming language implementation extensions: There is no need to be a compiler expert to be able to extend the JIT compiler. Moreover, we can build VMs for different languages by just providing different interpreters.

Generated JIT compilers face the well-known problem of phase ordering [START_REF] Almagor | Compilation order matters: Exploring the structure of the space of compilation sequences using randomized search algorithms[END_REF][START_REF] Amir Hossein Ashouri | Predictive modeling methodology for compiler phase-ordering[END_REF]: selecting a good optimisation sequence to apply to the compiled programs (Section 3). On the other hand, manual optimisation lists are hard to maintain and are one-size-fits-all solutions that assume that a single sequence is equally effective in all possible programs. Generating such a list automatically is still challenging nowadays. The interrelation between optimisations is hard to predict [START_REF] Almagor | Compilation order matters: Exploring the structure of the space of compilation sequences using randomized search algorithms[END_REF] and generates a large search space, raising the challenge of guiding the search and its stop condition [START_REF] Almagor | Finding effective compilation sequences[END_REF]. Figure 1 illustrates the problem by showing the number of instructions in the meta-compiled primitiveAdd of Pharo's VM as optimisations are applied. The figure shows three sets of optimisations:

1. optimisations that reduce the number of instructions, 2. optimisations that in addition to (1), keep the same number of instructions, and 3. optimisations that in addition to (1) and ( 2), increase the number of instructions.

This example shows that different sequences of optimisations produce different results and that if the search stops at a local minimum it can prevent the obtention of better results. Existing work identifies the search space as discrete with many local minima but rare good solutions. Many use machine learning to identify optimisation sequences e.g., genetic algorithms, predictive algorithms, biased random searches, or neural networks [START_REF] Almagor | Finding effective compilation sequences[END_REF]2003;[START_REF] Kulkarni | Mitigating the compiler optimization phase-ordering problem using machine learning[END_REF][START_REF] Amir Hossein Ashouri | Predictive modeling methodology for compiler phase-ordering[END_REF]. In this paper, we explore the phase-ordering problem in the case of the meta-compilation of Pharo VM interpreter primitives (Section 4). Until now, the meta-compilation scheme used a hand-written list that took months of iterated work by compiler developers. This list is hard to maintain and sometimes applies optimisations without effect in the compiled program. We use three alternative strategies to find goodenough optimisation sequences: a search-based approach, a predictive approach based on code shape, and an automatically-found fixed list approach. We compare them with the manual one by measuring the relative compiled code size and their rate of convergence.

Optimisations step Number of instructions

We evaluate this work over 17 of Pharo's language interpreter primitives (Section 5). We show that all strategies arrive at the same optimal result for most primitives. On average, the predictive strategy gives its optimal result before the rest with 21% fewer optimisations. The search strategy finds better results in complex cases. None of them found an optimisation sequence better than the hand-written version in complex cases.

Context: Meta-compilation of primitives

The present work is implemented for Pharo [START_REF] Ducasse | Pharo by Example 5[END_REF], a dynamically-typed object-oriented programming language. In the Pharo Virtual Machine (VM), primitive definitions are methods written in Slang, a subset of Pharo itself, inside an Interpreter class [START_REF] Miranda | Two decades of smalltalk vm development: live vm development through simulation tools[END_REF].

Our meta-compiler, called Druid, performs an ahead-of-time meta-compilation of VM interpreter primitives to generate a JIT compiler template for each of them. The meta-compilation works as a translation process that takes as input the language interpreter and produces JIT compiler code. Our meta-compilation approach uses meta-interpretation: it is implemented as an abstract interpreter interpreting the VM interpreter.

Internally, our meta-compiler uses an SSA-form Intermediate Representation (IR), a register-based control flow graph, where optimisations are applied. Optimisations perform mutations on the IR. We show a diagram of the architecture in Figure 2. In the end, the resulting IR is transformed into a JIT compiler template. This is a method inside a JIT compiler class. The template uses abstract registers and operations and, at run time the JIT compiler maps them to a specific target architecture and generates the expected machine code [START_REF] Miranda | The cog smalltalk virtual machine[END_REF].

Figure 3 illustrates the interpreter and JIT compiler code for the same primitiveAdd. This primitive performs the addition of integer objects with an overflow check:

• At the interpreter level, it will pop the last two values on the stack (argument and receiver), check that they are integer objects, calculate the addition of them, check it does not overflow, and finally push the result to the stack.

• The JIT compiler is activated when the system detects a hot-spot. It uses the template to generate machine code. The generated code performs the integer object checks on registers already loaded with the first argument and receiver values, calculate the addition checking overflow, and move the result to the expected register. In both cases, if any check fails, the primitive fails and the virtual machine falls back to execute a normal message send routine.

Motivation for optimal optimisation sequences

To have an optimised JIT compiler template an optimising meta-compiler is needed. Finding an optimal optimisation sequence for a program is not trivial because the interrelation between them is hard to predict [START_REF] Almagor | Compilation order matters: Exploring the structure of the space of compilation sequences using randomized search algorithms[END_REF]. Some optimisations open new opportunities to others, so different orders usually arrive at different results.

Figure 4 shows an example of the compiler phase-ordering problem with three optimisations. One removes unused code (R), the second performs constant propagation without removing dead code (P) and the third duplicates basic blocks increasing code size but uncovering optimization opportunities (D). The final number of instructions in the IR of a primitive depends on the order that they are applied.

This dependency also implies that one optimisation could be applied multiple times in different moments.

Solving this problem implies generating a large discrete search space with many local minima but few optimal solutions [START_REF] Almagor | Finding effective compilation sequences[END_REF]. Guiding the search in this space and deciding when a solution is good enough is a hard and not intuitive task.

Our research questions are as follow:

• Is there one heuristic that is good enough to optimise our set of primitives?

• How many optimisations are necessary, at least, to arrive at the optimal version of each primitive?

Comparing the four approaches

To answer our questions, we developed three different strategies to select the optimisation sequence for a primitive meta-compilation, that we compared to our pre-existent optimisation list hand-written by compiler developers (manual strategy). We analysed them by comparing the resulting IR version and optimisation sequences. The metric to compare IRs and to guide the search is the number of instructions of the Control Flow Graph. Thus, an IR is better than another if it can perform the same primitive computation with fewer instructions.

Manual strategy. This is a list of optimisations hand-written by experts based on their experience and knowledge about how optimisations work. This list took months of iterated work by compiler developers. Thus it is hard to maintain. This list is fixed, pre-calculated and the same for all primitives in a one-sizefits-all fashion. As this strategy is not guided nor profiled, there is no way to know when it arrives at its optimal result. This means that the optimisation list is always applied until the end, without effect in most cases. We will use this strategy as a baseline to compare evaluation results.

