AN EXTENTION OF HERMAN’S THEOREM FOR NONLINEAR CIRCLE MAPS WITH TWO BREAKS Текст научной статьи по специальности «Математика»

Russian Journal of Nonlinear Dynamics, 2018, vol. 14, no. 4, pp. 553-577. Full-texts are available at http://nd.ics.org.ru DOI: 10.20537/nd180409

MATHEMATICAL PROBLEMS OF NONLINEARITY

MSC 2010: 37E10, 37C15, 37C40

An Extention of Herman's Theorem for Nonlinear Circle Maps with Two Breaks

M. Herman showed that the invariant measure ¡h of a piecewise linear (PL) circle homeo-morphism h with two break points and an irrational rotation number ph is absolutely continuous iff the two break points belong to the same orbit. We extend Herman's result to the class P of piecewise C2+£-circle maps f with an irrational rotation number pf and two break points a0, c0, which do not lie on the same orbit and whose total jump ratio is Of = 1, as follows: if ¡f denotes the invariant measure of the P-homeomorphism f, then for Lebesgue almost all values of ¡f ([ao,co]) the measure ¡f is singular with respect to Lebesgue measure.

Keywords: piecewise-smooth circle homeomorphism, break point, rotation number, invariant measure

Received September 10, 2018 Accepted November 19, 2018

Akhtam Dzhalilov [email protected] Turin Polytechnic University

Kichik Halka yuli 17, Tashkent, 100095 Uzbekistan Dieter Mayer

[email protected]

Institut fUr Theoretische Physik

TU Clausthal, D-38678 Clausthal-Zellerfeld, Germany

Shukhrat Djalilov [email protected]

Samarkand Institute of Economics and Service A. Temura st. 9, Samarkand, 140100 Uzbekistan

Abdurakhmon Aliyev [email protected]

National University of Uzbekistan

VUZ Gorodok, Tashkent, 700174 Uzbekistan

A. Dzhalilov, D.Mayer, S. Djalilov, A. Aliyev

1. Introduction

One of the important problems of circle dynamics is that of absolute continuity of the invariant measure for circle homeomorphisms. For smooth circle homeomorphisms this problem is well understood. Fundamental results on this were obtained by V. Arnold, M. Herman, J. Moser, J.-C.Yoccoz, Ya. Sinai and K.Khanin, Y. Kaznelson and D. Ornstein and others. Let f be an orientation-preserving homeomorphism of the circle S1 = R/Z with lift F : R ^ R, which is continuous, strictly increasing and fulfills F(x + 1) = F(x) + 1, x £ R. The circle homeomorphism f is then defined by f (x) = F(x) mod 1 with x £ R a lift of x £ S1. The rotation number pf is defined by

Fn (x) — x pf := lim-=- mod 1,

J n—n

where F1 denotes the ith iteration of the map F. It is well known that the rotation number pf does not depend on the starting point xx £ R and is irrational if and only if f has no periodic points (see [4]). The rotation number pf is invariant under topological conjugations.

Denjoy's classical theorem states [5] that a circle diffeomorphism f with an irrational rotation number p = pf and log Df of bounded variation can be conjugated to the linear rotation Rp with lift Rp(x) = x + p, that is, there exists a homeomorphism p: S1 ^ S1 with f = po Rp op-1.

It is well known that a circle homeomorphism f with an irrational rotation number pf is uniquely ergodic, i.e., it has a unique invariant probability measure ¡f. A remarkable fact then is that the conjugacy p can be defined by p(x) = ¡f ([0,x]) (see [4]), which shows that the smoothness properties of the conjugacy p imply corresponding properties of the density of the absolutely continuous invariant measure ¡f for a sufficiently smooth circle diffeomorphism with a typical irrational rotation number (see [2, 12, 14, 16]). The problem of smoothness of the conjugacy for smooth diffeomorphisms is by now very well understood (see, for instance, [2, 12, 14, 16, 23]).

A natural generalization of circle diffeomorphisms are piecewise smooth homeomorphisms with break points (see [14]).

The class of P-homeomorphisms consists of orientation-preserving circle homeomor-phisms f which are differentiable except at a finite or countable number of break points, denoted by BP(f) = {xb £ S1}, at which the one-sided positive derivatives Df- and Df+ exist, but do not coincide, and for which there exist constants 0 < c1 <c2 < <x such that

• C1 < Df-(xb) < C2 and C1 < Df+(xb) < C2;

• C1 < Df (x) < C2 for all x £ S 1\BP(f);

• log Df has finite total variation v in S1.

Piecewise linear (for short PL) orientation-preserving circle homeomorphisms are the simplest examples of P-homeomorphisms. They occur in many different areas of mathematics such as group theory, homotopy theory or logic via the Thompson groups. A family of PL-homeo-morphisms was first studied by M. Herman [14] as examples of circle homeomorphisms with an arbitrary irrational rotation number which admit no invariant ^-finite measure absolutely continuous with respect to Lebesgue measure.

Indeed, Herman proved thereby

Theorem 1. The invariant measure of a PL-circle homeomorphism with two break points and an irrational rotation number is absolutely continuous with respect to Lebesgue measure if and only if these break points lie on the same orbit.

Herman's family of maps has been studied later by several authors (see, for instance, [3, 1921]) in the context of interval exchange transformations. Special cases are affine 2-interval exchange transformations, to which Herman's examples with break points a0 = 0 and c0 = c belong.

In [8] Dzhalilov et al. proved the following result. Let f be a P-homeomorphism with two break points. If f is C2 except at the two break points and its rotation number is of bounded type, then the unique f-invariant probability measure ¡f is equivalent to Lebesgue measure if and only if the two break points of f lie on the same orbit and its total jump ratio is Of = 1. Notice that the condition of bounded type is essential: if pf is not of bounded type, the homeomorphism f obtained by conjugating a C2 diffeomorphism with singular invariant measure (such diffeomorphisms exist by [13]) with a P-homeomorphism with one break point is a P-homeomorphism with two break points on the same orbit and with a trivial total jump ratio, but the f-invariant probability measure ¡f is singular.

The invariant measures of P-homeomorphisms f with a finite number of break points have been studied by several authors (see, for instance, [1, 3, 6-9, 11, 15, 22]). For such a homeomorphism the character of the invariant measure strongly depends on its total jump ratio Of being trivial or nontrivial, i.e., Of = 1 or Of = 1. A recent result of Dzhalilov et al. in [11] in the case Of = 1 is

Theorem 2. Let f e C2+£(S1 \{aiaam}), e > 0 be a P-homeomorphism with an irrational rotation number pf and a finite number of break points a1,a2,...,am. Suppose its total jump ratio is Of = Of (a1) • Of (a2) • ... • Of (am) = 1. Then its invariant probability measure ¡f is singular with respect to Lebesgue measure l.

