CC BY
5
DOI: 10.17516/1997-1397-2021-14-3-287-300 УДК 517.9+519.1
A Note on the Conjugacy Between Two Critical Circle Maps
Utkir A. Safarov*
Turin Politechnic University in Tashkent Tashkent, Uzbekistan Tashkent State University of Economics Tashkent, Uzbekistan
Received 10.11.2020, received in revised form 16.12.2020, accepted 04.02.2021 Abstract. We study a conjugacy between two critical circle homeomorphisms with irrational rotation number. Let fi,i = 1, 2 be a C3 circle homeomorphisms with critical point хСГ of the order 2mi + 1. We prove that if 2mi + 1 = 2m2 + 1, then conjugating between f1 and f2 is a singular function. Keywords: circle homeomorphism, critical point, conjugating map, rotation number, singular function.
Citation: U.A. Safarov, A Note on the Conjugacy Between Two Critical Circle Maps, J. Sib. Fed. Univ. Math. Phys., 2021, 14(3), 287-300. DOI: 10.17516/1997-1397-2021-14-3-287-300.
1. Introduction and preliminaries
Denjoy's classical theorem [4] states, that if the C2 circle diffeomorphism f and irrational rotation number p = pf then f is topologically conjugate to the linear rotation fp, that is, there exists a circle homeomorphism p with f = p-1 o fp o p.
It is well known that a circle homeomorphisms f with irrational rotation number is strictly ergodic, i.e. it has a unique f-invariant probability measure Vf. A remarkable fact is that the conjugacy p can be defined by p(x) = Vf ([0, x]), which shows, that the regularity properties of conjugacy p and the absolute continuity of invariant measure Vf are closely related. The problem of smoothness of the conjugacy p for diffeomorphisms is one of the important problems of circle dynamics. The fundamental results were obtained by V. I. Arnold [1] , J. Moser [15], M. Herman [9], J.Yoccoz [17], Ya.G.Sinai and K.Khanin [12], Y.Katsnelson and D.Ornstein [13]. Notice that for sufficiently smooth circle deffeomorphisms f with a typical irrational rotation number the conjugacy p is C^diffeomorphism. Consequently, the invariant measure Vf is absolutely continuous with respect to Lebesgue measure n on S1.
Since the works of Mostow, Margulis, Sullivan, and others, rigidity problems occupy a central place in the theory of holomorphic dynamical systems. This type of problems is classical in dynamics: a rigidity theorem postulates that in a certain class of dynamical systems equivalence (combinatorial, continuous, smooth, etc.) automatically has a higher regularity. The dynamical systems considered in this paper are critical circle maps, that is smooth homeomorphisms of the circle with a single critical point having an odd type. These maps have been a subject of intensive study since the early 1980's as one of the two main examples of universality in transition to chaos. Yoccoz in [17] generalized Denjoy's classical result, a critical circle homeomorphism with irrational rotation number is topologically conjugate to an irrational rotation.
*safarovua@mail.ru © Siberian Federal University. All rights reserved
Definition 1.1. The point xcr £ S1 is called non-flat critical point of a homeomorphism f with order (2m + 1), m £ N, if for a some S-neighborhood Ug (xcr), the function f belongs to the class of c2m+1(ug(xcr)) and
f '(xcr ) = f ''(Xcr ) = ■■■ = f (2m)(xcr ) = 0, f (2m+1)(xcr ) = 0.
The order of the critical point xcr is 2m + 1. By a critical circle map we define an orientation preserving circle homeomorphism with exactly one non-flat critical point of odd type.
An important one-parameter family of examples of critical circle maps are the Arnold's maps defined by
fe(x) := x + 9 +--sin2nx (modi), x £ S1.
2n
For every 9 £ R1 the map fe is a critical map with critical point 0 of cubic type.
