A nonhomogenizable linear transport equation in <span class="mathjax-formula">$\mathbb{R}^2$</span>

A nonhomogenizable linear transport equation inR²

ROBERTOPEIRONE

Abstract.In this paper I investigate the homogenizability of linear transport equa- tions with periodic data. Some results on homogenizability and on the form of the limit are known in literature. In particular, in [9], I proved the homogenizability in the two-dimensional case for nonvanishing functions, and, on the other hand I gave an example of a nonhomogenizable equation in the three-dimensional case.

In this paper, I describe an example of a nonhomogenizable equation in two di- mensions. As in [9], I study the problem using an equivalent formulation in terms of dynamical system properties of the associated ODEs.

Mathematics Subject Classification (2000):35B27 (primary); 37E45 (secondary).

1. Introduction

The subject of this paper is that of linear transport equations, and more specially, that of their homogenization. Let us consider the differential problem

∂u

∂t(t,x)+ f x

ε · ∂u

∂x(t,x)=0, u(0,x)=ψ(x) (P^ε_f,ψ) where ε > 0, f is a C¹ function from R^{N to} R^{N, which is}Z^{N-periodic, i.e.,} f(x +m) = f(x)for all m ∈ Z^{N, and} ψ is of classC¹. Also, ^∂_∂^u_x denotes the vector_∂u

∂x₁, ...,_∂^∂_x^u_N

, and·denotes the scalar product inR^N. The equation inP^ε_f,ψ, denoted by P^ε_f, is called linear transport equation and by homogenization of P^ε_f we mean in some sense, to find a sort of limit, forε→0⁺, of its solutions.

IfN =1 it is known and simple to prove that the solutions arestronglyconver- gent and if in addition, f does not vanish, we easily get a limit equation. However, the strong convergence appears to be a very strong requirement in the general case.

In fact, ifN >1 the solutions may not strongly converge even in simple and natural cases (cf.[3, Remark 2.2], and references therein). On the contrary, a sort of weak convergence appears to be a natural requirement. So, we are lead to say that P^ε_f is homogenizable if the solutionsu_ε,f,ψ ofP^ε_f,ψare convergent in the weak^∗topology ofL^∞_locfor everyψof classC¹.

Received April 20, 2007; accepted in revised form September 3, 2008.

Y. Brenier [1] proved the homogenizability of P^ε_f, under the hypothesis that the divergence of f is 0. Under the same hypothesis, in other papers, the form of the limit of the solutions has been also investigated. For example, T. Hou and X.

Xin in [5] treated the case where N = 2 and nonvanishing f, W.E [4] studied the problem in a much more general context, T. Tassa [10] studied the more general case div(p f) = 0 for some p : R^{2 →} R, positive, of classC¹, and periodic. In this paper, I merely consider the problem of the existence of the limit. E. De Giorgi conjectured that P^ε_f is homogenizable independently of f [3, Conjecture 1.1]. I previously investigated in [9] the homogenizability problem. The main results in [9]

state that in the case N = 2, f(x) = 0 for everyx ∈R² (with no conditions on the divergence of some function related to f), P^ε_f is homogenizable, and on the other hand, there exists an example with N = 3, f of classC^∞, f(x) = 0 for every x ∈ R^{3, in which Pε}f is not homogenizable. The case N = 2, f vanishing somewhere is not covered by the previous results. In [9, Remark 4.11], in fact, an example of f such that P^ε

f is not homogenizable in the two-dimensional case, is stated to exist but it is not described. Of course, such a function f has to attain the value 0 somewhere.

In this paper I will describe such an example. I will use a similar notation as in [9], in particular, fork∈N^{∪ {∞},Ck(}T⁾denotes the set of functions f of class C^k fromR^{N to}R^{N which are}Z^N-periodic. If f ∈C¹(T),x ∈R^{N, thenT}f^t(x), or simplyT^t(x), denotes the value attained att by the solution of

y(t)= f(y(t)) y(0)=x.

We will always mean N = 2 except in Section 2, whereN can also amount to 1.