Piecewise smooth P-homeomorphisms f with a finite number of break points and the trivial total jump ratio Of = 1 are more difficult to investigate. In the special case of piecewise C2+£ P-homeomorphisms f whose break points all lie on the same orbit, the invariant measure ¡f is absolutely continuous w.r.t. to Lebesgue measure for typical irrational rotation numbers (see [7]). Rather complicated is the case where the break points of such a homeomorphism f are not on the same orbit. In this case A. Teplinskii constructed in [22] examples of PL-homeomor-phisms f with four break points and the trivial total jump ratio Of = 1 whose irrational rotation numbers pf are of unbounded type and whose invariant measures ¡f are absolutely continuous w.r.t. Lebesgue measure l.

In the present paper we study C2+e P-homeomorphisms f with an arbitrary irrational rotation number pf and two break points not on the same orbit, whose total jump ratio Of = 1. Put

r := {p e [0,1]: p = [ki,k2,...,kn,...] e R \ Q: there exists N = N (p) e N, such that kn ^ 3, for all n ^ N}.

Our main result for these homeomorphisms is

Theorem 3. Let f e C2+£(S1 \ {b1,b2}) be a P-homeomorphism with an irrational rotation number p := pf e r and two break points b1,b2 on different orbits with the trivial total jump ratio Of = Of (b1) • Of (b2) = 1. Denote its invariant measure by ¡f. Then there exists a subset Mp c [0,1] of full Lebesgue measure, such that ¡f is singular w.r.t. Lebesgue measure if ¡f ([b1,b2]) e Mp.

Notice that, if ¡f ([b1,b2]) = qp+p, p,q e Z, then the two break points b1 ,b2 lie on the same orbit and the invariant measure ¡f is absolutely continuous w.r.t. Lebesgue measure (see [14]).

Remark 1. To every piecewise smooth circle homeomorphism f with two break points bi, b2, total trivial jump af := af (b1)-af (b2) = 1 and an irrational rotation number p := pf there corresponds a unique

PL-map of Herman with two break points a0 = 0, c0, such that ah(a0) = af (bi) and ([a0,c0]) =

fj

= fj,f([bi, 62]) (for details see Section 6). Moreover, ^([00, cq]) = --j- Hence, the condition in

Theorem 3 can be written also in terms of the parameter /3 of this unique PL-map as -—— G Mp.

JL

1 + /3

Remark 2. Theorem 3 can be extended to piecewise smooth homeomorphisms of the circle with finitely many break points and a trivial total jump ratio.

Remark 3. The condition p := pf e r in Theorem 3 is a technical one, and the theorem can be proved for all irrational rotation numbers.

2. Some definitions and well-known facts

Let f be an orientation-preserving circle homeomorphism with an irrational rotation number pf and an invariant measure /if. Then pf has a unique continued fraction expansion as pf = 1/(ki + 1/(&2 + •• •)) := [ki,k2, ...,kn,...). Denote by pn/qn = [ki,k2, ■■■,kn ], n ^ 1, its nth convergent. The numbers qn,n ^ 1, are called also the first return times of f and satisfy the recurrence relations qn+1 = kn+1 qn + qn-1, n ^ 1, where q0 = 1 and q1 = k1. Fix an arbitrary x0 e S1. Its forward orbit O+ (x0) = [xi = fi(x0), i = 0,1,2...} defines a sequence

of natural partitions of the circle. Namely, denote by l0n)(x0) the closed interval in S1 with endpoints x0 and xqn = fqn(x0). In the clockwise orientation of the circle the point xqn is for n

odd to the left of x0, and for n even to its right. Denote the iterates of the interval l0n)(x0) under f by I(n)(x0) := f i(l0n (x0)), i ^ 1. We call these intervals for fixed n intervals of rank n. If

An := /f (l0n) (x0))= if (If (x0)) (2.1)

denotes the measure of the intervals of rank n, then An = \qnpf — pn\.

It is well known that the set £n(x0) of intervals with mutually disjoint interiors, defined as

Cn(x0) = [l0n-1)(x0), /1n-1)(x0),...,/qn-1)(x0)}U[/0n)(x0), /1n)(x0),...,/qn!!-1(x0)},

determines a partition of the circle for any n. The partition £n(x0) is called the nth dynamical partition of S1 determined by the point x0 and the map f, (where we suppressed the dependence on the map f for simplicity).

Proceeding from £n(x0) to £n+1(x0), all the intervals ljn)(x0), 0 ^ j ^ qn-1 — 1, are

preserved, whereas each of the intervals I(n-1)(x0), 0 ^ i ^ qn — 1, is partitioned into kn+1 + 1 subintervals belonging to {n+1(x0) such that

4n-1)(x0) = /f+1)(x0) U U 4+L1+q(x0). (2.2)

s=0

Obviously, one has {1(x0) ^ £2(x0) ^ ... ^ £n(x0) ^____

Definition 1. Let K > 1 be a constant. We call two intervals I1 and I2 of S1 K-comparable if the inequality K-1£(h) < KI1) < K£(h) holds.

Next we introduce another partition An(x0). Define

Jn) := l(n)(xo) U l(n-1)(xo), 0 4 i 4 qn-1 — 1.

The system of intervals

A* (m) := /(n) /(n) ll I //" (n-1) T(n-1) r(n-1)\

?n(Xo) := ,J1 , ...,Jqn-1-1 j U l/qn-1 ,rqn-1+1, . . .,1qn-1 f

determines also a partition of the circle.

Definition 2. An interval I = [r,t] c S1 is said to be qn-small and its endpoints qn-close if the intervals f l(I), 0 4 i 4 qn — 1, are, except for the endpoints, pairwise disjoint.

Let f be a P-homeomorphism with an irrational rotation number pf and \BP(f)| < to. The following facts about P-homeomorphisms are standard:

• if the interval I = (x,y) c S1 is qn-small and f s(x), f s(y) e BP(f) for all 0 4 s < qn, then for any k e [0, qn) Finzi's inequality

e_„ DfHfl.')_ Dfk(y)

holds, where v is the total variation of logDf on S1;

• for any y0 with ys := fs(y0) e BP(f) for all 0 4 s < qn, the inequality

qn-1

e-v < J] Df (ys) 4 ev (2.4)

s=0

holds. This inequality is called Denjoy's inequality;

• for an arbitrary element I(n) of the dynamical partition £n(x0)

£(I(n)) 4 const \n (2.5)

_ l

holds, where A = (1 + e~v) 2 < 1;

• the homeomorphism f is topologically conjugate to the linear rotation Rp i.e., there exists a homeomorphism p such that p o f = fp o p;

• let ¡f be the unique invariant probability measure ¡f of f. Then (see [14, p. 71])

J log Df (x)df (x) = 0. (2.6)

s 1

3. Herman's family of PL-homeomorphisms with two break points

In [14, Section 7 of Chapter VI] M. Herman introduced a family of PL-homeomorphisms with two break points, for which he studied their invariant measures and the regularity of the

Fig. 1

maps conjugating them to linear rotations: given two real numbers A > 1 and f > 0, he defines for X e [0,1] the piecewise linear map Hp,x: [0,1] ^ [0,1] as

{Ax, if 0 ^ X ^ c,

A-P(X — 1) + 1, if c < X < 1,

such that Ac = A-p(c — 1) + 1.

