THE THERMODYNAMIC FORMALISM AND THE CENTRAL LIMIT THEOREM FOR STOCHASTIC PERTURBATIONS OF CIRCLE MAPS WITH A BREAK Текст научной статьи по специальности «Математика»

Russian Journal of Nonlinear Dynamics, 2022, vol. 18, no. 2, pp. 253-287. Full-texts are available at http://nd.ics.org.ru DOI: 10.20537/nd220208

MATHEMATICAL PROBLEMS OF NONLINEARITY

MSC 2010: 37C05, 37C15, 37E05, 37E10, 37E20, 37B10

The Thermodynamic Formalism and the Central Limit Theorem for Stochastic Perturbations of Circle Maps

with a Break

A.Dzhalilov, D.Mayer, A. Aliyev

Let T £ C2+£(S1 \ {xb}), e > 0, be an orientation preserving circle homeomorphism with rotation number pT = [k1, k2, ..., km, 1, 1, ...], m ^ 1, and a single break point xb. Stochastic perturbations ~zn+i = T(zn) + z0 := z £ S1 of critical circle maps have been studied some

time ago by Diaz-Espinoza and de la Llave, who showed for the resulting sum of random variables a central limit theorem and its rate of convergence. Their approach used the renormalization group technique. We will use here Sinai's et al. thermodynamic formalism approach, generalised to circle maps with a break point by Dzhalilov et al., to extend the above results to circle homem-orphisms with a break point. This and the sequence of dynamical partitions allows us, following earlier work of Vul at al., to establish a symbolic dynamics for any point z £ S1 and to define a transfer operator whose leading eigenvalue can be used to bound the Lyapunov function. To prove the central limit theorem and its convergence rate we decompose the stochastic sequence via

n—1 n—1

a Taylor expansion in the variables into the linear term Ln(z0) = £n + ^ £k H T'(zj), z0 £ S1

k=1 j=k

and a higher order term, which is possible in a neighbourhood An of the points zk, k ^ n — 1, not containing the break points of Tn. For this we construct for a certain sequence {nm} a series of neighbourhoods An™ of the points zk which do not contain any break point of the map Tqnm, qn the first return times of T. The proof of our results follows from the proof of the central limit theorem for the linearized process.

Keywords: circle map, rotation number, break point, stochastic perturbation, central limit theorem, thermodynamic formalism

Received November 30, 2021 Accepted May 05, 2022

Akhtam Dzhalilov adzhalilov21@gmail.com

Natural-Mathematical Science Department, Turin Polytechnic University Kichik Halqa Yoli 17, Tashkent 100095, Uzbekistan

Dieter Mayer

dieter.mayer@tu-clausthal.de

Institut für Theoretische Physik, TU Clausthal Leibnizstrasse 10, D-38678 Clausthal-Zellerfeld, Germany

Abdurahmon Aliyev aliyev95@mail.ru

V. I. Romanovsky Institute of Mathematics, Academy of Sciences Beruniy street 369, Tashkent 100170, Uzbekistan

1. Introduction

Dynamical systems theory is mostly interested in describing the typical behavior of orbits as time goes to infinity, and to understand how this behavior is modified under small perturbations of the system. In the present work we study stochastic perturbations of circle maps with one break point, using as the main tool the thermodynamic formalism. Ya. G. Sinai constructed in [19] the first example of a thermodynamic formalism for Anosov's flows, which was generalized later in the works of D.Ruelle [18], R. Bowen [2] and others for Smale's Axiom A systems. EB. Vul, Ya. G. Sinai and K. M. Khanin finally succeeded in establishing in [20] a thermodynamic formalism approach to Feigenbaum universality in families of critical interval maps, the first example of such an approach to nonhyperbolic systems, whereas the standard approach by Feigenbaum and others has been the renormalization group well known from statistical mechanics.

A natural generalization of smooth interval maps and circle diffeomorphisms are piecewise smooth circle homeomorphisms with break points (see [14]). Contrary to diffeomorphisms, the invariant measure of these circle homeomorphisms T £ C2+e(S1 \ {xb}), e > 0, with a break point xb and an irrational rotation number is singular w.r.t. Lebesque measure [6]. The renormal-izations of such maps are exponentially approximated by fractional-linear maps [13]. Consider two homeomorphisms T1 and T2 with the same irrational rotation number p = p(T1) = p(T2), and with identical breakpoint xb = x1 = x2. The question of the regularity of the conjugation $ between T1 and T2 is called the rigidity problem. It has been intensively studied in the works of [11, 12] and others.

Theorem 1 (see [11]). Let T1, T2 £ C2+e(S1 \ {xb}), e > 0, be circle homeomorphisms with break point xb. Suppose that

1) they have the same rotation number p(T1) = p(T2) = p;

2) p is irrational and has a periodic continued fraction expansion of the form

p = [k1i k2) ■■■ i ks) k1J k2J • • •) ■ ■ ■ ]) s ^ 1

Then the conjugating homeomorphism $ between T1 and T2 belongs to the class Cl+d(S1), where 9 > 0 depends only on the rotation number p.

J. Crutchfield et al. and B. Shraiman et al. considered in [1] respectively [15] heuristically a renormalization group respectively a field theoretic path-integral approach for weak Gaussian noise perturbing one-dimensional maps with period doubling at the onset of chaos. The main result in those papers was that, after appropriately rescaling space and time, the Lyapunov exponent at the transition satisfies some scaling relations. Vul et al. developed in [16] a rigorous thermodynamic formalism approach for critical maps with period doubling. Among many other results these authors studied the effect of noise on the ergodic properties of these maps and showed that for systems with weak noise at the accumulation of period doubling there is a stationary measure, depending on the magnitude of the noise, which converges for vanishing noise to the invariant measure of the attractor.

O. Diaz-Espinosa and R. de la Llave studied in [9] stochastic perturbations of several systems using the renormalization group technique. Among others they proved a central limit theorem for critical circle maps with a golden mean rotation number and some mild conditions on the stochastic noise.

Before turning to the formulation of the main results of our work we recall the general setup and more details of the two main results of O. Diaz-Espinosa and R. de la Llave in [9].

Let (Q, F, P) be a probability space and T: S1 ^ S1 a homeomorphism of the circle S1 ^ S1. Let the stochastic sequence be defined as

xn+1 = T(xn) + cr^n+i, x0 := x G S1, (1-1)

where (£n) is a sequence of independent random variables with p > 2 finite moments satisfying the following conditions:

Ein = 0; (1.2)

const < (E\in\2)1/2 < (E\UP)1/p < Const. (1.3)

In the following we set

Xn = Tn(x0), n > 1, where Tn denotes the nth iteration of T. The linearized effective noise is defined as

n-1 n-1

Ln(x)= L + E ^n T'(Xj), x e S1. (1.4)

k=1 j=k

Let wn(x, a) be the stochastic process defined by

un(x, a) = f" ~Xn . (1.5)

a^/ var(Ljx))

For an arbitrary z0 e S1 \ [Tl(xb), i = 0, —1, —2, ...} and each s ^ 0, n ^ 1 the Lyapunov functions As(z0, n) and A(z0, n) are defined as follows:

n- 1 n- 1

As(zo,n) = 1 + ^n |T'(Zj)|s, (1.6)

k=1j=k

i i

A(zo,n) = 1smanx-^n |T'(zj)|. (1.7)

k=1j=k

Using the renormalization group approach, O. Diaz-Espinosa and R. de la Llave established in [9] a sufficient condition for the following CLT to hold for the sequence of random variables wn(x, an) defined by certain one-dimensional maps.

Theorem 2 (see [9]). Let T: M ^ M be a C2 map for M = R1, I = [—1, 1] or S1 and let l£n, n = 1, 2, ...} be a sequence of independent random variables with p > 2 finite moments. Suppose that for some x e M there is an increasing sequence of positive integers nk such that

Ap(x, nk)

lim A /9 = 0. (1.8)

k^ (A2(x, nk))p/2

Let ak be a sequence of positive numbers. Furthermore, assume either of the following two conditions:

H1) the noise satisfies conditions (1.2) and (1.3) with p > 2 and the sequence ak satisfies

sup \T\xM max ^fp(A^ nk))6ak

xes1 1<j<nk j _ n

&-vm^-=0; (L9)

H2) the noise satisfies conditions (1.2) and (1.3) with p ^ 4 and the sequence ak satisfies

sup |T//(£)||| max ^\\p(A(x, Uk))a

lim ^-^-= 0. (1.10)

k\/A2(x, Uk)

Then there exists a sequence of events Bk £F such that M1) lim P (Bk) = 1;

M2) The two processes defined by

xx

Unkfr,<Tk) = -,k k , (1.11)

a^ var(LUk (x))

"nk(x, ak) = _ k (1.12)

^/var((.Trafc -xnk)lBJ converge in distribution to a standard Gaussian as k

If, furthermore, the sequence is supported on a compact set, then we can choose Bk = 0 for all k.

For the rate of convergence to the Gaussian in this theorem these authors got the following result, where denotes the distribution of the standard Gaussian on the real line.

Theorem 3 (see [9]). Let T, £n be as in Theorem 2 and let s = min(p, 3). Assume that condition (1.8) holds at some x £ M. If ak is a sequence of positive numbers such that

then we have

sup\P(u) (x, ak)lB < z) - *{z)\ < a A;(-T' W*} (1.14)

z£R k k (As(x, Uk))s/2

where the constant A > 0 depends only on x. Our main result of the present paper is

Theorem 4. Let T £ C2+e(S1 \ {xb}), e > 0, be a circle homeomorphism with a break point xb, T/(x) ^ const > 0, x £ [xb, xb+1] and a rotation number pT = [k1, k2, ..., km, 1, 1, ...], m ^ 1. Consider a sequence of independent random variables (£n) with p > 2 finite moments satisfying conditions (1.2) and (1.3) for some x £ S1 \ {Tl(xb), i = 0, —1, —2, ...}. Then

1) there exists a constant y > 0 such that, if

lim anuY = 0, (1.15)

the process wq (x, aq ) defined by (1.11) converges in distribution to the standard Gaussian;

2) furthermore, there are constants t > 0 and k > 0 depending on p and a constant C1 > 0 such that, if an ^ C1n-T, then

sup

zeR

PKn (x, ) < z) — $(z)

< Cqn

where qn is the first return time of T, the constant C > 0 depends only on x and $(z) is the distribution function of the standard Gaussian on R.

Remark 1. Our paper is strongly influenced by the work of O. Diaz-Espinosa and R. de la Llave in [9]. Following their ideas, we prove the analog of their Theorem 2 for circle maps with a break point with small modifications. For estimating the Lyapunov function, however, we are using the thermodynamic formalism for such maps.