Search strategy. This is a heuristic-based search using a hill climber algorithm that selects an optimisation that reduces the number of instructions in the IR. It is an automatic approach, no need for optimisations experts to find a good optimisations sequence. This strategy is configured to find the first, last or best optimisation in a list. It computes all possible optimisations to the current IR and compares the resulting IR. Building search heuristics is hard, especially with complex IRs. It increases compilation time since it has to try many possible options in each stage. The heuristic is based on a hill-climber algorithm with three levels of search presented in Table 1. In our approach, if there exist optimisations that do not improve the IR removing instructions but open new opportunities to others, they are selected. In the end, if the algorithm does not find any optimisation at any level, then the search is finished and a Null optimisation is selected. A Null optimisation does not perform any change on the IR, it is not necessary to continue the search, thus it has arrived at its optimal result.

Search-based heuristic

At level 1, it searches for optimisations that improve the IR for our metric. Only optimisations that reduce the number of instructions are selected here. If it does not find any optimisation, because no one can reduce instructions in the current state, the next configured levels are used.

At level 2, it searches for an optimisation that improves the IR for our metric after applying the optimisation and removing dead code. This allows the algorithm to select optimisations that do not remove instructions but left unused instructions, such as constant propagation and folding. When it does not find any optimisation, the next configured levels are used.

Level 3 takes into account optimisations that produce code duplication. As these optimisations increase the number of instructions, we consider the IR after copy propagation and folding and dead code elimination. This allows the selection of optimisations that duplicate code but open other opportunities.

Figure 1 shows the effect of each different level. Optimisations selected by level 1 always reduce the number of instructions. Optimisations selected by level 2 keep the same number of instructions but they decrease later. Optimisations selected by level 3 increase the number of instructions, where the graph move from 20 to 44 instructions (more than 100%), but at the end, the IR finishes with fewer instructions, in our example with 10 instructions (50% less). The stable value at the end represents the number of instructions of the program after applying all optimisations.

Predictive strategy. This strategy computes a list of possible optimisations based on IR form. It is an automatic approach that does not try every optimisation: the shape of the IR selects the corresponding optimisation to be applied. Table 2 shows the conditions for each optimisation that the IR should satisfy.

At first, this strategy evaluates the incoming IR and creates a list of possible optimisations. It applies all of them in any order. Once finished, it recomputes the list of possible optimisations using the current IR and repeats. This strategy ends when the list of possible optimisations to apply does not improve the current IR, or when the total number of optimisations arrives at a configurable limit. Automatically-found fixed list strategy. This strategy uses a fixed list of optimisations automatically generated by the previous search strategy. We selected the largest generated optimisation list and evaluate it with all primitives. As it is based on a fixed list, this strategy applies all optimisations until the end, similarly to the manual strategy. This strategy does not need optimisation experts to find a good optimisation sequence, making it a cheap strategy that is calculated once and reused for many cases.

Predictive strategy conditions

Evaluation

Currently, our meta-compiler supports the meta-compilation of the 17 primitives listed in Table 3. Table 4 shows all implemented optimisations with a small description and opportunities that it opens. The search-based strategy we use is always the level 3 heuristic in three different versions: the first, the last and the best result that improves the metric. We applied all strategies to all supported primitives tracing the number of instructions after each optimisation, similar to Figure 1.

Primitives

Optimal IR

For each strategy, we show the number of instructions after applying all optimisations. This number refers to the minimum number of instructions found by the strategy for each primitive. We see that all strategies arrive at the same number of instructions for all simple primitives. For complex primitives, small integer division and quotient and array access, all strategies arrive at an IR with more instructions than the manual approach. Figure 5 shows the number of instructions relative to the manual strategy for each primitive.

Out of the three variants of the search strategy, the Best configuration achieves better results, having on average 14% more instructions than the manual strategy. We will consider only this configuration in the rest of this paper.

Predictive and automatically-found fixed list strategies finish with 22% and 26% more instructions than manual strategy on average, respectively. It demonstrates that our heuristics are good enough for most supported primitives, they arrive at the same result, but they have problems taking decisions over complex scenarios.

Optimisation list

We measure how many optimisations were performed by each strategy to arrive at their optimal number of instructions in each trace, as illustrated in Figure 6. On average, Best search and automatically-found fixed list strategies arrive at each optimal IR by applying the same number of optimisations as the manual strategy. The predictive strategy arrives at its optimal IR with 21% fewer optimisations than the manual strategy. We identify that the manual strategy converges faster in the case of simple integer operations, this

Phi Simplication

Replace one operand phis with copy instruction

Better propagations

Redundant Copy Elimination

Remove copies of form x := x -

SCCP

Performs constants folding and propagation

Better dependency analysis and possible unused code Table 4: Optimisations implemented in our meta-compiler. For each optimisation, we present a description and possible opportunities that it opens to other optimisations.

is probably because the optimisation list was created based on these primitives. In the cases of complex primitives, where not all strategies arrive at the same IR, we have different results.

It is important to note that this analysis is good to compare the rate of convergence of each strategy, but it does not answer which strategy will finish before. Remember that manual and automatically-found fixed list strategies must apply all optimisations until the end, while search and predictive strategies have to test many possible options in each state.

Results

We found that all strategies arrive at the same optimal IR in most primitives compilation. For those simple cases, Predictive and Best Search strategies achieve the optimal IR with 21% fewer optimisations than the manual strategy, on average. In complex cases, the best search strategy arrives at IRs with 14% more instructions than the manual strategy, on average. It is the closest strategy to the manual approach, taking a similar number of optimisations to arrive at the optimal IR.

The best search and predictive strategies calculate the next optimisation(s) based on the current IR state. These strategies add a searching time to the optimisation time. This is a trade-off between automatic optimisation selection and fixed optimisation list.

The automatically-found fixed list strategy is the cheapest option measured by avoiding search time without manual selection by the developers. For complex primitives, it arrives with 26% more instructions than the manual strategy, on average, but it keeps the same rate of convergence. The manual strategy arrived at better optimal results, but it is also the hardest to maintain.

Related work

Cooper et al. describe the problem of building a well-ordered list of optimisations in optimising compilers [START_REF] Cooper | Adaptive optimizing compilers for the 21st century[END_REF]. They identify the search space as discrete with many local minima but rare good solutions. They suggest solutions based on genetic algorithms, predictive algorithms, or biased random searches to improve a simple hill climber [START_REF] Almagor | Finding effective compilation sequences[END_REF]. Kulkarni et al. explain how to improve the search time of optimisation sequences using genetic algorithms [START_REF] Kulkarni | Fast and efficient searches for effective optimization-phase sequences[END_REF].

Kulkarni and Cavazos describe some issues to solve the phase-ordering problem using genetic algorithms and propose a neuro-evolution technique to construct heuristics based on neural networks [START_REF] Kulkarni | Mitigating the compiler optimization phase-ordering problem using machine learning[END_REF] [START_REF] Guo | Evaluating the role of optimization-specific search heuristics in effective autotuning[END_REF]. Their work is close to the one presented here.

