L E Ha Vy Nguyen

Catherine Bonnet
email: catherine.bonnet@inria.fr

AND ANDR É Ricardo Fioravanti
email: fioravanti@fem.unicamp.br

H ∞ -STABILITY ANALYSIS OF FRACTIONAL DELAY SYSTEMS OF NEUTRAL TYPE *

Keywords: Fractional systems, neutral systems, delay effects, H∞-stability AMS subject classifications. 93C23, 93D25, 93C05

In this paper we consider linear fractional systems of commensurate orders and with commensurate delays, whose characteristic equation is a polynomial in the two variables s α (0 < α < 1) and e -sτ (τ > 0). These systems may have single or multiple chains of poles asymptotic to the imaginary axis. Location of poles of large modulus belonging to these chains are determined by approximation and simple necessary and sufficient H∞-stability conditions are derived.

1. Introduction. In the time domain, fractional models involve derivatives and/or integrals of orders which are not necessarily integers. Similarly, they involve in the frequency domain non (necessarily) integer powers of the Laplace variable s. These models have found applications in many fields, for instance electromagnetics [START_REF] Knospe | Performance limitations of non-laminated magnetic suspension systems[END_REF][START_REF] Westerlund | Capacitor theory[END_REF], mechanics [START_REF] Caputo | A new dissipation model based on memory mechanism[END_REF][START_REF] Koh | Application of fractional derivatives to seismic analysis of basedisolated models[END_REF], and biology [START_REF] Grahovac | Modelling of the hamstring muscle group by use of fractional derivatives[END_REF], since they have been showed to better model certain phenomena than integer-order models.

In control engineering, a lot of results are available on fractional controllers and their implementation. See for example [START_REF] Oustaloup | The CRONE control of resonant plants: Application to a flexible transmission[END_REF][START_REF] Podlubny | Fractional-order systems and P I λ D µ -controllers[END_REF][START_REF] Monje | Fractional-order Systems and Controls: Fundamentals and Applications[END_REF], and the references therein.

With the spreading of fractional systems including both plants and controllers, it is natural to think about fractional systems with delays since delays are commonly encountered in real systems. However, not all fractional systems with delays are obtained in this manner. One example is an acoustic system subjected to viscothermal perturbations, which was first modeled by generalized partial differential equations [START_REF] Matignon | Spectral and time-domain consequences of an integrodifferential perturbation of the wave PDE[END_REF] and then was modeled in some ranges of low frequencies by an approximated transfer function of fractional type with a delay [START_REF] Hotzel | On linear systems with a fractional derivation: Introductory theory and examples[END_REF].

Recently, an increasing interest is paid towards fractional systems with delays, both for stability aspects and also for stabilization problems. For stability aspects, most of the results are obtained in the frequency domain, i.e. by considering the spectrum of the characteristic equation. The characteristics of fractional systems with delays are similar to those of classical systems with delays, i.e. there are infinitely many poles in chains which can be classified in retarded, advanced and neutral types [START_REF] Hotzel | Some stability conditions for fractional delay systems[END_REF][START_REF]Stabilization of some fractional delay systems of neutral type[END_REF].

To the authors' knowledge, the stability of fractional systems with delays is first examined in [START_REF] Hotzel | Some stability conditions for fractional delay systems[END_REF] where some sufficient as well as necessary conditions for BIBO and H ∞ stabilities of the simple model of the aforementioned acoustic system are derived. In [4], the most general class of systems is considered. For systems of retarded type, it is proved that the classical stability condition "no poles in the closed right halfplane" is a necessary and sufficient condition for BIBO-stability. However, this is only a necessary condition for neutral systems since, in the critical case where poles approach the imaginary axis, the system may be unstable even though all poles are in the open left half-plane. This interesting phenomenon is also present in classical delay systems.

In such a delicate situation, H ∞ -stability of fractional systems with one delay is studied in [START_REF]Stabilization of some fractional delay systems of neutral type[END_REF] where simple necessary and sufficient conditions are derived. Within the same framework, [START_REF] Bonnet | Stability of neutral systems with commensurate delays and poles asymptotic to the imaginary axis[END_REF] studies H ∞ -stability of some classes of classical systems with commensurate delays and with single chains of poles asymptotic to the imaginary axis.

In this paper, we examine the H ∞ -stability of fractional systems of commensurate orders and with commensurate delays whose characteristic equation is a polynomial in two variables s α (0 < α < 1) and e -sτ (τ > 0). Specially, these systems may have multiple chains of poles asymptotic to the imaginary axis. We are interested in the effects of poles of large modulus on the stability and will not pay attention to poles of small modulus, which can be computed using numerical methods such as QPmR [START_REF] Vyhlidal | QPmR v.2 -Quasipolynomial rootfinder, algorithm and examples[END_REF] and YALTA [START_REF] Avanessoff | H∞-stability analysis of (fractional) delay systems of retarded and neutral type with the Matlab Toolbox YALTA[END_REF].

This paper extends the works presented in [START_REF] Fioravanti | Stability of fractional neutral systems with multiple delays and poles asymptotic to the imaginary axis[END_REF][START_REF] Nguyen | Stability analysis of fractional neutral time-delay systems with multiple chains of poles asymptotic to same points in the imaginary axis[END_REF] and makes some corrections for the latter which will be announced right after the related result.

The rest of the paper is organized as follows. Section 2 presents the fractional delay system of interest and some preliminaries. In §3 we approximate the characteristic equation around its poles of large modulus. The obtained expression is repeatedly used in the next two sections to determine pole location with respect to asymptotic axes and estimate the magnitude of the characteristic equation on the imaginary axis, which allows one to conclude about H ∞ -stability of the system. Section 4 examines single neutral chains of poles while §5 is dedicated to multiple chains. Illustrative examples are given in §6. We conclude the paper with §7.

Here, we denote

N N = {1, . . . , N }, which is a set of natural numbers, Z + = {x ∈ Z | x ≥ 0}, C + = {s ∈ C | s > 0},
G(s) = t(s µ ) p(s µ ) + N k=1 q k (s µ )e -ksτ
, where τ > 0 is the delay, t, p, and q k for all k ∈ N N are real polynomials in s µ , 0 < µ < 1, -π < arg(s) < π in order to have a single-valued transfer function, deg p ≥ deg t, deg p ≥ deg q k for all k ∈ N N , and deg p = deg q k at least for one k ∈ N N in order to deal with proper neutral systems. Note that degrees of the polynomials in this paper stand for the degrees in s µ . The function s µ where µ is a real number, 0 < µ < 1, has a branch point at s = 0. To study this function we make a cut in the complex plane at R -and consider the domain C \ R -.

In this paper, we consider stability of systems in a H ∞ -sense. We recall that a system is H ∞ -stable if and only if for input signals that are bounded in the norm L 2 it provides output signals that are also bounded in the same norm. Recall that a linear time-invariant system is H ∞ -stable if and only if its transfer function belongs to

H ∞ := {f : C + → C | f is analytic in C + and sup s∈C+ |f (s)| < ∞}.
Since deg p ≥ deg q k for all k ∈ N N , we can obtain the following expression for each k

(2.2) q k (s µ ) p(s µ ) = α k + β k s µ + γ k s 2µ + δ k s 3µ + k s 4µ + O(s -5µ ) as |s| → ∞.
The coefficient of the highest degree term of the denominator of the transfer function (2.1) can be written as a multiple of the following polynomial in z (which can be called formal polynomial as in the case of standard delay systems)

(2.3) c d (z) = 1 + N k=1 α k z k ,
where z = e -sτ . Let us denote s n a root of the denominator of the transfer function. According to Rouché's theorem, s n is asymptotic to a root r of c d (e -sτ ) as |s n | → ∞ and thus is approximated by (2.4)

s n τ = λ n + o(1), where (2.5
)

λ n = -ln |r| -Arg(r) + 2πn, n ∈ Z, n → ∞.
We can say that to each root r of c d (z) is associated a chain of poles and this chain is said to be of neutral type. As a consequence, when n → ∞ the neutral chain of poles asymptotically approaches the vertical line

(2.6) (s) = - ln |r| τ .
If the vertical line is on the right or on the left of the imaginary axis, which happens when |r| < 1 or |r| > 1, then poles asymptotic to this vertical line are respectively on the right or on the left of the imaginary axis, and then their effects on H ∞ -stability are easily concluded. Therefore, in these cases, we do not need to further analyze in order to determine on which side of the asymptotic axis the neutral chains of poles lie.

However, when a vertical line coincides with the imaginary axis, a more precise analysis is required in order to determine in which half-plane the neutral chains of poles is located.

Remark 1. Note that the basic results recalled in this section are also valid for the classical case, i.e. µ = 1.

3. Approximation of the characteristic equation. We already have the first approximation of neutral poles corresponding to a root r of c d (z) in (2.4). Our objective is to find the next non-zero approximation term of these poles, which are denoted by s n . Let us write

s n τ = λ n + ν n,1 + o(n -y1 ),
where

ν n,1 = ν 1 n y1 , ν 1 = 0, y 1 > 0, n ∈ Z, n → ∞,
and we recall that n y1 is defined on the branch where -π < arg(n) ≤ π. Remark 2. We will see later that in general y 1 is not an integer multiple of µ. For example, y 1 = µ in certain cases of single chains, but y 1 = µ/m in certain cases of multiple chains, where m is the multiplicity of r.

We have (s n ) = [ (λ n ) + (ν n,1 ) + o(n -y1 )]/τ . Therefore, the sign of (ν n,1 ) indicates the location of poles of the neutral chain with respect to the asymptotic axis.

