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Rational Rotation Numbers for Homeomorphisms with Two Break-type Singularities$^*$\footnote{$^*$üÔĎÔ ÔĹËÓÔ ĐĎĚŐŢĹÎ ÉÚ ĆÁĘĚÁ 2.tex}

Rational Rotation Numbers for Homeomorphisms with Two Break-type Singularities^*¹

D.Khmelev

1 Introduction

This work deals with the family of homeomorphisms of the circle T_a:S¹Ž S¹, where S¹=[0,1), a Î [0,1]. Each T_a is defined by its lift f_a:RŽR such that {f_a(x)}=Tx. We consider simpliest family f_a=f+a, where 0 Ł f(0) < 1.

Suppose that T={f} has two break-type singularities at points x_cr\11 š x_cr\12, i.e., the following conditions hold:

1) T Î C^2+e(S¹\{x_cr\11,x_cr\12}),

2) There exists T˘(x_cr\1i-) and T˘(x_cr\1i+), i=1,2, and

c_i= ć
Ö

T˘(x_cr\1i-)
T˘(x_cr\1i+)

š 1.

Denote by r(T_a)=r(f_a)=lim_nŽĽ fⁿ_a(x₀)/n the rotation number of the homeomorphism r(T_a).

Theorem 1 Suppose c₁c₂ š 1 and c₁ š c₂. Then the Lesbesgue measure of the set {a | r(T_a) Î [0,1]\Q} is zero.

This theorem for one break-type singualrity was proven in [1].

2 Notations and apriori estimates

Let us denote an interval with endpoints a and b as follows

I(a,b) =

ě
í
î

(a,b),

if a Ł b,

(b,a),

if a > b.

We also put I[a,b]=[`(I(a,b))]. A good description for Farey numbers is given in [1]. We shall need only the following properties of Farey interval A=(p₁/q₁,p₂/q₂) Ě (0,1):

p₂ q₁-p₁q₂=1.

All rationals inside A have the form (kp₁+lp₂)/(kq₁+lq₂). The rational number with smallest denominator is (p₁+p₂)/(q₁+q₂).

Consider homeomorphism T with two break-type singularities and corresponding lifting f. Choose arbitrary point x₀ on the circle and consider the trajectory {x_i=Tⁱ x₀, 0 Ł i Ł q₁+q₂}. Denote by D\11₀ and D\12₀ the closed intervals [x₀,x_q₁] and [x_q₂,x₀], respectively. Denote by D\11_i and D\12_j the images of these intervals under the action of T:

D\11_i=TⁱD\11₀, D\12_i=T^jD\12₀,

The following Preposition was proved in [1] and the proof works in our situation without any change.

Preposition 1 Suppose r(T) Î (p₁/q₁,p₂/q₂). The trajectory {x_i=Tⁱ x₀, 0 Ł i Ł q₁+q₂} forms a partition of the circle consisting of intervals D\11_i, 0 Ł i Ł q₂ and D\12_j, 0 Ł j Ł q₁.

Denote this partition by x(A,x₀). Denote

v=|lnc₁² |+|lnc₂² |+\Var_S¹lnf˘.

Consider now trajectory y_i=Tⁱ y₀ of point y₀, such that y_i š x_cr\11 and y_i š x_cr\12 for any 0 Ł i Ł q₂. Denote q=max(q₁,q₂) and p=max(p₁,p₂).

Preposition 2 Suppose r(T) Î I((p₁+p₂)/(q₁+q₂),p/q) or r(T)=p/q. Then

e^-v Ł q-1
Ő
i=0
f˘(y_i) Ł e^v.
(1)

Picture Omitted

Figure 1:

Picture Omitted

Figure 2:

Proof We follow the lines of proof given in [1] with corrections due to the second singularity point. To be definite, assume that q₂ > q₁. Then q=q₂, p=p₂. Let us consider the case r(T) š p₂/q₂ (the same arguments apply to case r(T)=p₂/q₂ because there exists a periodic trajectory x₀, ..., x_q₂-1, such that x₀, ..., x_q₂-1 form a proper partition of the circle). Consider partition x((p₁+p₂)/(q₁+q₂),p₂/q₂; x₀), where the trajectory x_i=Tⁱ x₀ of x₀ do not touch critical points. Notice that x_q₁+q₂ Î D\11₀, since r(T) Î ((p₁+p₂)/(q₁+q₂),p₂/q₂). Let us show that

ę
ę q₂-1
ĺ
i=0
lnf˘(y_i)- q₂-1
ĺ
i=0
lnf˘(x_i) ę
ę Ł \Var_S₁ lnf˘.
(2)
Indeed, assume that y₀ Î D\12_j, 0 Ł j Ł q₁. Let x_k+j correspond to y_k for 0 Ł k < q₁-j and let x_k+j-q₁ correspond to y_k for q₁-j Ł k < q₂. We have [x_k+j,y_k] Ě D\12_k+j, 0 Ł k < q₁-j for 0 Ł k < q₁-j and [y_k,x_k+j-q₁] Ě D\11_k+j-q₁ for q₁-j Ł k < q₂, i.e., the intervals do not intersect each other. Hence, we get estimate (2). The same arguments apply to the case y₀ Î D\11_i, 0 Ł i Ł q₂.

Notice now that

ó
ő
S₁=[0,1)
q₂-1
Ő
i=0
f˘(y_i) dy₀ = 1
ó
ő
0
ś
śy₀
f^q₂(y₀) d y₀ = f^q₂(1)-f^q₂(0)=1.

Notice that g=śf^q₂/śy₀ is not equal 1. It follows that g takes values above and below 1. Without loss of generality suppose that x_cr\11 = 0 and x_cr\12 Î (0,1). If g is arbitrary close to 1 in neighbourhood of some point y₀, then we are done (this case is shown on Figure 1).

Hence, assume that g never approaches 1, as shown on Figure 2. Notice that g is piecewise-continuous (and even of class C^1+e on each interval of continuity). All disconinuity points are formed by preimages T^-ix_cr\11 and T^-jx_cr\12 of singularities x_cr\11 and x_cr\12 for 0 Ł i,j Ł q₂. For some i, j we can have T^-ix_cr\11 = T^-jx_cr\12 and this imply that corresponding singularity of g has break c₁c₂. Other singularities have break equal c₁ or c₂. Notice that no other breaks can happen. Indeed, if for some 0 Ł i₁, j₁, i₂, j₂ Ł q₂ we have

T^-i₁x_cr\11 = T^-j₁x_cr\12 and T^-i₂x_cr\11 = T^-j₂x_cr\12,

then point y^*=T^-j₂x_cr\12 is periodic of period q^*=|(j₁-j₂)+(i₁-i₂) |, since

T^-j₂°T^j₁°T^-i₁°T^i₂y^*=y^*

Hence r(T)=p^*/q^* for some p^*. But q^*=|(j₁-j₂)+(i₁-i₂) | Ł 2 q₂, and this contradicts property (2) of Farey intervals, since

p^*
q^*
Î ( p₁+p₂
q₁+q₂
, p₂
q₂
)=B

and the rational with smallest possible denominator in B is (p₁+2p₂)/(q₁+2q₂), but

q₁+2q₂ ł 1+2q₂ > 2q₂ ł q^*.

Using the same arguments we arrive at contradiction if T^-i₁x_cr\11 = T^-i₂x_cr\11 or T^-i₁x_cr\11 = T^-i₂x_cr\11 for some 0 Ł i₁, i₂ Ł q₂.

The case r(T)=p₂/q₂ is even easier. If for some i, j we have T^-ix_cr\11 = T^-jx_cr\12, then x_cr\11 = T^i-jx_cr\12 and all q₂ singularities have break c₁c₂. If T^-i₁x_cr\1j = T^-i₂x_cr\1j for some i₁ < i₂, i₂-i₁ < q₂, j=1,2, then we have r(T)=p^*/q^*, where q^*=i₂-i₁. But q^* < q₂ and we arrive at contradiction with r(T)=p₂/q₂,

Hence the following inequality holds for every singular point y^*:

ę
ę
ę ln ć
Ö

g(y_*-)
g(y_*+)

ę
ę
ę Ł |lnc₁ |+|lnc₂ |=C_1,2.
(3)
Notice now that since g take values above and below 1, one can choose a singularity point y^* such that one of the following conditions holds.

1) g(y^*-) < 1, g(y^*+) > 1,

2) g(y^*-) > 1, g(y^*+) < 1.

Consider case 1). Then, by (3)

g(y^*-)
g(y^*+)
ł 1
e^{2 C_1,2}
.
(4)
Since g(y^*+) > 1,

g(y^*-) > 1
e^2C_1,2
,

and since g(y^*-) < 1,

e^2C_1,2 > g(y^*+).

The same argument works for case 2). Hence,

g(y^*-),g(y^*+) Î (e^-2C_1,2,e^2C_1,2).

Hence given e > 0 we can choose x₀ such that

|lng(x₀) | < 2C_1,2+e=|lnc₁² |+|lnc₂² |+e.
(5)
Estimates (2) and (5) imply

ę
ę q₂-1
ĺ
i=0
lnf˘(y_i) ę
ę Ł \Var_S₁ lnf˘+|lnc₁² |+|lnc₂² |+e.
(6)
Since e is arbitrary, inequality (6) holds for e =0 and taking exponents of both sides of (6) we obtain (1). q.e.d.