We restrict our discussion to the simplest case of circle maps with one break point and eventually golden mean rotation number. Since the thermodynamic formalism in our approach can be extended to rotation numbers with arbitrary eventually periodic continued fraction expansions, our constructions can be in principle generalized to such cases. However, technically these become more involved and would lead to a longer paper.

We emphasize that the limit theorems in this paper are considered in the setup of sums of random variables.

2. Preliminaries and notations

Let T be an orientation-preserving circle homeomorphism with an irrational rotation number pT. Then pT can be uniquely expanded as a continued fraction, i.e., pT = 1/(k1 + 1/(k2 + + •••)) := [fci, k2, • • •, kn, ...). Denote by ^ = [fci, k2, • • •, kn], n ^ 1, its ??th convergent.

qn

The numbers qn, n ^ 1 are called the first return times of T and satisfy the recurrence relations qn+1 = kn+1qn + qn-1, n ^ 1, where q0 = 1 and q1 = k1. Fix an arbitrary point z0 G S1. Its forward orbit O+(z0) = {zi = Ti(z0), i = 0, 1, 2, ...} defines a sequence of natural partitions of the circle. Indeed, denote by /0n) (z0) the closed interval in S1 with endpoints z0 and zqn = = Tqn (z0). In the clockwise orientation of the circle the point zq is then for n odd to the left

of z0, and for n even to its right. If I-n)(z0) = Ti(/0n)(z0)), i ^ 1, denote the iterates of the

interval l0n) (z0) under T, it is well known that the set Pn(z0) of intervals with mutually disjoint interiors, defined as

Pn(z0) = {I[in)(zc), 0 < i< qn+1} u {jn+1)(z0), 0 < j< qn},

determines a partition of the circle for any n. The partition Pn(z0) is called the nth dynamical partition of S1 determined by the point z0 and the map T.

Proceeding from partition Pn(z0) to Pn+1(z0), all the intervals ljn+1)(z0), 0 ^ j ^ qn — 1,

are preserved, whereas each of the intervals (z0), 0 ^ i ^ qn+1 — 1, is partitioned into kn+2 + 1 subintervals belonging to Pn+1 (z0), such that

kn+2-1

I(n)(z0)= Iin+2)(z0) ^ I^sqn+1 (z0).

s=0

Obviously, one has P^i(z0) -< P2(z0) -< ... < Pn(z0) -<

The intervals l(n)(z0), l(n+1) (z0) are called generators of the partition Pn(z0).

Later we will use also the so-called renormalization intervals J(n(z0) = (z0))

T(n)(^\ „• _ n 10 T(n)f^ \ - T(n)u \ I I An+1)i

— J¿^)(zi), i = 0, 1, 2, ..., where '(z0) — IQ'(z¿) U I¿n+l)(z0) and z = T*(z0). Define the Poincaré map nn: J¿n\z¿) ^ J¿n (z0) by

() f Tqn+1 x, if x G l0n)(zo),

nn(X ) — S / 1N

[Tqnx, if x G l0n+l) (z0).

The following lemma plays a key role for studying the metrical properties of the homeomor-phism T.

Lemma 1 (see [6]). Let T be a circle homeomorphism with one break point xb with jump

I Tlx —)

ratio cT(xb) = J T(xb+) ^ 1 and an irrational rotation number. Suppose T G C ([xb, xb + 1]) and var InT' = v < 00. Put v = v + 2| lncT(.T6)|. If y0 G S1 and Tl(y0) / xb, 0 ^ i < qn,

ze.[xb,xb+i]

then

qn-1

e-v < II DT(Vs) < (2.1)

s=0

holds.

Inequality (2.1) is called Denjoy's inequality.

It follows from Lemma 1 that the intervals of the dynamical partition Pn(z0) have exponentially small lengths. Indeed, one finds

Corollary 1. Suppose the circle map T satisfies the conditions of Lemma 1. Then for an arbitrary element I(n) of the dynamical partition Pn(z0) the following bounds hold:

£(I(n)) < const • dn, (2.2)

where 9 — (1 + e-v)-l/2 < 1 and l denotes Lebesque measure.

Corollary 1 implies that the trajectory of every point x G Sl is dense in Sl. This, together with monotonicity of T, implies that the homeomorphism T is topologically conjugate to the linear rotation Tp(x) — x + p mod 1.

Definition 1. Let K > 1 be a constant. We call two intervals Il nd I2 of the circle Sl K-comparable if the inequality K-l\I2\ ^ \Il\ ^ K\I2\ holds.

Lemma 2. Suppose the circle homeomorphism T satisfies the conditions of Lemma 1 and z0 G Sl. Then for an arbitrary interval I(n) of the dynamical partition Pn(z0) at least n — 1 elements of Pn(T, z0) are ev-comparable with I(n).

Proof. Let I(n G Pn(T, z0), 0 ^ j0 < qn+l. First we assume j0 — 0, i.e., j — I¿n. Applying Denjoy's inequality (2.1), we see that the intervals Tq0(I¿n)), Tqi(I¿n)), ..., Tqn(I0n)) of the partition Pn are ev-comparable with I¿n.

If 0 < j¿ < qn+l, then qÍQ < j¿ < qÍ0+v for some 0 < i¿ < n. In this case, the intervals

T-qio I(n) T-qio-i I(n) T-q0I(n) tq0I(n) Tqn-i0-iI(n) (23)

j0 ' j0 ' " ' ' j0 ' j0 ' " ' ' j0 '

are elements of the partition Pn(T, z0). Applying again Denjoy's inequality (2.1), we can see that each interval in (2.3) is ev-comparable with .

(n+1)

For intervals Ij^ , 0 ^ j0 < qn, the statement can be proved analogously. □

We recall the following definition introduced in [10].

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

It follows from the structure of the dynamical partition that an interval I = [t, t] is qn-small if and only if either t ^ t ^ Tq"— (t) or Tqn-1 (t) ^ t ^ t.

Lemma 3 (see [8]). Suppose the homeomorphism T with an irrational rotation number pT satisfies the conditions of Lemma 1 and the interval I = (x, y) C S1 is qn-small. Then for any 0 ^ k < qn Finzi's inequality holds:

v DTk(x) < DTk{y) < '

where v is the total variation of log DT on S1.

Next, we formulate the thermodynamic formalism for circle maps with a break point. Let Xbr be the set of strictly increasing pairs of functions (f (x), x G [—1, 0], g(x), x G [0, a] for some a > 0) satisfying the following conditions:

• f (0) = a, g(0) = —1;

• f (—1)= g(a);

• f (g(0)) = f (—1) < 0;

• f (2)(g(0)) ^ 0;

• f (x) G C2+e([—1, 0]), g(x) G C2+e([0, a]) for all e > 0;

• f+ (0)= g- (0).

These conditions allow us to construct a circle homeomorphism Gf g on [—1, a) from a pair (f, g) G Xbr with a break point xb = 0 (and possibly with a second break point xb = —1 if f '(—1) = g'(a)) as follows:

if (x) if x G [—1, 0), f g \ g(x) if x G [0, a).

Using the map I: [—1, a] —> S1 with l(x) = ^pj, we get a circle homeomorphism loG^gol~l on S1 = R mod 1, which we denote for simplicity also by G f g whenever its domain of definition is clear. We define the rotation number p(G f g ) of G f g by the rotation number of this circle homeomorphism when acting on S1. Denote by Xbr(w) the subset of (f, g) G Xbr with p(Gf g) =

= uj = the golden mean. Recall the jump ratio c = y^^^^y = \Zda(o+) ^f g ^

break point xb = 0.

It is clear that in the case c = 1 the homeomorphism Gf is a smooth map as long as 0 is its only break point. We assume that c = 1. Define a renormalization operator Rbr: Xbr(w) ^ Xbr(w) as follows:

Rbr(f (x), g(x)) = fx), x e [-1, 0]; g(x), x e [0, a']

where

f(x) = -a-1 f (g(-ax)), g(x) = -a-1f (-ax), a' = -a-1 f (-1)

IDG - )

Since Df( 0_) = Df(-l+)Dg(0+) and Dg( 0+) = Df( 0_), we find c( 0) = \/dg^(o~)

= c~ly/Df{-1+). On the other hand, £>/(-1+) = Df(f(-l))Dg(a_) and Dg{a'_) =

DG ~ (a'_)

= £>/(/(-!)), which leads to c(-l) = • 1-M - 1

DG~ (-1,)

\JDg(a_ )

, which in general is also differ-

ent from 1. This shows that Ghas in general two break points, one at x = xb = 0 and one

at x —

Gj^xb. For the product of these two jump ratios one finds c(0)c(—1) = 1

From the work of Khanin and Vul in [13] it is known that Rbr: Xbr (w) ^ Xbr (w) has a unique periodic orbit [fi(x, c) gi(x, ci), i = 1, 2} of period two, which means

Rbr (f1(x, c1), gl(x, c1)) = (f2(x, c2), g2(x, c2)), Rbr (f2(x, c2), g2(x, c2)) = (f1(x, c1), gl(x, c1)),

where the functions fi(x, ci) and gi(x, ci), i = 1, 2 have the following explicit form:

fi(x, Ci) gi(x, ci)

{a, + clx)!3l

Pi + (Ai + ai~ ci)x ' aA(xi- ci)

aiAc + (ci- ai- CA)x

(2.5)

(2.6)

with

a, —

c - A,2

0 . c-1 - A2 „ „ „ „-1

1 + Ao'

«2 —

1 + Ao

c1 — ci c2 — c , Al — A2 — Ao ,

A0 the unique root of the equation

A^ — A"^ — [32——————

A + 1 — 0,

belonging to the interval (0, 1).

By using the pairs of functions (fi, gi), i = 1, 2, we define circle homeomorphisms Gi: [-1, ai] ^ [-1, ai], i = 1, 2, as

Gi(x) —

fi(x, ci) if x G [-1, 0),

Ji(x, ci) if x e [0, ai), where we have used the fact that fi(0) = ai and gi(0) = -1 respectively fi(-1) = gi(ai) = -Ai. Since Dfl(0_) = and Dgi{0+) = ^¿^, one finds ^(0) = sj^ =

= \ In But this equals 1 only iff c = 1. Hence, Gi has a break point at xh = 0. On

y (1+Po)(1—Pq ) b

the other hand, one finds Df1(—1+) = /3°^_1Qh/3°'> respectively Dg1(a1) = , which leads

cG_ (-1) =

\j= ЩМ- ^^ eciua^s 1 iff с = 1. This shows that the map G1

has indeed two break points, namely, at x — Xfo

— 0 and at x — G 1 (xb) — —1. For the product 1-1 defines then a homeomorphism of the

of the jump ratios one finds cG (0)cG (6^(0)) = лfc. Using the function l^x) = the map l о Gi о Г

circle S1 with breakpoints xb = —W and x'b = 0.

i +

We rename the homeomorphism of the circle S1 corresponding to G1 by Gbr and denote by B(Gbr) the set of all circle homeomorphisms which are C1Qe conjugate to Gbr.