Remark 3. Note that for neutral chains of poles relative to a root r

(ν n,1 ) n>0 = (ν 1 ) n y1 , (ν n,1 ) n<0 = (ν 1 ) cos(y 1 π) + (ν 1 ) sin(y 1 π) |n| y1 .
Since the signs of (ν n,1 ) n>0 and (ν n,1 ) n<0 , which are determined by the signs of (ν 1 ) and ( (ν 1 ) cos(y 1 π) + (ν 1 ) sin(y 1 π)) respectively, may be different, so are the locations around the asymptotic axis of poles of large modulus in the upper and lower half-planes.

To determine the approximation term ν n,1 of neutral poles of the system, we will need the series expansion of the characteristic equation around s n as n → ∞.

Let us denote d(s) the characteristic equation of G(s), that is

d(s) := p(s µ ) + N k=1 q k (s µ )e -ksτ .
Since s n is a pole of G(s), we have d(s n ) = 0. We will now compute the series expansion as n → ∞ of d(s n )/p(s µ n ) which is

(3.1) 1 + N k=1 q k (s µ n ) p(s µ n )
e -ksnτ = 0 assuming that s n has the form

s n τ = λ n + ν n,1 + ν n,2 + . . . + ν n,M + o(n -4µ )
with ν n,i = ν i n -yi , i = 1, . . . , M where ν i = 0 and 0 < y 1 < . . . < y M ≤ 4µ. We choose indeed a development of order 4µ which will allow us to analyze in this paper several cases of interest.

In the next sections, we will show that by collecting terms of the same orders from the series expansion of d(s n )/p(s µ n ) and by matching the two sides of (3.1), we will be able to determine ν n,1 .

In the series expansion (see Appendix A for detailed computations), the following sums appear

A L r := N k=1 α k r k k L , B L r := N k=1 β k r k k L , C L r := N k=1 γ k r k k L , D L r := N k=1 δ k r k k L , E L r := N k=1 k r k k L ,
with L ∈ Z + . These sums will be shown to play an important role in the sequel.

In order to facilitate the collection of the terms of highest order of the development, we can decompose the series expansion of d(s n )/p(s µ n ) into three components as follows

d(s n ) p(s µ n ) = g 1 + g 2 + g 3 + o(n -4µ ) = 0 (3.2)
where

g 1 = 1 + A 0 r + τ µ B 0 r (2π) µ n -µ 1 + O(n -1 ) + τ 2µ C 0 r (2π) 2µ n -2µ 1 + O(n -1 ) (3.3) + τ 3µ D 0 r (2π) 3µ n -3µ + τ 4µ E 0 r (2π) 4µ n -4µ , g 2 = (l1,...,l M )∈L(4µ) (-1) M i=1 li M i=1 ν li i A M i=1 li r M i=1 l i ! n -M i=1 liyi , (3.4)
and

g 3 = τ µ (2π) µ n -µ 1 + O(n -1 ) (3.5) × (l1,...,l M )∈L(3µ) (-1) M i=1 li M i=1 ν li i B M i=1 li r M i=1 l i ! n -M i=1 liyi + τ 2µ (2π) 2µ n -2µ 1 + O(n -1 ) × (l1,...,l M )∈L(2µ) (-1) M i=1 li M i=1 ν li i C M i=1 li r M i=1 l i ! n -M i=1 liyi + τ 3µ (2π) 3µ n -3µ (l1,...,l M )∈L(µ) (-1) M i=1 li M i=1 ν li i D M i=1 li r M i=1 l i ! n -M i=1 liyi , where (3.6) L(x) := (l 1 , . . . , l M ) | l i ∈ Z + , M i=1 l i ≥ 1 and M i=1 l i y i ≤ x .
Note that for example in g 1 , the highest order in n is

-µ if B 0 r = 0. It is -2µ if B 0 r = 0 and C 0 r = 0.
To find the highest order in n for g 2 , note that

A M i=1 li r = 0 if M i=1 l i < m and it is zero if M i=1 l i = m (see Appendix B).
In this case we have M i=1 l i y i ≥ my 1 and the highest order in n is -my 1 . Hence, the highest order of the sum (g 1 + g 2 + g 3 ) may be a function of y 1 .

As g 1 + g 2 + g 3 + o(n -4µ ) = 0, the term of highest order of the left expression is thus zero and so is the term of highest order of the sum (g 1 + g 2 + g 3 ). As we will see in the sequel, this allows us to derive y 1 and ν 1 .

We have already seen that an important role is played by the coefficients A 0 r , B 0 r , . . .. In the following sections, we will derive y 1 and ν 1 for classes of systems which may have some of these coefficients vanishing.

We start with the case of single chains, i.e. m = 1, for which the analysis for systems with vanishing or non vanishing coefficients does not differ too much. More precisely, we will consider the case B 0 r = 0 and show that the results can then be expanded to other possible cases by similar arguments.

On the other hand, the analysis in the case of multiple chains, i.e. m ≥ 2, needs in each case (vanishing or non vanishing coefficients) an appropriate development to get y 1 and ν 1 from the highest order of development in the terms g 1 , g 2 , and g 3 . Three cases will be investigated:

• B 0 r = 0, • B 0 r = 0, B 1 r = 0, and C 0 r = 0, • B 0 r = 0, B 1 r = 0, B 2 r = 0
, and C 0 r = 0. 4. Single chains. To avoid complexity, we begin by studying thoroughly the simplest case, whose results can be exactly adapted to other cases.

4.1.

The case where B 0 r = 0. First, we give a more precise approximation of roots of large modulus of the characteristic equation than that given in (2.4).

Theorem 4.1. Let G(s) be a fractional neutral delay system defined by (2.1) and suppose that at least one root of the formal polynomial c d (z) defined in (2.3) has multiplicity one. If such a root, denoted by r, satisfies

B 0 r = 0, (4.1)
then for large enough n ∈ Z poles of the neutral chain relative to r are approximated by

s n τ = λ n + ν n,1 + o(n -µ )
with λ n given by (2.5) and

ν n,1 = τ µ K r (2π) µ n -µ , (4.2)
where

K r = B 0 r A 1 r . (4.3)
Proof. Under the condition (4.1), the highest order in n of g 1 is -µ. Obviously, the highest order of g 3 is smaller than that of g 2 , which is -y 1 . Therefore, y 1 = µ and (3.2) can be rewritten as

τ µ B 0 r (2π) µ n -µ -ν 1 A 1 r n -y1 + o(n -µ ) = 0,
which completes the proof. Given a more precise approximation of neutral poles of large modulus as above, now our interest is mainly on which side of vertical line the poles are, in other words, to find out the sign of (ν n,1 ) for n sufficiently large. This question is particularly important when the asymptotic axis is the imaginary axis.

Recall from Remark 3 that for a chain relative to a root of c d (s), its poles in the upper and lower half-planes may lie on different sides of the asymptotic axis. Fortunately, for the case considered in this subsection, these two parts of the chain may be on the same side. This behavior is characterized in the next two corollaries.

Corollary 4.2. Let 0 < µ < 1, ν n,1 be given by (4.2). Then (ν n,1 ) < 0 for all n ∈ Z if and only if

(K r ) < -tan µπ 2 | (K r )| (4.4)
with K r defined in (4.3).

Proof. Besides K r , the only term of interest is J = (n) -µ , as sgn( (ν n,1 )) = sgn( (JK r )). Since n can be both positive or negative, this term is given by

J = 1 |n| µ cos µπ 2 ∓  sin µπ 2 .
Multiplying J by K r and getting its real part leads to

(JK r ) = 1 |n| µ cos µπ 2 (K r ) ± sin µπ 2 (K r )
from which (4.4) follows from the fact that 0 < µ < 1. Some remarks can be drawn from this corollary. First, the condition (4.4) does not depend on the delay τ . This means that for all τ > 0 the chain of poles does not change side with respect to the vertical line in question when the delay varies. Second, the condition (4.4) still holds if r is replaced by its complex conjugate r, which is also a root of the formal polynomial c d (z). Therefore, the chain relative to r lies on the same side as the one relative to r.

As K r is independent of µ, we can reformulate the previous corollary to give the critical value of µ as follows Corollary 4.3. Let 0 < µ < 1, ν n,1 be given by (4.2) and its associated K r by (4.3). Then, if (K r ) < 0, all poles of the respective chain asymptotic to the vertical line (s) = -ln |r|/τ will be on the left of this line if

(4.5) µ < 2 π arctan - (K r ) | (K r )| .
Proof. This follows directly from Corollary 4.2.

In the case where (ν n,1 ) = 0, further analysis is needed to determine the location of poles. However, the procedure is similar to the one given in [START_REF] Bonnet | Stability of neutral systems with commensurate delays and poles asymptotic to the imaginary axis[END_REF] and therefore will be omitted. Now, we are interested in answering the question of stability of G(s) in the H ∞sense.

For systems without chains of poles asymptotic to the left of the imaginary axis, the stability can be concluded if there is no poles in the closed right half-plane.

On the other hand, if there exist neutral chains of poles approaching the imaginary axis from the left, we may have to consider the magnitude of the transfer function on the axis in order to answer the question of H ∞ -stability. This is the objective of the proposition below.

We will refer to poles in the closed right half-plane C+ as unstable poles. Proposition 4.4. Let G(s) be a transfer function given as in (2.1) and suppose that the formal polynomial c d (z) defined in (2.3) has at least one simple root of modulus one, the other roots being of modulus strictly greater than one. We also suppose that every root of modulus one which is denoted by r satisfies (4.1).