Now denote

q= max
(q₁,q₂);
q˘= min
(q₁,q₂)

p= max
(p₁,p₂);
p˘= min
(p₁,p₂).

Preposition 2 gives a-priori estimate for derivative of f^q for r(T) Î I((p₁+p₂)/(q₁+q₂),p/q) and r(T)=p/q. The following prepositions provide esitmates for derivatives of T^q˘ and T^q on the whole interval [p₁/q₁,p₂/q₂]. Both prepositions are simple corollaries of Preposition 2. They are analogous to Prepositions 3.3 and 3.4 in [1] and are proven in the same way.

Preposition 3 Suppose one of the following conditions hold:

1) r(T) Î (p₁/q₁,p₂/q₂);

2) r(T)=p˘/q˘;

3) r(T)=p/q, q ł 2q˘.

Then

e^-v Ł q˘-1
Ő
i=0
f˘(y_i) Ł e^v.

If the following condition 3') holds

3') r(T)=p/q and q < 2q˘,

then

e^-2v Ł q˘-1
Ő
i=0
f˘(y_i) Ł e^2v.
(7)

Proof Take 0 < q₃=q-lq˘ < q˘, l Î N, and, respectevely, p₃=p-lp˘. Applying of Preposition 1 to Farey interval I(p₃/q₃,p˘/q˘) we obtain that the following estimate

e^-v Ł q˘-1
Ő
i=0
f˘(y_i) Ł e^v
(8)
holds if r(T) Î ((p₃+p˘)/(q₃+q˘),p˘/q˘) or r(T)=p˘/q˘. Hence, we immidiately obtain our statement in case 2). Case 1) is also obtained from (8), since

( p₁
q₁
, p₂
q₂
) = I( p˘
q˘
, p
q
) Ě I( p˘
q˘
, p₃+p˘
q₃+q˘
).

Consider now case 3). Notice that if q=2q˘, then by property (1) of Farey intervals we have

1=|pq˘-p˘q |=|pq˘-2p˘q˘|=|p-2p˘|q˘,

and since q˘ is integer, we have q˘=1. Hence, our estimate is trivial. Therefore, assume that q > 2q˘. Then q₃=q-lq˘ with l ł 2 and hence

p
q
Î I( p˘
q˘
, p₃+p˘
q₃+q˘
).

It stays to conisder case 3'). We have q < 2q˘. Apply our statement to Farey interval I((p-p˘)/(q-q˘),p˘/q˘). Since q-q˘ < q˘ and r(T)=p/q Î I((p-p˘)/(q-q˘),p˘/q˘), we fall in case 1) considered before. Hence, the following estimate hold:

e^-v Ł q-q˘-1
Ő
i=0
f˘(z_i) Ł e^v.

Applying Preposition 1 to Farey interval I(p˘/q˘, p/q), and remembering that r(T)=p/q, we obtain

e^-v Ł q-1
Ő
i=0
f˘(z_i) Ł e^v.

Hence,

e^-2v Ł q-1
Ő
i=0
f˘(z_i) / q-q˘-1
Ő
i=0
f˘(z_i) Ł e^2v

or

e^-2v Ł q-1
Ő
i=q-q˘
f˘(z_i) Ł e^2v.

Choosing z_q-q˘=y₀ we obtain (7). q.e.d.

Preposition 4 Choose q₃=q-lq˘ such that 0 < q₃ < q˘, l Î N.

1) If r(T) Î I(p˘/q˘, (p₁+p₂)/(q₁+q₂)), then

e^-(l+1)v Ł q-1
Ő
i=0
f˘(y_i) Ł e^(l+1)v.
(9)

2) If r(T)=p˘/q˘, then

e^-(l+2)v Ł q-1
Ő
i=0
f˘(y_i) Ł e^(l+2)v.
(10)

3) Denote

Y_k=I( (k+1)p˘+p
(k+1)q˘+q
, kp˘+p
kq˘+q
).

If r(T) Î Y_k or r(T)=(kp˘+p)/(kq˘+q) for k ł 1, then

e^-(k+1)v Ł q-1
Ő
i=0
f˘(y_i) Ł e^(k+1)v
(11)

Proof Consider Farey interval I(p₃/q₃,p˘/q˘). If r(T) Î I(p₃/q₃,p˘/q˘), then, by case 1) of Preposition 3 we have estimate

e^-v Ł q₃-1
Ő
i=0
f˘(z_i) Ł e^v.
(12)
Notice also, that q˘+q ł 2q˘. Hence Farey interval I(p˘/q˘,(p˘+p)/(q˘+q)) and rotation number r(T) Î I[p˘/q˘,(p˘+p)/(q˘+q)] satisfy conditions 1)-3) of Preposition 3. Therefore,

e^-v Ł q˘-1
Ő
i=0
f˘(z_i) Ł e^v.
(13)
Since q=lq˘+q₃, combining (12) and (13) we obtain (9).

If r(T)=p˘/q˘, then for Farey interval I(p₃/q₃,p˘/q˘) conditions of cases 3) or 3') of Preposition 3 holds. In worst case 3') we have estimate

e^-v Ł q₃-1
Ő
i=0
f˘(z_i) Ł e^v.

Combining the last inequality with (13), we get (10).

Now consider case 3), i.e., r(T) Î Y_k or r(Tf)=(kp˘+p)/(kq˘+q). Apply Preposition 2 to Farey interval I(p˘/q˘,(kp˘+p)/(kq˘+q)). It follows that

e^-v Ł kq˘+q-1
Ő
i=0
f˘(z_i) Ł e^v for r(T) Î Y_k or r(Tf)= kp˘+p
kq˘+q
.
(14)
Application of Preposition 3 to Farey interval I(p˘/q,(kp˘+p)/(kq˘+q)) yields

e^-v Ł q˘-1
Ő
i=0
f˘(z_i) Ł e^v for r(T) Î Y_k or r(Tf)= kp˘+p
kq˘+q
,
(15)
since kq˘+q ł q˘+q ł 2q˘ and estimates for cases 1)-3) of Preposition 3 hold. Combining (14), (15) and q=lq˘+q₃ we get (11). q.e.d.

As one can see, if q < l q˘, then we have effective estimates for derivatives of T^q and T^q˘. In fact, this is one of reasons for introducing so-called ``good'' Farey intervals used by G. \'Swiatek [2] and K. Khanin [1] in similiar setting. A Farey interval A=(p₁/q₁,p₂/q₂) is called ``good'' if

q= max
(q₁,q₂) < 2q˘= max
(q₁,q₂).

It follows from statement 3) of Preposition 4 that if rotation number is not too close to the ends of Farey interval (p₁/q₁, p₂/q₂), then we also have effective estimates for derivatives. In particular, the following corollary holds.

Corollary 1 Consider Farey interval (p₁/q₁,p₂/q₂). Suppose r(T)=(p₁+p₂)/(q₁+q₂). Then

e^-v Ł q₁+q₂-1
Ő
i=0
f˘(y_i) Ł e^v,
(16)

e^-2v Ł q₁-1
Ő
i=0
f˘(y_i), q₂-1
Ő
i=0
f˘(y_i) Ł e^2v
(17)
for any trajectory y_i=Tⁱy₀, i=0, ..., q₁+q₂-1 such that y_i š x_cr\1j, j=1,2.

Corollary 2 Consider Farey interval (p₁/q₁,p₂/q₂). Suppose

r(T) Î [ p₁+p₂
q₁+q₂
, p₁+2q₂
q₁+2q₂
].

Then

e^-3v Ł q₁-1
Ő
i=0
f˘(y_i), q₂-1
Ő
i=0
f˘(y_i) Ł e^3v
(18)

Proof Let us prove an estimate for śf^q₁/śx. If q₁ Ł q₂, then Preposition 7, gives an estiamate

e^-2v Ł q₁-1
Ő
i=0
f˘(y_i) Ł e^2v.

If q₁ > q₂, then q˘=min(q₁,q₂)=q₂, q=max(q₁,q₂)=q₁ and we can apply Preposition 4 for

Y₁=I( 2p˘+p
2q˘+q
, p˘+p
q˘+q
) = ( p₁+p₂
q₁+q₂
, p₁+2p₂
q₁+2q₂
),

and we obtain that

e^-2v Ł q₁-1
Ő
i=0
f˘(y_i) Ł e^2v for all r(T) Î Y₁.

Application of case 3) of preposition 4 for r(T)=(p₁+2p₂)(q₁+2q₂) yields estimate

e^-3v Ł q₁-1
Ő
i=0
f˘(y_i) Ł e^3v for all r(T)
-

Y

k
.q.e.d.

Notice that estimate (2) imply the following important Lemma about decay of lengths of intervals D\11_i and D\12_j, that could be proven like Preposition 3.5 in [1].

Lemma 1 Take l =(1+e^-v)^-1/2 < 1. Suppose that r(T) Î [p₁/q₁, p₂/q₂] and the expansion of p₁/q₁ to the continuous fraction has length n:

p₁
q₁
=[k₁,ź, k_n], k_n ł 2.

