CHERRY MAPS WITH DIffERENT CRITICAL EXPONENTS: BIFURCATION OF GEOMETRY Текст научной статьи по специальности «Математика»

Russian Journal of Nonlinear Dynamics, 2020, vol. 16, no. 4, pp. 651-672. Full-texts are available at http://nd.ics.org.ru DOI: 10.20537/nd200409

MATHEMATICAL PROBLEMS OF NONLINEARITY

MSC 2010: 37E10

Cherry Maps with Different Critical Exponents: Bifurcation of Geometry

B. Ndawa Tangue

We consider order-preserving C3 circle maps with a flat piece, irrational rotation number and critical exponents (£1,£2)-

We detect a change in the geometry of the system. For (£1,£2) E [1,2]2 the geometry is degenerate and becomes bounded for (£1,£2) E [2, to)2 \ {(2,2)}. When the rotation number is of the form [abab...]; for some a,b E N*, the geometry is bounded for (£1,£2) belonging above a curve defined on ]1, +to[2. As a consequence, we estimate the Hausdorff dimension of the nonwandering set Kf = S1 \ (J°=0 f -%(U). Precisely, the Hausdorff dimension of this set is equal to zero when the geometry is degenerate and it is strictly positive when the geometry is bounded-

Keywords: circle map, irrational rotation number, flat piece, critical exponent, geometry, Hausdorff dimension

To TIEMGNI DEFFO's family: the mother Carine and the children Abigail, Joakim, and Helza after the death of Richard, the father of the family.

Received September 01, 2020 Accepted October 27, 2020

The author was partially supported by the Centre d'Excellence Africain en Science Mathématiques et Applications (CEA-SMA).

Bertuel Ndawa Tangue [email protected]

Institute of Mathematics and Physical Sciences Avakpa, Porto-Novo, 613 Benin National Advanced School of Engineering Route Melen, Yaounde, 8390 Cameroon

1. Introduction

We study a certain class of weakly order-preserving, noninjective (on an interval called a flat piece) circle maps which appear naturally in the study of Cherry flows on the two-dimensional torus (see [8, 11, 14, 15]), noninvertible continuous circle map (see [7]) and of the dependence of the rotation interval on the parameter value for one-parameter families of continuous circle maps (see [16]). The dynamics of circle maps with a flat interval has been intensively explored in the past years, see [4, 5, 7, 14, 17, 18].

We discuss the geometry of the nonwandering set (a set obtained by removing from the circle all preimages of the flat piece). Where the geometry is concerned, we discover a dichotomy which generalizes the one found in [5]. Some of our maps show a "degenerate geometry", while others seem to be subject to the "bounded geometry".

Before we can explain more precisely our results, we introduce our class, adopt some notations and present basic lemmas.

1.1. The class of functions

We fix ¿i, ¿2 ^ 1 and consider the class L of continuous circle maps f of degree one for which an arc U exists so that the following properties hold:

1. The image of U is one point.

2. The restriction of / to 51 \ U is a C3-diffeomorphism onto its image.

3. Let (a, b) be a preimage of U under the projection of the real line to Sl. In some right-sided neighborhood of b, f can be represented as

hr((x - b)£2),

where hr is a C3-diffeomorphism in a two-sided neighborhood of b. Analogously, in a left-sided neighborhood of a, f equals

hi((x - a)£l).

In the following, we assume that hl(x) = hr(x) = x. In fact, it is possible to make C3 coordinate changes near a and b that will allow us to replace both hi and hr by the identity function.

Let F be a lift of f on the real line. The rotation number p(f) of f is defined (independently of x and F ) by

Fn(x) _ x

p{f) := lim -^—-(modi).

Let (qn) be the sequence of denominators of the convergents of p(f ) (irrational) defined recursively by qi = 1, q2 = a1 and qn+i = anqn + qn-\ for all n ^ 3, with

p(f) = [o-oC'i •••]:= oo H--

1

a\ H--

1

a2 H--

Additional Assumption

Let f G L. We say that Sf (the Schwarzian derivative of f) is negative if

with Dnf being the nth derivative of f, for n G N.

We will assume in the proof of the first part of Theorem 1 that f has a negative negative Schwarzian derivative.

1.2. Notations and Definitions

The fundamental notations are established in [5, pp. 2-3]. Let f G L.

1. For every i G Z, the symbol i means fl(U).

2. Let I and J be two intervals. (I, J) is the interval between I and J. [I, J) := I U (I, J) and (I, J] := (I, J) U J. \I\ is the length of the interval I, \[I,J)\ := \I\ + \(I,J)\ and \(I, J]\ := \ J\ + \(I, J)\. We say that I and J are comparable when \I\ and \ J\ are comparable. This means that there is k > 0 such that -\I\ < |J| < k\I\.

k

3. For any sequence rn and for any real d, we adopt the following notation:

rd{£l,e,) ( rf1 if n = 0[2], <<£,£) _ I r„¿1 if n = 0[2], " " '"ir?3 if n = l[2], 1 r^ if nsi[2].

1.3. Discussion and statement of the results Scaling ratios

The sequence

„ ,_Kz3nM_ Kf~qn(u),u)\

Otn,

\U~HU),u)\ + \f-HU)\

measures the geometry near a critical point- In fact, it serves as scaling relating the geometries of successive dynamical partitions.