Other research explores feedback-driven searches. Some use strategies to explore the optimisation space based on iterative compilation and many optimised versions of the same code [START_REF] Triantafyllis | Compiler optimizationspace exploration[END_REF].

Others explore a profile-based approach based on an execution profiler [START_REF] Pohua | Using profile information to assist classic code optimizations[END_REF].

Conclusion

In this paper, we compared four different strategies to select the list of optimisations to apply in a metacompiler of primitive methods: manual, best search, predictive and automatically-found fixed list. We measured the number of instructions for each optimised IR and the number of optimisations necessary to arrive at it by each strategy.

We found that all strategies arrive at the same IR in most primitive compilations. On average, the automatic predictive strategy achieves its optimal IR with 21% fewer optimisations than the manual strategy built by compiler developers. In complex cases, on average, the best search strategy arrives at IRs with 14% more instructions than the manual strategy.

We have shown that good-enough compiled code for Pharo's primitives can be achieved by automatically selecting optimisations. Searching time is the trade-off between automatic searches and a fixed optimisation list maintained by compiler developers. But long-time analysis is not a big problem for this work, as it is an ahead-of-time task. We can search for a good solution without time constraints. We want to work in better heuristics for new search-based approaches.

As we are looking to increase the number of supported primitives and implemented optimisations, we are interested in an automatic approach to applying the optimisations. The next primitives will be more complex than the current ones, and we will need to implement new optimisations for them. With an automatic approach, VM developers will be free of maintaining the current manual list of optimisations for the metacompiler, which is less trivial to understand in each development iteration.

Correlations between the number of instructions and the number of optimisations expose the presence of primitives with similar behaviour, thus similar IR. It suggests that a clustering-based approach (Martins et al., 2014) will allow reusing one optimisation sequence for many primitives. Maybe a mix of our predictive and best search strategies can be an option also.

Figure 1 :

 1 Figure1: Trace of primitiveAdd with a search-based approach using the three heuristic levels. It shows the change in the number of instructions in the IR as selected optimisations are applied for each level of heuristic.

Figure 3 :

 3 Figure 3: Interpreter vs. JIT compiler primitive for integer addition. The interpreter definition works with values on the stack. The JIT compiler uses registers and machine code operations.

Figure 4 :

 4 Figure 4: Optimisation order. Applying the same optimisations in different orders produces different results.

Figure 5 :

 5 Figure 5: Number of instructions by strategy after applying all selected optimisations compare to manual strategy. Lower is better.

Figure 6 :

 6 Figure 6: Number of optimisations to arrive at its optimal version. Lower is better.

Table 1 :

 1 Different search-based heuristics. Each level describes the optimisation target and the postoptimisations to be selected by the hill climber algorithm.

	Level	Optimisation target	Post optimisations
	1	Reduce the number of instructions	-
	2	Propagation and constants folding	Dead Code Elimination
	3	Code duplication	Copy Propagation, SCCP, Dead Code Elimi-
			nation and Clean Control Flow

Table 2 :

 2 Predictive strategy IR conditions for each optimisation.

	Optimisation	IR Condition
	Branch Collapse	If there is a conditional jump without an inlined condition
	Clean Control Flow	If there is a simple jump to a block with a unique predecessor
	Copy Propagation	If there is a copy instruction
	Dead Block Elimination	If there is a block without predecessors
	Dead Branch Elimination	If there is a dead branch
	Dead Code Elimination	If an instruction (with a result computation) has no users
	Dead Edge Splitting	If there is a dead path
	Failure Code Tail Duplication If the exit primitive block has more than one predecessor
	Phi Simplication	If there is a phi
	Redundant Copy Elimination	If there is a copy between same physical register
	SCCP	If an instruction lattice results in a constant value (a constant
		folding success)

Table 3 :

 3 Pharo VM interpreter primitives supported by our meta-compiler.

	Name	Description
	primitiveAdd	Small integers addition with overflow check
	primitiveSubtract	Small integers subtraction with overflow check
	primitiveMultiply	Small integers multiplication with overflow check
	primitiveLessThan	Small integers comparison
	primitiveGreaterThan	Small integers comparison
	primitiveLessOrEqual	Small integers comparison
	primitiveGreaterOrEqual	Small integers comparison
	primitiveEqual	Small integers comparison
	primitiveNotEqual	Small integers comparison
	primitiveDivide	Machine integers division
	primitiveQuo	Machine integers quotient
	primitiveBitXor	Bits xor
	primitiveBitShift	Bits shift
	primitiveFail	Failing primitive
	primitiveMod	Small integers quotient with overflow check
	primitiveDiv	Small integers division with overflow check
	primitiveAt	Array access with bound check

  Tail duplicate block with resulted code Divide fail and success paths

		Optimisations
	Name	Description	Open opportunities
	Branch Collapse	Inline conditions in conditional jumps	Dead path analysis
	Clean Control Flow	Merge instructions in consecutive	Better local analysis
		blocks to avoid unnecessary jumps
	Copy Propagation	Replace copy instructions in operands	Better dependency analysis and
		by real value	possible unused code
	Dead Block Elimina-	Remove inaccessible blocks	-
	tion
	Dead Branch Elimina-	Remove branches with only dead paths Better propagations
	tion
	Dead Code Elimina-	Remove unused instructions	-
	tion
	Dead Edge Splitting	Duplicate blocks with dead and not-	Create branches with only dead
		dead paths	paths
	Failure Code Tail Du-
	plication

  . The survey[START_REF] Amir | A survey on compiler autotuning using machine learning[END_REF] is a good description of state-of-the-art techniques to solve these issues using Machine Learning. Most of the work in this area is based on machine learning techniques.

	Ashouri et al. propose a predictive trained model for speedup predictions based on a greedy Depth First
	Search heuristic (Ashouri et al., 2016). It can select the next-best optimisation improving the default LLVM-
	generated code by 2%.
	Guo et al. developed an optimisation-specific search heuristic, based on specific knowledge about optimisa-
	tions, and compares it with other generic searches

Acknowledgments

This work was funded by Inria's Action Exploratoire AlaMVic.