Then

|D\11_i |,|D\12_j | Ł const lⁿ

for all 0 Ł i Ł q₂, 0 Ł j Ł q₁.

3 Periodic rotation numbers

Here and further we shall denote by const any universal constant that depends on homeomorphism f only and do not depend on a. In this section we consider f_a for some a such that r(f_a)=p/q=[k₁,ź,k_n], k_n ł 2. For sake of shortness we shall omit index a till the end of the section.

It is well-known that for f=f_a there exists a periodic trajectory of period q. Denote by D₀=[y₁,y₂] an interval of this trajectory, contaning point x_cr\11. Denote D_i=fⁱ D₀. For some r we have D_r ' x_cr\12. Let us introduce relative coordinates of x_cr\11, x_cr\12 and f^-rx_cr\12 on intervals D₀, D_r and D₀, resp. Clearly,

d₁
= x_cr\11-y₁
y₂-y₁
= x_cr\11-y₁
|D₀ |
,

d₂
= x_cr\12-f^ry₁
f^ry₂-f^ry₁
= x_cr\12-f^ry₁
|D_r |
,

~

d

2

= f^-rx_cr\12-y₁
y₂-y₁
= f^-rx_cr\12-y₁
|D₀ |
.

Let us introduce

-

f

(w)= 1
y₂-y₁
(f^q(y₁+w(y₂-y₁))-y₁),

where w Î [0,1]. Clearly, [`f](w) is a rescaled return map f^q:D₀ŽD₀.

In order to study [`f](w) we need the following notations:

f^l_c,d(w)
= c² w
1+d(c²-1)
,

f^r_c,d(w)
= w+d(c²-1)
1+d(c²-1)
,

F_M(w)
= w
M(1-w)+w
.

One can see that function

f^lr_c,d(w) = ě
ď
í
ď
î

f^l_c,d(w),
w Î [0,d],

f^r_c,d(w),
w Î [d,1],

is a continuous piecewise-linear map [0,1]Ž [0,1] such that

f^lr_c,d(0)=0, f^lr_c,d(1)=1, ś
św
f^lr_c,d(w) ę
ę

w=d-
/ ś
św
f^lr_c,d(w) ę
ę

w=d+
=c².

This map will be used to approximate mapping f on intervals D₀ and D_r, since by Lemma 1 they are exponentially small.

As was shown in Preposition 4.3 in [1] the mapping F_M(w) is a very good approximation for mapping f^r₂-r₁:D_r₁ŽD_r₂ if intervals D_i do not cover critical points x_cr\11 and x_cr\12 for i=r₁, ..., r₂-1. Denote

M(r₁,r₂)=exp ć
č r₂-1
ĺ
i=r₁
ó
ő

y Î D_i
f˘˘(y)
2f˘(y)
dy ö
ř .

Also, define

f^r₂,r₁(w)= 1
|D_r₂ |
(f^r₂-r₁((1-w)f^r₁y₁+wf^r₁y₂)-f^r₂y₁)

Lemma 2 Suppose intervals D_i do not cover critical points x_cr\11 and x_cr\12 for i=r₁, ..., r₂-1. Then

||F_M(r₁,r₂)-f^r₂,r₁||_C²([0,1]) Ł const lⁿ,

where l is a universal constant, given in Lemma 1.

This lemma is proved as Preposition 4.3 in [1]. The only prerequisite required in proof of Preposition 4.3 [1] is an apriori estimates for geometrical decay of size of intervals. In our setting, this estimate is provided with Lemma 1. Hence we omit the proof for sake of saving space.

Let us define

M₁
=M(1,r)=exp ć
č r-1
ĺ
i=1
ó
ő

y Î D_i
f˘˘(y)
2f˘(y)
dy ö
ř ,

M₂
=M(r+1,q)=exp ć
č q-1
ĺ
i=r+1
ó
ő

y Î D_i
f˘˘(y)
2f˘(y)
dy ö
ř .

Functions f^l_c,d, f^r_c,d and F_M have the following useful properties:

F_N°F_M
=F_MN,

f^l_c,d(d)=f^r_c,d(d)
=F_1/c²(d)

for all M, N, c, d. Indeed,

F_M(w)= w
M(1-w)+w
, 1-F_M(w)= M(1-w)
M(1-w)+w

(19)
and

F_N°F_M (w)
=
w
M(1-w)+w

N M(1-w)
M(1-w)+w
+ w
M(1-w)+w

= w
MN(1-w)+w
=F_MN(w),

as required for (19). To see why (20) holds, one needs very simple algebra:

f^l_c,d(d)=f^r_c,d(d)
= c² d
1+d(c²-1)
= c² d
(1-d)+c²d

= d
1
c²
(1-d)+d
=F_1/c²(d).

Now let us introduce function

G_{c₁,d₁,c₂,[d\tilde]₂,M₁,d₂}(w):[0,1]Ž [0,1]

by the following rules. If 0 Ł d₁ Ł [d\tilde]₂, then

G_{c₁,d₁,c₂,[d\tilde]₂,M₁,d₂}(w) = ě
ď
ď
ď
í
ď
ď
ď
î

F_{[(c₁c₂)/(M₁)]}°f^l_c₂,d₂°F_M₁°f^l_c₁,d₁(w)
for 0 Ł w < d₁,

F_{[(c₁c₂)/(M₁)]}°f^l_c₂,d₂°F_M₁°f^r_c₁,d₁(w)
for d₁ Ł w <
~

d

2
,

F_{[(c₁c₂)/(M₁)]}°f^r_c₂,d₂°F_M₁°f^r_c₁,d₁(w)
for
~

d

2
Ł w Ł 1.

If [d\tilde]₂ < d₁ Ł 1, we have

G_{c₁,d₁,c₂,[d\tilde]₂,M₁,d₂}(w) = ě
ď
ď
ď
í
ď
ď
ď
î

F_{[(c₁c₂)/(M₁)]}°f^l_c₂,d₂°F_M₁°f^l_c₁,d₁(w)
for 0 Ł w <
~

d

2
,

F_{[(c₁c₂)/(M₁)]}°f^r_c₂,d₂°F_M₁°f^l_c₁,d₁(w)
for
~

d

2
Ł w < d₁,

F_{[(c₁c₂)/(M₁)]}°f^r_c₂,d₂°F_M₁°f^r_c₁,d₁(w)
for d₁ Ł w Ł 1.

Lemma 3

||
-

f

(w)-G_{c₁,d₁,c₂,[d\tilde]₂,M₁,d₂}||_{C²([0,1]\{d₁,[d\tilde]₂})} Ł const l^ne.
(20)

ProofClearly,

-

f

(w)=f^r+1,q°f^r,r+1°f^1,r°f^0,1.

We can use Lemma 1 to approximate f^0,1 with f^lr_c₁,d₁ and f^r,r+1 with f^lr_c₂,d₂. Lemma 2 imply that f^1,r is close to F_M₁(w) and f^r+1,q is close to F_M₂(w). To obtain formula for G we only need to notice that M₁M₂ ť c₁c₂. More precisely,

lnM₁+lnM₂
= q-1
ĺ
i=0
ó
ő

D_i
f˘˘(y)
2f˘(y)
dy- ó
ő

D₀
f˘˘(y)
2f˘(y)
dy- ó
ő

D_r
f˘˘(y)
2f˘(y)
dy

=lnc₁ c₂- ó
ő

D₀
f˘˘(y)
2f˘(y)
dy- ó
ő

D_r
f˘˘(y)
2f˘(y)
dy.

By Lemma 1

ę
ę ó
ő

D₀
f˘˘(y)
2f˘(y)
dy + ó
ő

D_r
f˘˘(y)
2f˘(y)
dy ę
ę Ł const lⁿ,

as required. q.e.d. Let us introduce

^

d

2
(
~

d

2
) = ě
ď
ď
í
ď
ď
î

F_M₁(f^l_c₁,d₁(
~

d

2
))
0 Ł
~

d

2
Ł d₁,

F_M₁(f^r_c₁,d₁(
~

d

2
))
d₁ Ł
~

d

2
Ł 1.

(21)

Lemma 4 We have

||G_{c₁,d₁,c₂,[d\tilde]₂,M₁,d₂}-G_{c₁,d₁,c₂,[d\tilde]₂,M₁,[^d]₂([d\tilde]₂)}||_{C²([0,1]\{d₁,[d\tilde]₂})} Ł const l^ne
(22)

Proof Notice that

|
^

d

2
(
~

d

2
)-d₂ | Ł Cl^ne.

Both functions are continuous on the same intervals. On each of them they are just fractional-linear functions with exponentially close parameters. Moreover, all parameters are bounded away from 0 and infinity. Hence, these functions are close with both derivatives. q.e.d.

Therefore, the following preposition holds

Preposition 5

||G_{c₁,d₁,c₂,[d\tilde]₂,M₁,[^d]₂([d\tilde]₂)}-
-

f

||_{C²([0,1]\{d₁,[d\tilde]₂})} Ł const l^ne

For any monotonous function g:[0,1]Ž[0,1], satisfying g(0)=0 and g(1)=1 let us introduce relation R(g,d), given by formula

R(g,d)= 1-g(d)
g(d)
: 1-d
d
.