The geometry is said to be degenerate when an goes to zero and bounded when an is bounded away from zero.

The study of this geometry is parameterized by a rotation number and critical exponents. In [5], for pairs (£,£); £ > 1 and p the irrational number of bounded type (i.e., maxn an < to), the authors found a transition between degenerate geometry and bounded geometry. In fact, they show under (A1) that, if 1 < £ ^ 2 and p E R \ Q, then the geometry is degenerate and it is bounded (independently of (A1)) if £> 2 and p is an irrational number of bounded type. In [4], for £ > 1, the author proved that the class of function f of critical exponents (1,£) or (£, 1) has a degenerate geometry. In [15], the authors show that, for the maps in L with Fibonacci rotation number, when the critical exponents (£1 ,£2) belong to (1,2)2, the geometry is degenerate. Let us note that, in contrast to other previous works, information on the geometry of the system is obtained by the study of the asymptotic behavior of the renormalization operator.

In the present paper, we consider the cases where the critical exponents (l1,l2) belong to a subdomain (containing the previous domains) of [1, to)2; also, the rotation number is not necessarily of Fibonacci type and the results do not depend on the renormalization operator as in [9]. We use the formalism presented in [5] which is based on recursive inequalities analysis of an. For technical reasons, in the case of bounded geometry, we introduce the vector sequence vn := (— ln an, — ln an_i) and, the new recursive inequality is controlled by a 2 x 2 matrix. When the rotation number is bi-periodic (p = [abab...]; a,b G N), the 2 x 2 matrix has two eigenvalues (depending on the rotation number and critical exponents) As G (0,1) and Au > 0. The equation Au = Au((a,b); (l1 ,l2)) = 1 defines a curve C\u=1 (presented above) which separates the (£1,£2) plan into two components C\u>1 (below the curve) and C\u<1.

1HCa„ = 1

2 V CXu <1

1

1 2

1. Main result Let f G L with critical exponents (l1,l2). Then

1) the scaling ratio an goes to zero when (l1,l2) G [1,2]2, p G R \ Q and (A1) holds;

2) the scaling ratio an is bounded away from zero when (l1 ,l2) g [2, to)2 \ {(2,2)} and p is of bounded type;

3) the scaling ratio an is bounded away from zero when (l1,l2) G C\u<1 and p is bi-periodic.

2. Estimation of Hausdorff dimension of the nonwandering set. In the symmetric case (l1 = l2 = l), in [8], for f G L with a rotation number of bounded type, the author shows that, if l G (1,2], then the Hausdorff dimension of the nonwandering set is equal to zero, and that, if l > 2, then the Hausdorff dimension of the nonwandering set is strictly greater than zero. This result generalizes the one in [19] where the author treats the maps in L with critical exponents (1,1). Note that the results in [8] are more general (they only depend on geometry); precisely, if the rotation number is of bounded type, then the Hausdorff dimension of the nonwandering set is equal to zero when the geometry is degenerate and it is strictly greater than zero when the geometry is bounded; so we have the following result which is proved at the end of the paper.

Corollary. Let f G L with critical exponent (l1 ,l2) with a rotation number of bounded type. Then

1) the Hausdorff dimension of the nonwandering set is equal to zero when (l1,l2) G [1, 2]2 and (A1) holds and for the pairs (l, 1); l > 1;

2) the Hausdorff dimension is strictly larger than zero when (l1 ,l2) G [2, to)2 \ {(2, 2)};

3) the Hausdorff dimension is strictly larger than zero when (l1,l2) G C\u<1 and p is bi-periodic.

The following remark will simplify the statements and proof of the results.

Remark 1. Note that our setting has some inherent symmetry. This will simplify the statements and proof of our results.

2. Tools

2.1. Cross-ratio inequalities

Notation 1. We denote by R< the subset of R4 defined by

R< := {(x\,x2,x3,x4) G R4, such that x\ <x2 <x3 < x4}.

The following result can be found in [1] (Theorem 2) Proposition 1. The cross-ratio inequality (CRI).

»4

\b — a\\d — c

Let f G L. Let (a,b,c,d) G R<. The cross-ratio Cr is defined by

Cr (a, b, c, d) :=

\c — a\\d — b\ and the cross-ratio Poin is defined by

\d — a\\b — c\

Poin (a, b, c, d) :=

\c — a\\d — b\'

The distortion of the cross-ratio Cr and cross-ratio Poin are given, respectively, by

Cr (a, b, c, d)

and

VPom(aAc,d) := P°M/(a)./№.№)./№)

Poin (a, b, c, d)

Let us consider a set of n + 1 quadruples {ai, bi, Ci, di] with the following properties:

1) each point of the circle belongs to at most k intervals (ai ,di);

2) the intervals (bi,ci) do not intersect U.

Then there is Ck,C'k > 0 such that the following inequalities hold:

n

~\VCr (ai ,b

i, ci, dj) ^ Ck

i=0

and

n

DPoin (ai, bi, ci, di) ^ C'k.

i=0

Observe that Poin + Cr = 1. Thus, by Lemma 1.5 in [2] we have the following result. _RUSSIAN JOURNAL OF NONLINEAR DYNAMICS, 2020, 16(4), 651 672

Proposition 2. Let f be a C3 function .such that the Schwarzian derivative is negative. Then C'k > 1, that is, Ck < 1.