Then Herman considered for 0 ^ 9 ^ 1 the one-parameter family of PL-maps Hp,x,g of the unit interval with

Hp,x,g (X) = Hp, x (X) + 9 mod 1, and the induced piecewise linear homeomorphisms of the circle

hp,x,g(x) = Hp,x,g (X) mod 1. (3.1)

Obviously, a0 = 0 and c0 = c are break points of all these hp,x,g. Denote their rotation number for fixed A > 1 and f > 0 by pg. Continuity and monotonicity of pg as a function of 9 imply that for an arbitrary irrational number a e [0,1] there exists a unique 9 = 9(a) e [0,1] with pg = a. Herman then proved in [14]

Theorem 4. The following properties are equivalent:

(i) hp,x,g is conjugate to the linear rotation Ra through an absolutely continuous homeomorphism;

(ii) hp,x,g is conjugate to Ra through a Lipschitz homeomorphism;

(iii) hp,x,g can be conjugated to Ra by a piecewise C™ homeomorphism, which is not PL;

(iv) i G Z a mod 1;

(v) the break points a0 and c0 belong to the same orbit under hp,x,g. It can be easily checked (see [14]) that up to two points

log Dh^xAx) _ __/3_

(I+ /3) log A -KaoMW 1+/3> W

where X[a0,c0] is the characteristic function of the interval [a0,c0]. Obviously,

v y Dh+(0)

for h = hp,x,0, and hence log a = —(1 + () log A. Then we can rewrite (3.2) as

log Dhp, x, 0 (x) (

log a 1 + (

and hence also for any n ^ 1

— X[ao,co](x), (3-3)

log DKngx0(x) n( ^

-- = T+P ~ (3.4)

Therefore, the following useful lemma holds for Herman's homeomorphism hp,x,0 Lemma 1. For every n ^ 1

^ log Dh';3Xd{x) = n JL- mod 1. (3.5)

Furthermore, one has

Lemma 2. Let ¡i0 be the invariant measure of Herman's map hp,x,0 with break points a0, c0 and an irrational rotation number p0. Then

(

M[ao,co}) = (3-6)

„ , D P , r l°gDhp,\,e(x) i , n n T t logDhp,\,e(x) ,, Proof. By bq. (2.6) one has J --d^0{x) = 0. Inserting for -—- the

right-hand side of (3.3), we get

/ {T+J3 ~ | dfig(x) = - Co]) = 0,

s 1

and hence the lemma is proved. ■

Remark 4. In (3.6) the right-hand side does not depend on the parameter 0.

The uniform distribution of sequences is one of the classical problems of ergodic theory (see, for instance, [17]). Indeed, one has

Theorem 5 (see [17]). For [a,b] C R let un: [a,b] ^ R, n = 1,2,..., be a sequence of continuously differentiable real-valued functions. Suppose, for arbitrary m,n £ N, n = m, the function Dun(x) — Dum(x) is monotone with respect to x and that furthermore \Dun(x) — — Dum(x)\ ^ K > 0 for some constant K not depending on x, m and n. Then the sequence un(x),n = 1,2,... is uniformly distributed mod 1 for almost all x in [a, b].

Since un(x) = qn ■ x fulfills the assumptions of Theorem 5, it follows that the sequence

q ■ 3 (3

Yq—-j mod 1 is uniformly distributed for Lebesgue almost all Y~\~Jj' dearly> this sequence is

((

not uniformly distributed for all ^ since for ^ ^ = mpf/3xe mod 1 for some integer m,

qn ■ /3 l + !3

rotation numbers of bounded type one has indeed the following result.

lim

n

= 0, where ||x|| denotes the distance of x to the nearest integer. In the case of

Theorem 6 (see [17, 18]). Let a be an irrational number of bounded type with partial

lim \\qnx\\ = 0

quotients Then

qn

if and only if x £ Z a mod 1.

4. On the location of break points

In this section we recall four lemmas from [10] concerning the break points for an arbitrary P-homeomorphism f with an irrational rotation number pf and two break points ao = a0 and co = c_ not on the same orbit. They describe the locations of break points of the map fqn

1 P

and the values of the derivative Dfq" on S , where are the partial quotients of pf. Obviously

the map fqn has 2 qn break points denoted by BP(fqn) := BP(fqn; a_) U BP(fqn; c_) with BP(fqn; a_) := {aQ,a^1,...,a!_qn+1}, respectively, BP(fqn; c_) := {4,— ,...,c-qn+1}, where a-i = f—i(a__), respectively, c- = f—i(c__), 0 < i < qn — 1.

It is clear that the break points of the map fqn define a partition Pn(f) of the circle S1 into 2 qn intervals with pairwise non-intersecting interior.

Let (n(a_) be the nth dynamical partition determined by the break point a0 = a_ with respect to the map f. Then one has for the second break point c_ either c_ £ ii^a) for some 0 ^ io < qn-1, or c_ £ ljn—1) (ao) = fj0((ao, a—qn])Ufj0((a—qn ,aqn_1)) for some 0 ^ jo < qn, i.e., c_ £ fj0((ao,a-qn]) or Co £ fj0((a-qn,aqn-1)). The two last cases have to be treated separately. The following three lemmas describe the location of the break points of fqn in intervals of different n dynamical partitions (for the proofs see [10]).

Lemma 3 ([10]). Assume c_ £ (a_) for some io with 0 ^ io < qn-1. Then the break points {a_—i, c_—i, 0 ^ i ^ qn — 1} of fqn belong to the following elements of the dynamical partition £n(ao) (see also Fig. 2):

• ao £ ion)(ao);

• c—io+, = fs(c—io) £ lSn)(ao), 0 < s < io; ai q_+s = f s(a_q. ) e f s((ao,a_lie iSn—1)(a

—qn+s = fs(a—qn) £ fs((a*o,a—qn]) C I}/1 1)(ao), 1 < s < io;

a

—qn+s, C—qn—io +s = f s(c—qn — i0) £ fs(K ,a—qn ]) C ^ ^^ io + 1 < S < qn — 1.

Lemma 4 ([10]). Assume c_ £ fi0((a_,a—qn]) for some 0 ^ io < qn. Then the break points of fqn belong to the following elements of the dynamical partition (n(c—i0) of the break point c*_i0 (see Fig. 3):

• c—i0 ,ao £ ion)(c—i0)

• c—i0+s = fs(c—i0), a—qn+s = fs(a—q„) £ fs([c—i0,a—qn]) C isn—1 (c—i0), 1 < s < io;

• C—qn—i0+s = f s(c—qn—i0 ), a—qn+s = f s(a—qn ) £ f s([C— ^ ,C— q„ ]) C 10 (c— i0 ),

io + 1 ^ s ^ qn — 1.