The sequence of dynamical partitions Pn(xb), n ^ 1, allows us to introduce symbolic dynamics for the map Gbr. For this we take an arbitrary point x G S1 \ O+ (xb) where Oq (xb)

br br

denotes the forward orbit of the break point xb of Gbr. For n ^ 0 put an+1 =: an+1 (x) = a if x G ljn+1 (xb), 0 ^ j < qn. If, however, x G I(n)(xb), 0 ^ i < qn+1, we know from the construction of the partition Pn+1 from Pn in case p(Gbr) = ш = that, either x G 0 ^ i < qn+1 or x G ii+nQ^ (xb), 0 ^ i < qn+1. In the first case we put an+1 = 0 and in the second an+1 = 1. In this way we get a one-to-one correspondence

V: S1 \ OG (xb) о {(a!, ...,a.n,...), an G {a, 0, 1}; an+1 = a ^^ an = 0, n ^ 1} := вА,

br

where вА denotes the space of allowed infinite one-sided sequences of symbols from the alphabet A = {1, 0, a} with the transition matrix

/1 1 0\

A =

0 0 1 1 1 0 у

that is,

©a := {a = (a-1, • • •, an, ...), a,G A, A0.i0.+i = 1 for i G Z+j.

The triple (0A, A, a) with a: —y the shift map (a(a))i = a,i+l is called a subshift of finite type over the alphabet A with the transition matrix A. Notice that every interval I(n) of the dynamical partition Pn corresponds to the unique finite word (a1, a2, ..., an) of length n. In particular, for n even the words (a, 0, a, 0, ..., a, 0) and (0, a, 0, a, ..., 0, a) correspond to the atoms I0n+l1 (xb) respectively I0n\xb) of Pn.

In [5] A. Dzhalilov and J. Karimov constructed the thermodynamic formalism for maps in B(Gbr) by using a closely related subshift of finite type as follows:

Denote by A* — (At j),i,j G A the transposed matrix of A and by (6At ,A, a) the subshift of finite type with alphabet A and transition matrix A*. Obviously, b=(b1,---, bn, bn+1,...) G 6At if and only if bn+1 — 0 iff bn — a. Therefore, a finite sequence (a1, •••

, an) is a subsequence of

some a G ©^ iff (a,n, an_i, • • • , o^) is a subsequnce of some b G ©At. Since Af j = (A*)^ > 0 for all i,j G A, the map a is topological mixing in both ©A and ©At.

In the following we denote by a the vector (a1, ..., an) G An: Aa,— 1, 1 ^ i ^ n — 1,

and by V — (b1, ..., bn): Ab b — 1, 1 ^ i ^ n — 1.

i' i+1

Define for x E A

{(a, 0, a, 0, ..., a, 0, ...), x = 0, 1,

(0, a, 0, a, ..., 0, a, ...), x = a.

Theorem 5 (see [5]). For any G E B(Gbr), there exists a function Ubr: 0At ^ (-&>, 0) continuous in the product topology, such that the following properties hold:

1) for any b = (&!, ..., bk, bk+1, bn, ...), c = (bv ..., bk, ck+1, ..., cn, ...) e 6At there exist constants Cl > 0 and q E (0, 1), not depending on b, c and k, such that

IUbr{b) ~ Ubr{c)\ < C, • qk;

2) let An C ASJ, 1 < r <n, 0 < Sr < ^r+i - 1, 0 < sn < qn+1 - 1 and ^A^) = = (a1, ..., ar,..., an), ^>(Ar) = (a1, ..., ar), then

lAi"}| = (1 + Vv(<h> On)) I | exp | ^ Ubr (as, a8_1} ..., ar, ..., a1} 7(«-i))|,

where (a1, ..., an)| ^ Ck-qr, with the constant Ck > 0 not depending on r, n and (a1, ..., an).

Here A(n) are elements of the dynamical partition Pn(G, Xb) := {A(n) = Gi A0n), 0 < i < < qn+1 - 1}U {A^n+1) = Gj A0n+1), 0 < j < qn - 1} with A0n) = [x,n, Xb) respectively A0n+1) = = [xb, xq ) and xi = G%xb, where xb = 0 denotes the break point of G. The function Ubr is defined as follows: for b = (b1, ..., bk, bk+l, ...) E @At and any k ^ 1 denote for i = 1,2 by ak the finite sequence ak = (bk, ..., bi). Choose I(ak) E P^_i+1 such that y>(I(ak)) = ak. Obviously, one has I (ak) C I (a2k). Define for b k = (b1, ..., bk) the function Uk = Uk ( b k) as

it (t) ._wlJ(4)l

One can show that, the thermodynamic limit, k —> oo of the function Uk exists, namely, for b E BAt

Ubr(b) = lim Uk(bk)

is a well defined function on 6At with values in (-x>, 0).

According to a well-known theorem in [14, Proposition 5.13], the following limit exists:

In \H = lim — In

p n^tt n

Y^ exp J /3 ^ Ubr {e8, tt!, 7(t i))

€=(€1,...,€n) I «=1

(2.7)

where X^ is the leading eigenvalue of the transfer operators D^: C(BAt) ^ C(6At) respectively its dual Dl: M(6At) ^ M(6At)

(Dl3f)(b) = E e^^m, (2.8)

xea-Hb)neAt

(2-9)

a: —> Ois the shift, map cr(6)j = bi+1, i ^ 1, C(B^t) denotes the space of continuous functions and M(0^t) the space of Borel measures on 6At. We used also the formal expression dy(x) = /i(x) dx for the measure d^,.

n

3. Estimates for the Lyapunov function

In this section we estimate the Lyapunov functions. Let T £ C2+e(S1 \ {xb}), e > 0, be a circle homeomorphism with a break point xb, T'(x) ^ const > 0, x £ S1 = [xb, xb + + 1) and a rotation number pT = [fc1, k2, ..., km, 1, 1, ...] for some m ^ 1. In this case, obviously, qn+1 = qn + qn-1, n ^ m. Consider the nth dynamical partition Pn(xb) = {j (xb), 0 4 j < qn+\, 4n+1)(xb), 0 4 i < qn} determined by the break point xb under the map T. We denote by I(xb; z0) the interval of the partition Pn(xb) containing the point z0.

Theorem 6. For any /3 > 0 and zo £ S1 \ {T%(xb), i = 0, —1, —2, ..., -qn + 1}, there exists n0 := n0(/3) > m such that for every n > n0 the following inequalities hold:

C1\I(n)(xb; zo)\3\n-p 4 A3(zo, qn) 4 C1\I(n)(xb; Zo)f—, (3.1)

where \_p is the leading eigenvalue of the transfer operator D_3 in (2.8) and the positive constants c1, C1 depend on the map T.

Proof. Using the elements of the partition Pn(xb), we introduce the following sum:

iqn+1-1 qn-1

Sn,P(zo) = \l0n)(xb; zo)f £ \I{n(xb)\-3 + E \l(n+1)(xb)-

\ k=o s=o

Then one has the following

Lemma 4. For any zo £ S1 \{Tl(xb), i = 0, —1, —2, ..., —qn — qn+1 + 1}, /3 > 0 and n ^ 1

c2 ' Sn,3(zo) 4 A3(zo, qn + qn+1) 4 C2 ' Sn,3(zo), (3.2)

where the positive constants c2 and C2 depend only on the total variation of log T' and /3.

Proof. For any zo £ S1 \ {T-txb, 0 4 i < qn + qn+1} and /3 > 0, we consider

qn+qn+1-1 qn+qn+1-1

A3(zo,qn + qn+1) = 1+ E II |DT(zj)|3. (3.3)

k=1 j=k

We rewrite A3(zo, qn + qn+i) in the following form:

(qn+qn+i-1

1+ £ DTk (z1)" \. (3.4) k=1

Since

DTqn+q«+i (zn)

and using the bounds

K-1 4 DT(x) 4 K1, x £ [xb,xb + 1], K1 = K1(T) > 0, respectively Denjoy's inequality (2.4), we obtain

K-1e-2v 4 DTqn+qn+i-1(z1) 4 K1e2v, (3.5)

where v = var DT(x) + 2\ln cT(xb)\. Next, we estimate the sum in (3.4). Let Pn(T, xb)

and Pn(T, zo) be the nth dynamical partitions determined by xb respectively zo under the map T. Here two cases are possible

Case zo £ ^ (xb^ 0 4 io < qn+v

Case II. zo £ I^(xb), 0 4 jo < qn.

We consider only case I, case II can be treated similarly.

I\:\xb)

zin+i z° zqn

Obviously,

qn+i+qn-1 (qn+i qn+i+qn

-3

1+ E DTk(z1) = \DT(zo)\3 E \DTk(zo)\-3 + E \DTk(zo)

k=1 \k=1 k=qn+i + 1

= \DT (zo)\3 (S« (/)+ S® (/)). (3.6)

By assumption zo £ I(n)(xb), 0 4 io < qn+1. In the case 0 4 io 4 qn+1 — 2 we split the

First we estimate the sum S^^ (/).

By assumption zo £ I sum Sn:)(/) into two parts

qn+i-io-1 qn+i

S™ (/)= E \DTk (zo)\-3 + E \DTk (zo)\-3. (3.7)

k=1 k=qn+i-io

In the case io = qn+1 — 1 we put

qn+i

S«(3) = E \DTk(zo)\-3. (3.8)

k=1

If 0 4 io 4 qn+1 — 2 we estimate DTk(zo) for 1 4 k 4 qn+1 — io — 1 and qn+1 — io 4 k 4 qn+1 separately.