1. Suppose that (ν n,1 ) < 0 for all r and that G has no unstable pole of small modulus (which could exist only in a finite number), then G is H ∞ -stable if and only if deg p ≥ deg t + 1. 2. If there exists a root r for which (ν n,1 ) = 0, then the condition

deg p ≥ deg t + 1 is necessary for H ∞ -stability. Proof. Let s = s n + η ∈ R, we have p(s µ ) + N k=1 q k (s µ )e -ksτ ≈ |η| p (s µ n )µs µ-1 + N k=1 q k (s µ n )µs µ-1 -kτ q k (s µ n ) e -ksnτ ≈ |η| |p(s µ n )| p (s µ n ) p(s µ n ) µs µ-1 + N k=1 q k (s µ n ) p(s µ n ) µs µ-1 -kτ q k (s µ n ) p(s µ n ) e -ksnτ ≈ |η| |p(s µ n )| N k=1 kτ q k (s µ n ) p(s µ n ) e -ksnτ ≈ τ |η| |p(s µ n )| A 1 r as n → ∞, n ∈ Z.
Recall that A 1 r is non zero by assumption. If (ν n,1 ) = 0, then η is at least of order n -µ and a necessary and sufficient condition of H ∞ -stability is that deg p ≥ deg t + 1. If (ν n,1 ) = 0 the condition is still necessary.

The results of this subsection are illustrated later by Example 1 in §6.

In the next section, the same stability analysis will be realized for other cases of systems with single chains of poles, thus completing the analysis for this class of systems.

Other cases.

Returning to the approximation of the characteristic equation around poles of a single chain, we see that the terms of highest order are only constituted from those of g 1 and g 2 given in (3.3) and (3.4) respectively since the highest order of g 3 (3.5) is smaller than that of g 2 . While that term of g 2 remains the same, i.e. (-ν n,1 A 1 r ), for all cases of single chains, that of g 1 is decided by its non-zero terms. Some examples are cases where • B 0 r = 0 and C 0 r = 0 (the term of highest order of g 1 is τ 2µ C 0 r /(2πn) 2µ ), • B 0 r = 0, C 0 r = 0, D 0 r = 0 (the term of highest order of g 1 is τ 3µ D 0 r /(2πn) 3µ ), • and so on. Similarly, we obtain easily ν n,1 for the cases above by noting that in the development of the characteristic equation around a pole the coefficient of the highest order is zero. In general,

ν n,1 = τ 2πn xrµ Kr (4.6)
with Kr a function in r and the coefficients α k , β k , . . . in (2.2), and x r ∈ N. Note that we get a value of x r for each root r of multiplicity one of c d (z) (2.3), where comes the subscript. For example, if r satisfies the first case above, i.e. B 0 r = 0 and C 0 r = 0, then x r = 2 and Kr = C 0 r /(A 1 r ). Now, as in the previous subsection, we can realize a similar analysis about the location of the chain of poles relative to r. Here, sgn( (ν n,1 )) = sgn( ( -xrµ Kr )). Therefore, (ν n,1 ) < 0 for all n ∈ Z, i.e. the chain is on the left of the asymptotic axis, if and only if cos

x r µπ 2 ( Kr ) < -sin x r µπ 2 ( Kr ) . (4.7) However, (4.7) does not hold for x r µ = 2k + 1 for k ∈ N, 0 < µ < 1. In that case, we will be in the same situation as the case µ = 1 [START_REF] Bonnet | Stability of neutral systems with commensurate delays and poles asymptotic to the imaginary axis[END_REF], where either (ν n,1 ) = 0 (when ( Kr ) = 0) or (ν 1 ) = ±c (when ( Kr ) = 0), meaning that respectively more approximation terms are needed or we conclude to have both left and right chains.

From all the above analyzes about the location of poles of large modulus, the H ∞ -stability condition for systems with single chains asymptotic to the imaginary axis can be restated as follows.

Proposition 4.5. Let G(s) be a transfer function given as (2.1) and suppose that the formal polynomial c d (z) defined in (2.3) has at least one simple root of modulus one, denoted r, the other roots being of modulus strictly greater than one.

1. Suppose that (ν n,1 ) < 0 for all r and that G has no unstable pole of small modulus, then G is H ∞ -stable if and only if deg p ≥ deg t + max r {x r }, where, for each r, x r = y 1 /µ and -y 1 is the order in n of ν n,1 . 2. If (ν n,1 ) = 0 for any r, then the condition deg p ≥ deg t + max r {x r } is necessary for H ∞ -stability. Proof. As in the proof of Proposition 4.4, let us consider the numerator of G(s) at s on the imaginary axis near poles of the neutral chain relative a root r. Let s = s n + η ∈ R, we also have

p(s µ ) + N k=1 q k (s µ )e -ksτ ≈ τ |η| |p(s µ n )| A 1 r as n → ∞, n ∈ Z.
Here, if (ν n,1 ) = 0, then η is at least of order n -xrµ . Remark 4. Because the mentioned cases are defined by algebraic relations involving some equality ones, systems which belong to one of these cases may easily change category due to small changes in their coefficients. For example, in the case where B 0 r = 0 and C 0 r = 0, if the coefficients β k , k = 1, . . . , n change a little, it is likely that the equality B 0 r = 0 is no longer satisfied and the system now belongs to the case studied in the previous subsection. Hence, the stability of the system may as well change. This is the same situation for other cases that we will consider in the sequel since they will also be defined by the same type of conditions.

Multiple chains.

While the stability analysis of single chains under different conditions results in similar conclusions, the stability of multiple chains differs significantly from case to case.

In this section, we do not aim for a complete analysis of general cases but for a large class of systems. This analysis reveals interesting different behaviors.

5.1.

The case where m ≥ 2 and B 0 r = 0. Under the same condition, an exhaustive H ∞ -stability analysis for neutral chains relative to roots of multiplicity one of (2.3) has been conducted in Subsection 4.1. In this section, multiple chains will be studied. The first step is also to approximate the pole location. However, the analysis based on this approximation ends shortly.

Theorem 5.1. Let G(s) be a fractional neutral delay system defined by (2.1), and suppose that at least one root of the formal polynomial c d (z) defined by (2.3) has multiplicity m > 1. If for such a root, denoted by r, the condition (4.1) is satisfied, then for large enough n ∈ Z, poles of neutral chains relative to those m identical roots are approximated by

s n τ = λ n + ν n,1 + o(n -µ/m ),
with λ n given by (2.5) and

(5.1)

ν n,1 = ν 1 n -µ/m ,
where

(5.2) ν m 1 = (-1) m+1 m!τ µ B 0 r (2π) µ A m r ,
which gives rise to m different neutral chains, sharing the same λ n , the same

y 1 = µ/m, but different values of ν 1 .
Proof. Because of the condition (4.1), the highest order in n of g 1 is -µ, which is obviously higher than that of g 3 . Therefore, in order to vanish the highest order of d(s n )/p(s µ n ), those of g 1 and g 2 must be equal. Recall that the highest order of g 2 is -my 1 . Then y 1 = µ/m and from (3.2), we obtain

τ µ B 0 r (2π) µ n -µ + (-1) m A m r m! ν m 1 n -my1 + o(n -µ ) = 0, then (5.
2) holds, which completes the proof. It is interesting to see that, in this case, the order in n of ν n,1 is no longer a multiple of µ as in the cases of single chains but -µ/m. Also, note that (5.1) and (5.2) are identical to (4.2) for m = 1.

Although approximations of poles of single and multiple chains seem to share a similar form, we will show in the next corollary that they have a different position relative to their asymptotic axis.

Corollary 5.2. Let G(s) be a fractional neutral delay system defined by (2.1). If a root r of multiplicity m > 1 of the formal polynomial c d defined in (2.3) satisfies (4.1), then there exist neutral chains of poles on both sides of the corresponding asymptotic axis (s) = -ln |r|/τ .

Proof. Under the assumptions, ν n,1 is given by (5.1) and (5.2) for neutral chains relative to r.

Recall from Remark 3 that the location of poles of large modulus around the asymptotic axis is decided by the sign of (ν 1 ) in the upper half-plane, i.e. n > 0, and by the sign of ( (ν 1 ) cos(µπ/m) + (ν 1 ) sin(µπ/m)) in the lower half-plane, i.e. n < 0.

First, we consider (ν 1 ). Note that the equation of ν m 1 (5.2) has m distinct roots that are equally distributed on a circle centered at the origin in the complex plane.

If m ≥ 3, it is obvious that there exist both roots with positive and negative real part.

If m = 2, the two roots are symmetric with respect to the origin. Hence, there is always one root with positive real part and the other root with negative real part except for the case of two purely imaginary roots.

In that case, (ν 1 ) = 0 and (ν 1 ) = ±c = 0. Hence, (ν 1 ) cos(µπ/m) + (ν 1 ) sin(µπ/m) = ±c = 0 and thus in the lower half-plane there are one chain on the left and one chain on the right of the asymptotic axis.

In conclusion, if any multiple root of modulus one of (2.3) satisfies the condition (4.1), then the system is unstable. Clearly, this condition does not depend on τ and µ, with 0 < µ < 1.

In the next subsections, we progress in the analysis of the remaining cases and we start in Subsection 5.2 with the case of B 0 r = 0. 5.2. The case where m ≥ 2, B 0 r = 0, B 1 r = 0, C 0 r = 0. In the previous case, all neutral chains relative to the same root r of (2.3) approach the asymptotic axis at the same rate since the corresponding approximation terms have the same order in n. This may no longer occur for the current case as well as for other cases that we will study later.