Let us denote byE_f the ODE in the previous formula. In view of [9, Lemma 2.2], which in turn is a simple variant of results in [8], the problem of homogenizability can be seen as a dynamical system problem. More precisely, in order to construct the claimed example, it suffices to find a function f ∈C¹(T⁾such that the limit

t→+∞lim

[0,1]²

T^t

f,2(x)−x₂

t d x (1.1)

does not exist, whereT^t

f,2denotes the second component ofT^t

f. Note that the func- tionV_a(t,x)= ^Tft(x_t^)−x represents the average velocity in a time-interval of length t of a solution ofE_f starting fromx. In particular,V_a(t,x)cannot pointwise con- verge fort → +∞, and in fact cannot converge a.e., and thus our problem is related to the notion of rotation set introduced in [7]. The existence of a function with such a property is not completely trivial, although I guess it is known. However, in view of (1.1), we have to find a function with the stronger property that the average of V_a(t,x)in x ∈ [0,1]² does not converge fort → +∞. This is a similar idea to

that of the example in the three-dimensional case discussed in [9]. However, the construction of the example in the two-dimensional case is much harder, due to the fact that in two dimensions a solution ofE_f is forced to remain in a “strip” bounded by two solutions (see [9, Proof of Theorem 3.1]), so that it is not trivial to avoid that it approaches a periodic solution. Note that, in spite of what I announced in [9], we can choose the function f to be of classC^∞(see Remark 6.3 here).

The function f is defined in terms of two functionsg and, the former de- noting the section map for the flow on the torus on the manifold y = 0, and the latter being comparable to the time of the return (cf. for example [6, Section 0.3]).

The mapg is a diffeomorphism of the circle, and is constructed as the limit of a Cauchy sequence of functions in the C¹metric, and the approximating functions are defined recursively and their rotation numbers are rational numbers that ap- proximate very quickly the irrational rotation number ofg. Such a method is called Fast-approximation methodin [6, Section 12.6], where is used, for example, in the construction of diffeomorphisms of the circle with irrational rotation numbers and pathological conjugacies (cf. also [2, Chapter 3, Section 5]). Sections 2 and 3 de- scribe the plan of the construction. Section 4 introduces a class of functions basic in this construction. In Section 5, the key lemma (Lemma 5.5) is stated, which allows us to construct the(n+1)^{t h} function in terms of then^{t h} function in the sequence approximating g. In Section 6, assuming Lemma 5.5, I construct our example.

Sections 7 to 9 are devoted to prove Lemma 5.5.

2. Motivation of the construction

I start this section by recalling the main definitions concerning the homeomor- phisms of the circle. The reader can refer, for example, to [2] and to [6] for in- formation on this topic. Put

D⁰(T^):=g:R^→R^{: g−I d}is 1-periodic, gstrictly increasing and continuous}.

Ifg∈D⁰(T), then there existsρ∈R^{so that} gⁿ(x)−x

n −→

n→+∞ρ (2.1)

uniformly with respect tox ∈ R. Such a numberρis called therotation number ofg. Clearly, such a type of mapgcan be interpreted as a homeomorphism of the circle, where the circle can be defined asRquotiented by the equivalence relation that identifies two real numbersaandbiffa−bis integer. The simplest map having ρas its rotation number isx →x+ρ. I recall the following results.

Lemma 2.1. If p∈Z^{, q∈}N^{\ {0}, we have}

i) ρ(g)= _q^p if and only if there exists x ∈Rsuch that g^q(x)=x+ p.

ii) Ifρ(g) = _q^p, then for every x ∈ Rthere exists x ∈ Rsuch that g^nq(x)− np −→

n→+∞x, and g^q(x)=x+p.

iii) If for some x ∈R^{we have gq(x)≥x+ p(}respectively g^q(x)≤x+p), then ρ(g)≥ _q^p (respectivelyρ(g)≤ _q^p).

iv) If r ∈R^{and gq(x) > x +r (}respectively g^q(x) < x +r)for every x ∈ R^, thenρ(g) > _q^r (respectivelyρ(g) < ^r_q).

Thus, a map g ∈ D⁰(T⁾has periodic points (interpreted as a map on the circle) if and only if its rotation number is rational, and, in such a case, in some sense, every point is “asymptotically periodic”. In the next lemma and in the rest of this paper, Fr(a)will denote the fractional part ofa∈R^{,i.e.,a− [a].}

Lemma 2.2. Given p ∈ Z^{, q} ∈ N\ {0}with p and q relatively prime, suppose ρ(g) = _q^p, and put0_h = Fr

g^h(0)

. Then, the points0_h, h = 0, ...,q−1are mutually distinct and their order is independent of the particular map having the above properties (once p and q are given).