This relation satisfy the following important equality:

R(F_M,d)=M.
(23)
Indeed, by (21)

R(F_M,d)=
M(1-d)
M(1-d)+d

d
M(1-d)+d
: 1-d
d
=M.

Lemma 5 We have

| R(G_{c₁,d₁,c₂,[d\tilde]₂,M₁,[^d]₂([d\tilde]₂)},d₁)
R(
-

f

,d₁)
-1 |Ž0

as nŽĽ for any fixed f (and hence fixed c₁, c₂).

Proof For sake of shortness let us denote

G(w)=G_{c₁,d₁,c₂,[d\tilde]₂,M₁,[^d]₂([d\tilde]₂)}(w).

Notice now, that

R(G,d₁)
R(
-

f

,d₁)
= 1-G(d₁)
1-
-

f

(d₁)

-

f

(d₁)

G(d₁)
.

Fix some e > 0. There exists d > 0 such that for all d₁ > e we have G(d₁) > G(e) > d > 0. Let us consider fraction [([`f](d₁))/(G(d₁))]. Clearly,

|

-

f

(d₁)

G(d₁)
-1 | = |

-

f

(d₁)-G(d₁)

G(d₁)
|

Since ||[`f](d₁)-G(d₁)|| Ł lⁿ, we have

|

-

f

(d₁)-G(d₁)

G(d₁)
| Ł lⁿ
d
Ł e

for all n large enough. Hence, the only ``dangerous'' case is d₁ < e.

Picture Omitted

Figure 3:

The following three situation can occur (cf 3):

1) d₁ < [d\tilde]₂,

2) d₁ > [d\tilde]₂ and [d\tilde]₂ < d₁-[d\tilde]₂,

3) d₁ > [d\tilde]₂ and [d\tilde]₂ > d₁-[d\tilde]₂.

In case 1) we have

-

f

(d₁)-G(d₁)

G(d₁)
=

-

f

(d₁)/d₁-G(d₁)/d₁

G(d₁)/d₁
=

-

f

˘(x)-G˘(z)

G˘(z)
,

where x, z Î (0,d₁). Notice that second derivatives of [`f] and G are bounded by C₂ and first derivatives are bounded away from 0 and Ľ. Hence, G˘(z) > C₁ > 0 and

|
-

f

˘(x)-G˘(z) |=|
-

f

˘(0)-G˘(0)+
-

f

˘˘(x₁)x-G˘˘(z₁)z| Ł |
-

f

˘(0)-G˘(0) |+2C₂ e.

Hence, for all n large enough we have

-

f

(d₁)-G(d₁)

G(d₁)
Ł

-

f

˘(0)-G˘(0)

C₁
+ 2C₂e
C₁
Ł C₄ e,

where C₄ > 1 is independent of e (here we used that |f˘(0)-G˘(0) | < const lⁿ). Hence if d₁ < [d\tilde]₂, then for any fixed e for all n large enough we have

|

-

f

(d₁)-G(d₁)

G(d₁)
| Ł C₄e.

Therefore, this fraction converges to 0 as nŽĽ.

If case 2) or 3) holds we use Lagrange theorem twice for [`f] and for G:

-

f

(d₁)
=
-

f

(
~

d

2
)+
-

f

˘(x₂)(d₁-
~

d

2
)=
-

f

˘(x₁)
~

d

2
+
-

f

(x₂)(d₁-
~

d

2
),

G(d₁)
=G˘(z₁)
~

d

2
+G˘(z₂)(d₁-
~

d

2
),

where x₁,z₁ Î (0,[d\tilde]₂), x₂,z₂ Î ([d\tilde]₂,d₁). Again, it is enough to consider case d₁ < e and show that the following inequality holds:

|

-

f

(d₁)-G(d₁)

G(d₁)
| = |

-

f

˘(x₁)
~

d

2
+
-

f

(x₂)(d₁-
~

d

2
)

G˘(z₁)
~

d

2
+G˘(z₂)(d₁-
~

d

2
)
|=| P
Q
| Ł const e

Suppose [d\tilde]₂ Ł d₁-[d\tilde]₂, i.e., case 2) holds. Let us multiply P and Q by 1/(d₁-[d\tilde]₂). In this case

| P
d₁-
~

d

2
|
=|(
-

f

˘(x₁)-G˘(x₁))

~

d

2

d₁-
~

d

2
+
-

f

˘(x₂)-G˘(z₂) |

Ł |
-

f

˘(x₁)-G˘(x₁) |+|
-

f

˘(x₂)-G˘(z₂) |

=|
-

f

˘(0)-G˘(0) |+|f˘˘(x₁₁)x₁-G˘˘(z₁₁)z₁ |

+ |
-

f

˘(
~

d

2
)-G˘(
~

d

2
) |+ |f˘˘(x₂₁)x₂-G˘˘(z₂₁)z₂ |

Ł 2lⁿ+4 C₂e,

where x₁₁ Î (0,x₁), z₁₁ Î (0,z₁), x₂₁ Î ([d\tilde]₂,x₂) and z₂₁ Î ([d\tilde]₂,z₂). We also have

| Q
d₁-
~

d

2
|=G˘(z₁)

~

d

2

d₁-
~

d

2
+G˘(z₂) ł G˘(z₂) > C₁.

Hence,

| P
Q
| Ł 2lⁿ+4 C₂e
C₁
Ł 4C₄e

for all n large enough.

Case 3) is considered in the same way with multiplying P and Q by 1/[d\tilde]₂. q.e.d.

Picture Omitted

Figure 4:

Let us denote

R(d₁,
~

d

2
)=R(G_{c₁,d₁,c₂,[d\tilde]₂,M₁,[^d]₂([d\tilde]₂)},d₁).

Lemma 6 We have R(d₁,d₁)=1/(c₁ c₂) and R(d₁,0)=R(d₁,1)=c₂/c₁. For any fixed d₁ the function g([d\tilde]₂)=R(d₁,[d\tilde]₂) is monotonous for [d\tilde]₂ Î [0,d₁] and for [d\tilde]₂ Î [d₁,1]. Hence the ratio R(d₁,[d\tilde]₂) takes values between 1/(c₁c₂) and c₂/c₁ (cf 4).

Proof Notice that by (19)-(20) and (23),

G_{c₁,d₁,c₂,d₁,M₁,[^d]₂(d₁)}(d₁)
= F_{[(c₁c₂)/(M₁)]}°f^l_{c₂,[^d]₂(d₁)}°F_M₁°f^l_c₁,d₁ (d₁)

=F_{[(c₁c₂)/(M₁)]}°F_1/c₂²°F_M₁°F_1/c₁² (d₁)=F_1/(c₁c₂)(d₁).

Using (25), we obtain

R(d₁,d₁)=R(G_{c₁,d₁,c₂,d₁,M₁,[^d]₂(d₁)},d₁) = R(F_1/(c₁c₂),d₁)= 1
c₁c₂

as required.

If [d\tilde]₂ > d₁, then

G_{c₁,d₁,c₂,[d\tilde]₂,M₁,[^d]₂([d\tilde]₂)}(d₁) = F_{[(c₁c₂)/(M₁)]}°f^l_{c₂,[^d]₂([d\tilde]₂)}°F_M₁°f^l_c₁,d₁ (d₁).

Suppose 0 < d₁ < 1. Notice that [^d]₂([d\tilde]₂) = 1 if [d\tilde]₂=1 and hence

G_{c₁,d₁,c₂,[d\tilde]₂,M₁,[^d]₂([d\tilde]₂)} = G_{c₁,d₁,c₂,1,M₁,1}

when [d\tilde]₂=1. Using (19) and (20), we obtain

G_{c₁,d₁,c₂,1,M₁,1}(d₁)
= F_{[(c₁c₂)/(M₁)]}°f^l_c₂,1°F_M₁°f^l_c₁,d₁ (d₁)

=F_{[(c₁c₂)/(M₁)]}°F_M₁°f^l_c₁,d₁ (d₁)

=F_c₁c₂°f^l_c₁,d₁ (d₁)=F_c₁c₂°F_1/c₁²(d₁)=

=F_c₂/c₁(d₁).

Using (25) we obtain

R(d₁,1)=R(F_c₂/c₁,d₁)= c₂
c₁
.

If [d\tilde]₂ < d₁, then

G_{c₁,d₁,c₂,[d\tilde]₂,M₁,[^d]₂([d\tilde]₂)}(d₁) = F_{[(c₁c₂)/(M₁)]}°f^r_{c₂,[^d]₂([d\tilde]₂)}°F_M₁°f^l_c₁,d₁ (d₁).