Remark 2. Let I and J be two intervals of finite and nonzero length such that In J = 0. We

assume that J is on the right of I and we put I := [a, b] and J := [c, d], then

Cr (/'J) := = Cr (o'6'c'd)

and

Fact 1. Let f E L. Let l(U) and r(U) be the left and right endpoints of U (the flat piece of f), respectively. There are a left-sided neighborhood I1 of l(U), a right-sided neighborhood Ir of r(U) and three positive constants K1, K2, K3 such that the following holds:

1. If y E Ili with l1 := l, l2 := r, then

K1\li(U) - y\li < \f (li(U)) - f (y)\ < K2\li(U) - y\li, KiHU) - y\li~l < < K2\h(U) - y\li~l.

2. If y E (x, z) C Ili, with z the closest point to the flat interval U, then

\f(x)~f(y)\ < R \x - y| \f(x) - f(z)\ ^ 3|a:-*r

The first part of Fact 1 implies that

f^Iii & kixf'i, i = 1,2. We assume that k1 = k2. (A2)

We need this assumption to prove that an goes to zero (Lemma 1 and Lemma 2).

2.2. Basic Results

Proposition 3. Let n ^ 1.

• The set of "long" intervals consists of the intervals

An ■= {(1, qn + i); 0 < i < qn+i ~ 1}.

• The set of "short" intervals consists of the intervals

Bn ■= {{Qn+1 + i,i); o < i < qn ~ 1}.

The set Pn := An U Bn covers the circle modulo the end points and the flat piece and it is called the nth dynamical partition.

The dynamical partition produced by the first. qn+i + qn preimages of U is denoted by Vn. It consists of

Pn ■= O^ i^ qn+i + qn - 1} "j^_RUSSIAN JOURNAL OF NONLINEAR DYNAMICS, 2020, 16(4), 651-672_

together with the gaps between these sets. As in the case of Pn, there are two kinds of gaps, "long" and "short":

• The set of "long" intervals consists of the intervals

An := {(-qn-i,^i) =: if; 0 < i < qn+1 - 1}.

• The set of "short" intervals consists of the intervals

Bn := {(^j, -qn+i ~ i) =: i?+1; 0 < i < qn - 1}.

Proposition 4. The sequence |(0, qn)\ tends to zero at least exponentially fast.

Proposition 5. If A is a preimage of U belonging to Pn and if B is one of the gaps adjacent to A, then |A|/|B| is bounded away from zero by a constant that does not depend on n, A or B.

Lemma 1. The sequence

,, , 1(1+ 1)1

|(gra-l + 1,1)| is bounded.

The proofs of these results can be found in [5] (proof of Proposition 1, Proposition 2 and Lemma 1.3).

A proof of the following proposition can be found in [3, Theorem 3.1, p. 285].

Proposition 6 (Koebe principle). Let f G L. For every ç, a > 0, there exists a constant Z(ç, a) > 0 such that the following holds. Let T and M C T be two intervals and let S, D be the left and the right component of T \ M and n G N. Suppose that:

n— 1

1) £ fl(T) < ç,

i=0

2) f n : T —> f n(T) is a diffeomorphism,

\P(M)\ \P(M)\

' \fn(S)\ ' \fn(D)\ < •

Then

^-rT^rr^C^«), Vx,yeM,

that is,

where,

Z(ç, a) Df n(y)

< TTBT < C(Ç, «) • ¡4[, \/A, B (intervals) Ç M,

Z(ç, a) \Bp fn(B' \B

a

and C ^ 0 only depends on f.

Remark 3. Let f e L. Given n > 1, T and M as before, fn: T —> f n(T) is a diffeomorphism if and only if, for all 0 ^ i ^ n — 1, f'(T) n U = 0, where U denotes the closure of U.

3. Proof of results

Let us put together the (parameter) sequences which are frequently used in this section.

_ |(-gra,0)| 1(0, Qn)\ _ |[-gn-2,0]| 1(0, g»)I

"" " IbSa. A)| ' ^ " |(£2_i, 0| ' - |0| ' Tn - |(0, q^)|

an<^ l( k )l

Pn{k) = Ir .. , k = 0,1,... an-1-

I [-Qn + fcgra-i,Q)|

3.1. Proof of the first part of the main result 3.1.1. A priori bounds of an

Proposition 7. Let n G N and (i\,i2) G ^o = [1, 2]2. For all an,

an 2 < 0.55;

for at least every other an

If

then either

2

y.n 2

h ¿2 £iiÎ2

an 2 < 0.44 or a^ < 0.16.

Proof. For every n E N and k = 0,1,... an-1, we define the parameter sequences

71 n(k)

lhn(k) '■= \{~Qn + kqn-1, 0)|, 71,„-l := 7i,„_i(0), jn(k) = —-

--7l,ra-l

7(/i№{k) := l|iif n G 2Z and := if n G 2Z + 1.

71,n-1 71,n-1

These notations simplify the formalization of the following lemma which will play an important (essential) role in the proof of Proposition 7.