Theorem 5.3. Let G(s) be a neutral delay system defined by (2.1), and suppose that one of the roots of the formal polynomial c d (z) defined in (2.3) has multiplicity m > 1. If this root, denoted by r, satisfies

B 0 r = 0, (5.3)
B 1 r = 0, (5.4)
C 0 r = 0, (5.5)
then, for large enough n ∈ Z, poles of neutral chains relative to those m identical roots are approximated by

s n τ = λ n + ν n,1 + o(n -y1 ),
with λ n given by (2.5) and

ν n,1 = ν 1 n -y1 ,
where for m = 2, y 1 = µ and ν 1 satisfies the equation

A 2 r 2 ν 2 1 - τ µ B 1 r (2π) µ ν 1 + τ 2µ (2π) 2µ C 0 r = 0, (5.6)
and for m ≥ 3, (y 1 , ν 1 ) takes m different pair of values below

y 1 = µ, ν 1 = τ µ C 0 r (2π) µ B 1 r , (5.7)
y 1 = µ m -1 , ν m-1 1 = (-1) m m!τ µ B 1 r (2π) µ A m r , (5.8)
the latter gives m -1 different values of ν 1 corresponding to the same value of y 1 = µ/(m -1).

Proof. From the conditions (5.3)-(5.5), we deduce that the highest orders in n of g 1 , g 2 , and g 3 , which are given by (3.3), (3.4), and (3.5), are -2µ, -my 1 , and -µ -y 1 respectively. Example 2. Therefore, the complementary condition to ensure H ∞ -stability of the system in that situation is the objective of the following proposition.

Proposition 5.5. Let G(s) be a neutral delay system defined by (2.1), and suppose that the formal polynomial c d (z) defined in (2.3) has at least one root of modulus one of multiplicity two, the other roots being of modulus strictly greater than one. We also suppose that each root of modulus one of c d (z) satisfies (5.3)- (5.5). If G has no unstable poles of small modulus and

(ν n,1 ) < 0 then G is H ∞ -stable if and only if deg p ≥ deg t + 2.
Proof. Under the assumptions, all the poles of G(s) are in the open left halfplane. Now, G(s) is H ∞ -stable if and only if G(s) is bounded on the imaginary axis. Therefore, let us consider the magnitude of G(s) on the imaginary axis by first examining its denominator d(s).

Let s = s n + η n ∈ jR, where s n is one of poles of the neutral chain relative to a root r of modulus one and of multiplicity two of c d (z). Recall that s n = (λ n + ν 1 n -µ )/τ + o(n -µ ) and note that (λ n ) = 0. Since (ν n,1 ) = 0, then η n is at least of order n -µ . In this case, we can write η n = ηn -µ + o(n -µ ), and thus s = [λ n + (ν 1 + ητ )n -µ ]/τ + o(n -µ ), which is of the same form as s n recalled earlier if we replace

ν 1 = ν 1 + ητ .
Therefore, the developments of the denominator of G around s and s n are the same. Note that the development of d(s n ) as |s n | → ∞ is obtained from (3.2) by collecting terms of highest order of g 1 , g 2 , g 3 as follows

d(s n ) = p(s µ n ) f (ν 1 ) n 2µ + o(n -2µ )
where f (ν 1 ) is the left expression of (5.6). Similarly, d(s) as |s| → ∞, s ∈ jR near s n is given by

d(s) = p(s µ ) f (ν 1 + ητ ) n 2µ + o(n -2µ )
Now, we will prove that f (ν 1 + ητ ) = 0. Let us denote ν

(1) 1 and ν

(2) 1 two roots of f (ν 1 ) and first consider f (ν

(1)
1 + ητ ). We see that f (ν 2 ) = 0 under the assumption (ν n,1 ) < 0. Therefore, f (ν

(1) 1 + ητ ) = 0 if and only if ν (1) 1 + ητ = ν (2) 1 , which is in turn equivalent to η = ν (2) 1 -ν (1) 1 τ
(1)
1 +ητ ) = 0. Similarly, we can prove that f (ν

(2)
1 +ητ ) = 0. Hence, the order in n µ of the denominator of G(s) is deg p -2.

While systems considered in the previous subsection were all unstable, we have been able to find in this subsection systems with multiple chains asymptotic to the imaginary axis which are H ∞ -stable. We will then continue our analysis for another case in order to see in which situation there may exist H ∞ -stable systems.

• for 2 ≤ m ≤ 3, C 0 r = 0, • for m = 4, B 2 r = 0 and C 0 r = 0 • for m ≥ 5, B 2 r = 0. Then there exist neutral chains of poles on both sides of the asymptotic axis (s) = -ln |r|/τ .

Proof. For 2 ≤ m ≤ 3 and m ≥ 5, the proof is similar to that of Corollary 5.2. Now, we consider the case of m = 4. By replacing ν 2 1 = x in (5.18), we obtain

x 2 4! A 4 r + xτ µ 2(2π) µ B 2 r + τ 2µ (2π) 2µ C 0 r = 0
Let us denote x 1 and x 2 the two roots of the above equation. Equation (5.18) has at least one value of ν 1 with positive real part except the case where both roots x 1 , x 2 are negative. However, we will demonstrate that this case does not exist.

The two roots of the equation satisfy

x 1 + x 2 = - 12τ µ B 2 r (2π) µ A 4 r = - 12τ µ (2π) µ Kr
where Kr = B 2 r /A 4 r . We consider r ∈ R and r ∈ C\R. If r is real, then x 1 + x 2 is not real. Therefore, x 1 and x 2 cannot be both real. If r is not real, then r is also a root of (2.3). Denote x 1 and x 2 roots corresponding to r. Hence, they satisfy

x 1 + x 2 = - 12τ µ (2π) µ Kr .
Therefore

x 1 + x 2 + x 1 + x 2 = - 24τ µ (2π) µ ( Kr ),
indicating that x 1 , x 2 , x 1 , and x 2 cannot be all real. Corollary 5.7 shows that if (2.3) possesses a multiple root of modulus one satisfying the conditions in Theorem 5.6 then the system is unstable.

Examples.

The numerical examples in this section illustrate several results obtained in the preceding sections. They are however proposed for illustrative purpose only and do not come from models found in the literature.

Example 1. First, we consider the system with the transfer function given by

G 1 (s) = s µ + 1 s 2µ + (-1.9s 2µ + s µ )e -s + (s 2µ -s µ + 0.3)e -2s .
For this system, the fractional order is µ and the delay is τ = 1. It is easy to see that the coefficients of the development q k (s µ )/p(s µ ) are α 1 = -1.9, β 1 = 1, α 2 = 1, β 2 = -1, and thus the formal polynomial is c d (z) = 1 -1.9z + z 2 , which has two complex conjugate roots r = (19 ±  √ 39)/20 of multiplicity m = 1. Since |r| = 1 for each r, then the asymptotic axis defined by (s) = -ln |r|/τ is the imaginary axis. As Therefore, in the upper half-plane, i.e. n > 0, the two neutral chains of poles are on the left of the imaginary axis. So are the chains in the lower half-plane since poles of G(s) are symmetric about the real axis, which is due to the fact that the denominator of G(s) is a quasi-polynomial with real coefficients.

The same conclusion about the location of neutral poles can be drawn using Corollary 4.3. The critical value of µ is

µ c = (2/π) arctan(-(K r )/| (K r )|) = 0.8989 with K r = 2 k=1 β k r k / 2 k=1 kα k r k .
Recall that µ c is the same for both r. Since µ = 0.5 < µ c , then the two neutral chains of poles relative to r are on the left of the imaginary axis as we can see in Figure 1a. In addition, all poles of small modulus of the system are the open left half-plane. Then Proposition 4.4 shows that G is H ∞ -stable since deg t = deg p -1. Indeed, G(s) is bounded on the imaginary axis, which can be seen in Figure 2. If µ = 0.9 > µ c , then chains of poles go to the right of the imaginary axis (see Figure 1b).

Example 2. The system is given by G 2 (s) = (s + (-2s + s 0.5 + 0.25)e -s + (s -s 0.5 )e -2s ) -1 .

We see that the delay is τ = 1 and the fractional order is µ = 0.5. The polynomial c d (z) given in (2.3) has root r = 1 of multiplicity two, then the system has two chains of poles asymptotic to the imaginary axis. The system satisfies 2 k=1 β k r k = 0, 2 k=1 kβ k r k = 0, and 2 k=1 γ k r k = 0, then Theorem 5.3 is applied. Equation (5.6) has a double root, which gives ν n,1 = (-0.1410 + 0.1410)/n 0.5 for n → +∞. Therefore, the two neutral chains are on the left of the imaginary axis.

If some parameters of G 2 (s) change slightly, the system might fail to satisfy the condition 2 k=1 β k r k = 0, and thus is no longer stable due to Corollary 5.2. This remark fits in the following system G 2 (s) = (s + (-2s + s 0.5 + 0.25)e -s + (s -(1 + ∆)s 0.5 )e -2s ) -1 .

If ∆ = 0, then 2 k=1 β k r k = 0, thus Corollary 5.2 states that the system has a chain of poles in the right half-plane.

We observe the chains of poles of G 2 (s) and G 2 (s) with ∆ = 0.01 in Figure 3a and3b. The unstable chain of G 2 crosses the imaginary axis from left to right. shows that G 2 (s) is stable in the sense of H ∞ -stability. Indeed, the system does not have unstable poles and is bounded on the imaginary axis (see Figure 4). Clearly, the system defined by (s 0.5 + 1)G 2 (s) is unstable since the order of the numerator is too high making the transfer function unbounded on the imaginary axis (see Figure 5). Example 3. We consider the system with the transfer function given by G 3 (s) = (s 0.8 + (-3s 0.8 + 3s 0.4 + 1)e -s + (3s 0.8 -5s 0.4 + 2)e -2s +(-s 0.8 + 2s 0.4 + 3)s -3s ) -1 .