Graczyk and Swiatek in [7] proved that if f is C3 smooth circle homeomorphism with finitely many critical points of polynomial type and an irrational rotation number of bounded type, then the conjugating map < is singular function on S1 i.e. <p'(x) = 0 a.e. on S1. Consequently, the invariant measure of critical circle homeomorphisms is singular w.r.t. Lebesque measure on S1. Hence the problem of regularity of the conjugacy between two critical maps with identical irrational rotation number arises naturally. This is called the rigidity problem for critical circle homeomorphisms. For the critical circle maps the rigidity problem is developed by de Faria, de Melo, Yampolsky, Khanin and Teplinsky, Guarino among others.
The first result concerning on rigidity for critical maps was proven by de Melo and de Faria [6].
Theorem 1.1 (see [6]). If f1: f2 are C3 critical circle mappings with the same irrational rotation number of bounded type and the same power-law at the critical point, then there exists a C1+a conjugacy h between f1 and f2 for some universal a > 0.
The following result of D. Khmelev and M. Yampolski [14] seemed to indicate that the analytic case could be different.
Theorem 1.2 ([14]). There exists a universal constant a > 0 such that the following holds. Let f1 and f2 be two analytic critical circle maps with the same irrational rotation number. Denote h : S1 ^ S1 conjugacies between f1 and f2 fixing the critical points. Then h is C1+a at the critical point.
K. Khanin and A. Teplinskii [11] proved that any two f1 and f2 analytic critical circle maps with the same order of critical points and the same irrational rotation number are C1 -smoothly conjugate to each other. Later, A. Avila [2] showed, that there exist f1 and f2 analytic homeomorphisms with the same irrational rotation number such that h is not C1+a for any a > 0. Next we formulate the result of P. Guarino, M. Martens, and W. de Melo [8].
Theorem 1.3 ([8]). Let f1 and f2 be two analytic C4-circle homeomorphisms with the same irrational rotation number and with a unique critical point of the same odd type. Then they are C1 -smoothly conjugate to each other. The conjugacy is C1+a for Lebesgue almost every rotation number.
The present work continuous and completes the above results. Namely we show that if the rotation numbers of two critical homeomorphisms coincide but the orders of critical points are different then the conjugacy h is a singular function. Now we formulate our main result.
Theorem 1.4. Let f1 and f2 be C3 critical circle maps with the same irrational rotation number. Suppose that the orders of critical points of f1 and f2 are different i.e. 2m1 +1 = 2m2 + 1. Then the conjugacy h between f1 and f2 is a singular function on S1.
2. Notations, terminalogy, background
Let f be a circle homeomorphism that preserves orientation, i.e. f (x) = F(x)(mod 1), x G S1 ~ [0,1), where F is continuous, strictly increasing on R1 and F(x+1) = F(x)+1 for any x G R.
F is called lift of homeomorphism f. The important characteristic of the circle homeomorphism
Fn (x)
f is it's rotation number (see for instance [6]) pf which defined by pf = lim -(mod 1),
n^^ n
here and later Fn denotes the n-th iteration of F. The rotation number pf is rational if and only if f has periodic orbits.
2.1. Dynamical partition. Let f be an orientation preserving homeomorphism of the circle with lift F and irrational rotation number p = pf. We denote by {an,n G N} the sequence of entries in the continued fraction expansion of p, i.e. p = [a1,a2,... ,an,... ]. Denote by pn/qn = [a1,a2,... ,an] the convergents of p. Their denominators qn satisfy the recurrence relation, that is qn+1 = ari+1qn + qn-1, n > 1, qo = 1, qi = ai.
For an arbitrary point x0 G S1 we define A0n)(x0) the closed interval on S1 with endpoints x0 and xqn = fqn (x0). Note that for odd n the point xqn lies to the left of x0 and for even n to the right. Denote by A(n)(x0) the iterates of the interval A0n)(x0) under f :A(n)(x0) := fi(A0n)(x0)), i > 1.
Lemma 2.1 (see [12]). Consider an arbitrary point x0 G S1. A finite piece {xi, 0 ^ i < qn + qn-1} of the trajectory of this point divides the circle into the following disjoint (except for the endpoints) intervals: A(n ^(xci), 0 ^ i < qn, Ajn)(x0), 0 ^ j < qn-1.