Lemma 2. For n large enough and for every k = 0,1,... an-1 — 1, the following inequality holds:

(1+ai^HmMk)^ + 7-s,Mk +1}- (3-1}

Proof. Let n be an even nonnegative large enough integer. For fixed k = 0,1,... an-1 — 1, according to assumption (A2), the left-hand side of (3.1) is equal to the cross-ratio

Pom (~qn + kqn-i + 1, -qn-i + 1)• Applying fqn-1-1, by expanding the cross-ratio property, we get the inequality. □

1 ,'-'2

The left-hand side is a function of the three variables ¡3n(k)£l '('2, ol^I and y^1 ^. Observe that the function increases monotonically with each of the first two variables. However, with respect to the third variable, the function reaches a minimum. To see this, take the logarithm of the function and check that the first derivative is equal to zero only when

(,Jii|«3h2 = Pn(kYl42

an—1

By substituting this value for y^1 ^ in (3.1), we get

< snfin(k + 1). (3.2)

Pn(k) 2

£1^2

e-, ,tn 2

n-

l + Pnik^a^

Put:

xn{k) := min j/?„(fc) 2", CK„i! j and yn(k) := fJn{k) 2'

Since (3n(k+1) ^ yn(k+1), substituting the above variable into (3.2) gives rise to a quadratic inequality in xn(k) whose only root in the interval (0,1) is given by

\/ Snyn{k + 1) 1 + \J1 - Snyn(k + 1)'

that is,

*.<*) = mii ^ + (3.3)

I J 1 + V1 " snyn(k + 1)

Lemma 3. There is a subsequence of an including at least every other an, such that

limsupato2 ^0.3.

Proof. We use the following elementary lemma in [5]. Lemma 4. The function

hn{z) =

1 + a/1 - snz

moves points to the left, h(z) < z, if z ^ 0.3 and n is large enough. We select the subsequence.

1. The initial term: there exists n — 2 e N such that. an22 ^ 0.3. This comes directly from the properties of the function hn (Lemma 4) and from (3.3).

2. The next element: suppose that an-2 has been selected. If

e-i,e-2 ¿1,12

xn(k) = or ^ 0.3,

for some k = 0,1,... an-1 — 1, then we select an-1 as the next term. Otherwise, an is the next term. Thus, the sequence is constructed. □

Corollary 1. For the whole sequence (an) we have

limsupato2 ^0.3.

Moreover,

if an-1 does not belong to the subsequence (an) defined by Lemma 3, then either

an2l <0.44 or an 2 <0.16.

Proof. Observe that the function

H : (s, t) G R+2 ^ F (s, t) =

2 ^ S + t

1 + St

is symmetric and, for fixed s, the function F (s, ■) reaches its minimum in zero by taking the value s. Therefore, for every s,t ^ 0,

s +1 s,t^-.

' 1+St

So,

Cl,C2 ¿1,12 £1^2 a„2 + a 21

n n ^ n ' — 1 /Q a \

, <-fl fl ■ (3.4)

1 + an 2 anli

n-

Hence, an-2 is an element of the sequence and suppose that an-1 do not belong to the previous subsequence of Lemma 3, then it follows from (3.2) that the right member of (3.4) is estimated as follows:

ttn2 + an-i _ / - . . / - . .

< ysnfU « \J/3n{vo.3. (3.5)

2 ™ 2

1 + an

Also

1 + an2 an2l

«ii£2 £II£2

min | an 2 ,an21 j = an 2 ^0.3. (3.6)

Thus, if an 2 ^ 0.16, then by combining this with (3.5) and (3.6), we obtain the desired estimate. □

Proposition 7 is proved. □

3.1.2. Recursive formula of an

Proposition 8. Let n be a large enough integer, 1) if ¿x, ¿2 > 1, we have

a2n ^ M2n(¿i)a^n-2 and (x^n+x ^ M2n+i(¿2)a^n-i, (3-7)

where 2 , ,

Mn(£) = 4-1

¿

2) if ¿i = ¿2 = 1, then

t / 2(I - 1) 1 - a«-2 On-2

1 + \/ 1--1,-Sn-lCXn-l

Oin < ■ —an-2. (3.8)

On-2

1 ,'2

Proof. We treat the case n even and the case n odd is treated in a similar way. Recall that

_ KzSnM ^'ItskM'

For all even n large enough, by Proposition 4 and point 2 of Fact 1, applying f to the equality, we get

£l \i~Qn + 1;1)|

a = —-,

" \\~Qn +1,1)1'

which is certainly less than the cross-ratio

Poin(—gra + 1, (1, —qn-i + 1]) • Since the cross-ratio Poin is expanded by fqn-1-1, then

an < 5n (1)sn(1) (3.9)

with

(k) _ \(-Qn + kqn-i,kqn-i)\

| [~qn + kqn-1, kqn-\)\

and

_ \[-qn + kqn-i,0}\

Sn[k): |(-fr + fcgw-i,o]|-

If an-1 = 1, by multiplying and dividing the right member of (3.9) by a2n-2, we obtain directly (3.11). n

Suppose that an-1 > 1 and estimate 5n(k). By the mean value (Lagrange) theorem, f transforms the intervals defining the ratio 5n(k) into a pair whose ratio is

—dn(k) Vk

with uk being the derivative of f (x£l) at a point in the interval

Uk ■■= (~qn + kqn-i,kqn-i),

and vk being the derivative of f (x£l) at a point in the interval

Vk := [~qn + kqn-i,kqn-i).