If 1 4 k 4 qn+1 — io — 1 we have

n)

Iffio (xb) = J DTk (t)dt.

in (xb)

It is clear that

= f DTk(t) DTk(z0) J DTk(z0) ■

Applying Finzi's inequality (2.4), we obtain

e~v ^^ < (DT'Xzo))-1 < ev ^^ . (3.9)

These inequalities imply

qn+i-io-1 qn+i-io-1

e-3v\I(n(xb)\3 E \k+)lo(xb)\-3 4 £ \DTk(zo)\-3 4

o k+io k=1 k=1

qn+i-io-1

4 e3v\Iin)(xb)\3 £ \I^o(xb)\-3. (3.10)

o k+io k=1

Next, consider the case qn+1 — io < k 4 qn+]_. Then, obviously,

DTk (zo) = DTqn+i-io (zo)DTk-(qn+i-io)(zqn+i-i0). (3.11)

As for the estimate (3.9), we can show that

11+1 0

Since

e-v\IiVi-io)(xb)\ 4 ^(xb)\ = j DTqn+im 4 ev

I(n) (X )

k-(qn+i-io) b

one finds

e~'2v tl^{Xb)l < {DTk-^+))-i ^ e2v |J^(Xb)l . I(n) (r )l n+i I(n) (r )l

\Jk-(qn+i-io) (xb) \ \ Jk-(qn+i-io) (xb ) \

Inserting k = qn+1 — io into (3.9), we get

„ < (DTqn+1-i0(~))-i < Ir(") ( M 1 1 Ir(") ( \\

\4+ i (xb)\ \4+ i (xb )\

The last inequalities together with (3.11) imply for qn+1 — io 4 k 4 qn+1

e~3v lIZ){Xb)l < (DTk(z0))-' 4 e3v 'ff^1 . (3.12)

I(n) (x ) o I(n) (x )

\Jk-(qn+i-io) (xb) \ \1k-(qn+i-io) (xb) \

In the case io = qn+1 — 1 one shows for 1 4 k 4 qn+1 as above the bounds

aX'^-iK)! k ! 2,1411-1(^)1

M M ( ojj ^ i r(n) , • \Ik^1(xb)\ \Ik^1(xb)\

From (3.10), (3.12) and the last inequality we get the bounds for S-(^1)(/). Similarly, we can derive the following estimate:

qn-1 qn-1

e-3v3I^y £ ^(xb)\-3 4 Sn\3) 4 e3v3\%)(Xb)\3Y: \I(n+1)(xb)\-3.

s=o s=o

Using (3.5) and the estimates for S^(/) respectively the ones for Sla) (/), we get the assertion of Lemma 4. □

Next, we estimate the sum Sn/3(z0).

Let n > m > 0. Consider the mth renormalization interval Jm(xb) = [xq , xq ) =

= 4m+1) (xb) U (xb).

Define

Tn-m(xb) := Pn(T, xb) ^ Jm(xb), Tn-m (xb) := Pn(T, xb) ^ [xqm+1, xb), Tn—m(xb) := Pn(T, xb) ^ [xb, xqm).

Obviously, rn-m(xb) is a partition of Jm(xb). We will compare Sn/3(z0) with the sum of the lengths of the intervals of this partition Tn-m(xb).

(

Sn,8(z0) = |I0 ) (xb; z0)\

\

e \Ik)(xb)\-8 + E \Ij:+1)(xb)\-8

I(n)(x )<FT(m+1)

1ik (Xb)^Tn-m

I(n+1)(x )Gr(m+1)

1jk (xb)^Tn-m

+

/

+ \ I0n)(xb; Z0) \8

E \ I(k)(xb) \-8 + E \ If+1) (xb) \-8

I(n)(x )cT(m)

1ik (Xb)^Tn-m

I(n+1)(x )GT(m) 1jk (Xb)^Tn-m

+

/

qm-1

+ \ I0n) (xb; Z0) \8 E

s=1

E \ TS(iV (xb)) \-8 + E \ TS (It1 (xb)) \-8

I(n)(x )^T(m+1)

1ik (Xb)^Tn-m

T(n + 1 (x )gT(m+1) 1jk (Xb)^Tn-m

+

qm+1-1

+ \ l0n)(xb; Z0) \8 E

s=1

\

E \ T s(l(n (xb)) \-8 + E \ Ts (l(n+1) (xb)) \-8

I (n)(x )c T(m)

Tik (Xb )^Tn-m

I^^ (X )FT(m)

Tjk (xb)^Tn-m

J

(3.13)

We denote the first two sums in (3.13) by \10n) (xb; z0) \8Dn-m(xb, f) and the remaining part

by \ I0n) (xb; z0) \8Dn-m(xb, f). Since 0 < const ^ T'(x) ^ Const fOT x E [xb, xb + ^ we have

.(1)

0 < const ^ min inf DTs(x) ^ max sup DTs(x) ^ Const.

S1 1<s<qm S1

Hence, one finds for some constants c and C depending on n, m and z0

c ' \ I0 ) (xb; z0) \8 Dn-m(xb, f) ^ Sn,8 (z0) ^ C ' \ I0 ) (xb; z0) \8 Dn-m(xb, f).

(3.14)

To estimate the sum Dn-m(xb, f), we make use of the thermodynamic formalism. For this consider the first return map nm: J0m) (xb) ^ J0m) (xb) given by

nm(x) =

Tqm (x), if x E l0m+1) (xb),

Tqm+1 (x), if x E l0m) (xb)

8

We can pass from [xq , xq ) to the unit circle S1 = [0, 1) by the affine map z: [xq , xq ) —

i1 =

qm + 1' qm' L ' ' " * qm + 1' qm'

— [0, 1) given by

rp _ rp

•As t_tj q

Z — i f* ^ 3/ ^ f-j ♦

x _ x qm+1 qm

xqm xQm+1

We rewrite the maps Tqm and Tqm+1 in normalized coordinates

Tqm (x(z)) — xq Tqm (xq + z(xq — xq )) — xq

r / \ V V '' %-,-, +1 _ V %-,-, +1 V 1m %-n + l" %-,-, +1

x — X x — X '

qm qm+1 qm qm+1

Tqm+1 (x(z)) — xq Tqm+1 (xq + z(xq — xq )) — xq

/ \___V V " qm+1 _ _V qm+1 V 1m qm +1" qm +1

ilm\A'J ' x _ x x — X

qm qm+1 qm qm+1

where 0 ^ z < 1.

With am := x b we define the map

qm qm+1

(fm(z), if 0 ^ z < am,

jmy j, ^ m,

gm (z), if am < z< 1

Since Tm(0) = fm(0) = gm(1) = Tm(1) and Tm(am — ) = fm(am) = 1 resPectively Tm(am+) = = gm(am) = 0, the map Tm defines a homeomorphism of the circle S1 = [0, 1). Notice that the rotation number pT of Tm equals the rotation number pT which is the "golden mean", i.e.,

PTm = ^\1 = [ 1, 1, !,•••]•

Since DTm(am—) = Dfm(am—) = DTqm(xb—) and DTm(am+) = DTqm+1 (xb+), one finds for the jump ratio c% (am) = °f the map T at the point am e S1:

qm-1-1

CTm (am)= cT(xb) n DT(xqm+s)-1 •

s=0

On the other hand, since Dfm(0+) = DTqm (xq +) respectively Dgm(1—) = DTqm+1 (xq —) =

qm

= DTq™{xq -)DTq™-i-{xqni-), one finds for the jump ratio 0) = ^"(0+) of at t,he point z = 0 = 1:

qm-1-1

„2

^ (0)= ll DT(xqm+s)^

m

s=0

Hence, we get cT (am)cT (0) = cT(xb), which means that the map Tm: S1 — S1 in general has two break points, namely, at z = am and z = 0 = 1.

Hence, for the homeomorphism Tm £ B(Gbr) the statements of Theorem 5 hold true and therefore to Tm there corresponds the same potential Ubr.

Denote by rn-m(am) the (n — m)th dynamical partition of the circle S1 determined by the point am and the map Tm-m, n > m. We denote the elements of the partition Tn-m(am) by A(n-m+1) and Ajn-m), i.e.,

Tn-m(am) = {A(n m+1), 0 < *< Qn-m, Aj ^, 0 < j < Qn-m+1} •

The affine map 2: [xq +, xqm] ^ S1 induces a 1 — 1 correspondence 2: rn-m(xb) ^ Tn-m(am) of the intervals

A(n-m)

of the partition Tn-m(xb) and the intervals I(n m)) of the dynamical partition rn_m(am) of S1 such that

I t(n-m) I

| [xqm_1, xqm ) |

Define Dn-m (fi) as the following sum:

qn-m

Dn-m(fi) := E I AT^I + £ I AT~m)\

Then, obviously,

Qn-m— 1 Qn-m + 1 1

(n-m+l) | -ß + ^ | A(n-m) | -ß i=0 j=0

Dn-miß) — ir n I Dn_m(xb, ß). (3.15)

l[XQm-l, XQm )l

Using the same arguments as for an analogous result in [3, Theorem 2.4], it can be shown that

lim = r(ß) > 0. (3.16) -ß

Summarizing (3.14), (3.15) and (3.16), we obtain

c • r(ß)|/0n)(x6; Zo)|ßX— 4 S^(zo) 4 C ■ r(ß)lI(n\xb] zo)|ßX--m (3.17)

The last inequalities and Lemma 4 imply the statement of Theorem 6. □

Lemma 5. For arbitrary real numbers ß and ö with 1 4 ö < ß the following inequality holds:

V5 < Xß X-ß < Xs,

where Xt is the leading eigenvalue of the transfer operator Dt defined in (2.8). Proof. Using the bounds (3.1), we obtain

q AM< (A„(*, ft,))' < £ AM .. ...

* (A,«,„))" < c, I^J ■ (3'18)

We will show

(Aß(x, qjy lim ——-—~ = 0.

(A5 (x, Qn))ß

This together with (3.18) proves X—ß < X—s and hence Lemma 5. Using Lemma 4, we have

(Agfo qn))s ^ „ {Sntß{x))s

0 4 lim —L--—~ 4 C onst, lim ' =

(A5(x, qn))ß (Sn,5(x))ß

= Const lim

n^tt

t \ (qn+1-1 f \ qn-1 , , -n N

l^0n)(xb, zo)|8M £ \I(n)(xb)- + E |lf+1)(xbT13

\ s=0 k=0 ,

1 \ (qn + 1-1 i \ qn-1 / , ^

|i0n)(xb; E \i(n)(xb)-6 + E |/f+1)(xb)|-6

s=0 k=0

(

= Const lim

n^tt

qn+1-1

E

s=0

qn-1

E ^(xbT3+ E I^x)-13

k=0

/qn+1-1 , . qn-1 ,

E ^(xb)- + E |lkn+1)M-6

ls (xb)l + ^ ^k s=0 k=0

• (3-19)

For any / £ R1 we denote

qn+1-1 qn-1

Sn,3 = £ lSn) (xbT3+ E I^xr3 •

s=0

k=0

We have

lim

S,

n,3

n^tt I o o'

Sn,6 ' Sn,6

(3-6)/6

lim

n

qn+1-1 , s qn-1 , . -i\

E ^(xb)-3 + Z l4n+1)(Xb)Г3

s b | | k s=0 k=0

n+1-1

s=0

n-1

I

k=0

E | lSn)(xb) | -^ + £ | I^+1)(xb) | -

Next, for every 0 ^ s < qn+1

(3-20)

^xt3

^(xb)-6 S^

(3-6)/6

•|r( n), 3-6)/6 1 )(xb)| 6 \

S,

n,6

^(xb)-6

qn + 1 1 / \ in-1 /

E ^ xT6 + E |ikn+1) xT6

s=0

<

<

\

k=0 (3-6)/6

\^n)(xbTS

1n +1"1 , .