In

-I—

-*o

C-io-qn a-qn

+

-K-

Qn-1

+

V+i

"¿0+1

'-¿o-gn+l

a

-qu+1

a9n-l+l

-+-

-+-

a.

10

9n a~qn+io -•-X-

a.

'qn-i+io

V+»o+l

^o+l

¿0 1 Fig. 2

a,

'9n-l+9n—1

-+-

C-»0+9n

-iO

a-9n C-io-qn

c-io+qn-i

C-i0+qn+l al clio+1

a*-qn+1 C-io-9n+l

c-i0+gn-i+l

-1- Cqn -1- -a- r* C0 -X- a-qn+io -1- C-qn -1- cqn-1

Cqn+1 -1- ai0+l -1- Cl -1- * -X- C-qn+l -a- C9n-1+1 -1-

c2<jn-i0-l a9n-l cgn-i0-l a!_1 c-i0-l Cgn-l+9n-io-l

Fig. 3

Lemma 5 ([10]). If c0 £ fi0((a-qn,aqn-1 ]) for some i0 with 0 ^ i0 < qn, the break points of fqn are located in the following elements of the dynamical partition £n(a—qn+1) of the break point a*-qn+1 (see also Fig. 4):

• a*-qn+i+s £ lin-1\a-qn+i), c-zo+i+s £ lin-1\a-qn+i), 0 < S < i0 - 1;

• a-qn+i0 + 1+s £ 40V1 (a-qn + 1), c*-q„ + 1+s £ ^O+s^ (a-q„ + 1), 0 ^ S ^ qn - io - 1

—\-X-•-1-1-1-

a-qn+1 C~io+l al+«n-l C-io-9n+l a-qn+qn-i+l

aio a-qn+i0 C0 aio+qn-i C-9n a-gn+9„-i+i0

-h

«¿0+1 a* +i+1 C1 ai0+qn-l+l c*-qn+l a-qn+qn-l+i0+l

qn

Cqn~io aQn-i+9n

»0

yn—1

Fig. 4

Next we recall the case of a P-homeomorphism f with an irrational rotation number pf and two break points a0 := a0, a*0 := fi0(a0), ¿0 > 0, on the same orbit. Put ni0 := min{n: qn ^ ¿0}. Assume that n > ni0. If the total jump ratio is Of = 1, the map fqn has 2i0 break points

a-qn + 1 := a-qn + 1, a-qn+2 := a-qn+2, ■ ■ ■ , a-qn+i0 : = a-qn+io

and

a1 := al, a2 := a2, ■ ■ ■ , a*0 : = aio ■

If Of = 1, the map fqn has qn + i0 break points

H< H< H< H<

a-qn+1 := a-qn+1, a-qn+2 : = a-qn+2, ■ ■ ■ , ao : = a0, ■ ■ ■ , ai0 : = aio■

Then one has

Lemma 6 ([10]). Assume f is a P-homeomorphism with an irrational rotation number pf and two break points a0 := a0, a*0 := fi0(a0), i0 > 0, on the same orbit. Choose n > nio.

1) If Of = 1, then one finds for the break points a*-c^n+s+1, a*+1 of fqn

• a-qn+s+1, aS+1 G fs([a1 ,a-qn+1]) C /¡+-1)(a0) G £n(a0), 0 < s < ¿0 - 1 ;

(see Fig. 5);

2) if Of = 1, we have

• a0 C /0n-1)(a0);

• a-qn+1+s, a*1+s G fs([ai0+1,a-qn^J) C 40+1+s^, 0 < s < qn - ¿0 - ï.

• as+1 G f s([a1,a-qn+1]) C /1+-1)(a0), ¿0 < s < qn - ¿0 - L

-1-X—

aqn+s+1 a*+1

aqn-i+s+l

Fig. 5

Lemmas 3 to 6 show the location of the break points of fqn on elements of different nth dynamical partitions determined by the map f, respectively, their order along the circle. Indeed, these lemmas hold true also for any pure rotation Rp with p irrational and any two points a0, c0 £ S1, whose preimages under Rqpn correspond to the break points of the P-homeomor-phism fqn.

Lemmas 3-5 imply the following corollary.

Corollary 1. Let h be a PL-homeomorphism with two break points a0 and c0. Then each interval of the partition £n contains exactly one z £ BP(hqn; a0) and one z £ BP(hqn; c0).

5. Denjoy equality for piecewise linear circle homeomorphisms with two break points

In this section we recall the explicit expressions derived in [10] for the derivatives Dhqn of the piecewise linear homeomorphism h with two break points a0, c0 in terms of the jump ratio ah(a(0)) and the ^-measures of certain intervals of the partition Pn(h) of S1 determined by the break points of hqn.

They follow from Lemmas 3-5 applied to a PL-circle homeomorphism h with an irrational rotation number ph and two break points a0 = a0 = 0, c0 = c0 not on the same orbit and with the total jump ratio ah = 1.

In the case of Lemma 3 and Lemma 5 the break points PB(hqn; a0) of hqn, associated with ao = 0, and PB(hqn; cO), associated with c0, alternate in their order along the circle S1.

Let n be odd. Obviously, the break points of hqn define a system of disjoint subintervals of the circle, given in the case of the assumption in Lemma 3 by (see Fig. 2)

[c-io+s,a-qn+s], 1 < s < io,

(5.1)

respectively,

[c-i0-qn+s, ^-q^ i0 + 1 < s < Qn.

(5.2)

We combine these subintervals to the subsets

An(i0) := U[c-Í0+s,a-çn+s], Bn(i0) := (J [c-i0-qn+s,a-qn+s]. (5.3)

— io+s, a-qn+s

In the case of the assumption in Lemma 5 the subintervals are given by (see Fig. 4)

[a*-qn+s,c*-io+sL 1 < S < i0,

(5.4)

respectively,

[a*-qn ^ c-io-qn+s], i0 + 1 < s < Q^

(5.5)

which we combine to the subsets

io qn

An(i0) := [J[a-qn+s>C-io+s]> Bn(io) : = U [a*_qn+s,C*_io-qn+s\- (5-6)

s=1 s=io+1

For n even, the orientation of the above intervals has to be reversed. Therefore, in the case of Lemma 3 we have the following system of disjoint intervals:

[a_qn+s,c_io+s\, 1 < s < io, (5-7)

respectively,

[a_qn^ C_io_qn+s\, i0 + 1 < s < Qn- (5-8)

In the case of Lemma 5 one finds

[C_io+s,a_qn+s\, 1 < S < i0, (5.9)

respectively,

[C*-io-qn+s,a*-qn+s], i0 + 1 < s < Qn- (5-10)

In the case of Lemma 3 and n even, respectively, in the case of Lemma 5 and n odd, the subsets An(i0) and Bn(i0) can be defined as before. The above constructions show that the boundaries of every interval in the subsets An(i0) and Bn(i0) consist of break points from PB(hqn; a0), respectively, PB(hqn; c0). In the following we abbreviate the jump ratio of h at the break point a0 by

Dh-( 0)

* := <7ft(0o) = DK{0) *

Next, we recall the first result on the values of (Dhqn(x)) from [10].