Hence, taking the mapx →x+_q^p, we have 0h <0_hif and only ifFr_hp

q

<

Fr_hp q

.

Note that, as simple examples show, the behaviour of a map in D⁰(T^{)is not} characterized by its rotation number. For example the map x → x + _q^p has the property that itsq^{t h} iterate has the formx →x+ p, while for other maps having rotation number equal to _q^p too, their iterates only have isolated periodic points on the circle. However, under some relatively mild hypothesis, this does not happen when the rotation number is irrational. This is the statement of the following well known theorem (Denjoy’s Theorem).

Theorem 2.3. Suppose that a map g ∈ D⁰(T⁾has the irrational rotation number ρ. Suppose further that it is of class C² and has positive derivative. Then g is topologically conjugate to the map g_ρ defined as g_ρ(x) = x +ρ. In other words there existsφ ∈D⁰(T⁾such that g=φ⁻¹◦g_ρ◦φ.

Note that in general even if gis of classC^∞ or even analytic, the conjugacy φ is not necessarily regular (neither of classC¹). The regularity ofφdepends on arithmetic properties of ρ, in particular the mapφ has been proved to be regular whengis sufficiently regular andρis a Diophantine number, that roughly speaking means that it cannot be approximated too fast by rationals. I refer to [2] or to [6]

for more details.

When we have a differential equation (E_f) as in Introduction, but with f : R → R1-periodic, we can define the rotation number of(E_f)as lim_t→+∞^y(_t^t) whereyis any solution of(E_f), or in other words, lim_t→+∞^T

t f(x)

t wherex ∈R^{. It} amounts to the rotation number of the Poincar´e map of(E_f), which is the mapT_f¹.

If, instead of the 1-dimensional case, we consider maps fromR^2toR^{2of the} formI d plus aZ²periodic function the situation is more complicated. In general, the limit in (2.1) does not necessarily exist, and even when exists, it can depend on x. An example of this type is the differential equation(E_f)where f(x)has the form (g(x₂),0). In such a case, a simple argument shows that ^T

t f(x)

t −→

t→+∞(g(x₂),0). In view of what stated in Introduction, we have in fact to find f such that it is false that the limit of the integrand in (1.1) exists a.e. In order to better understand the construction of next sections, I here describe the difficulties in obtaining it, in particular I will explain why simpler constructions fail. Recall that we have to define a function f Z²-periodic and of classC¹such thatP^ε_f is not homogenizable.

For simplicity, I will only consider f such that at any pointx ∈R^{2we have} either f₂(x) >0 or f(x)=0 (2.2) so that every solution of E_f, except for those passing through (or approaching) zeroes of f, tend to+∞in the second component. Suppose for the moment that f₂(x) > 0 for every x. Since f never vanishes, we know from the results of [9]

that ^T

t f(x)

t tends to a (finite) limit ast → +∞, for everyx. The reason for which this happens can be roughly explained in the following way: the solution passing through (x,0) will reach a point (g(x),1) after a time (x). The mapg has a rotation number ρ. Now, puttingn(x) = _n−1

i=0 (gⁱ(x))for n = 1,2, ..., we have that for everyx∈R, there existsc_x ∈R^{such that}

1

nn(x) −→

n→+∞c_x. (2.3)

To deduce (2.3) note that, ifρ∈Q, then this follows from Lemma 2.1ii. If, on the contrary,ρis irrational, and for simplicity we suppose we are in the hypothesis of Denjoy’s Theorem, we have for someφ∈D₀(T^),

1

nn(x)= 1 n

n−1

i=0

◦φ⁻¹(φ(x)+iρ) (2.4)

that tends to

[0,1] ◦φ⁻¹. In general, for any ρ, rational or irrational, it is not difficult to deduce from (2.3) that

T^t

f(x)

t −→

t→+∞

1 c_x(ρ,1).