We denote the obtained partition by £n(x0) and call it n-th dynamical partition of the circle. We now briefly describe the process of transition from £n(x0) to ^^^x^. All intervals Ajn\x0), 0 < j < qn-1, are preserved, and each of the intervals A(n ^(x^) is divided into an+1 + 1 sub intervals:
a>n+\ -1
A(r-1)(x0) = A(T+1)(x0) U U Atf»-^(x0).
s=0
Obviously one has ^1(x0) < & (x0) < ... < £n(x0) < ....
Definition 2.1. Let K > 1 be a constant. We call two intervals I1 and I2 of S1 are K-comparable, if the inequalities K-1p(I2) ^ MA) ^ Kp(I2) hold.
Next we formulate the lemma, that is proved in the similar way as in [16]. Let xcr G S1 be a critical point of homeomorphism f. For any x0 G S1, consider the dynamical partition £n(x0). For definiteness we assume that n is odd. Then xqn ~< x0 xqn_1. The structure of the dynamical partition implies that xcr = f -p(xcr) G [xqn ,xqn_1 ], for some p, 0 < p < qn. Let I1 and I2 be any elements of a dynamical partition £m(xcr), m > n having a common endpoints.
Lemma 2.2. Let f G C3(Sr) be a critical circle homeomorphism with irrational rotation number. Then there exists a constant K > 1 depending only on f such that the intervals I1 and I2 are K-comparable.
It follows from the Lemma 2.2 that the trajectory of each point is dense in S1. Hence it follows that there exists conjugation map p between f and fp, i.e. p(f (x)) = fp(p(x)) for any x e S1.
We assume that is element of partitioning £m+k (xcr), while is an element of
partitioning £m(xcr) that contains A(m+k).
Lemma 2.3 (see [10]). There exist constants X1(f) < X2(f) < 1 such that
£(A(m+k)) < const Xk (f )£(A(m)), £(A(0m)) > const X]n(f).
2.2. Cross-ratio tools. In the proof of our main theorem the tool of cross-ratio plays a key role.
Definition 2.2. The cross-ratio of four points (z1,z2,z3,z4), z1 < z2 < z3 < z4 is the number
(z2 - zi)(zi - z3)
Cr(z1,z2, z3, zA) =
(z3 - z1)(z4 - z2)'
Definition 2.3. Given four real numbers (z1 ,z2,z3,z4) with z1 < z2 < z3 < z4 and a strictly increasing function F : R1 ^ R1. The distortion of their cross-ratio under F is given by
n.t( . F) Cr(F(z1),F(z2),F(z3),F(z4))
Dist(z1,z2, z3, z4; F ) = -—---.
Cr(z1, z2,z3, z4)
For m > 3 and zi £ S1, 1 < i < m, suppose that z1 ~< z2 ~< ■ ■ ■ ~< zm ~< z1 (in the sense of the ordering on the circle). Then we set := z1 and
zi if z1 < zi < 1,
zi := i
[ 1 + zi if 0 < zi < z1.
for 2 ^ i ^ m.
Obviously, i < i2 < ... < im. The vector (i1, i2,..., im) is called the lifted vector of (z1, z2,..., zm) £ (S 1)m.
Let f be a circle homeomorphism with lift F. We define the cross-ratio distortion of (z1,z2,z3,z4), z1 -< z2 -< z3 -< z4 -< z1 with respect to f by Dist(z1, z2, z3, z4. f) = = Dist(z1, i2, i3,z4; F), where (z17 i2, i3, i4) is the lifted vector of (z1,z2,z3,z4). We need the following lemma.