Theorem 7. Let h be a PL-circle homeomorphism with an irrational rotation number ph and two break points a0 = 0 and c0, whose total jump ratio is ah = 1, and which lie on different orbits. Assume c0 fulfills the assumptions of Lemma 3 with 0 ^ i0 < qn-1, respectively, Lemma 5 for some i0 with 0 ^ i0 < qn. Then in the case of Lemma 3

, i a^h(Ari(io)uBn(io))-1, if x e An(io) U Bn(io),

(Dhqn (x))(-1)n = 1 T (5-11)

V y " \^h(An(io)UBn(i0)), if x e S1 \ (An(io) U Bn(io));

respectively, in the case of Lemma 5,

q (-1)n+i ( a^h(An(io )uBn(io ))-1, if x e An(io) U Bn (io), (Dhq(x)) = j ^(An(io)uBn(io)), if x e S1 \ (An(io) U Bn(io))- (5-12)

Theorem 7 shows that Dhqn is constant on every element of the partition Pn(h) and takes only two values under the assumptions of Lemmas 3 and 5. Moreover, the values of Dhqn are determined by the jump ratio a = ah(a0) and the ^-measure of An(i0) U Bn(i0). In the case of the assumption on c0 in Lemma 4 we can define again a system of disjoint subintervals determined by the elements in Pn(h). Let n be odd. Then these subintervals are as follows (see Fig. 3):

^-io^ a-qn+sL 1 < S < i0, (5-13)

respectively,

[a*_qn +s,c*_i0_gn+8], io + 1 < s < Qn- (5.14)

For n even the orientation of the above intervals has to be reversed. To determine in the case of Lemma 4 the values of Dfgn, we define

io gn

An(i0) := U^-io+s^-gn+sh Bn(i0) := U [a_gn+8,C*_io_gn+s]. (5-15)

8=1 8=i0 + 1

Then the following theorem holds.

Theorem 8 ([10]). Let h be a PL-circle homeomorphism with two break points aS and c0 with oh = 1, which lie on different orbits. Assume cS fulfills the assumption of Lemma 4 for some i0 with 0 ^ i0 < qn. Then for all n ^ 1

a^h{An(io))_Vh(Bn(io))_1 if x e An(i0),

(Dhgn(x))(_1)n = <( a^h(An(io))_^h(Bn(io))+1 if x e Bn(i0), (5.16)

a^h(An(io(Bn(io)) if x / An(i0) U Bn(i0).

It remains to discuss the case of a PL-homeomorphism h with an irrational rotation number ph and two break points a0 = 0 and a*0 = hi0 (aS), i0 > 0, on the same orbit. In this case the break points of hgn alternate in their order along the circle S1. Denote by Un(a*s), 1 ^ s ^ i0, the closed intervals with endpoints a*s and a_gn+8. Obviously, these subintervals are disjoint.

Lemma 6 implies that Un(a*s) C I(n_1 (a0), 1 ^ s ^ i0. Next, we define for every n ^ 1

io

Un = U Un(aS). (5.17)

s=1

Then one has

Theorem 9 ([10]). Let h be a PL-circle homeomorphism with two break points a0 = 0 and aS0 = hi0(a0), i0 > 0, with oh = 1, which lie on the same orbit. Put nio := min{n: qn ^ i0}. For n > nio one finds

( i)n+i ( a^h(Un) if x e Un, (Dhgn (x))(_1)n+1 = J (U ) , n, (5.18)

v y JJ | avh(Un)_1 if xeS1 \ Un.

Theorems 7 and 8 lead to the following

Theorem 10. Let h be a PL-circle homeomorphism with an irrational rotation number ph and two break points a0 = 0 and c0, which lie on different orbits. Assume cS fulfills the assumptions in Lemma 3 for some i0 with 0 ^ i0 < qn_1, respectively, in Lemma 4 or Lemma 5 for some i0 with 0 ^ i0 < qn. Then in the case of Lemma 3 or Lemma 4

1 0

log Dhg" (x) = (-1)" (MMio)) + MBa(io))) = qn • Y^J3 mod !> (5-19)

respectively, in the case of Lemma 5,

1 ^ogDW»(x) = (-l)n+H»h(Bn(io))-»h(Mio)))=qn-T^15 modi. (5.20)

log O V v v v v 1+0

6. Proof of Theorem 3

Let f e C2+e(S1 \ {b1, b2}) be a P-homeomorphism of the circle with an irrational rotation number pf and two break points b1 and b2 not on the same orbit, whose total jump ratio is af = af (b1) • af (b2) = 1. Denote by /if its unique invariant probability measure. Define the parameters 3 and A through

13 := HfdhM), A-1"^^ <7/(61). (6-1)

1+3

Let h = be Herman's PL-homeomorphism of S1 with break points a0 = 0 and c0 = c such that Ac = A_/3(c — 1) +1. Since the rotation number pf is irrational, we can find a unique 9 such that the rotation number p^ of h^ = h^ x g coincides with pf. Denote by ^ the invariant measure

3

of /?.£. By Lemma 2 ^([a,o,co]) = Since pf = the homeomorphisms / and h^ are

topologically conjugate via some homeomorphism p. We can choose p such that a0 = p(b1) and c0 = p(62), because ¿t/([6i,62]) = ^([a,0,c0]). Then one has also (p* ^)([bi,b2\) = ¿t/([6i,62]) = = ig([ao, Co]), since the invariant probability measure of / is unique. Hence, we have proved the following fact, which will play a key role in our proof of the main theorem.

Theorem 11. Consider the P-homeomorphism f : S1 ^ S1 with an irrational rotation number pf and two break points b1,b2 with jump ratios a := af (b1) and af (b2) = a-1. Let

hj := h^ Xg be Hermans PL-homeomorphism with two break points a,o = 0, Co, parameters ¡3, A as defined by (6.1) and an irrational rotation number p^ = pf. Then

(I) aj(ao) = (J and ^([a0, c0]) = ¿t/([6i, 62]);

(II) the maps f and h^ are topologically conjugate by some circle homeomorphism <p with p(b1) = ao and p(b2) = co.

Since the rotation number pf is irrational, the invariant probability measure /f has no discrete ergodic component. Indeed, one knows that every such P-homeomorphism is ergodic also w.r.t. Lebesgue measure l (see [14]). Suppose /f has an absolutely continuous component /J-0 with support A and /i°fC(A) > 0. Then also l(A) > 0. If p(x) is the density of f-0, then on A obviously p(x) ^ 0 and on S1 \ A one has p(x) = 0. Since p(x) satisfies the functional equation

p{fix)) = —x S S1 and Df(x) ^ const > 0, the subset A+ = {x: p{x) > 0} is Df (x)

f-invariant. Ergodicity of f with respect to Lebesgue measure l implies that either fC(A) = 1 or fc (A) = 0. Hence, the invariant measure /f is either pure absolutely continuous or pure singular on S1.