Return now to the more general case where (2.2) holds. In this case we can mimic the previous considerations. Clearly, for x such that the trajectory through(x,0) meets a zero of f, whose y-coordinate belongs to [0,1[, g is not defined and

could only be defined to be +∞. I will call thosex 1-singularand the other 1- regular. So, we have to slightly modify our program. Suppose for simplicity f only attains the value 0 on [0,1[×[0,1[ at(0,0)andg, defined as above on the regular points, can be extended to a sufficiently regular diffeomorphism onR^{. In} this case (2.3) is still valid whenρis rational, by essentially the same argument as in the previous case. Whenρis irrational, it is not so simple to extend the argument of the previous case. However, we can use the Birkhoff ergodic Theorem, to conclude that, if◦φ⁻¹is in L¹, then the limit in (2.4), puttingy in place ofφ(x), exists for almost all ywith respect to Lebesgue measureµ. This implies that the limit in (2.4) existsµ-a.e. provided the mapφis regular, and this happens, as told above, if ρis Diophantine. It is possible and not very difficult to prove that the case where ◦φ⁻¹is not inL¹can be reconducted to the previous considerations.

In conclusion, in order to construct our counterexample, the rotation numberρ ofgneed be an irrational Liouville (that is, not Diophantine) number. First of all, I will construct f, by using a sort of inverse process, in other words, I will construct two functions gand fromR^to R, and then, basing on them, I will recover f. In other words, I use the standard idea of defining a continuous dynamical system in terms of a discrete dynamical system. In order to avoid the value+∞, in place of it would be better to define its reciprocal. However, for technical reasons, I prefer to define a functionrepresenting the second component of the velocity f₂ of the solution of E_f at a point related to(x,0). Then, I will construct f in such a way thatis comparable with ¹

, say 1

2 ≤≤ 1 .

3. Plan of the construction

In this section, I describe the general idea behind the construction of the pair(g, ). First of all, I introduce the set of functions suitable to this aim.

Definition 3.1. LetD¹(T⁾be the space

g∈ D⁰(T^{), gof classC1,g>0.} We equipD¹(T)with a metricd₁defined byd₁(g₁,g₂)=sup{|g₁(x)−g₂(x)|,

|g₁(x)−g₂(x)| :x∈R}. The following result will be used in the sequel.

Lemma 3.2. The map g→gⁿfrom D¹(T^{)to D1(}T⁾is continuous for every n∈Z^. For some consideration it is also useful to equipD¹(T⁾with the metricd₀(g₁,g₂)= sup{|g₁(x)−g₂(x)| :x∈R^}. Note that the subspaceD¹_c(T^{)= {g∈D1(}T^{):g ≥} c} is complete with respect tod₁ for every c > 0. If g ∈ D¹(T^{), let xh},g = Fr(g^h(x))forh ∈Z^{,x ∈ [0,1[}. Moreover, whenx ∈ [0,1[, we callg-orbit ofx the set {x_h,g : h ∈ N^{}and(n,g)-orbit ofx the set {xh},g : 0 ≤ h ≤ n}for every n∈N^.

By the discussion in Introduction, in order to construct the counterexample, it suffices to find f satisfying the following

There exist two sequences t_n, t_n tending to+∞such that

n→+∞lim T^tn

f,2(x)−x₂

t_n =0, lim inf

n→+∞

T^tn

f,2(x)−x₂

t_n ≥ A (3.1)

for almost all x ∈ [0,1]², where A is a suitable positive constant.

It is possible to express (3.1) in terms of the pair (g, ), related to f as in the previous section. Namely, as I will prove in details in Section 6, (3.1) is related to the following condition:

There exist two sequences m_n,wnof naturals tending to+∞such that

n→+∞lim 1 m_n

mn−1 h=0

1

(g^h(x)) = +∞, lim sup

n→+∞

1 wn

wn−1 h=0

1

(g^h(x)) ≤ B (3.2) for almost all x ∈ [0,1], where B is a suitable positive constant.