Lemma 2.4 ([5]). Let zi £ S1, i = 1, 2, 3, 4, z1 -< z2 -< z3 -< z4. Consider a circle homeomor-phism f with f £ C2+£([z1, z4]), e > 0, and f'(x) ^ const > 0 for x £ [z1, z4]. Then there is a positive constant C1 = C1(f) such that
| Dist(z1, z2, z3, z4; f) - 1 C1\z4 - z!|1+e,
where (z1, z2, z3, z4) is the lifted vector of (z1, z2, z3, z4).
We now consider the case when the interval [z1,z4] contains a critical point xcr of the home-omorphism f. More precisely, suppose that z2 = xcr. We define numbers a, ¡3, y, £ and n as follows:
z z 3 z z z z £ 3 3
a := i2 - i1, ¡3 := Z3 - Z2, Y := Z4 - Z3, £ := —, n := —,
a y
where (z1, z2,z3, z4) is the lifted vector of (z1,z2, z3, z4). Thus we need the following lemma.
Lemma 2.5. Suppose that a homeomorphism f with lift F has a critical point xcr with order 2m + 1, m £ N. Then for any e > 0, there exist 6 = 6(e) > 0, such that for all Zi £ Ug(xcr), i = 1,n, z1 -— z2 = xcr — z3 -— z4 one has
1 e2mV2m + e2m-iV2m-1 + ••• + em + 1
DiSt(Zh Z2Z4; f) - Т-^Тё-^+ё^ X + cmrfm-1 + • •• + d^-in +1
< Roe,
where the constants e2m = 2m + 1, ei = C2m + C%2ra_i + ■ ■ ■ + and R0 depends only on
function f.
Proof. Fix a number e. It is easy to check that for any zi € Si, i = 1,n, zi ~< z2 ~< z3 ~< z4 one has
F(2m) (2 ) 1 f Z2
F (zi) = F z) - F'(z2)(z2 - zi) + ■■■ + 2m 2) (¿2 - 2i)2m - — J F (2m+i)(y)(y - 2i)2mdy,
F(2m) ( 2 )
F (Zs) =F (£2) + F'(Z2)(Zs - £2) + ■■■ + 0 ( 2)(2s - 22)2m+
1 2ml (2.1) + 2m. I F(2m+1)(y)z - y)2mdy, s = 3, 4-
By the assumption of the lemma, z2 = xcr, and using the (2.1) we write Cr(f(zi),f (z2),f(z3),f (z4)) as follows
Cr(f (zi)J(z2)J(z3)f (z4))=(F(Z2) - F(2i))(Fz) - Fz)) =
(FZ) - F(Zi))(FZ) - F(£2)) JF(2m+1)(y)(y - zi)2mdy
Z'3 Z2
— '2m
J f(2m+1)(y)(z3 - y)2mdy +1 F(2m+1)(y)(y - zi)2mdy (2.2)
Z2 Zi
Z4 Z3
J F(2m+1)(y)(z4 - y)2mdy - / F(2m+1)(y)(z3 - y)2mdy
x Z2-
Z4
j F(2m+i)(y)(z4 - y)2mdy
Z2
where (2i, 22, 23,24) is the lifted vector of (zi,z2, z3, z4). Since F(2l+i) € C(Uu(xcr)), there exist S(e) > 0, such that for any x,y € (xcr - w,xcr + u) the inequality |F(2m+i)(x) - F(2m+i)(y) | < e is true.
Hence from (2.2) we have
Cr(f (zi),f (z2),f (z3 ),f (z4)) = JF(2m+i)(xcr)(y - 2i)2mdy(1 + O(e))
(j3 F (2m+1)(xcr )(£з - y)2mdy + J F (2m+1)(xcr )(y - Z1)2mdy] (1 + O(e))
\&2 Zi J
J4 j \
I F(2m+1)(xcr )(Z4 - y)2mdy - / F(2m+1) (xcr )(£з - y)2mdy (1 + O(e))
Z2
j F(2m+1)(xcr)(Z4 - y)2mdy(1 + O(e))
x
^ + ^ " *2m+1(1 + 0(e)).