The idea for the proof of Theorem 3 is to construct for the homeomorphism f a sequence of measurable subsets Gnm C S1 such that l(Gnm) > const > 0 and

liminf l {x e Gnm-\Dfqnm (x) - 1| >5} > 0

for some 5 > 0. On the other hand, one knows that under the assumption of absolute continuity of the invariant measure /f w.r.t. Lebesgue measure l, Dfqn(x) tends, as n ^ to, to 1 in probability with respect to the probability measure l.

Hence, let us start with the following proposition shown in [14].

Proposition 1. Let f be a P-homeomorphism of the circle with an irrational rotation number pf. If its invariant probability measure ¡f is equivalent to Lebesgue measure l, then for all 5 > 0 the sequence l({x: \Dfgn(x) — 1| > 5}) tends to zero as n ^^.

Important for the construction of the above mentioned sets Gnm will be

Lemma 7. For arbitrary 5 e (0,1) and n ^ 1 consider three points z1,z2 ,z3 e S1, z2 e BP(fgn), with z1 — z2 — z3 — z1 such that the intervals [z1 ,z2] and [z2,z3] are qn-small. Assume the P-homeomorphism fgn e C2+s(S1 \ {z2}) has a jump ratio Ofqn (z2) = A at the break point z2. For v the total variation of log Df on S1 and ti e (z1,z2) and tr e (z2,z3) with

l([tl,Z2]) = l([tr,Z3})

l{[zi,z2]) /([22,23])

=5

(6.2)

one has either

for all x e [tl, z2), or

log Dfqn(x) < + Kev§}

log Dfqn(y) > -~Kev5

2

for all y e (z2,tr], when A > 1.

In the case A < 1 one has either

(6.3)

(6.4)

log Dfqn(x) > _ Kev6,

(6.5)

for all x e [ti, z2), or

for all y e (z2,tr].

log Dr»(y)^^+Kev6

Df(z2)

(6.6)

Proof. Assume log A = log Then log Df_n (z2) = log A + log Df+ (z2), and hence

Dfl"(z2)

> 0, the case log A < 0 can be treated analogously.

log Dff (z2) < if and only if log Dfin(z2) < l0gA

2

(6.7)

respectively,

Hence, either

or

log Dfl'(z2) > —!^ if and only if logD/!"(z2) >

■9»/~ ^ lo§A

log Dfg (z2) ^

(6.8)

(6.9) (6.10)

2

Then for arbitrary x £ [ti,z2) one finds

Dfqn (Z2) ■ ■

I log „ " ■ I < £ \ logDf.(f(z2))-logDf(f(x))\ < K K[f3(x),f3(z2)]) <

Df (x) j=o j=0

< * g «[t'MM < K*t g^g«[/'<*>, fiM) = j=0 Dfj ($) l([z1 ,z2))

for certain ( £ [ti,z2), $ £ [zi,z2) and an universal constant K > 0 depending only on f. According to inequality (2.3),

e- c Em i e"

Df'(H) and therefore

Kqy dp{o m,z2)) KeV

since '' ^ = 5. We have also used the fact that the interval [21,22] is (fo-small and hence h[z1,z2j)

the intervals [fj(z1), fj(z2)], 0 ^ j ^ qn — 1 are disjoint. This leads finally to the bound

Dfqn(z2)

In the same way it can be shown that

Df+ (z2)

for all y £ [tr, z2). Inserting the bounds (6.9)-(6.12), we get the bounds (6.3)-(6.6) in Lemma 7.

■

An important role in the proof of Theorem 3 is played by certain neighborhoods of the break points of the P-homeomorphisms fqn, which we define next. For this, recall the partition Pn(f) of the circle defined by the breakpoints BP(fqn) of the map fqn in Section 4. For z £ BP(fqn) we denote by Vn(z), respectively, Vr(z) the interval in Pn(f), whose right, respectively, left boundary point is the break point z. Given some 5 £ (0,1), using Lemma 7 we can then construct left and right subintervals Vln(z; 5) C V,n(z), respectively, Vr(z; 5) C Vr(z), both with the break point z as a boundary point, such that

l(Vt[(z-,6)) = KV^z-6)) = i{v\{z)) ' imz)) •

Definition 3. The subintervals Vln(z; 5), Vr(z; 5), respectively, the interval Vn(z; 5) = = Vn(z; 5) U Vr(z; 5) are called the left normalized 5-neighborhood, the right normalized S-neighborhood, respectively, the normalized 5-neighborhood of the break point z.

Assume now there exists for p = pf = pg, p £ T & set Mp such that the invariant measure p/ is absolutely continuous w.r.t. Lebesgue measure l if pf ([61,62]) £ Mp. We will show that this assumption contradicts Proposition 1.

For this, we derive first some properties of the distribution of log Dh^ (x) with respect to the partition Vn(h^) of Herman's PL-homeomorphisms h^ defined in Theorem 11.

For this, recall the sets An(i0),Bn(i0) defined in (5.3), (5.6), respectively, (5.15). Later we will need also the closely related sets An(io), B*n(io), defined according to Lemmas 3-5 as follows:

• if the break point c0 of Dhqn fulfills the assumption in Lemmas 3 or 5, we set (see Fig. 2 and Fig. 4)

io

An(i0) := U[a*-qn+s,aqn_1+s}, (6-13)

s=1

and

qn

Bn (i0):= U [a-qn+s,aqn-1 +s}; (6-14)

s=io+1

• if the break point c0 of Dhqn, on the other hand, fulfills the assumption in Lemma 4, we set (see Fig. 3)

io

An(i0) := \J[a*-qn+s,c-io +qn-1+s}, (6-15)

s=1

and

qn-io-1

K (i0):= U [c-qn+s,cqn-1+s }■ (6-16)

s=1

Consider first the case when the break point c0 of Dhqn fulfills the assumption of Lemma 3 or Lemma 4. We begin our discussion with the case of Lemma 3. By Theorem 10

Dhq» (x) = (-1 {Pe(An(i0)) +Pg(Bn(io))) = qn • ^ mod 1 (6.17)

for all x £ An(i0) U Bn(i0) as defined in (5.3). Therefore,

log Dh*> (,) mod 1 = ( 1 " + if W " °dd' (6.18)

logo- p,g(An(io)) + pj(Bn(io)) if n is even.