The relation between (3.1) and (3.2) is related to the fact that the quantity _m−1

i=0 1

(gⁱ(x)) is comparable to the time necessary for a point (x,0), under the equation E_f, to reach a point of the form (u,m). In view of the considerations in the previous section, the functiongshould have an irrational Liouville rotation numberρ, andshould amount to 0 at 0 and should be positive on]0,1[. In Sec- tion 6, I will prove in fact a variant of (3.2) that is Theorem 6.1, which is more complicated than (3.2), but has the advantage that it implies (3.1) in a more direct way. I will define(g, )as the limit of a sequence(g_n, φn)withρ(g_n) ∈ Q^{, de-} fined recursively. This is convenient since in view of Lemma 2.1, the maps having a rational rotation number can be better investigated. More precisely, asρis the limit of a sequence _q^pn

n of rationals, I will define g_n having _q^pn

n as rotation number and having 0 as a periodic point, namely g^qn(0) = p_n (cf. Lemma 2.1). The function φn will be defined on intervals of the form[αn,1], withαn −→

n→+∞0, and extending φn−1, andwill be the 1- periodic function extending everyφn. In order to define the pair(g, ), we need some preparatory definitions.

Definition 3.3. We say that a continuous functionφ: [α,1] →R^{with 0}< α <1, is anα-caliber ifφ(1)=0 and 0< φ(x)≤1 ifx∈ [α,1[.

If φ is anα-caliber andg ∈ D¹(T^{), h ∈} N, we say thatx ∈ O(g,h, α) if x∈ [0,1[,x_i,g ∈]α,1[fori =0, ...,h−1. In such a case, we put

T_φ(g,h,x)=^h−1

i=0

1 φ(xi,g).

The definition ofT_φ(g,h,x)and the following definition are suggested by formula (3.2).

Definition 3.4. Supposeg ∈ D¹(T^{), andφis an α}-caliber. Letλ, ∈ R^{, letw} andm be positive integers and let J be the union of finitely many disjoint closed intervals. Put = (λ, ,J, w,m)and letAbe the set ofso obtained. We say that the pair(g, φ)isoscillating if J ⊆O(g,m, α)∩O(g, w, α)and

T_φ(g, w,x)≤λw, T_φ(g,m,x)≥m,d∀x ∈ J.

Remark 3.5. Note that in Definition 3.4 J could well be empty. In such a case, clearly,(g, φ)isoscillating independently of∈A^.

Proposition 3.6. If(g, φ) is (λ, , J, w,m) oscillating, then there exists ε = ε(g, φ) >0such that if g₁ ∈ D¹(T^{), d1(g,g1) ≤ε, then(g1, φ) is(λ+1, −} 1,J, w,m)oscillating.

In such a case, ifd₁(g,g₁) ≤ ε andφ1 is anα1-caliber extendingφ, we say that(g₁, φ1)iscloseto(g, φ). Now, the idea of the construction consists in taking a sequence(g_n, φn) (λn, n,J_n, wn,m_n)oscillating,φnαn-caliber, each close to the previous ones and such thatαn tends to 0 asntends to infinity. Ifg_n is a Cauchy sequence in D_c¹(T^{)for somec >} 0, and the function on]0,1]extending eachφn

can be 1-periodically extended onR^{to aC∞ function, settingg = limgn, we} then have

m_n−1

h=0

1

(g^h(x)) ≥(n−1)m_n, ^wn−

1

h=0

1

(g^h(x)) ≤(λn+1)wn, forx∈ J_n. Suppose now we can choosen→ +∞,λnbounded,µ(J_n) >1−₂^Bn, m_n → +∞,wn → +∞. PuttingJ=

k∈N

n≥k

J_n, we haveµ(J)= 1. Therefore, if we construct f related togandas above, we can expect that for all points in a trajectory passing through J×0, possibly translated by a point inZ², hence for almost allx ∈ [0,1]², (3.1) holds.

In conclusion, in order to prove (3.1) (or (3.2)), one can proceed in the follow- ing way: first we construct a particular pair(g, φ). Then, we have to find a sequence of(g_n, φn), inductively. In order to avoid problems in extending periodically onR^, we require that the derivatives ofφof every order tend to 0 at 1, and the derivatives get smaller and smaller in the new intervals[αn+1, αn[.