For [d\tilde]₂=0 we have

G_{c₁,d₁,c₂,0,M₁,0}(d₁)
=F_{[(c₁c₂)/(M₁)]}°f^r_c₂,0°F_M₁°f^l_c₁,d₁ (d₁)

=F_{[(c₁c₂)/(M₁)]}°F_M₁°f^l_c₁,d₁ (d₁)=F_c₁c₂ f^l_c₁,d₁ (d₁)=F_c₂/c₁(d₁),

and hence,

R(d₁,0)=R(d₁,1)= c₂
c₁

Finally, let us study the derivative of R(d₁,[d\tilde]₂) w.r.t. [d\tilde]₂. Using chain rule we obtain

ś
ś
~

d

2
R(d₁,
~

d

2
)= d₁
1-d₁
1
(G(d₁))²
ś
ś
~

d

2
G(d₁).

Now notice that if d₁ < [d\tilde]₂,

ś
ś
~

d

2
G(d₁)
= ś
ś
~

d

2
F_{c₁c₂/M₁} °f^l_{c₂,[^d]₂([d\tilde]₂)}(F_M₁°f^l_c₁,d₁(d₁))=

=F˘_{c₁c₂/M₁}(f^l_{c₂,[^d]₂([d\tilde]₂)}(w)) ś
ś
~

d

2
f^l_{c₂,[^d]₂([d\tilde]₂)}(w),

where w=F_M₁(f^l_c₁,d₁(d₁)). Notice now that

ś
ś
~

d

2
f^l_{c₂,[^d]₂([d\tilde]₂)}(w)= ś
ś
~

d

2
ć
ç
č c₂² w
1+
^

d

2
(
~

d

2
)(c₂²-1)
ö
÷
ř = - c₂²w(c₂²-1)
(1+
^

d

2
(
~

d

2
)(c₂²-1))²
ś
ś
~

d

2

^

d

2
(
~

d

2
).

Hence the sign of the derivative śR(d₁,[d\tilde]₂)/ś[d\tilde]₂ conicide with the sign of -(c₂²-1) when [d\tilde]₂ > d₁. Proceeding in the same way we obtain that for [d\tilde]₂ < d₁ the sign of the derivative śR(d₁,[d\tilde]₂)/ś[d\tilde]₂ is the same as the sign of derivative

ś
ś
~

d

2
f^r_{c₂,[^d]₂([d\tilde]₂)} = (c₂²-1)(1-w)
(1+
^

d

2
(
~

d

2
)(c₂²-1))²
ś
ś
~

d

2

^

d

2
(
~

d

2
),

i.e., the sign of śR(d₁,[d\tilde]₂)/ś[d\tilde]₂ is the same as the sign of (c₂²-1).

Hence, if c₂ < 1, then g([d\tilde]₂)=R(d₁,[d\tilde]₂) is an increasing function for [d\tilde]₂ Î [0,d₁] and g([d\tilde]₂) is a decreasing function for [d\tilde]₂ Î [d₁,1]. Finally, g(0)=c₂/c₁, g(d₁)=1/c₁c₂, g(1)=c₂/c₁.

If c₂ > 1, then g([d\tilde]₂)=R(d₁,[d\tilde]₂) is a decreasing function for [d\tilde]₂ Î [0,d₁] and g([d\tilde]₂) is an increasing function for [d\tilde]₂ Î [d₁,1]. Again, g(0)=c₂/c₁, g(d₁)=1/c₁c₂, g(1)=c₂/c₁. q.e.d.

4 Critical periodic trajectory

Picture Omitted

Figure 5:

Picture Omitted

Figure 6:

Picture Omitted

Figure 7:

Picture Omitted

Figure 8:

Suppose now that d₁=0 or d₁=1. One can easily check that

G_{c₁,0,c₂,[d\tilde]₂,M₁,[^d]₂([d\tilde]₂)}=G_{c₁,1,c₂,[d\tilde]₂,M₁,[^d]₂([d\tilde]₂)}=G

and

G(w) = ě
ď
ď
í
ď
ď
î

F_{c₁c₂/M₁}°f^l_c₂,d₂°F_M₁ (w),
0 Ł w <
~

d

2
,

F_{c₁c₂/M₁}°f^r_c₂,d₂°F_M₁ (w),

~

d

2
Ł w < 1,

Notice that

(F_a(w))˘_w=0= 1
a
and (F_a(w))˘_w=1=a.

Let us introduce m_-=G˘(1) and m₊=G˘(0). Since

F_{[(c₁c₂)/(M₁)]}(0)=f^l_c₂,[^d]₂(0)=F_M₁(0)=0 and F_{[(c₁c₂)/(M₁)]}(1)=f^l_c₂,[^d]₂(1)=F_M₁(1)=1,

we have

m_-
=G˘(1)= c₁c₂
M₁
1
1+
^

d

2
(
~

d

2
)(c₂²-1)
M₁=c₁c₂ 1
1+
^

d

2
(
~

d

2
)(c₂²-1)
,

m₊
=G˘(0)= M₁
c₁c₂
c₂²
1+
^

d

2
(
~

d

2
)(c₂²-1)
1
M₁
= c₂
c₁
1
1+
^

d

2
(
~

d

2
)(c₂²-1)
.

Notice that, as one could expect

m_-
m₊
=c₁².

Let us also consider G([d\tilde]₂). Using (19) and (20), we obtain

G(
~

d

2
)
=F_{[(c₁c₂)/(M₁)]}°f^l_{c₂,[^d]₂([d\tilde]₂)}°F_M₁(
~

d

2
) = F_{[(c₁c₂)/(M₁)]}°f^l_{c₂,[^d]₂([d\tilde]₂)}(
^

d

2
(
~

d

2
))

=F_{[(c₁c₂)/(M₁)]}°F_{[ 1/(c₂²)]}(
^

d

2
(
~

d

2
)) = F_{[(c₁c₂)/(M₁)]}°F_{[ 1/(c₂²)]}°F_M₁(
~

d

2
) = F_c₁/c₂(
~

d

2
),

i.e.,

G(
~

d

2
)=

~

d

2

(c₁/c₂)(1-
~

d

2
)+
~

d

2
.
(24)

It follows from Lemma 6 and (26),(27) that

R(d₁,
~

d

2
)
Î I[ 1
c₁c₂
, c₂
c₁
],

m₊
Î I[ 1
c₁c₂
, c₂
c₁
],

m_-
Î I[c₁c₂, c₁
c₂
].

Now we would like to enumerate critical points in such a way, that segment I[[ 1/(c₁c₂)],[(c₂)/(c₁)]] have not touched point 1. As one can see, only the following two cases are possible.

A) If c₁c₂ > 1, then we have to enumerate x_cr\1i in such a way, that c₂/c₁ < 1.

B) If c₁c₂ < 1, then we have to enumerate x_cr\1i in such a way, that c₂/c₁ > 1.

Let us study the graph of G(w). In both cases it would be a graph of piecwise-fractional-linear function.

In case A) m_-=G˘(1) > const > 1 and m₊=G˘(0) < const < 1. Using (28), we obtain that for [d\tilde] Î [d,1-d] we have

G(
~

d

2
) Ł

~

d

2

1+e

for some e, since

c₁
c₂
(1-
~

d

2
)+
~

d

2
Î [1, c₁
c₂
].

Hence, the graph of G(w) is completely below the diagonal (cf Figures 5, 6).

In case B) m_-=G˘(1) < const < 1 and m₊=G˘(1) > const > 1. Using (28), we obtain that for [d\tilde] Î [d,1-d] we have

G(
~

d

2
) ł

~

d

2

1-e

for some e, since

c₁
c₂
(1-
~

d

2
)+
~

d

2
Î [ c₁
c₂
,1].

Hence, the graph of G(w) is completly above the diagonal (cf Figures 7, 8).

Let us denote by [`I](p/q)=[a₀(p/q),a₁(p/q)] a closed interval of values a such that r(f_a)=p/q.

Theorem 2 Suppose that critical points x_cr\1i are enumerated in such a way that

A) for c₁c₂ < 1 we have c₂ < c₁ and

B) for c₁c₂ > 1 we have c₂ > c₁.

In case A) for all n > const there exists a unique critical periodic trajectory for a=a₀(p/q). This is a trajectory of the critical point x_cr\11.

In case B) for all n > const there exists a unique critical periodic trajectory for a=a₁(p/q). This is a trajectory of the critical point x_cr\11.

Proof Without loss of generality, consider case A). If d₂ is bounded away from 0 and 1, then the result follows from Preposition 5, since the whole graph of [`f] would lie below the diagonal. If d₂ is close to 0, then [`f](d₁)=G˘(0) d₁+o(d₁) < d₁, since G˘(0)=m₊ < const < 1. The same argument works for d₂ close to 1. Using estimates of Preposition 5, we obtain that the whole graph of [`f] lie below the diagonal. q.e.d.

5 Non-linearity of f^q

Notice that it is enough to consider the case c₁c₂ < 1, c₂/c₁ < 1. Consider interval I(p/q)=(a₀,a₁). Suppose that p/q is obtained from Farey interval (p₁/q₁, p₂/q₂). It follows from Theorem 2 that T^q_a₀x_cr\11 = x_cr\11 and

T^q_a₀

for any point

Î (T^q₂_a₀x_cr\11, x_cr\11).

(25)

Denote now by R the ratio

x_cr\11-

~ x	-T^q₁_a₀x_cr\11

=R.

It follows from the following lemma that there exists a unique [a\tilde] Î I(p/q) such that T^q_[a\tilde][x\tilde] =[x\tilde].

Lemma 7 Suppose r(f_a₁) < p/q < r(f_a₂). Then for any x Î [0,1] there exists a unique a(x) Î [a₁,a₂] such that r(f_a(x))=p/q and f^q(x)=x+p.