Next, we estimate pg(An(io)) + Pg(Bn(io)). By Lemma 3 (see Fig. 2) we have obviously

io io

An(i0) = |J [c-io +s, a*-qn+s\ C |J [aqn+s, a*-qn+s} (6^19)

s=1 s=1

and

qn qn

Bn(i0 )= |J [c-io-qn+s ,a-qn+s} C U [as,a-qn+s}■ (6^20)

s=io+1 s=io+1

It is easy to see that

Hg(An(i0)) ^ 2i0 • A„, Hg(Bn(io)) ^ (qn ~ io) • A„, Using these bounds and the recurrence relations

(6.21)

we obtain

qn = knqn-1 + qn-2, An = knAn+1 + An+2, n ^ 1

p$(An(io)) + /J,e(Bn(i0)) ^ (qn + io) • An ^ ^ +

qnAn-1

= 1 +

Qn-l Qn J A„_

<(l+1

n1

kn J kn+1

Since kn ^ 3 for n ^ Np, we hence get

Pg(An(i0)) +p-g(Bn(io)) < g

(6.22)

for n ^ Np. Summarizing Eqs. (6.17), (6.18) and the last bound, we conclude that in the case of Lemma 3 and n > Np

and

0 ^ -—-— log DhSp (x) (mod 1) < for n even, log a 0 9

5 < -r^— log Dh%'(x) (mod 1) < 1, for n odd.

9 log <7

Consider now the case of Lemma 4.

Using the second assertion of Theorem 10, it follows that

1 logDh^(x) (mod 1) e {\^(Bn(io)) - ^(An(i0))\, 1 - \^Bn{i0)) - ^(An(i0))\}

(6.23)

(6.24)

logo-

for all x £ An(i0) U Bn(i0) defined in (5.15). Lemma 4 and (5.15) imply for n ^ Np

\Hg(An(i0)) - Hg(Bn(i0))\ ^ Hg(An(i0)) + Hg(Bn(i0)) < < io ■ An + (qn - io) ■ An = qn ■ An < Qn A" = < i.

qn ■ An-1 An-1 3

(6.25)

This and (6.25) lead, in case the break point c0 of Dhqn fulfills the assumption of Lemma 4, to

1

log a

log Dh~l (x) (mod 1) e

1

U

2

3'

This together with the bounds (6.23) and (6.24) show that, in case the break point c0 of Dhqn fulfills either the assumption of Lemma 3 or of Lemma 4, y^— log Dhtj?(x) (mod 1) takes values

only in [0,1} \

4 5 9' 9

i.e., for n ^ Np and x £ An(i0) U Bn(i0)

-r^— log M" (x) mod 1 e log a 0

u

1,1

(6.26)

1

0

3

From Theorem 10, however, we know that the sequence

{l^logDh^{x) = qn-TTp mod1}

is uniformly distributed on [0,1] and hence its values are dense in [0,1]. From (6.26) we therefore conclude that the cases where the break point c0 of Dhqn fulfills either the assumption of Lemma 3 or of Lemma 4 can happen only for a finite number of qn's. This means, on the other hand, that the values of ^ log Dh^ are dense on [0,1] only in case the break point Co of Dh

fulfills the assumption of Lemma 5.

Hence, we have to discuss next this case. By Theorem 10 we have for x £ An(i0) U Bn(i0), defined in (5.6),

^ log Dhq" (x) = (-1)" 0%(A„(io)) + (io(Bn(i0))) mod 1. (6.27)

But this means that

Dh*>(x) mod 1 = ( + **iBM) , ^ W " ^^ (6.28)

[ 1 - (Hg(An(i0)) + Hg(Bn(i0))) if n is ockl.

We can assume that for some subsequence {qnm }

lim Dhq2nm (x) = lim (M.B2nm (i0)) + ^h(An2nm (i0))) = w (6.29)

m^tt m^tt m

with w £ [0,1] . The case

lim Dhq2nm+1 (x) = hm (1 — (M.B2nm+i(i0)) + ^h(An2nm+1 (i0)))) = w

with w £ [0,1] can be treated analogously.

We have to consider two possible cases, namely, when 0 ^ ?'o < -y and when ^ io < qn-

Consider the case 0 ^ io < -y.

We start with several bounds and estimate first the ratio . It is clear that

(J,g{Bn{l ojj

qn

tig(An(i0)) __io^([a*_qn+1,c*_io+1})_ ^ -¿o < 2 _

l*o(Bn(io)) (qn - io)ttg([a*_qn+1,c*_io+1}) + (qn ~ ¿o)%([ci, c*_qn+1}) ^ Qn ~ io ^ Qn In

2

where we have used 0 ^ ?'o < -y. Consequently,

fioiMh)) < fJ$(Bn(io)). (6.30)

Using this bound, we obtain 1 ^ ^(An(io)) + Hg(Bn(io)) ^ 2^(An(io)), which implies

trtMio)) < 1 (6.31)

From the definition of Bn(i0) and Bn(i0) in (5.6) and (6.14) it follows that

VeiBniio)) = (Qn ~ '¿o)A„_i - /¿g(Bn(io)). (6.32)

Next, we estimate (qn — i0)An-i. It is obvious that

qn An-i + qn-iAn = 1. (6.33)

It is clear that

qnAn-i knqn-i + qn-2 k,n+iAn + An+2 , qn-2\ f, , An+A

= I kn + 77—7 kn+i +

qn-iAn qn-i An V qn-i J V + An J'

This and (6.33) imply

qnAn-i > 1 - 1 = i ■ (6-34)

knkn+i + 1 knkn+i + 1

Next, we compare qnA„_i and (qn — ?'o)An_i. Since 0 ^ ?'o < -y, we have

(gra - -¿0)Ara_i ^^ i

qnAn-i 2

Together with (6.34) this implies that

(qn - -i0)A„_i ^ }rqnA„_i ^ 1

2 2 knkn+\ + 1 Since kn ^ 3 for n ^ we therefore have shown

(qn-io) = (6.35)

Suppose now n¡m, m = 1,2,... is some sequence of even numbers such that

lim (pg(Bnm(¿o)) + (¿o))) = wi (6.36)

with £ [0,1]. Then, for any e > 0 and sufficiently large nm,

Ui - Pg(Anm(io)) - £ < Pg(Bnm(i0)) < Wi - ^(^(io)) +1-This together with (6.31) implies

loi - i - e ^ /j,g(Bn(io)) < toi + e.

This together with (6.32) implies for sufficiently large nm

VoiBZJioVZ^-Ui-e. (6.37)

The last two relations show that Pg(Bnm(io)) and Pj(B*m(io)) are comparable with some constant C > 1.

In the next step of the proof of Theorem 3 we consider two copies of the unit circle. Suppose the homeomorphism / acts on the first, circle, and Herman's homeomorphism h.j, on the second one. It is obvious that

l*f(<P (-B„m(io))) =%(-B„m(i0)), Vf{<P (-B*m(i0))) = fJ>g{B*m{io)). These relations together with (6.37) imply that

Ci < Vf (Anm (i0))) < C2, Ci < Vf (A*nm (i0))) < c2.

The last relations, the arrangement of the break points of fqn and absolutely continuity of the invariant measure Vf w.r.t. Lebesgue measure l imply for sufficiently large nm the following bounds:

C3 < Kp-1 (Anm (i0))) < C4, C3 < l(^~1(A*nm (i0))) < C4 (6.38)

for some constants C3,C4 £ (0,1).