David Chapron
email: david.chapron@universite-paris-saclay.fr

Jean-Philippe Michel
email: jean-philippe.michel@universite-paris-saclay.fr

Philippe Fontaine
email: philippe.fontaine@synchrotron-soleil.fr

Jérémy Godard
email: jeremy.godard@unilim.frfrederique.bregier@unilim.fr

Frédérique Brégier

Vincent Sol
email: vincent.sol@unilim.fr

Véronique Rosilio
email: veronique.rosilio@universite-paris-saclay.fr

Thermodynamic and structural properties of lipid-photosensitizer conjugates mixed with phospholipids: impact on the

Keywords: Phenalenone, phospholipid, Langmuir monolayer, vesicle, grazing incidence X-ray diffraction, DSC. DC: Methodology, Investigation, Validation, JPM: Investigation, Formal analysis, PF: Methodology, Validation, Conceptualization, Methodology, Validation, Supervision, Writing, Review

Introduction

Phenalenone (PN) is a natural photosensitizer (PS) well known for its ROS-mediated protective phytochemical activities [START_REF] Feng | Phytoalexin phenalenone derivatives inactivate mosquito larvae and root-knot nematode as type-II photosensitizer[END_REF] but also for its anticancer and antimicrobial photodynamic properties. [START_REF] Späth | Improving photodynamic inactivation of bacteria in dentistry: highly effective and fast killing of oral key pathogens with novel tooth-colored type-II photosensitizers[END_REF][START_REF] Salmeron | Phenalenone-photodynamic therapy induces apoptosis on human tumor cells mediated by caspase-8 and P38-MAPK activation[END_REF][START_REF] Kaye | A green-absorbing, red-fluorescent phenalenone-based photosensitizer as a theranostic agent for photodynamic therapy[END_REF] To favor an application in antimicrobial photodynamic therapy (aPDT), a PS should be water soluble, be stable in solution, and have a high singlet oxygen quantum yield. PN is considered a universal reference molecule for singlet oxygen production, with a quantum yield close to unity in a large variety of solvents including water. [START_REF] Schmidt | Phenalenone, a universal reference compound for the determination of quantum yields of singlet oxygen O2( 1 ∆g) sensitization[END_REF][START_REF] Segado | Mechanism of the photochemical process of singlet oxygen production by phenalenone[END_REF] Depending on the medium in which it is solubilized, PN may produce either ROS (reaction I) or singlet oxygen (reaction II), the latter reaction being favored in a hydrophobic environment. [START_REF] Lorente | A non-singlet oxygen mediated reaction photoinduced by phenalenone, a universal reference for singlet oxygen sensitization[END_REF][START_REF] Espinoza | Theoretical study on the photosensitizer mechanism of phenalenone in aqueous and lipid media[END_REF] Whether it is solubilized in membranes or in the cytosol of targeted cells is thus of paramount importance for its therapeutic effectiveness. With a log P close to 3, PN is lipophilic [START_REF] Freijo | Design, synthesis and evaluation of amino-substituted 1H-phenalen-1-ones as antileishmanial agents[END_REF] and spontaneously aggregates in aqueous media. [START_REF] Godard | Synthesis and supramolecular arrangement of new stearoyl acid-based phenalenone derivatives[END_REF] Its poor solubility makes it difficult to apply in therapeutics and can lower its singlet oxygen quantum yield by selfquenching effect at the excited state. [START_REF] Tsai | Improved photodynamic inactivation of gram-positive bacteria using hematoporphyrin encapsulated in liposomes and micelles[END_REF][START_REF] Pereira | Mechanism of hypericin incorporation to explain the photooxidation outcomes of phospholipid biomembrane models[END_REF] PS efficiency is usually improved when incorporated into lipidic supramolecular assemblies such as micelles, oil-in-water emulsions, or lipid vesicles. [START_REF] Tsai | Improved photodynamic inactivation of gram-positive bacteria using hematoporphyrin encapsulated in liposomes and micelles[END_REF][START_REF] Boyle | Structure and biodistribution relationships of photodynamic sensitizers[END_REF][START_REF] Chen | Liposome-encapsulated photosensitizers against bacteria[END_REF] Micelles and lipidic nanostructures allow to increase the concentration of the PS in aqueous media, without unwanted aggregation. The small size of these systems, especially micelles, seems advantageous for their efficacy. [START_REF] Tsai | Improved photodynamic inactivation of gram-positive bacteria using hematoporphyrin encapsulated in liposomes and micelles[END_REF] However, phenalenone is difficult to insert in micelles and vesicle bilayers.

Previous works have shown that the conjugation of lipid chains with a hydrophilic or hydrophobic drug may lead to the formation of supramolecular assemblies, highly concentrated in the compound of interest. [START_REF] Lepeltier | Self-assembly of squalene-based nucleolipids: relating the chemical structure of the bioconjugates to the architecture of the nanoparticles[END_REF][START_REF] Lepeltier | Self-assembly of polyisoprenoyl gemcitabine conjugates: influence of supramolecular organization on their biological activity[END_REF] Nano-assemblies of lipid-drug conjugates can contain up to 50 mol% of a drug. They are often nanostructured, which may confer them greater stability and, in the case of lipid-PS conjugates, enhances their photosensitizing properties. [START_REF] Komatsu | Molecular energy and electron transfer assemblies made of self-organized lipid-porphyrin bilayer vesicles[END_REF][START_REF] Lovell | Porphysome nanovesicles generated by porphyrin bilayers for use as multimodal biophotonic contrast agents[END_REF] The formation and stability of nano-assemblies made of lipid-PS conjugates are dependent on the geometry and thermodynamic properties of the lipid-PS molecules. [START_REF] Massiot | Newly synthesized lipid-porphyrin conjugates: Evaluation of their selfassembling properties, their miscibility with phospholipids and their photodynamic activity in vitro[END_REF][START_REF] Bronstein | Influence of the porphyrin structure and linker length on the interfacial behavior of phospholipid-porphyrin conjugates[END_REF] When these compounds cannot form pure nano-assemblies, they can be mixed at high molar ratios with other lipids like phospholipids or cholesterol to form vesicles. [START_REF] Massiot | Newly synthesized lipid-porphyrin conjugates: Evaluation of their selfassembling properties, their miscibility with phospholipids and their photodynamic activity in vitro[END_REF][START_REF] Bronstein | Phospholipidporphyrin conjugates: deciphering the driving forces behind their supramolecular assemblies[END_REF] One of the major difficulties when treating microbial infections is the destruction of the biofilms which consist of polymeric matrices protecting bacterial colonies against external conditions. These biofilms are an increasing cause of bacterial resistance against conventional antibiotics. [START_REF] Sharma | Antibiotics versus biofilm: an emerging battleground in microbial communities[END_REF] It has been shown that lipidic nano-assemblies favor the permeation of photosensitizers into biofilms, helping them to diffuse inside the matrix and reach bacteria. [START_REF] Kim | The delivery of benzyl penicillin to Staphylococcus aureus biofilms by use of liposomes[END_REF][START_REF] Rukavina | Current trends in development of liposomes for targeting bacterial biofilms[END_REF] In recent work, two approaches were studied to modify phenalenone by lipid chains. [START_REF] Godard | Synthesis and supramolecular arrangement of new stearoyl acid-based phenalenone derivatives[END_REF] Two conjugates with a stearic chain (PN-C18, conjugates 1A and 1B) differing only in the nature of the linker to phenalenone (amide or amide + triazole), and PN-(C18)2 (conjugate 2) in which PN is conjugated with two C18 chains were synthesized. Conjugate 2, whose chemical structure resembles that of a phospholipid proved able to self-assemble into large bilayer stacks at 40°C (Figure 1). When mixed with stearoyl oleoyl phosphatidylcholine (SOPC) at molar fractions ranging from 10 to 40% at the same temperature, the three compounds 1A, 1B, and 2 formed vesicles. However, these vesicles were unstable over time. CryoTEM images showed heterogenous mixtures of small and oligolamellar vesicles, sheets, stacks of bilayers, and ribbon-like structures, all of which appeared to result from the destabilization of initially formed multilamellar vesicles. 10 To better understand the behavior of these conjugates in mixtures with phospholipids and identify the optimal conditions for the formation of stable nano-assemblies, like the temperature and lipid/PN derivative ratio, the nature of the linker or the number of lipid chains, we analyzed, in this work, the thermodynamic and structural properties of the three PN conjugates 1A, 1B, and 2 in mixtures with two phospholipids with the same chain length, SOPC (18:1) and distearoylphosphatidylcholine, DSPC (18:0).