Here, we see that µ = 0.4, τ = 1, and c d (z) = 1 -3z + 3z 2 -1 with a root r = 1 of multiplicity m = 3. Therefore, the chains of poles approach the imaginary axis. Since

Conclusion.

Fractional delay systems of neutral types where poles approach the imaginary axis is delicate for stability analysis. In this paper, we answer the stability question in the sense of H ∞ -stability for a large class of systems, in particular systems with multiple chains asymptotic to the imaginary axis, and the necessary and sufficient conditions obtained are related not only to the location of poles w.r.t. the imaginary axis but also the relative order between the numerator and the denominator of the transfer function. These results will also be of use to decide on H ∞ -stabilizability of several classes of fractional delay systems by rational or fractional controllers (with delays). The deployed method can be used for other cases which are not examined here. However, it requires time and effort for each particular system. With the aim of integrating the results into a computation software, we currently attempt to find approximations of neutral poles for general cases, both fractional and classical systems. Some small changes in the fractional orders and the delays may lead to the class of systems with non-commensurate fractional orders and non-commensurate delays. The distribution of poles of these systems is complicated and different from the systems considered in this paper. For example, for integer-order systems with noncommensurate delays, poles are located in some vertical strips and no longer asymptotic to some vertical lines [START_REF] Avellar | On the zeros of exponential polynomials[END_REF]. Therefore, other techniques will be needed to handle these systems.

  ln(•) denotes the real logarithm function, Arg(x) the argument of x ∈ C satisfying -π < Arg(x) < π, and [x] the integer part of x ∈ R. 2. A class of fractional neutral time-delay systems. We consider fractional neutral time-delay systems with transfer function of the form (2.1)

  τ since s ∈ R and (ν[START_REF] Avellar | On the zeros of exponential polynomials[END_REF]

2 k=1

 2 β k r k = 0 for both r, Theorem 4.1 is applied. If µ = 0.5, from Theorem 4.1 we obtain ν n,1 = (-0.1636 + 0.1185)/n 0.5 for r = (19 +  √ 39)/20 and ν n,1 = (-0.1185 + 0.1636)/n 0.5 for r = (19 - √ 39)/20.

9 Fig. 1 :

 91 Fig. 1: Neutral chains of poles for G 1 (s)

FrequencyFig. 2 :

 2 Fig. 2: Bode diagram for G 1 (s)

Fig. 3 :

 3 Fig. 3: Neutral chains of poles

FrequencyFig. 4 :FrequencyFig. 5 :

 45 Fig. 4: Bode diagram for G 2 (s)

3 k=1 β k r k = 0, 3 k=1 kβ k r k = - 1 , and 3 k=1 γ k r k = 6 ,Fig. 6 :

 331366 Fig. 6: Neutral chains of poles for G 3 (s)

in(5.8), and thus the existence of a neutral chain on the right of the asymptotic axis. Furthermore, the result in the corollary does not depend on τ and µ, with 0 < µ < 1.Under the conditions in Theorem 5.3, two chains relative to r of multiplicity two may both lie on the left of the asymptotic axis. We will see such a system later in §6,

Acknowledgment. We would like to thank Hitay Özbay for helpful discussions.

The following cases may occur in order to eliminate the terms of highest order of the denominator at s n 2µ = my 1 < µ + y 1 (5.9) 2µ = µ + y 1 < my 1 (5.10) my 1 = µ + y 1 < 2µ (5.11)

The case (5.9) is eliminated as it cannot be satisfied for m ≥ 2. The case (5.12) is equivalent to y 1 = µ, m = 2 and, from (3.2), we have

and then (5.6) follows immediately. When m > 2, it is easy to see that both (5.10) and (5.11) are satisfied. From (5.10), we deduce that y 1 = µ and thus (3.2) can be rewritten as

giving one value ν 1 in (5.7).

Other values of ν 1 are derived from the case (5.11), where

giving m -1 non-zero values of ν 1 in (5.8). Remark 5. A previous version of the above theorem was stated in [START_REF] Nguyen | Stability analysis of fractional neutral time-delay systems with multiple chains of poles asymptotic to same points in the imaginary axis[END_REF]. However, the result about ν n,1 for the case m ≥ 3 was incomplete. Indeed, only the value (5.7) of ν n,1 was given and the others values with different order in n were missing.

As we have seen from (5.7) and (5.8) in the above theorem, due to different order of ν n,1 , the chains of poles relative to a multiple root r with m ≥ 3 approach the asymptotic axis with different rates. An example of such a system is given in §6, Example 3.

We recognize that ν m-1

1

, m ≥ 3 in (5.8) has the same pattern as ν m 1 , m ≥ 2 in (5.2), leading to the same conclusion on stability.

Corollary 5.4. Let G(s) be a neutral delay system defined by (2.1), and suppose that at least one root r of the formal polynomial c d (z) defined in (2.3) has multiplicity m ≥ 3, satisfies (5.3) and (5.4). Then there exist neutral chains of poles on both sides of the asymptotic axis (s) = -ln |r|/τ .

Proof. The proof is similar to the one of Corollary 5.2. Remark 6. The condition (5.5) is omitted in the above corollary. Indeed, whether or not the condition holds does not affect ν m-1 5.3. The case where m ≥ 2, B 0 r = 0, B 1 r = 0, B 2 r = 0, and C 0 r = 0. As for the previous cases, pole location is considered first. Theorem 5.6. Let G(s) be a neutral delay system defined by (2.1), and suppose that one of the roots of the formal polynomial c d (z) defined in (2.3) has multiplicity m > 1. If this root, denoted by r, satisfies

then, for large enough n ∈ Z, poles of neutral chains relative to those m identical roots are approximated by

with λ n given by (2.5) and

where for 2 ≤ m ≤ 3, y 1 = 2µ/m and

for m = 4, y 1 = µ/2 and ν 1 satisfies

and for m ≥ 5, (y 1 , ν 1 ) takes one of m different pairs of values

)

Here again it is clear that (5.17) gives rise to m different complex values of ν 1 , (5.18) gives rise to four, (5.19) to two, and (5.20) to m -2.

Proof. Under the assumptions, the terms of highest order in n of g 1 , g 2 , and g 3 , which are given by (3.3), (3.4), and (3.5), are -2µ, -my 1 , and max{-µ -2y 1 , -2µy 1 } respectively. Obviously, we just need to compare the first three orders as -2µ > -2µ -y 1 . For further steps to determine y 1 and ν 1 , which are the same as in the proof of Theorem 5.3, see Appendix C.

Remark 8. Note that in the case 2 ≤ m ≤ 3, condition (5.15) is not necessary and in the case m ≥ 5 condition (5.16) is not necessary (and we may conclude as well on the presence of chains of poles in the right half-plane).

Some quick observation leads to the following conclusions on the stability of the system in the current case.

Corollary 5.7. Let G(s) be a neutral delay system defined by (2.1), and suppose that at least one root of the formal polynomial c d (z) defined in (2.3) has multiplicity m ≥ 2, satisfies B 0 r = 0, B 1 r = 0 and Appendix A. Series expansion of the characteristic equation.

As |s n | → ∞, by using (2.2), we develop (3.1) as follows

Note that e -λn = r and e -kνn,i = 1 +

Thus when n is large enough, (A.1) becomes

and we obtain

where L(x) is defined as in (3.6). After simple computations, we get (3.2).

Appendix B.

Lemma A.1. Let r be a root of multiplicity m > Proof. Under the assumptions, the terms of highest order in n of g 1 , g 2 , and g 3 , which are given by (3.3), (3.4), and (3.5), are -2µ, -my 1 , and max{-µ -


François Béguin

Sylvain Crovisier

Tobias Jäger

Dmitri V Anosov

A dynamical decomposition of the torus into pseudo-circles

We build an irrational pseudo-rotation of the 2-torus which is semiconjugate to an irrational rotation of the circle in such a way that all the fibres of the semi-conjugacy are pseudo-circles. The proof uses the well-known 'fast-approximation method' introduced by Anosov and Katok.

Introduction

It is well known that continua (connected compact metric spaces) with complicated structure naturally appear in smooth surface dynamics. A striking example is provided by the pseudocircle, introduced by Bing [START_REF] Bing | Concerning hereditarily indecomposable continua[END_REF] and characterized by Fearnley [Fe]. It is a continuum which:

-can be embedded in S 2 and separates, -is circularly chainable: it admits coverings into compact subsets (A i ) i∈Z/nZ whose diameter are arbitrarily small, such that A i ∩ A j = ∅ if and only if if i = j ± 1 or i = j,

-is indecomposable: it cannot be written as the union of two proper continua, -and whose non-trivial proper subcontinua are indecomposable, homogeneous (any point can be sent on any other point by some homeomorphism) and all homeomorphic to the same topological space (called the pseudo-arc).

Handel [Ha] has built a smooth diffeomorphism of S 2 preserving a minimal invariant set homeomorphic to the pseudo-circle. Later, Prajs [Pr] has constructed a partition of the annulus into pseudo-arcs, and likewise his method could be used to produce partitions of the torus into pseudo-circles. It was not known, however, if such a pathological foliation could be 'dynamical', that is, invariant under the dynamics of a torus homeomorphism or diffeomorphism that permutes the leaves of the foliation. Conversely, if a homeomorphism of the two-torus is semiconjugate to an irrational rotation of the circle, one may wonder whether most, or at least some, of the fibres of the semi-conjugacy must have a simple structure or even be topological circles. We give a positive answer to the first and a negative to the second of these questions. Denote by T d = R d /Z d the d-dimensional torus and by Diff ω vol,0 (T 2 ) the space of real-analytic diffeomorphisms of T 2 that are isotopic to the identity and preserve the canonical volume. Theorem. There exists a minimal diffeomorphism f ∈ Diff ω vol,0 (T 2 ) whose rotation set is reduced to a unique totally irrational vector and which preserves a partition C of T 2 into pseudo-circles.