Consider next the set of break points

BPl(fqnm ) := {C-q„ + i,C-q„+2,..,C-i0 } C BP (fq"m ,C-tQ)

and for z £ BPi(fqnm) the left and right normalized 5-neighborhoods Vn (z,5), respectively, Vrnm (z, 5). Put

VL (S) :=

u

Vl(z,S), vm (S):=

u

vm (z,s).

zÇ.BPl(fq"m )

Obviously,

Vlm (S) C ^ (Bnm (io)), Km (S) C ^(B^ (io)).

-1

zEBPl(fqnm )

1

We conclude that the length l of each of these intervals covering the normalized 5-neighborhoods by definition is 5-:i times the length of the latter ones. Using the definitions of Bnm (i0), B^m (i0) (see also Fig. 4) and the normalized one-sided 5-neighborhoods Vn(z, 5), we obtain

l(p-1(Bnm (i0) H Vim (S))) = S • 1(P-1 (Bnm (i0))),

l(p-1 (B*nm(io) H K (S))) = S • l(p-1 (Bnm(io)))

(6.39)

(6.40)

and consequently

l((p-1(Bnm (i0)) U p-1(B*nm (i0))) H (Vlm (S) U VI (S))) = S • l^^ (io) U B*nm (io))). (6.41)

1

1

From this we can now derive bounds on the values of Dfqnm. Put 5 =

log A

log A

aKev

with A as defined

in Lemma 7 and a >

Kev

. From relations (6.3) and (6.4) of Lemma 7 it follows that either

on the left or on the right normalized 5-neighborhood of every break point z of fqn

| log Df qnm (x) | ^

(a - 2) log A

2a

(6.42)

Define

Gnm (S) = U {z E BP1 (fqnm ) : | log Dfqn (x)| ^

(a - 2) log A

2a

for all x G VL (z,S),s E {l,r}}.

It is clear that

l(Gnm(5)) > min{5 ■ l(Bnm(io)), 5 ■ l(B*nm(io))} > c5 ■ 5 with some constant c5 > 0. Finally, we obtain

| log Df qnm (x)| ^

(a - 2) log A

2a

for all x E Gnm (5). But this contradicts convergence of Dfqn(x to 1 in probability with respect to normalized Lebesgue measure.

In the second case when ^ < io < qn we will show only that either An (io) and A*n (i0) or Bn(i0) and Bn(i0) are comparable. The construction of the sets Gnm in these two cases proceeds in complete analogy to the case when 0 ^ ¿o < -77--

In the case -y ^ io < qn assume first

^(An(i0)) > ^(Bn(io))- (6.43)

We will then show that l~ig(Anm (io)) and ^(A*m (io)) axe comparable for some subsequence {nm}. Using the definition of An(i0) and An(i0) and Lemma 5, we get

^(A*(-i0)) = i0A„_i - ^(An(i0)). (6.44)

Assume again we are given a sequence nm, m = 1,2,... of even numbers such that

lim (^(Bnm (io)) + Lig(Anm (io))) = w2 (6.45)

with w2 E [0,1]. Then for any e > 0 and for sufficiently large nm

U2-e< ^g(Anm (io)) + tíg(Bnm (¿o)) < W2 + £. (6.46) This together with (6.43) implies

< (io)) <co2 + e. (6.47) Hence, together with (6.35) and (6.43) we get

/%(A*m('¿o)) = ¿oA„m_i ~ iig(Anm(io)) ^

1 19 9

(6.48)

The bounds (6.47) and (6.48) show that, choosing w2 less than one finds for sufficiently large n and e > 0 small

^g(Anm(io)) > d and ^(A*nm(io)) > c2 (6.49)

for some constants ci, c2 > 0. Hence, under the assumptions (6.43) l~ij(Anm(io)) and ^g(A^m(io)) are comparable with some constant K > 1.

Assume next in the case -y ^ ?'o < qn

Hg(An(i0)) < Hg(Bn(i0)).

This assumption and (6.46) imply:

U - e

< ^e(Bam(io)) < w2 + e.

(6.50)

(6.51)

Using the definition of An(i0), Bn(i0), Eq. (6.44) and Lemma 5 (see Fig. 4), we obtain

w2-e < %(4im(io)) + %(-Bnm(io)) = qnPe([a*_qnm+1,c*_io+1}) + (qnm -i0)A„m < w2 + e. (6.52) Put io := io(nm) = lo(nm)qnm, 0 ^ lo(nm) ^ 1. Then we have

- io)Anm = (1 - lo)qnm • Anm.

(6.53)

It easy to show that

and hence

Assume now

9 a 1

^Q ^ QnmAnm < g)

_9_

40

(1 - lo) <

- ¿o)A„m < i (1 - lo).

id

(6.54)

(6.55)

u2 - e

then by (6.52) and (6.54) we obtain

qnm^e{[a*-qnm+i,c*_io+l\) > 2 ,

which together with (6.52) and qnm — i0 < i0 implies

< QnmVeda-qnn+i, c-i0+i]) =

= ¿oM[a-«nm+l>C-io + l]) + - ¿o)^([a-«nm+l>C-io + l]) ^

< 2ioPg([a*_qnm+1,c*_io+1}) = 2pg(Anm(i0))

or

U2 - e

nm (io)).

(6.56)

Using this and (6.44), we can see that

20 4 4

which together with (6.56) shows that, under assumptions (6.55), Pg(Anm(io)) and №()(A*m(io)) are comparable for sufficiently large nm and sufficiently small e > 0. It remains to discuss the case where

(6.57)

n

m

n

m

In this case we obtain, using qnA„_i e lj,

3 27

(qnm - i0)Anm = (1 - lo)qnmAnm ^ -(w2 - e)qnmA„m ^ ^(uj2 ~ e).

Together with (6.51) we then get

27 7 47

= (qnm - io)A„m_i - Hg(Bnm(i0)) ^ 2q(w2 - e) - (w2 + e) ^ ^Wa - —e,

which together with (6.51) shows that, under the assumptions (6.50) and (6.57) for sufficiently

large nm, ^g(Bnm(io)) and tij(B*lm(io)) are comparable if w2 is small and e is sufficiently small.

Finally, we conclude that, under the assumptions of Theorem 3, either tle(Bnm(io)) and Hg(B*m(io)) are K-comparable or ^g(Anm(io)) and tij(A^m(io)) are K-comparable, with some constant K > 1, depending on u.

7. Acknowledgments

This work was partially performed during the first author's (A. D.) visit in 2016 as a senior associate of ICTP, Italy, and a visit in 2018 to the Institute of Theoretical Physics of Clausthal University, supported by the German Academic Exchange Service DAAD and the Friends of Clausthal University. A. D. wants to thank K.Khanin for a very fruitful discussion and the Institute for Theoretical Physics and its director P. Blochl for the kind hospitality.

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