Materials and methods

Materials

The phenalenone-lipid conjugates, namely N-((1-((1-oxo-1H-phenalen-2-yl)methyl)-1H-

Thermal analysis of hydrated phospholipid-phenalenone conjugates mixtures

Thermal analysis was carried out using a MicroCal VP-DSC instrument (Northampton, MA, USA) at 1°C/min scan rate between 1°C and 60°C under approximately 25 psi positive cell pressure. Phospholipid-PN-C18 mixtures (2 mM) were dissolved in a mixture of chloroform and methanol (9:1, v/v) and the solvents were evaporated under reduced pressure. The film obtained was hydrated by pure water at 60°C, and the resulting lipid suspension was vortexed for few minutes with glass beads until the film was detached from the flask wall.

Prior to the analysis of each lipid mixture and after cell cleaning, one heating was performed with pure water in the sample and reference cells. Three heating/cooling cycles were then recorded for each sample. The feedback mode/gain was set on high. Thermograms were analyzed using the Trios software (version 4.4) and enthalpy variations (ΔH) were calculated by integrating the area under transition peaks.

Pure conjugate 2 was also analyzed using a DSC Diamond calorimeter (PerkinElmer) calibrated with Zinc and Indium. Three heating/cooling cycles were performed between 20

and 75°C, at 5°C/min. A solution of conjugate 2 in chloroform and methanol (9:1, v/v) was deposited into a 50 µl DSC pan and evaporated overnight under reduced pressure to form a film covering the bottom of the pan. The sample was accurately weighted (1.10 ± 0.5 mg) and 15 µl of water were added before the pan was sealed. Several heating/cooling cycles were then performed between 25 °C and 75 °C until heating scans were reproducible.

Surface pressure measurements

Surface pressure-surface area (π-A) measurements were performed using a thermostated Langmuir film trough (775.75 cm 2 , Biolin Scientific, Finland) enclosed in a plexiglas box, protected from light. All experiments were performed at 22 ±1°C. After spreading, each system was left for 15 min to allow complete evaporation of the organic solvents. Monolayer compression was performed at 6 Å 2 .molecule -1 .min -1 speed. Results are mean values of at least 3 measurements. The experimental uncertainty was estimated to be 0.2 mN/m. The surface compressional moduli (Cs -1 ) of monolayers were calculated from the equation ( 1):

𝐶 𝑆 -1 = -𝐴 𝑑𝜋 𝑑𝐴 (1)
The free energy of mixing is the sum of the excess free energy and the ideal free energy calculated using equations ( 2) and (3):

∆𝐺 𝑒𝑥𝑐 = ∫ (𝐴 𝑚𝑖𝑥 -𝑥 1 𝐴 1 -𝑥 2 𝐴 2 )𝑑𝜋 𝜋 0
(2)

∆𝐺 𝑖𝑑𝑒𝑎𝑙 = 𝑛𝑅𝑇(𝑥 1 𝐿𝑛𝑥 1 + 𝑥 2 𝐿𝑛𝑥 2 ) (3)
With Amix, the experimental molecular area for the mixture at a given surface pressure, A1 and A2, the molecular area of the pure components at the same surface pressure and x1, and x2, the molar fraction of each component in the mixture. [START_REF] Ambike | Interaction of selfassembled squalenoyl gemcitabine nanoparticles with phospholipid-cholesterol monolayers mimicking a biomembrane[END_REF]

GIXD measurements under synchrotron radiation

X-ray diffraction measurements were taken on the SIRIUS beamline at the SOLEIL Synchrotron (Saint Aubin, France). [START_REF] Fontaine | Soft Interfaces and Resonant Investigation on Undulator Source: A Surface X-ray Scattering Beamline to Study Organic Molecular Films at the SOLEIL Synchrotron[END_REF] The incident X-ray energy was 8 keV (λ = 1.55 Å), and the beam size was 0.11 mm 2 . Grazing incidence X-ray diffraction (GIXD) was performed on monolayers, made of conjugate 2, pure or mixed with a phospholipid, to obtain in-plane and out-of-plane information about the molecular ordering at the air/water interface. Monolayers were spread on the water subphase of a custom-built Langmuir trough enclosed in a temperature-controlled sealed chamber (21°C) and flushed with helium during data collection to reduce the gas scattering and avoid beam damage to the monolayer. After spreading, the lipid film was left for 15 minutes to allow for solvent evaporation and film relaxation, and then compressed at 3 Å 2 .molec -1 .min -1 . The water surface was illuminated at an incident angle of 2.0 mrad below the critical angle of the air-water interface (2.8 mrad at 8 keV), so that the incident wave was totally reflected, while the refracted wave became evanescent, exploring a layer several nanometers deep beneath the interface. The diffraction pattern was collected Further information on data processing is provided in the Supporting Information.

Formation and characterization of phospholipid-phenalenone conjugate nanoassemblies.