E ^xT6,

s=0

<

1 \ (3-6)/6 ev(3-6)

ne

—v6

n(3-6)/6

Here we have used the assertion of Lemma 2. Thus, we have

1<

n

Analogously, we can show, for 0 ^ k < qn

(,3.21)

(3.22)

6

3

6

6

6

Using the bounds (3.21) and (3.22), we have

-1 \ *

^ I T(n+1)^ w-13

evS(P-S)

i qn+1-1 i \ qn-1 f UN \ 0

' E I I(n) (xb) I+E! I[n+l) (Xb) IX

s=0 k=0

v q E-1 i(n) x ) i - sn?-s)/s+QrE-1 ! 4n+i) (Xb) i - sn-s)/s,

\ s=0 k=0 /

<

This bound and relations (3.19) and (3.20) prove

lim

(A^(x, qn))s

n^ (AS(x, Qn)Y

and hence also Lemma 5. □

Lemma 6. For any zo G S1 \ {Ti(xb), i = 0, —1, —2, ..., —qn — qn+1 + 1}, and n ^ 1

A(zo, qn+1) 4 Const • n ■ X-1, (3.23)

where the positive constant Const depends only on the total variation of log T'. Proof. One can easily see (see [9, Remark 2.1]) that

A1(zo, n + t) = | DT(zn)IA1(zo, n) + A1 (zn, t). (3.24)

Let 1 4 in 4 qn+1 realize the maximum of

A (zo, qn+1) = 14imax -Xn \T'(Zj )| = E n \T'(z3 )| = A1 (zo, in) — 1, and decompose it as

i^Kq + 1-i ■

n+1 k=l j=k k=l j=k

in = b, q, + b, q, +-----+ b, q, ,

where 1 4 bj 4 kj, 0 4 l1 <l2

< ■■■ <lm 4 n, since p = [ko, k1, ..., kn, ...] is of bounded type.

A1 (zo, in) = A1 (zo, bm qm + bm—i qm—i + ■■■ + \ %) =

= I DTln-qim (zqt ) I A1(zo,qm )+A1(zq, ,in — qlm ) =

bi

lm

E 1 DTij (Zj ,qim ) 1 Ai (zu-i).q[m , qlm ) + Ai(Zbim .q^ , V " km • ) = j=l

bi im

= EI Drj (Zj ) I Ai (zu-i).q[ ,qlm ) + j=i

+ L -""m-1 (Zbmq.m K„q,,„ +j-i)q, <«--1 )+ ' +

in-bl m qi m -jqi

. . . bl

m m m- 1 m 1

j=i

b

1

+ E I DTbl 1jl 1 (zin-{h -j) • qh) I Ai(Zin-{h -j+iH < qh). (3.25)

j=i 1 1 1

m

Applying Theorem 6, we obtain

bi

im

A1 (z0, in) < C1 DT in-jqim z )||I(lm )(xb, z(j- 1) ■qi )| +

lm lm

j=1

blm1 m-1

+ 1 £ DT^qim-jqim-1 (zb .qm +j J-m-1) (xb, % q + j) + ••• +

m m m 1 m m m 1

j=1

b 1

+ C1A-1 £ dT'1 1-j)qi 1 (zin-(bi -)qi )|I(l1)(xb; zin-(b -j+1^ )|- (3-26)

1- 1 1 1

j=1

Next, we estimate

DTin-N(zN)(xb, zN-qi )| =

j

DTin-N(z )| T(lj) (x ; z )l 1 j) (xb; zN-qi )| r

f DTln~N (t)dt \I^(xb,zN)\ J

I( ij)(xb; zN) / j (xb; zN)

for in < qn+1, N < qn+1 and 1 ^ j ^ m. Using Finzi's inequality, we get

m*»-»{zn) —xb] zN)| _ < r .

/ j y- ; * t//

( i ) J D1 zn N (z„,)

1 j (xb; zN) I(j)(xb; ZN)

DT'n~N {t)

b; ZN

Using Denjoy's inequality, we obtain

^(lj )(x; z

b N-q |

.. .--— ^ Const?

(li h 2

—j)(xb; zn)|

and

J DTin-N(t)dt < 1

i( i j )(xb; zn )

where Const 1 and Const2 only depend on the total variation of ln DT. Since p is of bounded

type, bl ^ kl ^ Const3.

j j

From these inequalities and (3.26) one obtains

m

A1(z0, in) < X-C1 • Const1 • Const2 y^ bl ^ Xlm1C1 • Const1 • Const2 • Const3 • m ^ Const • n • X-1.

j=1

□

4. Barycentric coefficients

A universal bound for Tn : Si ^ Si is a constant that does not depend on n and the point y G Si.

Let [a, b] be an interval in Si and c G [a, b]. The barycentric coefficient of c in [a, b] is the

ratio

a,b] I

||, where |/| denotes the length of the interval I.

The intervals [a, b], [b, c] and [a, c] are comparable with each other if and only if the barycentric coefficient of c in [a, b] is universally bounded in (0, 1).

We denote by I(n)(z) the interval of the nth dynamical partition Pn(T, x0), x0 E S1, containing z.

Proposition 1. Let T be a circle homeomorphism with an irrational rotation number p of bounded type satisfying the conditions of Lemma 1. Suppose x0 := xb E S1 is the unique break point of T. Choose a point z0 E S1 whose orbit OT(z0) is disjoint from the one of x0. There exists a subsequence of integers {nm, m = 1, 2, ...} such that the barycentric coefficient of the points zk := Tkz0 in I(n,m\zk), 0 4 k < qn + is universally bounded in (0, 1), in other words, there exists a universal constant 0 < K1 < 1 such that the length of each of the two connected components of I(nm)(zk) \ zk is at least K11 I(n™\zk) |.

Let Tpx = x+p mod 1 be the rotation of S1 by the irrational angle p E (0, 1). Suppose that the orbits OT (x0), OT (x0) of x0 := x0 = xb and z0 = z0 are disjoint. Consider the dynamical

partition Pn(Tp, x0) of S1 determined by x0 and Tp and let xk = TpkX0, xk = Tpkx0, 0 4 k < qn+1. Then X0 E Xkn +1)(x0) = TP° [xqn+1, X0) for some 0 4 k0 <qn or X0 E X(:) (X0) = TP° ([X0, x-qn+1 )U

U\X-gn+1, xqn)) for some 0 4 Í0 < qn+^ i.e., x0 E TP° ([X0, x-qn+1)) or X0 E TP° ([X-qn+1, xqn)).

In the following we need some lemmas describing the location of the points xk, 0 4 k < qn+1, in intervals of Pn(Tp, x0) which we formulate next.

Lemma 7 (Case 1). If x0 E Iko+11 (x0) for some 0 4 k0 < qn, then the points zk, 0 4 k < < qn+1, belong to the following intervals of the dynamical partition Pn(Tp, x0):

• xk E in^k)(x0), 0 4 k<qn - k0,

• xk E l(-qT +kn (x0), qn - k0 4 k < qn+1.

SC

Xn

J-qn+l zqn-ko

XC

ko+qn+i-1

kn -1

XC

k0-qn+1-1 zqn-1

ko +qn-1

Xz

ko+qn+i

ko-qn

hk0+qn

Xqn+qn+l-1 zqn-k -1 Xqn-1

Xqn-qn+l-1 X2qn-k0-l ^n-1

q

n

k

0

0

q

n

x x —' x

qn qn-qn+1 z2qn-k0 2qn

qn+1-qn+ko-1 xZ-qn+ko-1 zqn+1-1 xqn+1+ko-1

xqn+1-qn+ko x-qn+ko xqn+1+ko

Lemma 8 (Case 2). Let z0 £ Tp°([x0, x-q 1)) for some 0 ^ i0 < qn+1 — qn. Then the points Zk, 0 ^ k < qn+1, belong to the following intervals of the dynamical partition Pn(Tp, z0):

• zk £ Tk+i0([x0, x-qn+1)) C Z+i0(x0), 0 < k < qn+1 — i0;

• xk £ xfc-+n+1+i0 (x0), qn+1 — i0 ^ k < qn+1.

Lemma 9 (Case 2'). If z0 £ Tp0([z0, z-q +1)) for some qn+1 — qn ^ i0 < qn+17 then the points xk, 0 ^ k < qn+\, belong to the following intervals of the dynamical partition Pn(Tp, z0):

• xk £ Tk+i0 ([x0, x-qn+1)) C xii0 (x0); 0 < k < qn+1 — i0> x(n+1)

zk £ xk(-+ + 1+i0 (x0), qn+1 i0 ^ k < qn+1 + qn i0

zk £ ixk-)qn+1-qn+i0 (x0), qn+1 + qn — i0 ^ k < qn+P

qn+1+i0

qn + 1-qn

Lemma 10 (Case 3). If z0 £ Tlp0([x-q 1, xqn)) for some 0 ^ i0 < qn+17 then the points of zk, 0 ^ k < qn+1, belong to the following intervals of the dynamical partition Pn(Tp, z0):

• zk £ Tk+i0 ([x-qn + 1, xqn)) C ^ (z0); 0 < k < qn+1 — i0,

k+i0 z(n)

zk £ TP 0 ([x-qn+1, xqn)) C I^~qn+1+i0 (x0), qn+1 — i0 ^ k < qn+1.

-qn+1' In" k-1n+1+i0

Denote by z(n)(zk) the interval of Pn(Tp, z0) which contains zk.

Lemma 11. Let Tpx = x + p mod 1 be the rotation of the circle with p irrational of bounded type. Choose a point z0 £ S1 such that the orbits O^ (z0) and Ot (z0) are disjoint. Then there exists a subsequence {nm} of integers such that the barycentric coefficient of the points zk in z(nm)(zk) £ Pn (Tp, x0), 0 ^ k < qn +1 is universally bounded in (0, 1).