Moreover there exists a continuous map p : T 2 → T 1 which semi-conjugates f to an irrational rotation. The elements of C are the pre-images p -1 (x). This result has implications for a number questions that naturally come up in the rotation theory on the torus and, more specifically, the dynamics of irrational pseudo-rotations. We discuss these issues in more detail in Section 4, alongside with the uniqueness of the semiconjugacy.

Idea of the construction. The diffeomorphism f is obtained as limit of a sequence of diffeomorphisms f n that are conjugated to rational rotations R αn by diffeomorphisms H n isotopic to the identity, following the celebrated Anosov-Katok method, see [START_REF] Fayad | Constructions in elliptic dynamics[END_REF]. As a side effect, this means that its dynamics can be made uniquely ergodic, although we will not expand on this. Note that most of the constructions using this method deal with the C ∞ category; some cases, as [START_REF] Fayad | Analytic uniquely ergodic volume preserving maps on odd spheres[END_REF] allow to work in the real-analytic category. The sequence

f n = H -1 n • R αn • H n is obtained inductively. At stage n, the diffeomorphism f n preserves the foliation V n by vertical circles H -1 n ({x} × T 1 )
. The main requirement is to have the circles of the foliation V n+1 arbitrarily close to the circles of V n in the Hausdorff topology, but more crooked. Then the partitions V n will converge to the partition into pseudo-circles. The foliation V n+1 is built inductively in the chart defined by the conjugacy H n , as the preimage of the foliation into vertical lines under a new homeomorphism h n+1 . A sketch of the construction is given in Figure 1. Since we then let

H n+1 = h n+1 • H n , this ensures that the foliation H n+1 (V n+1
) is again the one given by vertical circles. In order to obtain h n+1 , one first builds a leaf of V n+1 in the chart given by V n which is crooked with respect to the vertical circles in such a way that the leaf is transverse to a linear flow ϕ n+1 . This first leave is the image of some vertical circle. One then obtains the complete foliation (and thus the definition of h n+1 ) by pushing this initial leave by the flow.

h -1 n+1 ϕ n+1
Figure 1: In the coordinates defined by H n (given by the dashed lines), h -1 n+1 maps straight lines (solid lines on the left) to crooked leaves of V n+1 (on the right). The latter are translates of each other along the flow lines of ϕ n+1 .

We note that A. Avila has announced recently the construction of an element of Diff ω vol,0 (T 2 ) whose rotation set is a non-trivial compact interval contained in line with irrational slope and which does not contain any rational point (a counter-example to one case of a conjecture by Franks and Misiurewicz [FM]). His construction may somewhat be compared to ours: the diffeomorphism is obtained as the limit of diffeomorphisms acting periodically on the leaves of a foliation by circles. In his case however the homotopy class of the leaves is modified at each stage of the construction.

2 A criterion for the existence of a partition into pseudo-circles 2.a -Crooking. The construction of the pseudo-circle uses the following notions.

Definitions. A circular chain is finite family of sets D = {D , ∈ Z/N Z} such that D k intersects D if and only if k -∈ {-1, 0, +1}. A circular chain D = {D i , i ∈ Z/N Z} said to be crooked inside another circular chain D = {D , ∈ Z/N Z} if there exists a map : Z/N Z → Z/N Z with the following properties.

-

D i ⊂ D (i) for each i ∈ Z;
-if i < j are such that for all i < k < j the element (k) belongs to the same interval bounded by (i) and (j) (either positively or negatively oriented in Z/N Z) and the length of this interval is greater than 4, then there exists u, v with i < u < v < j such that d( (u), (j)) ≤ 1 and d( (v), (i)) ≤ 1. (Here d denotes the canonical distance on Z/N Z.)

The pseudo-circle can then be obtained as follows. For the sake of consistency with the later sections, we work in the torus instead of R 2 and require that the circular chains -and thus the resulting pseudo-circle -are homologically non-trivial.

Theorem ( [START_REF] Bing | Concerning hereditarily indecomposable continua[END_REF][START_REF] Fearnley | Classification of all hereditarily indecomposable circularly chainable continua[END_REF]). Consider a sequence (D n ) n≥0 of circular chains of open topological disks in T 2 . Assume that -D n+1 is crooked inside D n for each n, -the closure of D∈D n+1 D is contained in D∈Dn D for every n, -the maximal diameter of the elements of D n goes to zero as n → +∞.

-the union D∈Dn D contains homotopically non-trivial loops of a unique homotopy type v ∈ Z 2 \ {0}.

Then the compact set X := n≥0 D∈Dn D is homeomorphic to the pseudo-circle. Moreover, X is an annular continuum of homotopy type v (see [JP, JT]).

At some point later on, we will have to speak about lifts of circular chains in the torus to the universal covering R 2 , and similarly about lifts of circular chains of intervals in the circle to R. Suppose that D is a circular chain of topological disks in T 2 as above and denote by π : R 2 → T 2 the canonical projection. Note that for each D ∈ D, the preimage π -1 (D) consists of a countable number of connected components, each of which is a topological disk homeomorphic to D and disjoint from all its integer translates. Suppose in addition that Diam(D ) < 1/4 for all ∈ Z/N Z, so that none of the unions D ∪ D +1 is essential in the torus (contains a homotopically non-trivial loop).

Definitions. A lift D of D is a sequence of topological disks ( D ) ∈Z of R 2 such that -for all ∈ Z the disk D is a connected component of π -1 (D ); -D k intersects D if and only if k -∈ {-1, 0, 1}.
The disc D k+N is the image of D k by translation by a vector v ∈ Z 2 which does not depend on the lift, nor on k, and is called the homotopy type of D.

Note that if D and D are circular chains of topological disks with homotopy type v ∈ Z 2 \{0} as above, D is crooked inside D and D, D are lifts in the above sense, then there exists a function ˆ : Z → Z (to which we refer as a lift of :

Z/N Z → Z/N Z) such that -D i+N = D i + v and D i+N = D i + v; -ˆ (i + N ) = ˆ (i) + N ; -D i ⊆ D (i) for all i ∈ Z;
-if i < j < i + N are such that for all i < k < j the integer ˆ (k) belongs to the interval bounded by ˆ (i) and ˆ (j) and

| ˆ (j) -ˆ (i)| > 4, then there exists u, v with i < u < v < j such that | ˆ (u) -ˆ (j)| ≤ 1 and | ˆ (v), ˆ (i)| ≤ 1.
All these remarks apply in an analogous way to circular chains of intervals in the circle and their lifts to R.

During the construction, we will also use another notion of the crooking. Note that ε-crooked maps exist for any ε (see [START_REF] Bing | Higher-dimensional hereditarily indecomposable continua[END_REF]).

2.b -Elements of the construction. Let π : T 2 → T 1 be the projection on the first coordinate. Let B(N ) be the covering of the circle by N open intervals defined as follows

B(N ) = {B i , i ∈ Z/N Z} where B i = i -5/4 N , i + 1/4 N .
(2.1)

We will build inductively:

-a sequence of integers (N n ) n≥0 , -a sequence of positive real numbers (ε n ) n≥0 , -a sequence of conjugating diffeomorphisms (H n ) n≥0 in Diff ω vol,0 (T 2 ),
-a sequence of rational rotations (R αn ) n≥0 of T 2 .

To N n , ε n , H n , R αn , we will associate:

-for each x ∈ T 1 , the annulus A n,x which is the image under

H -1 n of the vertical annulus (x -ε n , x + ε n ) × T 1 , -for each x ∈ T 1 , the covering D n,x of the annulus A n,x defined by D n,x = H -1 n ((x -ε n , x + ε n ) × B i ) | B i ∈ B(N n ) (note that D n,x is a circular chain with N n elements), -the projection p n = π 1 • H n : T 2 → T 1 (where π 1 : T 2 → T 1 is the projection to the first coordinate), -the diffeomorphism f n = H -1 n • R αn • H n ∈ Diff ω vol,0 T 2 .
We will denote by rn qn , sn qn the coordinates of α n , with r n , s n ∈ Z and q n ∈ N -{0}.

2.c -Inductive properties. The torus T 2 is embedded as the subset

{(z 1 , z 2 ) ∈ C 2 , |z 1 | = |z 2 | = 1} of C 2 .
One will consider homeomorphisms f of T 2 such that both f and f -1 extend as holomorphic functions defined on a neighborhood of ∆

= (z 1 , z 2 ) ∈ C 2 , |z 1 |, |z 2 | ∈ 1 2 , 2 .
One then introduces the supremum norm . ∆ on ∆ and the metric

d 0 (f, f ) = max( f, f ∆ , f -1 , f -1 ∆ ) .
The sequences (N n ), (ε n ), (H n ), (R αn ) will be constructed inductively so that the following properties hold.