Pure conjugates 1A, 1B, or 2 and SOPC were dissolved in a mixture of chloroform and methanol (9:1, v/v) and the solvents were evaporated under reduced pressure to remove all residual solvents. The resulting film was hydrated by pure water at 40°C for conjugates 1A and 1B, and at 65°C for conjugate 2. The obtained lipid suspension was sonicated twice for 5 min by a series of 10 s-on/10 s-off cycles using a Branson SFX150 probe sonicator (150 W, The thermograms for hydrated conjugate 1A-SOPC and conjugate 1B-SOPC mixtures are shown in Figure 2, and the corresponding phase transition temperatures and enthalpies are presented in Table S1 and Figures S2 andS3. In the studied temperature range, conjugates 1A and 1B do not display any specific thermal properties. SOPC exhibits a phase transition temperature at 5.4°C, without any pretransition, in agreement with the literature. [START_REF] Kaneshina | Barotropic phase transitions of dioleoylphosphatidylcholine and stearoyl-oleoylphosphatidylcholine bilayer membranes[END_REF] This phase transition temperature slightly decreases as the molar fraction of compound 1A increases, and the enthalpies are gradually reduced from 5.3 to 0.6 kcal/mol lipid for the pure SOPC and the mixture containing 65 mol% conjugate 1A, respectively. Above 65 mol% 1A, the phase transition peak of SOPC completely disappears. Conjugate 1B also affects the thermal behavior of SOPC, with a weak increase of the phase transition temperature of the phospholipid with the molar fraction of conjugate 1B (Figure 2b, Table S1, Figure S3). The enthalpy peak of SOPC is enlarged up to 50 mol% 1B, then gradually fades and virtually disappears at 70 mol% 1B. Surface pressure vs. molecular area isotherms for monolayers of SOPC mixed with the two PN conjugates show that compounds 1A and 1B alter the interfacial behavior of the phospholipid, however in a different manner (Figures S4 andS5). Conjugate 1A tends to increase the compressibility of SOPC monolayers, whereas conjugate 1B rigidifies them.

Excess free energies are positive at all SOPC-1A ratios and all surface pressures while for conjugate 1B, positive and negative excess free energies are obtained depending on the molar fraction of the compound (Figure 3 and Tables S2 andS3). SOPC mixes better with 1B than with 1A.

Conjugate 2

Pure conjugate 2 exhibits a phase transition temperature at 47°C (Tm2). At high SOPC molar fractions, the observed transition is shifted to a slightly higher temperature, around 48°C

(Figure 4a, Table S4). The peak is thinner, and better defined than at the highest concentrations of the PN derivative. Like conjugates 1A and 1B, conjugate 2 affects the thermal behavior of SOPC: an increase in the phase transition temperature and gradual lowering of the enthalpies values for the phospholipid are observed as the molar ratio of the conjugate increases (Figure 4a and Figure S6). Although the deviation of experimental enthalpy data from theoretical ones is small, there is evidence that conjugate 2 penetrates and disorganizes the SOPC domains. Above 50 mol% of conjugate 2, the phase transition peak of SOPC (Tm1) is no longer visible. From 5 to 50 mol% of conjugate 2, both peaks appear on the thermograms, suggesting a phase separation, with the coexistence of conjugate 2-SOPC mixed domains and quasi-pure conjugate 2 domains (very thin peaks).

To analyze the effect of a phospholipid bearing more resemblance to conjugate 2, DSC measurements were also performed with the saturated distearoylphosphatidylcholine (DSPC)

(Figure 4b, Table S4, and Figure S7). DSPC, like conjugate 2, forms structured gel phase mono-and bilayers at 22°C and rigid monolayers. The thermograms of the fully hydrated DSPC bilayers exhibit a weak peak at 49.3°C assigned to the L' gel phase -> P' ripple phase pretransition and a sharp peak at 54°C (Tm3) corresponding to the P' -> L transition (lamellar liquid crystalline phase), in agreement with the literature. [START_REF] Koynova | Phases and phase transitions of the phosphatidylcholine[END_REF] At 5 mol% of conjugate 2 and above, the pre-transition of DSPC vanishes. At 20 mol% of conjugate 2, the phase transition peak of the conjugate appears (Tm2) indicating a phase separation. Figure S8). Compared to pure SOPC and DSPC monolayers, [START_REF] Dynarowicz-Latka | Influence of a neoglycolipid and its PEO-lipid moiety on the organization of phospholipid monolayers[END_REF][START_REF] Botet-Carreras | Effect of cholesterol on monolayer structure of different acyl chained phospholipids[END_REF] monolayers of conjugate 2 exhibit a smaller molecular area and higher surface pressure at collapse. The profiles of the isotherms for the mixtures suggest an interaction between the conjugate and the two phospholipids. This interaction, favorable, is confirmed by the negative values of the calculated excess free energies (Figure 6, Tables S5 andS6). Contrarily to conjugates 1A and 1B, Conjugate 2 forms stable mixtures with the two studied phospholipids at all molar fractions and surface pressures.

Structure of the mixed monolayers of conjugate 2 and phospholipids

To get a better insight into the structure of the mixtures of conjugate 2 with SOPC and DSPC, GIXD measurements were performed on monolayers of pure conjugate 2 as well as mixtures of conjugate 2 with both phospholipids. The pure SOPC monolayer does not diffract X-rays whereas GIXD spectra for the pure DSPC monolayer exhibit one in-plane and one out-ofplane peaks at surface pressures higher than 5 mN.m -1 when the layer is in a Liquid Condensed phase. [START_REF] Brezesinski | X-ray diffraction ad foam film investigations of PC head group interaction in water/ethanol mixtures[END_REF][START_REF] Desroches | Incorporation of glycoconjugated porphyrin derivatives into phospholipid monolayers: A screening method for the evaluation of their interaction with a cell membrane[END_REF][START_REF] Ropers | Lipid ordering in planar 2D and 3D model membranes[END_REF] DSPC diffraction pattern could be indexed using the rectangular centered lattice with NN orientation and significant chain tilt from 30° at low surface pressure to 26° at 30 mN.m -1 (Table 1).

Monolayers of the pure conjugate 2 could diffract at all surface pressures studied, even at 0 mN/m (Table 1, Table S7, Figure S9). As the surface pressure increases, the cell parameters, as deduced from the position of the peaks, evolve from a rectangular structure to pure hexagonal packing. The full width at half maximum of the in-plane Bragg peaks is almost stable with surface pressures and corresponds to in-plane correlation lengths of 16 nm on average, indicating an ordering which extends up to the second nearest neighboring molecules.

Compound 2 forms structured mixed monolayers with both SOPC and DSPC. Conjugate 2-SOPC monolayers exhibit a pure hexagonal cell (a/b = 3) with chains in an upright position (t = 0°) (Table 1 and Figure S10). When DSPC and conjugate 2 are mixed, an intense in-plane peak with a weak out-of-plane component is observed at all surface pressures (Figure S10, Table S7). All diffraction peaks can be indexed in the centered rectangular lattice with NN orientation. At a low conjugate 2 molar ratio (10 mol%), the structural parameters of the mixed monolayer are very close to those of pure DSPC. At higher conjugate 2 molar ratios (20 and 50 mol%), the lattice symmetry evolves from a rectangular to a more hexagonal one as the surface pressure increases in the film. For DSPC-conjugate 2 mixtures, GIXD measurements demonstrate the predominance of the major component in the mixture on the structural properties of the mixed monolayer. 7). The DLS report showed significant polydispersity with three peaks at 141.4 ± 23.9, 537.9 ± 65.