Proof. Denote by Pn(T p, z0) the partition generated by the points {z-q +1, z-q +1+1, •.., z0, ..., zq +q +1-1}. First, we prove that there exists a subsequence of positive integers {nm},

such that the barycentric coefficient of the point z0 in the interval A(nm) (z0) £ Pn (T p, z0) is universally bounded in (0, 1).

Suppose on the contrary that there is no such subsequence.

Fact 1. For sufficiently large n the point Xo is always at least in one of the two intervals A-n+4) and A +n+4 of P(n+4) (Tp, Xo) contained and intersecting the boundary of A(n) (Xo). But if zo does not belong to both these intervals, the point Xo splits the interval A(n (Xo) into two intervals A^(Xo) respectively A-1-1 (Xo), the length of both these intervals being larger

, , , , iT (n+4) i I ^ I A" (n+4) |

than |A(ra+4)| respectively jA^I. Hence, ~A± 4 4 1 - and there-

1 - ' H ^ ' + ' ' \A(n)(z0)\ ^ \A(n)(z0)\ ^ \^(n)(zo )|

(n+4)\

fore 0 < c < _ ,, < C < 1 where these constants, c and C, do not depend on n. This shows

\A(n)(z0)\ ^ p

that the barycentric coefficient of any yo G A(n\zo) is universally bounded in (0, 1).

Fact 2. The point Xo cannot be contained for all n in the same of the two intervals A±n+4 of the partition Pn+4(Tp, xo) contained and intersecting the boundary of A(n\zo). Otherwise the point zo must converge as n ^^ to the limit boundary point of the intervals A(n)(zo), which, however, belongs to the orbit OT (Xo) in contradiction to our assumption that the orbits of zo and xo are disjoint. Hence, there exists a subsequence nm such that zo belongs alternatively to an interval in p(n+4)(p, xo) contained in and intersecting the left respectively the right boundary of A(n)(zo). Therefore, the point Po must be separated from both the boundary points of A(a\zo) by at least one interval in P(n+6) (p, Xo) contained in and intersecting the boundary of A(n)(zo). Otherwise there would be another subsequence n, with zPn approaching the limit boundary point

of the intervals A(ni\Po) which, however, is on OT (x^), contrary to our assumption. That shows

that the barycentric coefficient of the point Xo in A(n")(Xo) is universally bounded in (0, 1). This holds true also for the interval A(n(Xo) replaced by the interval I(n) G Pn(Tp, xo) with xo = xo.

Next, we prove the barycentric coefficient of Xk in I(n) (Xk) g Pn(Tp, xo) for all 0 4 k 4 qn — 1 to be universally bounded in (0, 1).

For the point Xo the following three cases are possible:

1) lo G lln+i)(Xo), 0 4 ko 4 qn - 1;

2) Xo G [T> (Zo), T^1 (Xo)) C Iff (Xo), 0 4 ko 4 qn+i - 1;

3) Z G [Tkp0-qn+1 (Xo), Tp0+q"(Xo)) C Zno)(Xo), 0 4 ko 4 «n+i - 1.

Case 1. Io G (Xo), 0 4 ko 4 qn — 1 and Io is separated from both boundary points of ZO+^X). Then, for 0 4 k 4 qn — ko — 1, Ik is also separated from the boundary points of Ik+k^Xi). For qn — ko 4 k 4 qn+i — 1, Ik is clearly separated from the boundary points of Tk(Xo). But since Tk(XQ) C lIn^k-qn-ko(XQ), the barycentric coefficient of Ik in I>n^k-qn-ko(Xo) is universally bounded in (0, 1).

Case 2. Io G T0(XQ), Tkp0-qn+1 (Xq)) C Tj^X), 0 4 ko 4 qn+1 — 1 and Io is separated from the points Tp0(XQ) and Tk q"+1 (XQ). Then, for 0 4 k 4 qn+i — ko — 1, zk is separated from the points Tkp+ko(XQ) and Tkp+ko-qn+1 (XQ). Since [Tkp+ko(X0), Tkp+ko-qn+1 (XQ)) C T^^(XQ), the barycentric coefficient of Ik in (XQ) is universally bounded. For qn+i — k0 4 k 4 qn+i — 1, Ik

is separated from the points Tkp+k°(X0) and Tkp+ko q"+1 (X0), but [Tkp+k°(X0), Tkp+k° q"+1 (X0)) = = q (x0). Then the barycentric coefficient of Xk in îk+fc1-q (%0) is universally bounded

in (0, 1).

Case 3. X0 G [Tkp0-qn+1 (x|), Tk°+q"(x|)) C (X0), 0 < k0 < qn+1 — 1 and X0 is separated from the points Tk° qn+1 (x0) and Tko+q" (x0). For 0 ^ k ^ qn+1 — k0 — 1, 2k is separated from the points Tk+k0-q"+1 (Xo) and Tk+qn(x0). Since [TTk+ko-q"+1 (S|), Tk+ko-q"+1 (x|)) C X+k(X|), the barycentric coefficient of Xk in 2+ko ) is universally bounded. For qn+1 — k0 ^ k ^ qn+1 — 1, Xk is separated from the points Tkp+ko-qn+1 (X0) and T^+q"(x|), but [Tkp+ko-qn+1 (X0), T^+q"(X0)) =

2(n) ^^ ^ ^^^^^ ^ ^^ ^^ ^^ ^^ûn-i-VT/"* PAûffipmnt- /~vf* z m z(n+1)

ko-qn+1y

in (0, 1). □

= Z+ko-q (x0). Then the barycentric coefficient of zk in -q (x0) is universally bounded

We proved in Lemma 11 that there exists a subsequence of positive integers {nm}, such that the barycentric coefficient of the point P0 in the interval A(nm)(P0) E Pn (Tp, P0) is universally bounded in (0, 1).

This means that there exists a positive integer l := l(z0), such that the point P0 cannot be in the two intervals p—'m+1) and A +m +1) of P(n +l) (T p, P0) contained in and intersecting the boundary of p(nm)(P0). There are then three possible cases:

Case 1. P0 E pin;m+1)(Z0) := Tk°[P +1, P0] E Pnm(Tp, P0), for some ^ 0 < ^ < Qnm -1.

A (nm + l) A (nm + l)

+

Xk°+qnm+i z0 Xk0 Xk0-qnm+i XK

2—+1)(Z0)

Case 2. X0 G A^^ := Tk [X0, X-q^] G Pnm (T p, X0) for some k0, 0 < k0 < qnm+i — 1.

Obviously, A-o(nm+1) C Ak1™).

X (nm +l) X (nm + l)

+

Xfc0+qnm+i Xk0 z0 Xk0-qnm+i Xk0+qn„

A -(nm+1) (*o)

i-►

Case 3. X0 G ^J^1^™ := rf0 [X-qn +, Xqnm ] G (T p, X0) for some ^ 0 < k0 <

< qnm+1 — 1. Obviously, X-(nm+1),nm C Xknom).

A ("m + l) A ("m + l)

'-K-Q

A0

(-(nm + 1),nm) I

A k v m I Kq

One also can say that there exists at least one interval of the partition P(n +l)(Tp, x0) between the points Ak and A—, 0 ^ j ^ qn +1 — 1.

Lemma 12 (see [7]). Let T be a circle homeomorphism with an irrational rotation number of bounded type satisfying the conditions of Lemma 1. Put 9± := (1+e±v)_1/2 and let n and l ^ 2 be positive integers. Then there exist universal positive constants C1, C2, C1 < C2 such that for arbitrary I(n) e Pn(T, x0) and I(n+l) e Pn+l(T, x0) with I(n+l) c I(n) the following bounds hold:

I (n+l) l

°ie+ ^ 7— ^ a2e~-

Proof of Proposition 1. Consider the dynamical partition Pn(T, x0) and the finite part OJ+1 (z0) = {zk, 0 ^ k < qn+1] of the orbit of z0 under the map T. I(n)(zk) denotes again the interval in Pn(T, x0) which contains zk. Since T and Tp are topologically conjugate, Lemma 11 shows that there exists for l ^ 4 a subsequence {nm} in N such that the point zk cannot be in the two boundary intervals of the partition Pn +l(T, x0) contained in I(n™)(zk). Hence, according to Lemma 12, the barycentric coefficient of zk in I(n™)(zk) is universally bounded in (0, 1). □

xkn + q

Ako+qn

k

0 1 +1

o

5. Proof of Theorem 4

Take any point z0 not belonging to the orbit OT(xb) of the break point xb. Before giving the main steps in the proof of our main theorem, we emphasize an important point: a key ingredient in the proof of this theorem is the Taylor expansion of the process lq +1(zo> a)

Zqn+1 (z0, = Tqn+1 (20) + aLqn+i (20) + ^Qq^ ^0, , (5-1)

where the linear term Lq + is the sum of independent random variables as defined in (1.4). However, we can use Taylor's formula (5.1) only in case the neighborhood U(n) of the point zq = = Tqn+1 (z0) does not contain any break point of Tqn+1.

Let us briefly sketch the main steps of the proof of Theorem 4, which will overcome this difficulty.

I. By Lemma 4.1, there exists an increasing sequence of natural numbers {nm, m = 1, 2, ...} such that for 0 ^ k < qn +1 the interval I(nm)(zk) of the partition P(nm) (T, xb) contains only the point zk := Tk(z0) from the finite orbit {zi = T%z0, 0 ^ i < qn +1}. Moreover, the barycentric coordinates of the points zk are strictly separated from 0 and 1 by universal constants, i. e., the points zk are uniformly separated from the boundaries dI(n,m\zk).

For every m ^ 1 we will construct a series of neighborhoods A— c I(nm)(zk), 0 ^ k < < qn +1, of the points zk, such that every interval A— does not contain any break point of Tqm

II. For the stochastic sequence

zk = T(zk_i) + a £k, z0 = z0

nm + 1

we show that the probabilities of the events

Bn :— \z i g Aim, zo g A,y7n, ..., z„ _i g A™ _i}

nm L 1 1 ' 2 2 ' ' qnm+1 1 +1-iJ

tend to 1 as m

III. For fixed zQ g S1 \ the Taylor expansion of the process zq +1(zo> aq +1-i) the variables (2, • • • ■> -i allows us to decompose it under the condition ~z- G Ajm,

1 4 j 4 «nm+i — 1, as

= Tqnm+1-1(z0) + a +1-1L (z0)+ a* Q Jz0, a. +-), (5.2)

where the linear term LQ is the sum of independent random variables defined by (1.4).

VI. We prove the CLT for this linear part LQ — 1 which finally leads to the proof of

nm + 1

Theorem 4.