Note that, for n sufficiently large,

U1 <V1 <U2 <V2 ... < Van-1.

We see that the image of 5n(k) by f is smaller than

Poin(-f/„ + kqn-i + 1, (kqn-1 + 1, -qn-i + !])• Once again, by expanding the cross-ratio property (fqn-1-1), it follows that

—Sn(k) < sn(k + 1) • 5n(k + 1). (3.10)

Vk

Multiplying (3.10) for k = 0,...,an-1 — 1, and substituting the resulting estimate of 5n(1) into (3.9), we obtain

v

ai1 ^ Snian-^-^-Snil)... sn(an-i).

U1

Observe that sn(1)... sn(an-1) < sn and

< iK-qn^MY1'1 < |(~fr-2,0)

ui \ |(qn-i, 0)|

Thus,

which can be rewritten as

with

|(-gra-2,0)| an ^ sn ■ I, nxi-• dn{an-i),

Vn-2 :=

p 2 an ^ SnVn-2V"n-2a,n-2

|[-<?n-2,0)| \[-qn-2,i

1(^-1,0)1 \[-qn-2,an-iqn-i)

and

l^n-2 : =

{—qn-2, an-1qn-1)

\{-qn-2,[

(3.11)

It remains to estimate vn-2 and /in-2 to complete this part. For vn-2, observe that

so that

I ( <?ra—2 , 0) | < \(qn-3,0)

Vn-2 ^

Vn-1Vn-2 1 — an-2

(3.12)

The estimation of /in-2 is facilitated by the following elementary lemma in [5].

Lemma 5. Let £ E (1, 2). For all numbers x > y, we have the following inequality:

p p x—y /X—y

xp

x

£

£(£ - 1) f x - y

x

Now apply f into the intervals defining the ratio /in-2. By Lemma 5, the resulting ratio is larger than

The cross-ratio Poin

that is,

li(li - 1) ^n-2t+l--2-^n-2)-

Poin {-qn-2 + 1, iqn-1 + 1,1)),

\{-q„-2 + l,q„-i + l)||[-gra-2 + 1,1)1 \[-q„-2 + l,q„-i + 1)|| ( Qn—2 + 1,1)1

v

an-1

1

1

is larger again. Thus, by expanding the cross-ratio property on fqn-2, we obtain

,, li(li-l)

¡j,n-2(ti----¡Mn-2) ^ sn-ianan-i. (6.16)

By solving this quadratic inequality, we obtain

1 + \/l - ~ l)sn-ianan-i

£l

Vn-2 > -1--* (3-14)

Thus, by combining (3.13) and (3.14), we obtain

2 1

Hn-2 < J- ' -, . Sn-i(Tn(Tn-\. (3.15)

£i L 2(t\ — 1)

1 + w 1--j-sn-ianan-i

Since anan-i < an-\, the first inequality in (3.7) follows by combining inequalities (3.11), (3.12) and (3.15). Likewise, the second inequality in (3.7) is obtained by following suitably the same reasoning as previously. □

3.1.3. an go to zero

If £i = ¿2 = 1, then by Proposition 7, nk=n Wk

goes to zero; thus, by composing the inequality obtained by f, since f (an) is bounded (Lemma 1), the result follows.

Note that the cases where the critical exponents are of the form (1,£) or (¿, 1), with ¿> 1, are treated in [4].

Now let us suppose that £l ,£2 > 1. Technical reformulation of Proposition 8. Let Wn be a sequence defined by

Mn(e) = wn(£) an

Let

n\ .

On-2

Mn(£):= Mn(£)a2n-e2 and W,n(£) := Wn^atl■ The recursive formula (3.7) can be written for n even in the form

a„

°n-2

So,

k=n

On .„ -a0

k=2 00

ci1 < n wtiei)?rcft.

nk=n Wk(£i) goes to zero.

Observe that the size of W'n(£l) is given by the study of the function

-I__o

1 vcl

2 2y £i

The meaning of the variation of W'n(x, y, £l) relative to the third variable is given by the following lemma in [5].

Lemma 6. For any 0 < y < x G (0,1) and l\ G (1,2] the function W'n(x,y,ii) is

Ve

increasing with respect to l1.

Analysis of the asymptotic size of W!(2)

Since the hypotheses of Lemma 6 are satisfied (Proposition 7), the only remaining point is the verification of the convergence of Yïi=i W'(2).

• If an-2 < (0.3)£l, then W'(2) < W'(0.55, 0.16, 2) < 0, 9.

• If not, then by Proposition 7, W'(2) < W'(0.3,0.44,2) < 0,98 or else, Wil+1(2)Wi(2) <

W'(0.55, 0.16, 2)W'(0.16, 0.55, 2) < 0, 85.