Discussion

The formation of nano-assemblies of phenalenone-lipid conjugates with single or double C18 chains is not straightforward. Conjugates 1A and 1B cannot form micelles. We have shown previously that when mixed with SOPC they formed unstable vesicles which transformed into bilayer sheets and ribbons. [START_REF] Godard | Synthesis and supramolecular arrangement of new stearoyl acid-based phenalenone derivatives[END_REF] In this work, DSC and surface pressure measurements demonstrate that conjugates 1A and 1B affect SOPC thermodynamic properties and tend to segregate. This segregation can be partially explained by the unfavorable interaction observed by other authors between the stearic acid chain and an unsaturated phospholipid like DOPC, [START_REF] Hac-Wydro | Effect of saturation degree on the interactions between fatty acids and phosphatidylcholines in binary and ternary Langmuir monolayers[END_REF] but not only. The conjugate 1A is clearly more disruptive to monolayers and bilayers than the conjugate 1B. Apparently, this disruption results more from the bulkiness of the conjugate 1A linker and the complex interactions developed in the plane of the interface, than from hydrophobic interactions between the C18 chains. Indeed, in addition to π-π interactions between PN molecules, the amide and triazole groups may form hydrogen bonds with water and neighbor molecules which would rigidify the molecules and contribute to phase separation. According to Doiron et al. [START_REF] Doiron | Structural consequences of the 1,2,3-triazole as an amide bioisostere in analogs of cystic fibrosis drugs VX-809 and VX-770[END_REF] , a triazole group influences the shape of a molecule and makes it less planar than when the linker is an amide group. The amide group is a stronger hydrogen bond donor and acceptor than the triazole. It promotes intramolecular Hbonds that are lost with the triazole cycle. The triazole also weakens intermolecular hydrogen bonds. For conjugate 1A, whose linker contains both groups, interactions with neighboring molecules could be particularly complex at the level of the polar head region.

For conjugate 2, DSC and GIXD results suggest that instead of solubilizing the PN derivative, SOPC has a structuring effect on its domains. The phospholipid molecules increase the symmetry of the cell and participate in the structuring of conjugate 2 domains, preserving the extent of the 2D order of these domains. Monolayers are rigidified above 25-30 mN/m, a surface pressure considered as mimicking the lateral pressure in bilayers. [START_REF] Marsh | Lateral pressure in membranes[END_REF] The disappearance of the pre-transition of DSPC in thermograms of its mixtures with conjugate 2 indicates that PN moieties interact with the phospholipid headgroups. They are thus likely to be partially immersed in the water phase.

The double interaction of conjugate 2 with the phospholipids, at the level of the polar heads and at the level of the hydrophobic chains has a strong influence on the formation and stability of liposomes. Below 20 mol%, the effect of conjugate 2 is limited. This explains that multilamellar and unilamellar vesicles are predominant in the sample with 10% conjugate 2.

However, at 20 mol% and above, molecules of conjugate 2 get closer and strongly interact with one another, forming rigid domains. It is well known that when liquid-crystalline and gel phases coexist in a bilayer, the molecules at the domain boundaries have different packing. [START_REF] Lee | Lipid phase transitions and phase diagrams. I. Lipid phase transitions[END_REF] Molecules of conjugate 2 in the gel phase are more tightly packed than those of SOPC in the liquid-crystalline phase. The polyhedral (faceted) morphology of vesicles results from the existence of these lipid domains with different molecule packing and orientation, and defects at domain boundaries. [START_REF] Ickenstein | Disc formation in cholesterol-free liposomes during phase transition[END_REF] The increase in bilayer curvature leads to the reduction in the size of liposomes, breaking of vesicles, and formation of small bilayer discs with two or more stacked bilayers.

The formation of planar bilayers is generally associated with a difference in the packing parameter of the vesicle-forming components. This may be the case for conjugate 1A and 1B which must have a lower packing parameter than the phospholipid, but this is less expected from conjugate 2 which packing parameter is probably close to that of SOPC. For the three conjugates, it seems that the inter-bilayer stacking of PN molecules would be ultimately responsible for the disintegration of vesicles. CryoTEM images show that vesicles adhere to each other and to bilayer discs. A similar bridging phenomenon based on polar head interactions between liposome bilayers has been described for aescin-containing liposomes. [START_REF] Sreij | Temperature dependent self-organization of DMPC membranes promoted by intermediate amounts of the saponin aescin[END_REF] The increase in conjugate 2 concentration up to 75 mol% enhances this phenomenon and leads to small discs forming (about 100 nm in diameter).

Conclusion

Phenalenone-lipid conjugates have been synthesized with the objective to form pure nanoassemblies or mixed vesicles with SOPC containing a high concentration of PN, for its use in aPDT. However, when pure, these derivatives either do not self-assemble (conjugates 1A and 1B) or form large cohesive bilayer stacks (conjugate 2). When mixed with SOPC, conjugates 1A and 1B form multilamellar vesicles that rapidly disintegrate into large bilayer sheets and ribbons. Interestingly, conjugate 2 above 10 mol% leads to the formation of altered vesicles that evolve into small bilayer discs predominating at 75 mol%. The destabilization of vesicles and formation of multi-bilayer discs result from phospholipid-conjugate segregation in bilayers and lateral and inter-bilayer π-π interactions of phenalenone heads. The stability over time of these discs remains to be assessed, before testing the antimicrobial efficacy of these small bilayer discs on bacteria strains and their biofilms.

  Thermodynamic and structural properties of lipid-photosensitizer conjugates mixed with phospholipids: Impact on the formation and stability of nano-assemblies David Chapron, Jean-Philippe Michel, Philippe Fontaine, Jérémy Godard, Frédérique Brégier, Vincent Sol, Véronique Rosilio To cite this version: David Chapron, Jean-Philippe Michel, Philippe Fontaine, Jérémy Godard, Frédérique Brégier, et al.. Thermodynamic and structural properties of lipid-photosensitizer conjugates mixed with phospholipids: Impact on the formation and stability of nano-assemblies. Colloids and Surfaces B: Biointerfaces, 2023, 231, pp.113565. 10.1016/j.colsurfb.2023.113565. hal-04237933

Figure 1 :

 1 Figure 1: Chemical structure of compounds 1A, 1B, and 2, and cryoTEM images of nano-assemblies

  using a Pilatus3 detector (Dectris, Switzerland) associated with a Soller slit collimator (JJ-Xray, Denmark, resolution 0.06 deg) by scanning in the in-plane angle 2. Qxy-Qz images were generated by integrating horizontally the window defined by the Soller slit collimator on the original raw 2-D file of the detector to obtain the vertical distribution of the intensity.

27 3. Results 3 . 1 .