The natural tool for obtaining such a result would be a result stating that for every oscillating pair(g, φ)and for everyε >0 and for every, λ, δwe can find a pair(G, ) (, λ,J, w,m)oscillating close to(g, φ)such thatd(g,G) < ε, withw andm sufficiently large, andµ(J) > 1−δ. In such a case I will say that(G, ) derives from(g, φ). However, even restricting δ, w,m, it does not seem simple to prove such a result. So, I will require some technical additional properties on a pair in order to complete the inductive step. In other words, I will prove the inductive step in the sense that, if we have aregularpair then we can find a new regular pair deriving from it, where by regular I mean having some specific properties. In the next two sections, I will introduce the regularity property and state the recursive lemma.

4. Functions of₍m,q,l,p)type

In order to have a functiong∈D¹(T⁾as above, we have to require that its rotation number is irrational, and is approximated very quickly by rational numbers. So, I will construct a sequence of functions g_n, whose rotation numbers differ by a quantity which tends to 0 very rapidly. This clarifies the reason of the following definition. In fact, given g_n with rotation number _q^pn

n, I will construct g_n+1 with rotation number _q^pn+1

n+1 such that _q^pn+1

n+1 − _q^p_nⁿ = _qnq¹_n+1 andq_n+1sufficiently large. In other words, I use the standard way, used for example in continued fractions, of constructing a sequence ^p_qⁿ

n tending to an irrational (Liouville) number.

Definition 4.1. Given a function f ∈ D¹(T⁾, we say that f is of(m,q,l,p)type if f^q(0)= pand

m,q,l,p∈N^{\ {0}, q >m+1, mp =lq+1. (4.1)} Note that in such a case, the rotation number of f is _q^p. Moreover, it is easy to see that, once f is fixed, then m, q, l and p are unique. The functions of (m,q,l,p) type have a special importance for the following considerations, so we will investigate them in detail in this section. Note also that, if (4.1) holds, then the function x → x + _q^p is of (m,q,l,p) type, and for this function f, 0_h+m,f = 0_h,f + ¹_q (unless 0_h+m,f = 0), so that 0_h+m,f is the “successor” of 0_h,f (if different from 0). In view of Lemma 2.2 this happens foranyfunction of (m,q,l,p)type. Thus, the following definition ofU_h,f is, in some sense, natu- ral. Defineaˆfora ∈ [0,1]as aˆ =

aifa=0

1 ifa=0 . PutU_h,f = [0_h,f,0_h+m,fˆ[, µh,f =µ(U_h,f)=0_h+m,fˆ−0_h,f.

Lemma 4.2. Let f be of(m,q,l,p)type. Then

ø

−/ U_h,f ⊂

−/[0,1[. Moreover, i) 0<0_m,f < ... <0_(q−1)m,f <1=0_qm,fˆ.

ii) If h ≡hmod q, then0_h,f =0_h,f and U_h,f =U_h,f. iii) If h ≡hmod q, then0_h,f =0_h,f and U_h,f ∩U_h,f =

ø

iv) For every r ∈ Z^{the sets Uh},f, h = r, ...,r +q−1, are mutually disjoint and their union amounts to [0,1[. The same holds for the sets U_mh,f, h = 0, ...,q−1.

v) We have0h,f ≤0q−m,f. Also, if0h,f > 0, in particular if0< h < q, then 0_h,f ≥0_m,f.

vi) If h and r are natural numbers, r >0, and h+r m<q, then0_h,f <0_{h+r m,f}. vii) Let h,k ∈ Z^{. Then fk maps Uh},f to[f^k(0_h,f)] +U_h+k,f. More precisely, x_k,f = f^k(x)−[f^k(0_h,f)] ∈U_h+k,f for every x ∈U_h,f, hence x_k,f −x_k,f =

f^k(x)− f^k(x)if x,x∈U_h,f.

Proof. The proof is a not very complicated consequence of Lemma 2.2. The details are left to the reader.

Remark 4.3. Suppose f is of (m,q,l,p) type. Then, in general, f^k(0_h,f) = f^h+k(0)− [f^h(0)] for every h,k ∈ Z. As particular cases, we easily see that f^q(0_h,f)= p+0_h,f for eachh∈Z. Moreover, f^m(0_h,f)=0_h+m,f +lprovided 0≤h<q−m. Indeed, on one hand,x+l< f^m(x) <x+l+1 for everyx ∈R^{, as} if not, eitherρ(f)≤ _m^l < _q^p orρ(f)≥ ^l+_m¹ > _q^p, on the other, 0_h,f <0_h+m,f <

0_h,f +1. We deduce from these considerations that f ^{j q}(0_h,f) = j p+0_h,f for every j ∈Z^{, and f j m}(0_h,f)=0_{h+j m,f} + jlif j ∈N^{and 0}≤h<q− j m.