Proof Note that there exists K such that for all k > K we have

f^kq_a₁(x)-x-kp < 0 and f^kq_a₂ (x) -x -kp > 0.

Since f^kq_a(x) is a strictly monotonous function of a, there exists a unique a=a(x), a(x) Î [a₁,a₂] such that we have f^kq_a(x)-x-kp=0.

Now note that if f^q_a(x)-p > x for all x Î R then, by f_a(x)-1=f_a(x-1), we have

f^2q_a(x)-2p=f^q_a(f^q_a(x)-p)-p > f^q_a(x)-p > x

and by induction we obtain that f^lq_a(x)-lp > x for all l ł 1. Therefore, we arrive to contradiction with f^kq_a(x)-kp=x. The same arguments works for f^q_a(x)-p < x. Therefore, f^q_a(x)-p=x. q.e.d.

Let us show that at moment [a\tilde] the relative coordinate of x_cr\11 on critical trajectory passing through [x\tilde] is bounded away from 0 and 1 by constant.

Preposition 6

1
2
e^-2v
1+e^4v
R
R+1
Ł
x_cr\11-
~

x

T^q₁_[a\tilde]
~

x

-
~

x

Ł R
R+e^-2v
.
(26)

Proof Notice that p/q is obtained from some Farey interval A=(p₁/q₁,p₂/q₂), i.e., p/q=(p₁+p₂)/(q₁+q₂). Preposition 1, monotonicity of T_a and (29) imply that the following order of points takes place:

T^q₂_a₀x_cr\11 < T^q_a₀
~

x

<
~

x

< x_cr\11 < T^q₁_a₀
~

x

< T^q₁_a₀x_cr\11

and

T^q₁_a₀
~

x

< T^q₁_[a\tilde]
~

x

.

Since r(T_a₀)=p/q, it follows from (17) that

e^-2v Ł ( ś
śx
T_a^q₁)(x₀), ( ś
śx
T_a^q₂)(x₀) Ł e^2v
(27)
for each a Î [a₀,a₁] and for each point x₀, such that the derivative of T_a^q w.r.t. x₀ is well-defined.

Let us prove upper bound in (30). Notice that

x_cr\11-
~

x

T^q₁_[a\tilde]
~

x

-
~

x

Ł
x_cr\11-
~

x

T^q₁_a₀
~

x

-
~

x

=
x_cr\11-
~

x

x_cr\11-
~

x

+T^q₁_a₀
~

x

-x_cr\11
.

Using (31), we obtain

T^q₁_a₀
~

x

-x_cr\11
= ó
ő [x\tilde]

T^q₂_a₀x_cr\11
ś
śx
T^q₁_a₀(x) dx

ł e^-2v(
~

x

-T^q₂_a₀x_cr\11).

But

~

x

-T^q₂_a₀= 1
R
(x_cr\11-
~

x

).

Hence

x_cr\11-
~

x

x_cr\11-
~

x

+T^q₁_a₀
~

x

-x_cr\11

Ł
x_cr\11-
~

x

x_cr\11-
~

x

+ e^-2v
R
(x_cr\11-
~

x

)

= R
R+e^-2v
,

as required.

In order to prove the lower bound in (30) we need an estimate

T^q₁_[a\tilde]
~

x

-T^q₁_a₀
~

x

Ł e^2v (T^q_[a\tilde]
~

x

-T^q_a₀
~

x

).
(28)
To see why it is true, notice that by (49) (proven in Appendix),

ś
śa
T^q_a
~

x

= ś
śa
T^q-q₁_a°T^q₁_a
~

x

q-1
Ő
j=q₁
f˘(
~

x

j
(a)) ś
śa
T^q₁_a
~

x

+( ś
śa
T_a^q-q₁)(T_a^q₁
~

x

),

where [x\tilde]_j(a)=T^j_a [x\tilde]. Since the second summund in r.h.s. is non-negative,

ś
śa
T^q_a
~

x

ł q-1
Ő
j=q₁
f˘(
~

x

j
(a)) ś
śa
T^q₁_a
~

x

or

ś
śa
T^q_a
~

x

ł q₂-1
Ő
j=0
f˘(z_j(a)) ś
śa
T^q₁_a
~

x

,

where z_j(a)=T^j_a z₀(a), z₀(a)=[x\tilde]_q₁(a). But (31) imply that

e^-2v Ł q₂-1
Ő
j=0
f˘(z_j(a)) Ł e^2v

for all a Î I(p/q) such that [x\tilde]_i(a) š x_cr\1j for all 0 Ł i Ł q-1, j=0,1. Therefore

ś
śa
T^q₁_a
~

x

Ł e^2v ś
śa
T^q_a
~

x

(29)
for almost all a Î I[p/q]. To prove (32) it stays to notice that

T^q₁_[a\tilde]
~

x

-T^q₁_a₀
~

x

= ó
ő [a\tilde]

a₀
ś
śa
T^q₁_a(
~

x

) da

Ł e^2v ó
ő [a\tilde]

a₀
ś
śa
T^q_a(
~

x

) da = T^q_[a\tilde]
~

x

-T^q_a₀.

Notice that

T^q₁_[a\tilde]
~

x

-
~

x

=T^q₁_[a\tilde]
~

x

-T^q₁_a₀
~

x

+T^q₁_a₀
~

x

-
~

x

(30)
It follows from (32) that

T^q₁_[a\tilde]
~

x

-T^q₁_a₀
~

x

Ł e^2v(T^q_[a\tilde]
~

x

-T^q_a₀
~

x

) = e^2v(
~

x

-T^q_a₀
~

x

).

Notice that

T^q₁_a₀(T^q_a₀
~

x

,
~

x

) Ě (x_cr\11,T^q₁_a₀
~

x

) Ě (
~

x

,T^q₁_a₀
~

x

).

Using (31), we obtain

e^-2v|(T^q_a₀
~

x

,
~

x

) | Ł |(x_cr\11,T^q₁_a₀
~

x

) | Ł |(
~

x

,T^q₁_a₀
~

x

) |.

To sum up

T^q₁_[a\tilde]
~

x

-T^q₁_a₀
~

x

Ł e^4v(T^q₁_a₀
~

x

-
~

x

).

Hence, (34) yields

T^q₁_[a\tilde]
~

x

-
~

x

Ł (1+e^4v)(T^q₁_a₀
~

x

-
~

x

)

Therefore,

x_cr\11-
~

x

T^q₁_[a\tilde]
~

x

-
~

x

ł 1
1+e^4v

x_cr\11-
~

x

T^q₁_a₀
~

x

-
~

x

(31)
Now notice that

x_cr\11-
~

x

T^q₁_a₀
~

x

-
~

x

ł
x_cr\11-
~

x

T^q₁_a₀x_cr\11-T^q₂_a₀x_cr\11
.

The r.h.s. fraction could be estimated by the following lemma.

Picture Omitted

Figure 9:

Lemma 8 Given a, b, c > 0 such that

a
b
ł d and a
c
ł g,

the following inequality holds:

a+a
b+c
ł min
(d,g), i.e., a
b+c
ł 1
2
min
(d,g).
(32)

ProofConsider points B=(b,a), C=(c,a), and A=(b+c,a+a) on the plain xOy (cf Figure 9). The slope of ray OA lies between slopes of rays OB and OC. Hence,

a+a
b+c
ł min
( a
b
, a
c
) ł min
(d,g)

as required. q.e.d.

Let us use (36) with a=x_cr\11-[x\tilde], b=T^q₁_a₀x_cr\11-x_cr\11 and c=x_cr\11-T^q₂_a₀x_cr\11. We have

a
c
=
x_cr\11-
~

x

x_cr\11-T^q₂_a₀x_cr\11
= R
R+1

and

a
b
=
x_cr\11-
~

x

T^q₁_a₀x_cr\11-x_cr\11
.

Since

T^q₁_a₀(T^q₂_a₀x_cr\11,x_cr\11)=(x_cr\11,T^q₁_a₀x_cr\11),

we can apply (31) to obtain

e^-2v R
R+1
Ł
x_cr\11-
~

x

T^q₁_a₀x_cr\11-x_cr\11
Ł e^2v R
R+1

Therefore,

x_cr\11-
~

x

T^q₁_a₀x_cr\11-T^q₂_a₀x_cr\11
ł e^-2v
2
R
R+1

Combining our estimates, we obtain

x_cr\11-
~

x

T^q₁_[a\tilde]-
~

x

ł 1
1+e^4v
e^-2v
2
R
R+1
,

as required. q.e.d.

Picture Omitted
Figure 10: Intervals D_c⁽¹⁾=[[`x]_c²,x_c], D_c⁽²⁾=[x_c,[`x]_c], D_c⁽³⁾=[[`x]_c, [`x]_c¹], where [`x]_c²=Tf_a₁^q₂ x_c, [`x]_c¹=Tf_a₁^q₁ x_c, and [`x]_c=Tf_a₁^q x_c

Now take

~

x

= T_a₀^q₂x_cr\11+x_cr\11
2
.