Corollary 2. Let £1,£2 G [1, 2]. If 1 < i1 < 2, respectively, 1 < i2 < 2, then a2n, respectively, a2n+1 goes to zero least double exponentially fast. And if i1 =2 or 1, respectively, l2 = 2 or 1, then a2n, respectively, a2n+1 goes to zero least exponentially fast

Proof. Let n := 2pn G N. From the analysis of the asymptotic size of W'(2), it follows that, as n goes to infinity, Hn=0 W'i(l\) goes to zero and so does an. Therefore,

riM (h)

i=0

goes to zero as n goes to infinity. Thus, by Proposition 8, for n even, there is X0 such that • if 1 <h < 2,

an ^ A0vil

and if h = 1, 2,

'x-n ^ .

an < W

The case n odd is treated in the same way. □

3.2. Proof of the second part of the main result

In this section we find a bounded geometry domain.

3.2.1. Recursive affine inequality of order two on an

Let

l(Q,gn)l

l(0,-<7„-i)|

Remark 4. Since the point qn-2 lies in the gap between —qn-\ and —qn-i + qn-2 of the dynamical partition Pn-2, it follows by Proposition 5 that rn/an-i and Kn are comparable.

Proposition 9. For any rotation number of bounded type, there is a uniform constant K such that

1 — £d2„ +1 1 — £a2n+1+l

K2n

>K (a2n-1) l2 1 and K2n+1 >K (a2n) l1 1

2

1. Proof of the proposition: If an = 1, it comes down to showing that the sequence Kn is bounded away from zero, which becomes relatively very simple. In fact, suppose that

\(qn, -qn-1)| ^ |(0, qn)\, then

I (0, (¡n) I > 1

else, Kn is greater than

|(0,-ç„_i)| 2'

\-qn+i\

~qn+l, -qn-l

which by Proposition 5 is bounded away from zero.

In the following part of the proof, we suppose that an > 1 and the following lemma in [5] (Lemma 4.1) is used.

Lemma 7. The ratio

is bounded away from zero by a uniform constant for all i = 0,..., an-1.

Lemma 8. For every n £ N and for all i = 0,... ,an-1, there is a uniform constant K such that

Pn(i)£l 'h > (3n(i + 1).

Proof. For n large enough and for fixed i = 0,..., an-1, by Proposition 4 and Fact 1, we

have

íl/2 = \{-qn + kqn-i + 1,1)|

PnK,) \[-qn + iqn-i + 1,1)1 '

which is greater than

Cr([~qn-2 + 1, -qn + iqn-1 + 1), {~qn + iqn-1 + 1,1))•

By applying the cross-ratio inequality under fqn-2-1, by Fact 1 the resulting ratio is greater than

Cr([~qn-2 + 1, -qn + iqn-1 + qn-2 + 1),

{-qn + iqn-1 + qn-2 + 1, qn-2 +1))

times a uniform constant. We now repeat this sequence of steps an-2 — 1 times: Apply fqn-2-1, discard the interval containing 0 and use point 2 of Fact 1, and replace the result by a cross-ratio spanning the intervals —qn-2 + 1- At. the end, this will produce the cross ratio

Cr([-qn-2 + 1, ~qn + kn-1 + an-2qn-2 + 1), {-qn + iqn-1 + an-2qn-2 + 1, an-2qn-2 + !))•

And finally, by applying since by Lemma 7 the interval containing — g„_2 + qn-¿ is

bounded away from zero, the resulting ratio is

I{-qn + {i + l)qn-i, qn-1)I I [-qn + {i + l)g»-i, qn-1) I

times a uniform constant.. Thus, since —qn + qn-\ lies between —qn + {i + l)g„_i and qn-i, it. follows by Lemma 7 that this ratio is comparable to (3n(i + 1). □

K

n

2. Back to the proof of Proposition 9: Observe by Proposition 5 that

\{-qn + {i + l)gra_i,0)|

I[-Qn + iqn-i, 0)I are comparable. Therefore, Kn-1 is comparable to the product

fin(1). ..fin(an-1 - 1).

By combining this with Lemma 8, we have Proposition 9.

3. Recursive affine inequality of order two on an.

Proposition 10. If p(f ) is of bounded type, then there is a uniform constant K such that

Í2 1 - t-a2n

a2n > K (a2n-i)£l ¿2 - 1 (a2n-2)¿1

and

£i 1 - £-a2n+1

a2n+i > K (a2n)12 ll - 1 (a2n-iY2 " . 4. Proof of the proposition. If n is even and large enough, then

£l \(-qn + i,DI

Oin

\[-qg + 1,1)1' which in turn is larger than the product of two ratios

\{—Qn + 1,1)1 _ \(—qn + 1, —qn-i + 1)1

\(-qn + l,-qn-i + l)\ 42'w~ |[-gw + l,-gw-i + l)|-

Lemma 9. For all even n large enough

Cl,n ^ Krn-

Proof. Observe that £1>n is greater than

Cr((-gra + l,l),-gra_i + l).