To achieve this program, we formulate and prove in a first step several lemmas. For this take a point z0 G S1 \ OT(xb) and the sequence of increasing natural numbers nm, m = 1, 2, ... determined by Proposition 1. Consider the two partitions Pn (T, xb) (generated by the points [x-q +1, ..., x0, ..., xQ +q }) respectively Pn +l(T, xb). Each interval A(nm)(zk) g Pnm(T, xb), 0 4 k < qnm + 1 contains at least ql ^ q4 and hence at least three intervals of the partition Pn +l(T, xb). By Proposition 1, the point zk cannot be located in the intervals A±±n+l of the partition pn +l (T, xt) contained in and intersecting the interval A^nm)(zk).

Next, we introduce certain connected intervals Ajnm) composed of intervals of the partition p2n +i+1(xb) and containing the point zk. For this let A(2nm +l+1)(z0) be the element of the partition p2n +i+1(xb) which contains z0. As above, there are again three possible cases for this interval: A(2nm +l+1)(z0) = A2tnm+l+2 = [xt()+q2n +i+2, xto], A(2nm +l+1)(z0) = = A-(2nm +l+2) = [xt ,xt_q ] respectively A(2nm+l+1)(z™) = A-(2nm+l+2)'2nm +l+1 =

0 0 0 q2nm +1+2 0

= [xt0-q2nm ++2, xt0 +qnm + l+1 ]. TherebУ, to = to(ko, l) is chosen such that A(2nm +l+1)(z0) C

C Ankr^+1 (z0) \ (A—m+l U A+m+l). Then we define

. A0nm) := A(2nm+l+1)(zo) G p2nm +l+1(xb), • for every k, 1 4 k < qn +1, we set

1(nm)--A-l„ \\ \T i A(nm)

A^] :- A-(nm) U T(A^) u A+ (nm)

where Ak (nm) and A+(nm) are the left and right neighbors of T(A^!^) in the partition p2n +i+1(xb), respectively.

Lemma 13. A) c A(n™\zk), for all 0 < k< qnm+1.

Proof. Let the sequence {nm, m = 1, 2, ...} be determined by Lemma 4.1. Consider the partitions An (T, xb), An +l(T, xb) and A2n +l+1(T, xb). Recall that every interval of An (T, xb) contains at least three intervals of An +l (T, xb).

Furthermore, every interval of An +l (T, xb) contains at least qn +1 intervals of P2n +l+1(T, xb). This implies that

Aknm) c A(nm)(zk) e Pnm(T, xb), 0 < k< qnm+1.

□

The intervals A(nm) in red in case I.

- - e - -

.+1 -ko

xto

h -s-

H—h

X - e - -

kq

0 Hnm + 1

H - e - h

X —

H—I—h

xqnm +qnm + 1-1 z qn -ko-1 xqnm— 1

m 0

cq q 1 z2q —k— 1 x2qn —1

+1 —1 2qnm k0 1 nm

X —

-M—l-

x2q„

mm

z2q^ —ko

X -

-H—h

qnm + 1—qnm +k0 — 1

x — qnm +ko —1 zqnm + 1 —1 xqnm + 1+k0 —1

The intervals Ak""m) in red in case II.

-I — e - X

qnm + 1 V +1—k0 x0

qnm + 1+qnm —1

-H—H

-H-+

xko + qnm + i —1 zq„m+i—1 xk

ko+q^ —1

H-h

-e-

I—X

ko+q„

z0 x

kq

0 ^ n

k+q„

H - e - X

qn +qn +1—1

' m ' m + 1

Cqnm 1 ^ —k0 —1 xqnm —qnm + 1 — 1

2qn —1

m

x

x

x

q

-q

q

0

n„ +1

x

x

fc0 + qnm + 1

k

z

0

o

q

x

x

q

x

x

1

qnm + 1

x

1

x

x

x

qnm + 1

q

x

k0—qnm + 1 —1

x

o

x

k

o

H - : - X

— kr, xa

b2qn

H - : - X

Cqn +1 -1 za —k -1 x 1

nm + 1 qnm+1 k0 1 - 1

qn +1+9n -1

nm + 1 ' m

The intervals A?m in red in case III.

X^ : - h

qnm + 1

x0 zqn+1-ko

-qnm+1

-X

ko +qnn+1-1

X-1 z,

qn +1-1 xk0-q -1

ko+q^-1

x—I

í0 + q2nm + ¡ + 1

—I-h

"ko+qn

ko+q^ +1

ko-q™ +1

x - : - h

q1

qnm+1 1

x 1 zq -k -1 xqn +1+qn -1

— 1 qn +1 ko 1 nm + 1 nm

Lemma 14. Let T be a circle homeomorphism, be a sequence of independent random

variables satisfying the conditions of Theorem 4 and let 9+ = (1 + ev)_1/2 < 1.

Suppose that the sequence 1, n ^ 11 fulfills the condition

qn+i°qn+1

lim ,, . ,

Q++4

Then the probability of the event

' qnm+l 1 qnm+l-11

(5.3)

tends to 1 as m tends to infinity, where

zk = T(zk_i) + a 4, 1 < k < qn +1 - 1, z0 = z0.

nm + 1 '

Proof. It is clear that

p(^J = p n fee AM

k=i

qnm +1

-i

ki

= P(21gA^) n p {zk g A™"1} i P| {zi g A'¡m} . (5.4)

k=2

i=i

Next, we estimate the factors of the last product.

x

z

q

q

x

x

q

x

k

0

o

m

Applying Chebyshev's inequality, we obtain

G A"m) = P(T(z0) + ^ fi e ^r U TA'^ U A+) ^

nm + 1

< +1Var b

nm ' 1_

1nm+1^11 ^ —ll"l 1» I"1 \}j ^ - min{|^-|2) |A+|2}'

Since Ak and A+ are elements of P2n +7, we have min{ I A-1 2, I A+12 } ^ . Using the

last bound, we get

Co%

PfreA^l--^. (5.5)

For the other k, 2 4 k < qn +1, analogyously we have

P^^-gam j pi {zi G A'im} j

= P ({T(7fc) + %m+i4 G A" U TA^ U A+} / {7fc_! G A^j) >

al Var £k > P(\a„ 4 min{|Afc |A+|}) >1- —

min{ I A-12, I A+ I 2 }' and therefore, as before,

/ /k-1 N Ca2

P G AM j Oil, G A"m } J > 1 - (5.6)

Q2+Um

Summarizing (5.4), (5.5) and (5.6), we get

P(AO = P n ^ G A^\zk)} \> 1 - • (5-7)

jKJ

(^nm) = * I | | lzk ^ A \ k=1

By assumption,

qnm+1-1 \ f Cat

2 nm +

k=1 I \ 0+

lim = 0.

This, together with (5.6), implies that

P(Bnm ) ^ 1

as m ^ to. □

We fix a point zQ G S1 \ Assuming Ij G A"m, 1 ^ j 4 Qn +i — 1, we can use the

Taylor expansion for the random process ~zq +1(zo> aq 1-i)

^m +1-1(zo , aqnm + 1-1) = Tqnm + 1- (zo) + aLqnm + 1-1(zo) + alm + 1-1Qqnm + 1-1(zo, ^r^^,

(5.8)

where the linear term L is the sum of independent random variables defined by (1.4).

nm + '1

To the linear process

yn(zo, a) = Tn(zo) + aLn(zo) (5.9)

we can apply the straightforward extension of the central limit theorem as proved in [9, Lemma 3.1].

qnm + 1

Lemma 15 (see [9]). Let T e C2(S1 \ {xb}) be a circle homeomorphism with a break point xb and let {{n} be a sequence of independent random variables with p > 2 moments satisfying conditions (1.2) and (1.3). Assume condition (1.8) holds for some point z0 e S1 \ {OT(Xb)} and some increasing sequence {nk} of positive integers, then

^var Lnk(z0)

converges in distribution to the standard Gaussian as k ^ m. Moreover, there is a universal constant C such that

supip(in (z0) <t)~m\ ;?i/2. (5.io)

Condition (1.8) holds indeed in our case, namely, using (3.1), Lemma 5 and p > 2, we get

Ap(zo, qn) |4 )(xb; zo)|pX—p hm —-j-—t- = lim ——--— = 0. (5-11)

(A2(zo, qn))p/2 (\I(n)(xb; z0)|2A—2)p/2

Next, we treat the nonlinear part of the process (5.8). For this consider the process

+i-l ~ T9nm + 1~1 (ZQ)

W9„„+l-l -

vm+1-Vvar(V+1-i(Zo))

Lq +1-1(Z0)

Lemma 15 implies that. , converges weekly (in distribution) to the standard

'var(Lq +1 (z0))

var(L ,,_!(*())) 9,1,71+1 1 i/var(L„ ,,-1(^0))

nm +1

Gaussian.

Qqn +1-1(z0 'aqn +1-1)

Next., we will show that, the random process an —,. .. converges to 0

H q"m +1"1 /var(L ,(^0)) &

V nm +1

in probability. For this we introduce the following constants:

Ki = sup t^--:, K2 = exp (ii, sup |T>l(x)\) , K = Kl • K2 sup \T>l(x)\. xes1 \T (x)| V xes1 J xes1

Lemma 16. Suppose a circle map T satisfies the conditions of Theorem 4 and the sequence an satisfies relation (1.15). Let Dn be the event

Dnm = \K- %m+i max (Â(.0, 9nm+1)) < - . (5.12)

Under the condition of the event Bn in (5.3) the following inequality holds :

P

a a Qa (z0, aq )

qnm + 1^ anm +1K U' anm + 1J .

LDn

> £ 4

4

2Kaa

2/p\

p/2

A (xo,Qn+i)) ^ (^max 11 & Ip)

t<\/A2(-0> Qnm+1)

V

Proof. In a first step we estimate I a2 Qq (zo, aq

nm + 1 nm + 1 nm + 1

Using (1.4) we obtain the recurrence relation

Lk+1(zo) = Lk (zo )T'(zk) + £k+1. Let, ~zk G A'km, 0 4 k < qn +1, then we have

(5.13)

zfc+l(z0) %m + i) - T(zk) + ^

= T(zk + aqn +1 Lk (zo)+ aln +1 Qk (zo))+ aqn +1 £k+1 =

nm + 1 nm +1 nm + 1

= zk+1 + T '(Zk )(aqn +-, Lk(zo) + ag +-, Qk (zo)) + aq +-, £k+1, (5.14)

nm+1 nm+1 nm+1

where |zk - zk\ 4 \a Lk(z0) + a2 Qk(z0)\ = \zk - zk\ 4 K!m|.

nm+1 nm+1

On the other hand,

zk+1(zo, aqn +-, ) = zk+1 + aqn +-, Lk+1(zo) + aqn +i Qk+1(zo).

nm+1 nm+1 nm+1

The last relation, together with (5.14) and (5.13), implies that

Qk+i(zo) - aanm+1 Lk(z0)(T'(zk) - T'(zk)) + a^+1 Qk(z0)T'(zk).