By Lemma 7 there exists a unique [a\tilde], corresponding to passing of periodic trajectory through [x\tilde]:

r(T_[a\tilde])= p
q
, T_[a\tilde]^q
~

x

=
~

x

.

Denote y₀=T_[a\tilde]^q₂y₁, y₁=[x\tilde], y₂=T_[a\tilde]^q₁y₁, y₃=T_[a\tilde]^q₁y₂. Clearly, x_cr\11 Î (y₀,y₁).

It follows from (30), that relative coordinate d₁([a\tilde]) of x_cr\11 is bounded away from 0 and 1 for all n large enough:

1
2
e^-2v
1+e^4v
1
2
Ł d_a(
~

a

)= x_cr\11-y₁
y₂-y₁
Ł 1
1+e^-2v
.
(33)
Denote [`x]_c¹=Tf^q₁_[a\tilde]x_c), [`x]_c²=Tf^q₂_[a\tilde](x_c), [`x]_c=Tf^q=q₁+q₂_[a\tilde](x_c). Evidently, [`x]_c² Î (y₀,y₁), [`x]_c Î (y₂,y₁), [`x]_c¹ Î (y₁,y₀). Notice that [`x]_c > x_cr\11. Indeed, T^q_[a\tilde]x_cr\11 > T^q_a₀x_cr\11 = x_cr\11.

Points [`x]_c², x_cr\11, [`x]_c, [`x]_c¹ form three successive intervals. Denote this intervals by D_c\11, D_c\12, D_c\13, respectively. We have D_c\11 = [[`x]_c², x_cr\11], D_c\12 = [x_cr\11,[`x]_c], D_c\13 = [[`x]_c, x_c¹] (see Figure 10). We shall show that the length of each of these intervals is of the same order.

Preposition 7 There exists a constant C₄ such that for all n large enough

max
(l(D_c\11),l(D_c\12),l(D_c\13))

min
(l(D_c\11),l(D_c\12),l(D_c\13))
Ł C₄.

Proof Notice that (37) and Lemmas 5 and 6 imply that D_c\12/(x_cr\11-y₀) and D_c\12/(y₁-[`x]_c) are bounded away from 0 and 1 by universal constants. Hence it is sufficient to show that

(y₂-y₁) Ł const (y₁-y₂), (y₀-y₁) Ł (y₁-y₂).

But [y₃,y₂]=Tf^q₂[y₂,y₁], [y₁,y₀]=Tf^q₁[y₂,y₁] and these estimates follow from Corollary 1. q.e.d.

6 Proof of the theorem 1

Picture Omitted
Figure 11: Intervals I(p₁/q₁), I(p₂/q₂) and I_c(A)

Let A=(p₁/q₁,p₂/q₂) be a ``good'' Farey interval of rank n , i.e. a Farey interval such that q=max(q₁,q₂) < 2q˘=2min(q₁,q₂). Denote by a⁽¹⁾=a₀(p₁/q₁), a⁽²⁾=a₀(p₂/q₂) those ends of the intervals I(p₁/q₁), I(p₂/q₂) which correspond to the passage of the periodic trajectory through the singular point. By Theorem 2, these ends are left. Let I_c(A) be the interval (a⁽¹⁾,a⁽²⁾) and let [`I]_c(A) = [a⁽¹⁾,a⁽²⁾] be its closure (see Figure 11). Denote x^m (a)=Tf^m_a (x_c) where x_c=0 is a critical point and m ł 0.

Preposition 8 There exists 0 < g < 1 such that for all n large enough

l(I(p₁/q₁))
l(I_c(A))
ł 1-g
(34)

This statement implies theorem 1. Proof[Proof of Theorem 1] Suppose that r = r(a) is irrational and a is a condensation point for set {a | r(a) Î [0,1)\Q}. Then there exists an infinite sequence of ``good'' Farey intervals A_i, containing r. Since a Î I_c(A_i) and l(I_c(A_i))Ž 0 as iŽ Ľ, using (38) we arrive at contradiction with assumption that a is a condensation point. q.e.d.

Let us denote

M_1,2
=
max
a Î [`I]_c(A)
ć
č ś
śa
x^q_1,2(a) ö
ř

m_1,2
=
min
a Î [`I]_c(A)
ć
č ś
śa
x^q_1,2(a) ö
ř

Picture Omitted
Figure 12: Intevals w, v, u

Proof[Proof of Preposition 8] First notice, that the we have the following ordering:

a₀ ć
č p₁
q₁
ö
ř <
~

a

ć
č p₁
q₁
ö
ř < a₁ ć
č p₁
q₁
ö
ř < a₀ ć
č p₂
q₂
ö
ř < a₁ ć
č p₂
q₂
ö
ř (see Figure 11).

Let us denote

a=a₀ ć
č p₂
q₂
ö
ř -
~

a

ć
č p₁
q₁
ö
ř , b =
~

a

ć
č p₁
q₁
ö
ř -a₀ ć
č p₁
q₁
ö
ř .

For any function f(z) such that f˘(z) > 0 we have the following estimates for any z₁ < z₂:

f(z₂)-f(z₁)

max
z Î [z₁,z₂]
f˘(z)
Ł z₂-z₁ Ł f(z₂)-f(z₁)

min
z Î [z₁,z₂]
f˘(z)
.

It follows that

b ł
x^q₁(
~

a

ć
č p₁
q₁
ö
ř )-x^q₁(a₀ ć
č p₁
q₁
ö
ř )

M₁
.
(35)
Using the equality

x^q₁(a₀ ć
č p₁
q₁
ö
ř )=x_c,

and notation

v=x^q₁(
~

a

ć
č p₁
q₁
ö
ř )-x_c (see Figure 6)

we can rewrite (39) as follows:

b ł
x^q₁(
~

a

ć
č p₁
q₁
ö
ř )-x_c

m₁
= v
M₁
.
(36)
Also, we have

x^q₁(a₀ ć
č p₂
q₂
ö
ř )-x^q₁(
~

a

ć
č p₁
q₁
ö
ř )

M₁
Ł a Ł
x^q₁(a₀ ć
č p₂
q₂
ö
ř )-x^q₁(
~

a

ć
č p₁
q₁
ö
ř )

m₁
,

and after notation

u=x^q₁(a₀ ć
č p₂
q₂
ö
ř )-x^q₁(
~

a

ć
č p₁
q₁
ö
ř ) (see Figure 6),

we obtain

u
M₁
Ł a Ł u
m₁
.
(37)
Now, let us write

a Ł
x^q₂(a₀ ć
č p₂
q₂
ö
ř )-x^q₂(
~

a

ć
č p₁
q₁
ö
ř )

m₂
.

Since

x^q₂(a₀ ć
č p₂
q₂
ö
ř )=x_c,

this estimate with notation

w=x_cr\11-x^q₂(a₀ ć
č p₂
q₂
ö
ř ) (see Figure 6)

yields

a Ł w
m₂

(38)
Now let us notice that the following ordering takes place:

x^q₂(
~

a

ć
č p₁
q₁
ö
ř ) < x_c < x^q₁(
~

a

ć
č p₁
q₁
ö
ř ) < x^q₁(a₀ ć
č p₂
q₂
ö
ř ).

Suppose that p₁/q₁ is obtained from some Farey interval (p₂/q₂, p₀/q₀), i.e. q₁=q₀+q₁, p₁=p₀+p₂. In this case we can use Lemma 7, where

~

a

=
~

a

ć
č p₁
q₁
ö
ř , p/q=p₁/q₁, D_c⁽¹⁾=w, D_c⁽²⁾=v.

It follows from Lemma 7, that for some C we have

C Ł w
v
Ł 1/C.
(39)
Combining estimates (40), (41), (42) and (43), we get the following inequalities

l(I(p₁/q₁))
l(I_c(A))

ł b
b+a
= 1
1+a/b

ł 1
1+ u
m₁
M₁
v
= 1
1+ M₁
m₁
u
w
w
v

ł 1
1+ M₁
m₁
aM₁
am₂
w
v
= 1
1+ M₁
m₁
M₁
m₂
w
v
ł 1
1+C₁
< 1-g,

where we assumed that

M₁
m₁
, M₁
m₂
Ł const < Ľ.

These inequalities are proven in the following Preposition 9. q.e.d.

Preposition 9 Let A be a ``good '' Farey interval. Then

m₁
M₁
, m₂
M₂
, m₁
M₂
, m₂
M₁
ł const > 0.

Proof We use the following preposition (see [2] and [3,Preposition 6.3]). Denote

D_i = ć
č Tfⁱ_a⁽¹⁾(x_c),Tfⁱ_a⁽²⁾(x_c) ö
ř .

Preposition 10 The system of intervals D_i, 1 Ł i Ł q=max(q₁,q₂), covers any point of the circle at most twice.

Corollary 3 ĺ_i=1^q l(D_i) Ł 2.

Let [`q] Ł q=max(q₁,q₂), a Î I(A). It follows from (52) that

ś
śa
x^[`q](a)= [`q]-1
ĺ
i=0
[`q]-1
Ő
j=i+1
f_a˘(x^j(a)),
(40)
where

x^j(a)=Tf^j_a(x_c).