1. For each x ∈ T 1 , the circular chain D n+1,x is crooked inside the circular chain D n,x (in particular, the annulus A n+1,x is contained in the interior of the annulus A n,x ), and the supremum of the diameter of the elements of the coverings D n+1,x is less 1 n+1 , 2. The angle α n+1 is close, but not equal, to α n . More precisely: | r n+1 q n+1 -rn qn | and | s n+1 q n+1 -sn qn | are smaller than 1/(2 n+1 q n ) and the orbits of R α n+1 are 1 2 n+1 -dense in T 2 . 3. Every orbit of the diffeomorphism f n+1 is 1 n+1 -dense in T 2 . 4. The diffeomorphism f n+1 is (very) close to f n . More precisely, both f n+1 , H n+1 and their inverses extend holomorphically on (C \ {0}) 2 and satisfy:

(a) d 0 (f i n+1 , f i n ) < min( 1 2 d 0 (f i n , f i n-1 ), 1 n ) for i = 1, . . . , q n ; (b) d 0 (f n+1 , f n ) < ηn
2 where η n is choosen such that, for every homeomorphism g in the ball (for d 0 ) centered at f n of radius η n , the rotation set of g is contained in the ball centered at α n of radius 1 n . 5. The projection p n+1 is close to p n for the C 0 -topology. More precisely:

d 0 (p n+1 , p n ) < 1 2 n .
Remarks. The existence of the real number η n used in property 5.b is a consequence of the upper semi-continuity of the rotation set ρ(F ) with respect to F [START_REF] Misiurewicz | Rotation sets for maps of tori[END_REF]Corollary 3.7].

Property 1 (more precisely, the fact that D n+1,x is crooked inside D n,x ) implies that the sequence of conjugating diffeomorphisms (H n ) will necessarily diverge. Nevertheless, the sequence of diffeomorphisms

(f n ) = (H -1 n • R αn • H n ) will converge (Property 4
). This convergence is obtained by using the well-known ingredients of the Anosov-Katok method:

-one first chooses a conjugating diffeomorphism H n+1 of the form

H n+1 = h n+1 • H n ,
where h n+1 might be very wild, but commutes with the rotation R αn ; this implies that

f n = H -1 n+1 • R αn • H n+1 ; -then, choosing α n+1 close enough to α n is enough to ensure that f n+1 = H -1 n+1 •R α n+1 •H n+1 is close to f n = H -1 n+1 • R αn • H n+1 .
A specific point in our construction is that, although the sequence of diffeomorphisms (H n ) will diverge, we require that the sequence of maps (π • H n ) converges (Property 5). Indeed, we want that the fibers of π 1 • H n converge to pseudo-circles "foliating" T 2 .

2.d -Proof of the theorem. One can easily check that the theorem follows from the inductive properties stated above. Properties 1 imply that, for every x ∈ T 1 , the sequence of annuli (A n,x ) decreases and converges in Hausdorff topology to a pseudo-circle C x . Moreover, the collection of pseudo-circles C = {C x } x∈T 1 is a partition of T 2 ; this follows from the following fact:

-for every n, the collection of annuli {A n,x , x ∈ T 1 } covers T 2 , -for any x = x , the annuli A n,x and A n,x are disjoint if n is large enough. Property 4.a implies that the sequence (f n ) converges to an holomorphic function f on the interior of ∆. The same holds for (f -1 n ). Consequently, the restriction of

f to T 2 is a real-analytic diffeomorphism. Since each f n is volume-preserving, f belongs to Diff ω vol,0 (T 2 ). Given x ∈ T 1 , let x := x + π(α n ). Then f n (A n,x ) = A n,x .
From property 1, one deduces that A n+1,x is mapped by f n inside A n,x and A n+1,x is mapped by f -1 n inside A n,x . Hence, the annulus f (A n+1,x ) is contained in the d 0 (f, f n )-neighbourhood of the annulus A n,x and the the annulus f -1 (A n+1,x ) is contained in the d 0 (f, f n )-neighbourhood of the annulus A n,x . Since d 0 (f, f n ) tends to 0 as n goes to infinity, this implies that f preserves the partition in pseudo-circles

C = {C x } x∈T 1 .
Property 2 implies that the sequence (α n ) converges towards some α ∈ R 2 . It also implies that the q n first iterates of R α are 1/2 n+1 close to those of R α n+1 , hence are 1 2 n -dense in T 2 . Consequently α is totally irrational.

Properties 4 imply that the rotation set of f is reduced to {α} (indeed, they imply that d 0 (f, f n ) < η n and therefore the rotation set of f is contained in the ball of radius 1 n centered at α n for every n). Hence f is an irrational pseudo-rotation.

Consider a point z ∈ T 2 . For every n, according to property 3, the orbit of z under f n is 1 ndense in T 2 . But the orbit of z under f n is periodic of period less than q n (since f n is conjugate to the rotation R αn ). Using property 4.a, we obtain that the orbit of z under f remains at distance less than 2 n of the orbit of z under f n for a time q n . Combined with property 3, this means that the orbit of z under f is 3 n -dense in T 2 . Since n is arbitrary, f is minimal. Property 5 implies that the sequence of maps (p n ) converges in topology C 0 towards a continuous map p. For each n, the map p n semi-conjugates f n to the rotation of T 1 with angle π(α n ). Passing to the limit, it follows that the map p semi-conjugates f to the rotation of angle π(α). So we get all the conclusions of the theorem.

Inductive construction

Now we explain how to construct a sequence of integers (N n ) n≥0 , a sequence of real numbers (ε) n≥0 , a sequence of conjugating diffeomorphisms (H n ) n≥0 and a sequence of vectors (α n ) n≥0 , so that properties 1. . . 5 are satisfied. We assume that the sequences have already been constructed up to rank n. We will construct N n+1 , ε n+1 , H n+1 , α n+1 .

3.a -Preliminary constructions. Recall that we denote by pn qn , rn qn the coordinates of α n . We introduce a periodic linear flow ϕ n+1 : (t, (x, y)) → (x, y) + t • α n + t • (0, p n b n+1 ) on T 2 where b n+1 is an integer which will be specified below. Observe that the time 1 map of this flow is the rotation R αn . The first return map of ϕ n+1 on the vertical circle {x} × T 1 is the time qn pn map of ϕ n+1 . Denote by m n the period of this first return map, and observe that m n depends on α n , but does not depend on the choice of the integer b n+1 .

We also introduce a C ω map θ n : R → R satisfying the following properties.

-

θ n (m n x) -m n x is a trigonometric polynomial (hence θ n -Id is 1/m n -periodic), -θ n (0) = 0, θ n 1 2mn = 2, θ n 0, 1 2mn = [0, 2] and θ n is 1 4Nn -crooked on 0, 1 2mn , -θ n 1 mn = 1 mn , θ n 1 2mn , 1 mn = 1 mn , 2 and θ n is 1 4Nn -crooked on 1 2mn , 1 mn .
Note that θ n -Id induces a function on T 1 which extends holomorphically to C \ {0}. Recall that B(N ) denotes the covering of the circle by N compact intervals defined by (2.1). Moreover, θ itself induces a degree one map on the circle, which we denote by θ again for simplicity.

Claim. If N n+1 is large enough, then, for any ω ∈ T 1 , the circular chain of intervals {θ n (Bω) + ω, B ∈ B(N n+1 )} is crooked inside the circular chain B(N n ).

Proof. We work with a lift of the family B(N ), in the sense discussed in Section 2, obtained as a covering B(N ) of the real line by intervals of the form:

B i = i -5/4 N , i + 1/4 N , i ∈ Z. If N n+1 is large enough, each interval θ n ( B i -ω) + ω with B i ∈ B(N n+1
) has lenght strictly less than 1 2Nn and therefore is contained in an interval B ˆ (i) of the family B(N n ). Since θ n has degree 1, one can choose the function ˆ such that ˆ (i + N n+1 ) = ˆ (i) + N n .

Let i < j be two integers such that ˆ (k) belongs to the interval bounded by ˆ (i) and ˆ (j) for each i < k < j and such that 4

< | ˆ (j) -ˆ (i)| < N n . Let us choose i ≤ i < j ≤ j such that ˆ (i ) = ˆ (i), ˆ (j ) = ˆ (j)
I k = k 2mn , k+1 2mn . Since θ n is (4N n ) -1 -crooked on this interval, this implies that there exists a < c < d < b such that -θ n (a -ω) = θ n (a -ω) and θ n (d -ω) are (4N n ) -1 -close, -θ n (b -ω) = θ n (b -ω) and θ n (c -ω) are (4N n ) -1 -close. Hence d is contained in an interval B v such that | ˆ (v) -ˆ (i)| ≤ 1. Similarly, c is contained in an interval B u such that | ˆ (u) -ˆ (i)| ≤ 1. Since a < c < d < b and 2 < | ˆ (j) -ˆ (i)| one gets i < u < v < j.
In the projection to T 1 , the above shows that the circular chain {θ n (B-ω)+ω, B ∈ B(N n+1 )} is crooked inside the circular chain B(N n ) as announced.

Since m n only depends on α n , one can fix the map θ n but choose b n+1 and N n+1 arbitrarily large later in the construction.

3.b -Construction of H n+1 . Consider the map Θ n+1 : {0} × T 1 → T 1 defined by Θ n+1 : (0, y) → θ n (y)y. Since θ n -Id is 1/m n -periodic, and since the period of the return map associated to the linear flow ϕ n+1 on {0} × T 1 is equal to m n , this maps extends to a map Θ n+1 : T 2 → T 1 which is constant along the orbits of ϕ n+1 . We define H n+1 by setting

H n+1 := h n+1 • H n , where h n+1 (x, y) = ϕ n+1 - Θ n+1 (x, y) p n b n+1 , (x, y) = x, y -Θ n+1 (x, y) - Θ n+1 (x, y) q n b n+1 1, r n p n .
Since Θ n+1 is constant along the orbits of ϕ n+1 , one has

h -1 n+1 (x, y) = x, y + Θ n+1 (x, y) + Θ n+1 (x, y) q n b n+1 1, r n p n . (3.1)
Clearly, h n+1 (and hence H n+1 ) belongs to Diff ω vol,0 (T 2 ). Note also that both h n+1 and h -1 n+1 extend holomorphically to the domain (C \ {0}) 2 .