By applying CRI on fgn-1-1 and discarding the intervals containing 0, repeat this a,n-i — 1 times more: By Fact 1, the resulting ratio is larger than

Cr ((-qn-2 ~ qn-1 + 1, (an-1 ~ l)g»-i + 1), (1, ~qn-1 + !]))•

Apply fqn-1-1 and discard the intervals containing the flat interval. Apply f, replace the resulting by the cross-ratio

Cr ((-qn-2 + 1, qn-i + l), (1, ~qn-1 + 1]))•

2n-1

Thus, by CRI on fqn-2 l and the inequalities above, we obtain

Kgn-2, -qn-3]\

6,« > —r,-TTTn' (3-16)

The first factor on the right-hand side of (3.16) is greater than

I Qn—31 W-Qn-slV

which by Proposition 5 goes away from zero. The lemma is proved. □

Lemma 10. There is a uniform constant K so that, for all n,

(,-a2n-1 + 1 £-a2n + 1

&,2n > K (a2n-2) 1 and &,2n+i > K (ain-i) 1 .

Proof. If an-l = 1, then £2,n is greater than

Cr ([-qn-2 + 1, -Qn + 1), {-qn + 1, -qn-i + !))• By applying CRI (qn-2 — 1), the resulting ratio is greater than

Cr ([-qn-2 + 1, -qn-1 + 1), {-qn-1 + 1, ~qn-3 + 1))

times a uniform constant. Thus, by applying CRI to this ratio with fqn-3-l, the above inequalities and Proposition 5, the result follows.

Now suppose that an-l > 1, then £2,n is greater than

Cr {[-qn + qn-i + 1, ~qn + 1), {~qn + 1, ~qn-1 + !))• By applying fqn-1-A- to this ratio, it follows from Lemma 7 that

> K'Pn(1).

And the lemma follows from this by using Lemma 8 modulo the fact that

Pn (an-l) = an-2. □

Combining Lemma 9, Lemma 10, Proposition 9 and Remark 4, the result of Proposition 10 follows.

Remark 5. By the inequality obtained from Proposition 10, the sequence vn defined by

Vn = — ln an

verifies the following recursive affine inequalities of order two for every n > 0

V2n < T ■ i2(02„>2n-l + + K' (3.17)

£1

and

£1

V2n+i < J- • ti(a2n+l)V2n + f, "2>2„-i + K', (3.18)

£2

with

1 - £-j M = T^r-

3.2.2. Analysis of the recursive affine inequality

We will prove that the sequence vn is bounded. Let us consider the sequence of vectors (vn) defined by

( Vn \

Vn = ,

V Vn-1 J

the vector given by

and the sequence matrix

/ K'

k =

\ 0

h

A£u£2(2n) = I £i

t2(a2n) l-

1 0

and

A£l/2 (2n + 1) = l2

\ 1 0

say the associated matrix to the recursive affine tnequalities (3.17) and (3.18), respectively, in the sense that (3.17) and (3.18) can be rewritten, respectively, as

V2n ^ A£lj2(2n)v2n-i + « and v2n+i ^ A£2,£l(2n + l)v2n + (3.19)

Therefore, for every n := 2pn + rn, with pn £ N* and rn £ {0,1}, we have

vn ^A£u£2(n)A£u£2(n -2)... A£u£2( 2 + rn)v2-rn +

I Id + (n)4,i2 ("' - 2) • • • A£lM(i) J K',

\ i=4+rn J

where

A£u£2(n) = A£u£2(n)A£u£2(n - 1). Observe that, if (l1 ,l2) is very close to an element of the set

{(a, to), (<x>,b), (to, <x>), a,b £ R},

A£l,£2 (n) is diagonalizable with nonnegative eigenvalues, at most one of them being strictly positive, that is, l/l\ or l/l2 and as £\,£2 > 2, then A£l^2(n) contracts the Euclidean metric; therefore, vn (also vn and an) is bounded. In the following analysis, we suppose that 1 < l1, l2 < C < to for some C in R+.

Lemma 11. Fix (l1 ,l2) £ [2, to)2. The sequence

A-Zh := A£u£2(n)A£u£2(n - 2)... A£u£2(4) is bounded (uniformly) by max{l1 /l2,l2/l1}.

Proof. Observe that, for all n G N,

Aeue2 (n) ^

(1 - fe„(2))(l - bn-1(2>) |(1 - fe„(2))fe„_2(2)

bn-2(2)

=■ B£l/2{n)

with

b2n(t) = b2n := £-a2n and b2n+i(i) = bw := t-

-a,2n+l

Let be a sequence (xn)neN defined by

(xn)n€N := {1 - bn(2),bn(2); n e N}.

Remark that, for every n e N, xn e [2-a, 1 — 2-a], where, a := max{an,n e N}. Thus, by setting

( dhn ?_Ld3 ,n\ «1

~R°n —

B il /2 =

/

it follows that, for every i e {1,2,3,4}, there is xki ■, j = pn-2, ...,n — 2 such that

n- 2

d ' ^ ^^ -^Kii

j =Pn-2

xkii ■ ■ ■ xkij ^ 1

This proves the lemma.

□

Proposition 11. If (i 1,^2) £ [2, 00)2 \ {(2,2)}, then A™in contracts the Euclidean metric provided n is large enough. The scale of the contraction is bounded away from 1 independently of £\ and and the particular sequence bn, whereas the moment when the contraction starts depends on the upper bound of bn.