Consider one of the summands in (44),

B_i^q(a)= [`q]-1
Ő
j=i+1
f˘_a(x^j(a))

Lemma 9 There exists constant C such that for any a₁, a₂ Î [`I](A) we have

1
C
Ł B_i^[`q](a₁)
B_i^[`q](a₂)
Ł C

Proof It is easy to see that x_c Ď D_j, 1 Ł j Ł q. Hence,

|lnB^[`q]_i(a₁)-lnB^[`q]_i(a₂) |
Ł ę
ę [`q]-1
ĺ
j=i
(lnf˘(x_j(a₁)) -lnf˘(x_j(a₂))) ę
ę

Ł const
max
y Î S¹
ę
ę f˘˘(y)
f˘(y)
ę
ę [`q]
ĺ
j=i
l(D_j) Ł 2
max
y Î S¹
ę
ę f˘˘(y)
f˘(y)
ę
ę =lnC.

Here we used Corollary 3 and (lnf_a˘(y))˘=f˘˘_a(y)/f˘_a(y). q.e.d. Therefore, Lemma 9 implies that m₁/M₁, m₂/M₂ ł 1/C. Now let us compare [(śx^q₁(a))/(śa)] with [(śx^q₂(a))/(śa)]. Assume that q=max(q₁,q₂)=q₂. Using (49) for f(a)=f_a^q₂-q₁(x_c), g(a)=f_a^q₁(x_c), we obtain

ś
śa
x^q₂(a) = ć
č q₂-1
Ő
j=q₁
f˘_a(x^j(a)) ö
ř ś
śa
x^q₁(a)+ ś
śa
T f^q₂-q₁_a(x₀) ę
ę

x₀=x^q₁(a)
.
(41)
Applying Corollary 1 to the Farey interval (p₁/q₁, (p₂-p₁)/(q₂-q₁)), we get

e^-2v Ł q₂-1
Ő
j=q₁
f˘_a(x^j(a)) Ł e^2v, a Î I(A).

It follows from (45), that

śx^q₂(a)/śa
śx^q₁(a)/śa
ł ć
č q₂-1
Ő
j=q₁
f˘_a(x^j(a)) ö
ř ł e^-2v

and m₂/M₁ ł e^-2v/C. The following estimates could be obtained like Lemma 9.

1
C
Ł (ś/śa) Tf^q₂-q₁_a(x₀)|_{x₀=x^q₁(a)}
(ś/śa)x^q₂-q₁(a)
Ł C, a Î I(A).
(42)
As one can see, in order to prove (46), one have to consider

-

B

[`q]
i
(a)= [`q]-1
Ő
j=i+1
f˘(x₀^j(a)),

where x₀^j(a)=T_a^q₁x^j(a).

Lemma 10 There exists constant C such that for any a₁,a₂ Î [`I](A) we have

1
C
Ł

-

B

[`q]
i
(a₁)

-

B

[`q]
i
(a₂)
Ł C.

Proof

|ln
-

B

[`q]
i
(a₁)-ln
-

B

[`q]
i
(a₂) |
Ł | [`q]
ĺ
j=i
(lnf˘(x₀^j(a₁))-lnf˘(x^j(a₂))) |

Ł | [`q]
ĺ
j=i
(lnf˘(T_a₁^q₁x^j(a₁))-lnf˘(x^j(a₁)) |

+ | [`q]
ĺ
j=i
(lnf˘(x^j(a₁))-lnf˘(x^j(a₂))) |.

The second sum in r.h.s is estimated as in Lemma 9, and the first sum is bounded by constant, since the total length of intervals [x^j(a₁),T_a₁^q₁x^j(a₁)], j=0,ź[`q], do not exceed 1, i.e., the finite length of the circle. q.e.d.

Using (49) for f(a,·)=f_a^2q₁-q₂(·), g(a)=f_a^q₂-q₁(x_c), we obtain

ś
śa
x^q₁(a) = ć
č q₁-1
Ő
j=q₂-q₁
f˘_a(x^j(a)) ö
ř ś
śa
x^q₂-q₁(a)+ ś
śa
T f_a^2q₂-q₁(x₀) ę
ę

x₀=x^q₂-q₁(a)
.
(43)
We have

(ś/śa)x^q₁(a)
(ś/śa)x^q₂-q₁(a)
ł q₁-1
Ő
j=q₂-q₁
f_a˘(x^j(a)) ł e^-3v, a Î I(A),
(44)
where we applied [3,Preposition 3.4, p. 65] to the Farey interval ((2p₁-p₂)/(2q₁-q₂), (p₂-p₁)/(q₂-q₁)). Using (46) and (48) we obtain

ś
śa
T f^q₂-q₁_a(x₀) ę
ę

x₀=x^q₁(a)
Ł C ś
śa
x^q₂-q₁(a) Ł C e^3v ś
śa
x^q₁(a).

Substituting last estimate to (45) we obtain

ś
śa
x^q₂(a) Ł (e^2v+e^3v C) ś
śa
x^q₁(a).

Let [`a] be the value such that [(śx^q₁(a))/(śa)]|_a=[`a] is minimal. Combining our estimates, we obtain

m₁
M₂
=
(ś/śa)x^q₁(
-

a

)

(ś/śa)x^q₂(
-

a

)
·
(ś/śa)x^q₂(
-

a

)

M₂
ł ć
č 1
e^2v+C e^3v
ö
ř 1
C
q.e.d.

Appendix

Consider function f(x,y) Î C¹(R×R) and g(y) Î C¹(R). Denote

f_x(x,y)=

śz

f(z,y)

ę
ę

z=x

and f_y(x,y)=

śa

f(x,a)

ę
ę

a=y

Note that,

śy

f(g(y),y)=f_x(g(y),y) g˘(y)+f_y(g(y),y).

(45)

Denote by fⁿ(x,y) the nth iteration of function f with respect to variable x: fⁿ(x,y) = f(f(źf(f(x,y),y),ź,y),y)).

Lemma 11 For any n Î N

ś
śy
fⁿ(g(y),y)= n
ĺ
i=1
f_y(x_i(y),y) n
Ő
j=i+1
f_x(x_j(y),y)+ g˘(y) n
Ő
j=1
f_x(x_j(y),y),
(46)
where x_j(y)=f^j-1(g(y),y), x₁(y)=g(y).

Here and later the product with low limit higher than upper limit is equal to 1. Proof It follows from (49) that the statement is true for n=1. By induction, suppose that (50) is true for n=k-1. To obtain this formula for n=k we substitute g(y)Ž f(g(y),y), x_j(y)Ž x_j+1(y) in (50) for n=k-1.

ś
śy

f^k(g(y),y) = ś
śy
f^k-1(f(g(y),y),y)

= k-1
ĺ
i=1
f_y(x_i+1(y),y) k-1
Ő
j=i+1
f_x(x_j+1(y),y)+ ś
śy
f(g(y),y) ć
č k-1
Ő
j=1
f_x(x_j+1(y),y) ö
ř

= k
ĺ
i=2
f_y(x_i(y),y) k
Ő
j=i+1
f_x(x_j(y),y)

+ ć
č f_x(g(y),y) g˘(y)+f_y(g(y),y) ö
ř ć
č k
Ő
j=2
f_x(x_j(y),y) ö
ř

= k
ĺ
i=1
f_y(x_i(y),y) k
Ő
j=i+1
f_x(x_j(y),y)+ g˘(y) k
Ő
j=1
f_x(x_j(y),y).

Here we use (49). q.e.d.

Lemma 12 For any k Î N

ś
śy
f^k(x,y)= k-1
ĺ
i=0
f_y(x_i(y),y) k-1
Ő
j=i+1
f_x(x_j(y),y).
(47)
where x_j(y)=f^j(x,y), x₀(y)=x.

Proof Fix x and put g(y)=f(x,y), n=k-1 in (50). q.e.d. It follows from (51) that

ś
śa
fⁿ_a(x)= n-1
ĺ
i=0
n-1
Ő
j=i+1
f_a˘(x_j(a)),
(48)
where

x_j(a)=f^j_a(x).

References

[1]: K. M. Khanin and E. B. Vul. Circle homeomorphisms with weak discontinuities. In Dynamical systems and statistical mechanics (Moscow, 1991), pages 57-98. Amer. Math. Soc., Providence, RI, 1991.
[2]: Grzegorz \'Swiatek. Rational rotation numbers for maps of the circle. Comm. Math. Phys., 119(1):109-128, 1988.
[3]: K. M. Khanin. Rigidity for circle homeomorphisms with a break-type singularity. Dokl. Akad. Nauk, 357(2):176-179, 1997.

Footnotes:

¹^*üÔĎÔ ÔĹËÓÔ ĐĎĚŐŢĹÎ ÉÚ ĆÁĘĚÁ 2.tex

File translated from T_EX by T_TH, version 3.05.
On 16 Nov 2002, 17:11.

Rational Rotation Numbers for Homeomorphisms with Two Break-type Singularities*1

D.Khmelev

1 Introduction

2 Notations and apriori estimates

3 Periodic rotation numbers

4 Critical periodic trajectory

5 Non-linearity of fq

6 Proof of the theorem 1

Appendix

References

Contents

Footnotes:

Rational Rotation Numbers for Homeomorphisms with Two Break-type Singularities^*¹

5 Non-linearity of f^q