3.c -Choice of b n+1 , ε n+1 and N n+1 . Now, we explain how to fix the values of b n+1 , ε n+1 , and N n+1 so that properties 1 and 5 hold. From (3.1), if b n+1 is large enough, the image under h -1 n+1 of every point (x, y) ∈ T 2 is arbitrarily close to (x, y + Θ n+1 (x, y)). Now observe that, for each x ∈ T 1 , there exists

ω x ∈ T 1 such that Θ n+1 (x, y) = θ n (y -ω x ) -y + ω x . As a consequence, if b n+1 is large enough, the image under h -1 n+1 of every point z = (x, y) ∈ T 2 is arbitrarily close to (x, θ n (y -ω x ) + ω x ). Hence, if b n+1 is large enough and ε n+1 is small enough, the image under h -1 n+1 of a rectangle (x -ε n+1 , x + ε n+1 ) × B
is contained in an arbitrary small neighbourhood of the square {x} × (θ n (Bω x ) + ω x ). Using the claim above, this implies that the family D n+1,x is crooked inside D n,x , provided N n+1 is large enough. By continuity of H -1 n+1 , the diameter of the elements of the covering D n+1,x is less than 1 n+1 if b n+1 , N n+1 are large enough and ε n+1 is small enough. We have thus checked that property 1 holds, provided that b n+1 , N n+1 are chosen large enough and ε n+1 is chosen small enough.

By definition of p n , p n+1 , and H n+1 , in order to check that property 5 is satisfied, it is enough to check that π • h n+1 is close to π for the C 0 -topology. This is a direct consequence of the definition of h n+1 , provided that b n+1 is chosen large enough.

θ n (x) 3 2 1 0 x 0 1 2 1 h n+1 ({x} × R) ε n+1 0 1 2 1 3 2
Figure 2: Choice of the function θ (on the right) and the image of a vertical line {x} × R under the lift h n+1 of h n+1 (schematic picuture with m n = 2). Note that the preimages of vertical lines under h n+1 have a similar 'crookedness', which is the fact that is needed for our construction (see next section). The horizontal size of images (and preimages) of vertical lines under h n+1 is small compared to ε n+1 . This size is controlled by the flow lines of ϕ n+1 (dashed lines), whose direction is given by the almost vertical vector 1 b n+1 qn+(rn/pn) , 1 .

3.d -Choice of α n+1 . Clearly, one can choose α n+1 arbitrarily close to α n so that property 2 holds. By uniform continuity of H n+1 and H -1 n+1 , there exists η so that the orbits of

f n+1 = H n+1 • R α n+1 •H -1
n+1 are 1 n+1 -dense in T 2 provided that the orbits of the rotation R α n+1 are η-dense in T 2 . One can thus choose α n+1 arbitrarily close to α n so that property 3 is satisfied. By construction both f n+1 and f -1 n+1 extend holomorphically to (C \ {0}) 2 . Moreover the diffeomorphism h n+1 commutes with the flow ϕ n+1 , hence with the rotation R αn . Consequently

f n = H -1 n+1 • R αn • H n+1
. This shows that f n+1 is arbitrarily close to f n when α n+1 is chosen arbitrarily close to α n . In particular, property 4 holds provided that α n+1 is chosen close enough to α n .

The aim of this last section is to discuss, somewhat informally, the implications of our construction for some questions arising in the context of dynamics and rotation theory on the two-torus. Throughout this section, we assume f is an irrational pseudo-rotation with an invariant foliation of pseudo-circles, consisting of the fibres of a semi-conjugacy p : T 2 → T 1 to an irrational rotation R α . Moreover, we will freely add further assumptions on f if these can easily be ensured in the preceeding Anosov-Katok-construction. We first note that the semi-conjugacy in Theorem 1 is unique, modulo post-composition by rotations.

Proposition 4.1. The semi-conjugacy p in Theorem 1 is uniquely determined: any continuous map p which is homotopic to p and semi-conjugates f to the same circle rotation as p can be written as p = R • p where R is a rotation of the circle.

This follows directly by combining [START_REF] Jäger | On torus homeomorphisms semiconjugate to irrational circle rotations[END_REF]Corollary 4.3] (uniqueness of the semi-conjugacy provided the non-wandering set is externally transitive) with [Po, Theorem A] (external transitivity of the non-wandering set of irrational pseudo-rotations).

The uniqueness of the semi-conjugacy further allows to see that f does not admit any loop which is wandering (i.e. disjoint from all its iterates) and has the same homotopy type as the pseudo-circles (that is, homotopy vector v 2 = (0, 1) in our construction). The reason for this is the fact that the existence of such a loop Γ would allow to construct a semi-conjugacy p to the rotation R α such that Γ is contained in a single fibre of the semi-conjugacy. Details of this construction can be found in [START_REF] Jäger | On torus homeomorphisms semiconjugate to irrational circle rotations[END_REF]Lemma 3.2] (the fact that Γ is contained in a single fibre is not mentioned explicitly, but is obvious from the proof). Due to the uniqueness of the semi-conjugacy (modulo rotations) and the fact that none of the pseudo-circles of the foliation contains any non-degenerate curves, this yields a contradiction.

More generally, it is even possible to show that f does not admit any loops disjoint of all its iterates, regardless of the homotopy type. This is slightly more subtle, and we only sketch the argument. The crucial observation is the fact that we may construct f such that sup

| F n (z) -z -nρ, v | < ∞ iff v = (1, 0) , (4.1)
where F : R 2 → R 2 is a lift of f and ρ ∈ R 2 is the corresponding rotation vector. Now, if there exists a wandering loop of homotopy type w ∈ Z 2 \ {0}, then it is not hard to see that

sup F n (z) -z -nρ, w ⊥ < ∞ .
However, according to (4.1) this is only possible if w = (0, 1), and this is exactly the homotopy type of the pseudo-circles which was excluded before. Homotopically trivial wandering loops cannot exist by minimality, and therefore no loop of any homotopy type can be wandering. Roughly speaking, in order to prove (4.1) one has to use the fact that since the leaves of the foliations V k are increasingly crooked, connected fundamental domains of these circles in the lift become arbitrarily large in diameter. Since an iterate of f k acts as a rotation on these leaves, this allows to see that for suitable integers n k the vertical deviations F n k k (z)zn k ρ, w ⊥ become arbitrarily large. If the f k converge to f sufficiently fast, then this carries over to the limit and yields unbounded vertical deviations for f . At the same time horizontal deviations (that is, v = v 1 = (1, 0) in (4.1)) are bounded due to the existence of the semi-conjugacy (e.g. [START_REF] Jäger | Irrational rotation factors for conservative torus homeomorphisms[END_REF]Lemma 3.1]). Together, these two facts yield unbounded deviations for all v = v 1 .

The fact that f does not admit any wanding loops is of some interest in the context of the Arc Translation Theorem due to Kwapisz [Kw, BCL], which asserts that given an irrational pseudo-rotation and any integer n, there exist essential loops which are disjoint from their first n iterates. It is natural to ask under what additional assumptions this statement can be strengthened by passing from a finite number to all iterates. A natural obstruction is certainly to have unbounded deviations in all directions, as discussed above. However, our example shows that even if the deviations are bounded in some direction, the existence of a wandering curve is not guaranteed. Thus, in this sense the statement of the Arc Translation Theorem is optimal, and essential loops have to be replaced by more general classes of essential continua in order to obtain results in infinite time.

Finally, we want to mention a loose connection of our construction to the Franks-Misiurewicz Conjecture [FM]. The latter asserts that if the rotation set of a torus homeomorphism is a line segment of positive length, then either it contains infinitely many rational points, or it has a rational endpoint. As mentioned in the introduction, Avila has recently announced a counterexample to this conjecture for the case where the rotation segment has irrational slope, but the line it defines does not pass through a rational point. Conversely, Le Calvez and Tal have announced the first positive partial result on the conjecture: if the rotation set is a segment with irrational slope, it cannot contain a rational point in its relative interior. One case which is still completely open, however, is whether the rotation set can be a line segment with rational slope, but without rational points -for example, of the form {α} × [a, b] with α ∈ R irrational and a < b. Now, in the situation where there exists a semi-conjugacy p, homotopic to π 1 , to the rotation R α on the circle, the rotation set has to be contained in the line {α} × R [START_REF] Jäger | Irrational rotation factors for conservative torus homeomorphisms[END_REF]Lemma 3.1]. Hence, this is a natural class of maps to look for counterexamples to this subcase of the Franks-Misiurewicz Conjecture. However, it is known that in order to have a non-degenerate rotation interval, the fibres of the semi-conjugacy need to have a complicated structure -more precisely, they need to be indecomposable [JP]. Our example shows that such a rich fibre structure is possible in principle. Yet, whether a non-degenerate rotation interval can be achieved remains open. Here, the fact that the Anosov-Katok method typically leads to uniquely ergodic examples suggests that a different approach would be needed to produce such examples, if these exists at all.

1

  arXiv:1508.07022v1 [math.DS] 27 Aug 2015

Definition.

  For ε > 0, a continuous map g : I → R on the interval I is ε-crooked if for any a < b in I, there are a < c < d < b such that |g(d)g(a)| < ε and |g(c)g(b)| < ε.

  and some points a ∈ B i and b ∈ B j . One considers a ≤ a < b ≤ b such that a, a (resp. b, b ) have the same image by x → θ n (xω). One can assume that aω, bω belong to the same interval I k := k 2mn , k+1 2mn : indeed, for each k, the image under θ n of I k has length > 1 and the point θ n k 2mn is an end point of θ n (I k-1 ∪ I k ). Now the points aω and bω belong to the same interval