When f is of classC², we can control rather precisely the measure of f^h(I) when I is an interval. I here recall a standard lemma in this context, used for example for proving the Denjoy Theorem.

Lemma 4.4. Let f ∈ D¹(T⁾, of class C², and let V = V_f = V ar(log◦f) = 1

0

^f_f(⁽_x^x₎⁾d x. Let U be an interval in[0,1[, such that the sets fⁱ(U), i =0, ...,h− 1, are mutually disjoint on the circle, that is, x−x∈/Z^{if x∈ fi(U), x∈ fi(U),} with i,i=0,1, ...,h−1, i =i. Then, e^−V ≤₍⁽_f^fh^h)⁾(⁽x^x)) ≤e^V for every x,x∈U . Proof. See for example [6, Lemma 12.1.3.] Note that, although in the statement in [6], it is required that fⁱ(U)are mutually disjoint fori = 0, ...,h, in fact the proof there only requires this fori =0, ...,h−1.

Lemma 4.5. If f is of(m,q,l,p)type and of class C², and h ∈ Z^{, and x,x ∈} U_1,f, x <x, and|h| ≤q, then setting V =V_f, we have

i) µh+1,f

µ1,f

e^−V(x−x)≤x_h,f −x_h,f ≤ µh+1,f

µ1,f

e^V(x−x) , ii) µh+1,f

µ1,f

e^−V(0_m+1,fˆ−x)≤0_m+h+1,fˆ−x_h,f ≤ µh+1,f

µ1,f

e^V(0_m+1,fˆ−x) . Proof. i) Suppose for exampleh positive. Then we are in the hypothesis of Lem- ma 4.4, withU =U_1,f, by Lemma 4.2iv. The conclusion follows noting that both

x_{h, f}−x_h,f

x−x and^µ_µ^h+1,f

1,f amounts to(f^h)at some point inU. Ifhis negative, the same argument works withU =U_h+1,f. ii) This is obtained by i) taking the limit forx tending to 0_m+1,fˆ.

Corollary 4.6. If f is of class C²and of(m,q,l,p)type, then e^−Vf ≤ (f^q) ≤ e^Vf.

Proof. For everyh ∈ Z^{we have} U_h,f(f^q) = (p+0_h+m,fˆ)−(p+0_h,f) = µ(U_h,f), hence there existsv ∈U_h,f such that(f^q)(v) = 1. Since everyx ∈ R has the formn+vfor somen∈Z^,v∈Uh,f,h∈Z, we get the desired inequalities using Lemma 4.4 again.

5. The recursive lemma

I am now going to define the regularity property of an oscillating pair. To this aim, I will start by defining some specialC^∞ functions. Then, I will define more and more restricted classes of functions. Letψˆ be a function of classC^∞ fromR^toR such that

ψ(ˆ x)







=0 if x ≥1

=1 if x ≤0

∈]0,1[if 0<x <1.

Note thatψˆ^(j)≤L_j for each j ∈Nand for suitable positiveL_j (which of course depend onψˆ) withL₀=1. Fora≤b,u>0, letψˆa,b,u(x)= ˆψ(^x−_u^b)ψ(ˆ ^a−_u^x)and letψa,b,u be the 1-periodic extension onRof the restriction ofψˆa,b,u on[0,1[. If u<min{a,1−b}thenψa,b,uisC^∞.

Lemma 5.1. 0≤ ˆψa,b,u(x)≤1, and ψˆa,b,u(x)

=0 if x∈ [u+b,+∞[ ∪ ] − ∞,a−u]

=1 if x∈ [a,b].

Moreover, there exists L_j >0such thatψˆ_a⁽_,^j_b⁾_,u≤ ^L_u^jj if u <1, j=0,1,2,3, ...

As told in beginning of previous section, given a function f of (m,q,l,p) type, we are now interested in finding a function slightly bigger than f, with strictly bigger rotation number. Related to this is the following.