 2731 40 kHz, Branson Untrasonics TM , Emerson) at 20% amplitude. Suspensions obtained with pure conjugate 2 were used without further processing. Formulations containing mixtures of SOPC and the conjugates were centrifuged at 3000g for 15 min. All obtained nano-assembly suspensions were stored under nitrogen at 4°C in the dark. Dynamic light scattering (DLS) and zeta potential measurements were carried out in triplicate at 25°C, using a Zetasizer Nano-ZS 90 (Malvern Instruments Ltd, UK). The morphology of the nano-assemblies was analyzed using a ThermoFischer Scientific G3 Titan Themis 300 transmission electron microscope equipped with a Cryo-Box anticontamination system and a C-Twin objective lens (Cs = 2.7 mm, Cc = 2.7 mm, Focal length = 3.5 mm), using accelerating voltage of 300 kV, with the following illumination conditions: spot size of 5, a 150 µm condenser aperture, and no objective aperture. Images were recorded (defocus range: -0.3 to 0.4 µm) using the low electron dose system (10 e -/Å 2 per sec), with a magnification of 5,400x and 58,000x on a ThermoFischer Scientific TM Falcon 3 EC Direct Detection Electron 4Kx4K camera in linear mode. 4 µL of each suspension were deposited on a 300-mesh lacey carbon-coated grid. After removal of the excess liquid with a filter paper, the grid was quickly frozen in liquid ethane and inserted in the microscope using a nitrogen-cooled side entry Gatan Elsa TM (698) Cryoholder at -181°C. Thermodynamic properties of mixtures of phospholipid and phenalenone-lipid conjugates 3.1.1. Conjugates 1A and 1B

Figure 2 :

 2 Figure 2: Thermograms of SOPC mixtures with compounds 1A (a) and 1B (b) with increasing mol%

Figure 3 :

 3 Figure 3: ∆G exc values versus the molar fraction of the phenalenone-lipid conjugate for (a) for 1A-

Figure 4 :

 4 Figure 4: Thermograms of (a) conjugate 2-SOPC and (b) conjugate 2-DSPC mixtures. Tm1, Tm2 and

Figure 5 :

 5 Figure 5: π-A isotherms for (a) 2-SOPC and (b) and 2-DSPC mixed monolayers.

Figure 6 :

 6 Figure 6: ∆G exc values versus the molar fraction of the PN conjugate for mixtures of conjugate 2 with

Figure 7 :

 7 Figure 7: CryoTEM images showing the morphology of conjugate 2-SOPC nano-assemblies

CRediT authorship contribution statement DC:

 statement Methodology, Investigation, Validation; JPM: Investigation, Formal analysis; PF: Methodology, Validation; JG: Resources; FB: Resources; VS: Resources, Supervision; VR: Conceptualization, Methodology, Validation, Supervision, Writing, Review & Editing.

Table 1 :

 1 Characteristic GIXD parameters for monolayers of conjugate 2 mixed with DSPC or SOPC. *Below 15 mN/m. R: rectangular. H: hexagonal. NN: Nearest neighbor. Lxy:

	correlation length.
	System		0	5	15	30	40	Lattice
		(mN/m)						Tilt-
								Orientation
	Conjugate 2	A (Å 2 )	77.0	50.0	44.5	41.5	38.7
		a/b (Å) 4.89/8.35 4.86/8.33 4.77/8.27 4.87/8.25 4.84/8.25	R
		Tilt (°)	7.9	5.8	0	0	0	NN*
		Lxy (nm)	15.3	17.2	22.4	11.6	13.7
	DSPC	A (Å 2 )	77.7	51.0	49.0	46.3	-
		a/b (Å)	-	5.53/8.64 5.36/8.58 5.26/8.58	-	R
		Tilt (°)	-	>30	29.8	26.2	-	NN
		Lxy (nm)	-	20.6	15.1	23.9	-
	2-DSPC	A (Å 2 )	77.2	50.5	47.6	44.6	-
	50 mol% 2	a/b (Å)	5.05/8.5 5.05/8.48 4.97/8.43 4.99/8.4	-	R
		Tilt (°)	23.8	19.7	15.8	14.9	-	NN
		Lxy (nm)	13.7	12.5	12.9	10.1	-
	2-DSPC	A (Å 2 )	77.2	54.0	51.6	48.7	46.8
	20 mol% 2	a/b (Å)	-	5.53/8.62 5.25/8.56 5.13/8.53 4.99/8.47	R
		Tilt (°)	-	32.8	26.5	23.4	18.5	NN
		Lxy (nm)	-	19.3	13.0	18.1	16.2
	2-DSPC	A (Å 2 )	77.0	54.7	52.5	49.6	48
	10 mol% 2	a/b (Å)	-	5.63/8.64 5.28/8.59 4.88/8.88 5.13/8.52	R
		Tilt (°)	-	35.3	29.3	24	23	NN
		Lxy (nm)	-	27.4	21.9	22.0	18.8
	2-SOPC	A (Å 2 )	96.0	54.0	48.7	44.9	42.9	H
	80 mol% 2	a/b (Å)	-	4.81/8.33 4.79/8.29 4.77/8.26 4.76/8.25
		Tilt (°)	-	0	0	0	0
		Lxy (nm)	-	10.9	12.7	13.9	11.0
	2-SOPC	A (Å 2 )	96.5	61	53.6	48.3	46.0	H
	60 mol% 2	a/b (Å)	-	4.8/8.32 4.78/8.28 4.77/8.27 4.77/8.26
		Tilt (°)	-	0	0	0	0
		Lxy (nm)	-	12.7	14.5	17.0	14.9
	2-SOPC	A (Å 2 )	96.5	72.0	63.3	56.0	52.7	H
	50 mol% 2	a/b (Å) 4.83/8.37 4.81/8.34 4.79/8.3 4.78/8.29 4.78/8.29
		Tilt (°)	0	0	0	0	0
		Lxy (nm)	11.75	9.5	12.6	14.1	13.1
	2-SOPC	A (Å 2 )	96.8	82.0	68.4	59.0	55.0	H then R
	20 mol% 2	a/b (Å) 4.81/8.33 4.78/8.29 4.77/8.26 4.76/8.26 4.84/8.24
		Tilt (°)	0	0	0	0	0
		Lxy (nm)	20.2	22.4	21.0	17.1	28.9

  2, and 44.2 ± 4.7 nm (55.3%, 24.9%, and 19.8% intensity, respectively). The zeta potential values were negative (about -5 mV for pure SOPC and -20 mV for pure conjugate 2). At 25 mol% and above, nanoobjects with different morphologies were observed like faceted SUVs or stacked bilayers. At 75 mol% of conjugate 2, however, deformed MLVs and stacked bilayers appeared predominant.

Acknowledgments

The authors would like to acknowledge SOLEIL for the provision of synchrotron radiation facilities and the use of SIRIUS Beamline in projects 20200096 and 20210232. They also thank the CIMEX -Ecole Polytechnique, IP Paris, Palaiseau Cedex, France, and Dr. Eric Larquet for cryoTEM images and helpful discussions. This research was funded by CNRS.

Appendix A. Supporting information

Data Availability

The data presented in this study are available on request from the corresponding author.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Supplementary Material

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