Proof. For fixed n e N, putting

A°n -Ail/2 =

( d1'n(zl,z2) ^d3'n(Zl,Z2)\

¿2 j2,n

Ml

d2'n(Z!,Z2) d4'n(z1, Z2)

/

with zl = 1/(ll — 1) and z2 = 1/(t2 — 1), we find that di'n(zl,z2), i = 1,2,3,4 are polynomials of respective degree n — 2, n — 3, n — 3 and n — 4, whose coefficients belong to the interval [i-a, 1 — i-a], with I = m&x{li,l2). For fixed i G {1,2,3,4}, we denote by djn the coefficients of di'n(zl ,z2). Set

( di(2) = 1 and di(i) = 0; i = 1,3,4, \d2(3) = 1 and d2(i) = 0; i = 1,2,4.

Then by Lemma 11, the sums

Pn-2

^di'jz1+dl(l)Zj+d2(l) ; i = 1, 2, 3, 4 j=0

are uniformly bounded. Therefore, for every i G {1,2,3,4}

oo

djn4+dlW4+d2M 0 as k .

j=k

Lemma 12. For every i G {1, 2, 3, 4}, the sequence (diJn) tends to zero at least exponentially. Proof. By a simple calculation, we have

Pn-2 Pn-2

dln(0, 0) = [] bn-2i+i, d4n(0, 0) = [] bn-2i, d2n(0, 0) = d3n(0, 0) = 0.

i=l i=l

Now suppose that, for given 0 < n — 1 and 0 < j < n — 1, all the coefficients djn-1 i G {1,2,3,4} tend to zero at least exponentially fast. Then, since

A°£U£2 = A£uh(n)A£iX2

and by the form of the coefficients of Aelte2(n), the lemma and hence Proposition 11 is proved. □

□

3.2.3. Particular case of Bounded Geometry

Proposition 12. Let f G L with critical exponents (il,i2) and rotation number p(f ) = = [abab...] for some a,b G R. If the inequality

sJ{e-b_e-a) 2 + (il(6)i2(a) + 2{l~b + qa))tl{b)t2{a)+

+ti(b)t2(a) + — + — — 2 < 0

holds, then the geometry of f is bounded. Proof. If p(f) = [abab...], then

Ah,Î2 =

i«» «* /

and the eigenvalues As and Xu are defined as follows:

2A, = - ^r6 " ¿2°)2 + Mb)t2(a) + 2{l\h + qa))tl{b)t2{a) +

+ti(b)t2(a)+ — + £-a

and

2A„ = - qa)2 + Mb)t2(a) + 2{qb + ¿ïa))Ub)t2{a) +

+ti(b)t2(a)+ — + l-a.

Observe that. Xs G (0,1). Thus, if Xu < 1, then A^^ contracts the Euclidean metric.

This proves the proposition. □

3.3. Proof of the Corollary Lemma 13. Let

\(-Qn + (i + l)qn-i, -Qn + iQn-i)

Wn (i) =

\-Qn + iQn-i\

be a parameter sequence with i = 0 ...an-i — 1. wn (i), i = 1 ...an-1 — 1 and w^ 'f'2 (0) are comparable to an-1.

Proof. Suppose that an-1 > 1 and let i = 1,..., an-1 — 1. We apply Proposition 6 to

• T = [~qn + (i + l)qn-i, ~qn + (?- ~ l)<?n-i],

• M = (~qn + (i + l)qn-i, ~qn + (?- ~ l)<fo-i),

• S = -qn + (i + l)qn-i,

• D = -qn + (i - l)qn-i,

• f qn-(i-i)qn-i

1) For every j < qn — (i — l)qn-\, fj{T) n U = 0; so f9n-^-i)qn-1 js a diffeomorphism on T (Remark 3);

2) the set IJqr=-('l-1"iqn-1 (t) covers the circle at most two times;

3) for n large enough, by Proposition 5, we have

fqn-(i-l)qn-l(M) f<ln-(i-l)qn-l(M) | (0, -2gra_j)|

fqn-(i-l)qn-l(S) < f<ln-(i-l)qn-l(D) ~ \-'2qn-i\ <

Therefore, it follows from Proposition 6 and Proposition 5 (|— qn-i\ and |(0, — qn-i}\ are comparable) that wn(i) and an-1 are comparable.

For i = 0 (which is the only case where a = 1), we apply Proposition 6 to

• T = [~qn + qn_i + 1, -qn - qn_i + 1],

• M = (~qn + qn-1 + 1, -qn - qn-1 + 1),

• S = -qn + qn-1 + 1,

• D = -qn - qn-i + 1, f qn-qn-1-1

As before, the hypotheses are satisfied. And for n large enough,

!(0) =

\(-qn+qn-i + l,-qn + l)\

I-Qn + 1|

which is also uniformly comparable to an-1.

This concludes the proof. □

The rest of the proof of the corollary is as in [8] (Theorem 1.4 and Theorem 1.5).

Acknowledgments

I sincerely thank Prof. M. Martens and Prof. Dr. L.Palmisano for introducing me to the subject of this paper, for valuable advice, continuous encouragement and helpful discussions. I thank Prof. Dr. Carlos Ogouyandjou for participating in the presentations related to this work.

Conflict of Interest

The author declares that he has no conflict of interest.

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