Lemma 5.2. The map f → ρ(f)is continuous with respect to the uniform con- vergence and increasing.

Proof. Cf.for example [6, Proposition 11.1.6 and Proposition 11.1.8].

However, rather surprisingly, in general, the map f → ρ(f)is not strictly increasing, in other words there exist functions f with rational rotation numbers such thatρ(g)=ρ(f)for everyg ∈ D¹(T^{)with f < g< f +ε}for sufficiently small positive ε. This is a well known property of the rotation number (cf. for example [6, Proposition 11.1.10]). The reason is that if f^q −I d− pattains both positive and negative values, then, in view of Lemma 2.1, every sufficiently small perturbation of f has rotation number equal to _q^p. This does not happen if f^q(x)≥ x+ pfor allx. In [2, Definition 1 in Chapter 3, Section 5], a functiongis called (p,q)-stable forward ifg^q(x)≥x+ pfor allx and the equality holds for at least onex. Here, we need a variant of such a definition.

Definition 5.3. We say thatgof(m,q,l,p)type is

(m,q,l,p)

sf-stable if i) g^q(x)≥x+pd ∀x ∈R^,

ii) ∃z₁ ∈Int(U_1,g)such thatg^q(z₁)=z₁+p.

Here, sf stands forstrongly forwardand Int denotes the interior of a set. For func- tions of(m,q,l,p)type, the equality necessarily holds for 0 (and its orbit). In ii) I require that the equality holds for somexnot in the orbit of 0. I will constructG derived bygin several steps. At the first step, I will construct a function g, sum-ˆ ming a very small positive functionstog. Thus, i) assures thatρ(gˆ) > ρ(g). I will now introduce the definition of a “regular” sf-stable function, and in such a class of functions I will construct(G, )derived by(g, φ)(Lemma 5.5). I will now explain the idea of such a definition. However, the actual construction ofGcould differ in some details from the ideas hinted below.

As the function G will turn out to be a very small perturbation of g, weˆ will implicitly assume that G has similar properties to g. We want thatˆ (G, ) is(λ, ,J, w,q)oscillating for arbitraryand suitable J, w, J filling[0,1[ apart from a set of small measure. LetIbe a compact interval contained in[0_1,g,z₁[, and letx ∈ I. On one hand, if we require thatis positive but very small near 0, as the(q−1,G)-orbit ofx contains the pointx_q−1,G, which is very close to 0, we haveT(G,q,x)≥q; on the other, if we could haves(z₁)=0, then, thanks to ii), the gˆ^q-orbit ofz₁would amount toz₁, hence, using also i),Fr

ˆ g^hq(I)

would tend toz₁ashtends to infinity.

Thus, if we require thats(z₁)is positive but small in comparison to the values attained bysfar fromz₁, thenFr

ˆ g^hq(I)

approachesz₁, forhsufficiently, but not too, large. It follows that, if we require T_φ(g,q,z₁) < λq, then T(G,hq,x) <

λhq for suitably large h, independent of x ∈ I, and we can take w = hq. Of course, the same properties hold for x in the set J formed by the union of the (q−1,G)-orbit of the points inI. In Definition 5.4,[0_1,g, α1−q,g]stands forI, so thatJis strictly related to the set in (5.4).

Definition 5.4. Suppose g is(m,q,l,p)sf-stable, φ is aC^∞ α-caliber. We say that the(λ, ,J, w,m)oscillating pair(g, φ)is(η, δ)-regular ifq> p,η ∈]0,1], δ∈]0,1[,λ≥1,≥1 andα∈Int

U_0,g and

α < (z₁)₋1,g, (5.1)

φ=η on[α,0_m,g], (5.2)

T_φ(g,q,z₁) < λq, (5.3)

µ q

h=1

0_h,g, αh−q,g

>1−δ. (5.4)

Lemma 5.5. Suppose g is (m,q,l,p)sf-stable and C^∞, ¹₂ < g < ³₂,φ C^∞ α- caliber,(g, φ) (λ, ,J, w,m)oscillating,(η, δ)-regular. Then for each

ε >0, δ₁> δ, δ∈]0,1[, ≥1, there exist

η >0, q, w, p, J, α, G, r, _η∀η∈]0, η[