Iterating the last recurrence relation, we obtain

k

ain +1 Qk+1(z0)

nm + 1

aa

i=1

(5.15)

Next, we estimate the right-hand side of (5.15). For 1 4 i 4 k one finds

I T'(Zi) — T'(zi) I 4 sup I T+ (z) 11 Z — zt I 4 sup I T+ (z) 11 aq +1 L^o) + a] Qi(zo) I . (5.16)

n +1 nm+1

^es1

zes1

But

Therefore,

IT '(Z ) |

4 Ki

(5.17)

m

3

m

m

1

IlT'(zs) = fjT'fe) fl ^ = l\T>(zs) fl f 1 + n%±.. nZ'Y\

T'(zs) fj Vs'f*V T'(zs)

s=l s=i s=l

k k k

k k k k ^11 T' (za ^ ( 1 + K1 sup T+ (z)\zi - zi\) < exp ^ Kx sup T+ (z)z - zi | [] T'(zs) <

s=i s=i V zes1 J \s=i zes1 J s=i

k k K sup) (z)|J [] T'(zs) = K2W T'(zs). (5.18)

zes1 ' s=i s=i

Using (5.15) and (5.16)-(5.18), we have for 0 ^ k < qn +1

+1 Qk+1(z0) <

kk

< K1K2 sup |T+ (z)W V |Li(zo)||a,n +1 Li(zo) + < +1 Q^I]T'(zs) <

,-q1 nm + 1 f * nm + 1 nm + 1

zes i=1 s=i

k k k k

^ K°ln Mil T'(zs) + K^ V |a2n +1 Qi(«,)|Li(«) )| II T'(zs) <

i=1 s=i i=1 s=i

2kk

^ mm(zo, k)) EnT'(zs)+

^ ' i=1 s=i

kk

+ Kaq +1 max |aq2 Qi(z,)| max |^i|A(zo,k) Z II T'(zs). (5.19)

nm + 1 nm + 1 -

i=1 s=i

It is then clear that

1<max+1 Qi(zo)| < KaL+1 maxk ^|2 (A (zo,k))3+ +maxk kl+1 Qi(zo ^r^ maxk ^ (A (zo,k))2 ^ K<m+1 maxk |ei|2 (A (zo,k))3+

+ max |ct? Qi(zo)Kaq max (A(zo, k)j . (5.20) Hence, we have the following bound:

3

max , |a2 +1 Qi(zo)| < Kal +1 max K^2 (A(zo, k)) +

Xi<k + 1 nm + 1 nm + 1 1<i<k \ /

1<i<k + 1 qnm + 1 i qnm + 1

+1mx+1 |<tm+l Qi(zo ^r^ maxk ^ (A (zo,k))2

Now we can prove Lemma 16.

Using (5.12) and (5.21), we get

K +1 Qqn +1 (zo)1Dn | < max +1 Qi(zo)lDn | <

^nm +1 *nm +1 nm \<i<q _ ^nm + 1 nm

1<^qnm,+1

^ 2K*l +1 ma* Ki|2 (A(zo, qnm+1^3 . (5.22)

nm + 1

Consequently,

(e I a* +i Qq +i (zo )1d„ I p/2)2/P 4 2Ka2q + (A (z0, gnm+i))3 (f( m ax | & | p ))

V Hnm + 1 nm + 1 nm / Hnm + 1 \ m / \ I4i4a +1 /

nm +1

2/p

z0, qnm+i)

Letting e > 0 and using Chebeshev's inequality, we obtain

P

a

anm+1 Qanm+1(zo ' aanm+1).

var La

(zo )

Dn

> £ 4

4

2Kaa fA(zo, 9nm+i)J F(^max | &| p)

2/p\

p/2

£, /varL

anm+^ 0

(zo )

Conditions (1.2) and (1.3) imply

constA2(zo, qnm+i) 4 var Lqn+1 (zo) 4 Const A2(zo, qnm+i). This and the previous bounds imply the assertion of Lemma 16.

□

Lemma 17. Suppose a circle map T satisfies the conditions of Theorem 4 and the sequence an satisfies the relation

lim ap qn+1 X2npn2p — 0.

n^œ an+1H n+1 -1

Then the probabilities of the events Dn defined in (5.12) tend to 1 as m ^ to. Proof. Consider the probability

(5.23)

P(Q \ DnJ = P C%m+i max (Ä(z0, g^))' > \

I ^ ^ nm + l

= P ^ max ^ --- 2

14i4anm+1 2Caa +1 (A(zo, Qnm+i)]

4

< ^^ I ti IP )\2°aan-+1 (A (zo ^+1)) J . (5.24)

A p

Condition (1.3) shows that

const 4 E( max I £iI p) 4 qn+iConst,

i4i4qn+1

and Lemma 6 implies A(zo, qn +1) 4 Const ■ nm ■ X"_mr{.

Using these bounds, we get the following estimate for P(Q \ Dn ):

P(Q \ Dn) 4 Consti • q,nm+i • n2m • X-?m • ap

anm. + 1^

which implies Lemma 17.

□

m

m

3

Lemma 18. Assume a circle map T satisfies the conditions of Theorem 4 and the sequence an satisfies the relation

lim a ■ n3 ■ a2/p ■ \5n/2 ■ O^n = 0

^ n qn+i A-i O+

Then

3' '2/p

p)

aqnm+i (A(xo, qnm+i)) E(i<max №)

Äo--^ ---- =

m °° V A2(z0, qnm + l)

Proof. In the proof of Lemma 17 we showed

const < E( max |{.|p) < qn+1Const

1<i<qn+i

respectively

A(zo, qnm+i) < C 'nm- A-m■ Since A_1 < A_ _2, Theorem 6 shows

CiI—+l\xb; Zo)|2A-m < A2(zo, qnm+i) < ClO^^^b; Zo)|2—. On the other hand, Lemma 12 implies

^+i)(xb; zo)| > CiO+m+i.

Using these inequalities, we find

aqn +1 (A (x0 >qnm + i^ (E (i<.r^ax |Ci|P n _ n3 A3nm 2/p

nm+1 K m J \ i<.<qnm + i I a +_ • nm • A-i qn+i

lim --v ^-J--< Const m+ r -<

a ,n3 . \3nm q2/p

^n ±qnm + l 'm A-i qn+i n + 3 \5nm/2 2/p a-2nm-2

^ Const,———-J--= Constaa ■ nL ■ X_{n/ qJ m

O2nm+2xnm/2 qnm +1 m i m + i +

□

For the proof of Theorem 4 we make use of

lim = inp-i,

n^m n 1

where pT = [k1, k2, ■ ■ ■, km, 1, 1, ...], m ^ 1. Thus, for arbitrary 0 < t < ^ there exists JVeN such that for all n > N

-n/2 „ -n(1-e) „ „ —n(1+s) —3n/2

pT < pt < qn < pt < pt .

Define

i 2 5 ln X_ 1 + 4 ln 0~1 + 6 1 2ln0,l

7 := max <-------, —h --— >.

I p ln pT 2 ln Pt I

In order to prove the first part of Theorem 4, it is enough to verify the conditions of Lemmas 14, 17 and 18.

1. Condition of Lemma 14. For sufficiently large n

2 ln e-1

^ra+l qn+1 _ 2 /i-2ra-4 _ 2 n lnPr ^

g,2ra+4 — (in+laqn+1a+ — Qn+l'7 qn+1PT ^

1

n — 2

2 In pT (n+l)(l-£) 1 / 7

< 1 qn+i T ^n+J < [qn+i%n+1

2. Condition of Lemma 17.

yP

apn+iqn+iX-nPn2p = (aqn+1 • «O+I • A2-- • n2)P < • q^i • X-/2 • n3 • 0-+2n)P .

3. Condition of Lemma 18. For sufficiently large n

ln e ,1

2 n- +

2/p x5n/2 3 /1—2'« 2/p 1 npT 2 lnpT lnpT

°qn+1 • qO+i • X-i •n • = °qn+1 • qO+i • pt t • pt t • pt t <

-1__5n___3 ln n___2n_ ®

2/p In PT 2(n + l)(l-s) lnpj.ln+Dd-E) (n+l)(l-£) In pT

< aqn+1 • qn+i • qn+i • qn+i • qn+i <

2 In- ln A-1 . 5n. _ in . ln 9+1 _ 6 In n In prj, (il + l) (il + l) In prj, ( 77 + 1) In

• qn+i <°qn+1 • qn+i.

tn + 1

Thus,

aqn+1 ■ n3 ■ X-/2q2n/+10-+2n-2 < aqn+1 ■ q^. (5.25)

For the second part of Theorem 4 it is enough to show that the rate of convergence to 0

Qq +1-1(z0'aq +1-1)

of ua _i—, ,T ""' „ - is bounded by the right-hand side in the estimate in (5.10),

nm+1

which means

3 \5u/2 2/j) n-2n-2 ^ Amin(p,3)(x, qn) <Tqn+1-n ■ A_! qn+1V+ < ^^ ^))min(pi3)/2.

To prove this inequality, we use (5.25) and Theorem 6 to show

^ (min(p, 3)\ O Amin(p, 3)(x, qn)

const, • -. , 4

xmm(p, 3) (A2 (X, qn))min(p' 3)/2'

\_2 j Hn

It is therefore enough to get

/ x2 \n/2

A

^.Ciieoost-l^gf , (5.26)

-2

where aq 4 C\ ■ qn+v Let, us choose s = min(p, 3) and r ^ 7 + 21nA3jn*lnA 2. Then

Pt

^ 2 In X_s —5 In À_2 2 1nA_s-slnA_3 3(11+1)

„ _r „ ^ 31n pT 3 1npT 2

aqn+1 4 Ci • q-+1 4 Ci • qn+i T 4 Ci • qn+i • pt t ^

from which (5.26) follows. □

2

1 ln e

2

To finish the proof, we can apply Lemma 3.2 in [9] by noting that there the quantity \\f "\\c0 has to be replaced in our case by sup \T"(z)\ which exists for T G C2+e(S1 \ {xb}). Then,

ze[xb,xb+i]

following the arguments of the proof of the CLT in Section 3.3 of [9] leads finally to the convergence of the process wn (z0, ank ) to the standard Gaussian distribution.

Acknowledgments

The authors thank a referee for several very helpful remarks.

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