a2m+1 + * 2m+1 (y + *)2m+1
From the last equality it follows that
1
Dist(zi,z2, Z3, Z4; f) =
1 - 5 + 5- -----1-
+ (1 + n)n2m-1 + n2m
(1 + 0(e)) =
1 - 5 + 52-••• + 52m (1 + n)2m + (1 + n)2m-1n + ••• + (1 + n)n2m-1 + n2m
(1 + n)2m
=_1_ e2mn2m + e2m-1n2m-1 + ••• + em + 1
1 - 5 + 52-••• + 52m x n2m + C1mn2m-1 + ••• + C2*m-1v +1( +
Thus Lemma 2.5 is proved. □
Next suppose the interval [z1,z4] is a subset of the interval Uu (xcr) but does not contain a critical point xcr of the homeomorphism f. Let d = min l([zs,xcr]). We now state an assertion
from [10].
Lemma 2.6 (see [10]). Suppose that a homeomorphism f satisfies the conditions of Lemma 2.5. Then the following equality holds
Dist(zuz2,z3,z4; f ) = 1 + j.
3. Proof of Theorem 1.4
In order to prove Theorem 1.4 we need several lemmas which we formulate next. Their proofs will be given later. We consider two copies of the unit circle S1. The homeomorphism f1 acts on the first circle and f2 acts on the second one. Assume that fi, i = 1,2 satisfies the conditions of Theorem 1.4.
Let y1 and y2 be conjugations of f1 and f2 to linear rotation fp, i.e. y1 o f1 = fp o y1 and y2 o f2 = fP o y2. It is easy to check that the homeomorphisms f1 and f2 are conjugated by h = y2 o y-1, i. e. h o f1(x) = f2 o h(x), Wx £ S1. Recall that every yi, i = 1, 2 is unique up to an additional constant. This gives us a possibility to choose h with initial condition h(xCr>) = x£).
Notice the conjugation h(x) is continuous function on S1. It suffices to show that h'(x) = 0 for almost all x with respect to the Lebesgue measure. The derivative h'(x) = 0 exists for almost all x with respect to the Lebesgue measure because the function h is monotonic. Let us show that h'(x) = 0 at all points where the derivative is defined.
Lemma 3.1 (see [5]). Assume, that the conjugating homeomorphism h(x) has a positive derivative h'(x0) = ^o at some point x0 £ S1, and that the following conditions hold for the points zi £ S1, i = 1, . . . , 4, with z1 — z2 — z3 — z4, and some constant R1 > 1 :
(a) the intervals [z1,z2], [z2, z3], [z3, z4] are pairwise R1-comparable;
(b) max^ l([zi,xo]) < R1i([zu z2]).
Then for any e > 0 there exists 6 = 6(e) > 0 such that
\Dist(z1,z2, z3, z4; h) - 1| < C2e, (3.1)
if zi £ (x0 - 6, x0 + 6) for all i = 1, 2, 3,4, where the constant C2 > 0 depends only on R1, w0 and not on e.
a +1
Suppose that h'(xo) = wo, where xo G S1. Let ^n(xo) be its n-th dynamical partition. Put to := h(x0) and consider the dynamical partition rn(t0) of t0 on the second circle determined by the homeomorphism f2, i.e.
Tn(to) = {I(n 1](to), 0 < i < qn - l]U{I(jn)(to), 0 < j < qn-1 - 1}
with I(n)(to) the closed interval with endpoints to and f2qn(to). Choose an odd natural number n1 = n(fi,f2) such that the n1-th renormalization neighborhoods [xqni ,xq x] and [tqni ,tq x] do not contain critical point of f1 and f2 respectively. Since the identical rotation number p of f1 and f2 is irrational, the order of the points on the orbit {fk (xo), k G Z} on the first circle will be precisely the same as the one for the orbit {f2(to), k G Z} on the second one. This together with the relation h(f1(x)) = f2(h(x)) for x G S1 implies that
h(Ajjni-1)) = ljni-1), 0 < i < qm - 1, h(Ajni)) = Ijni), 0 < j < qni-1 - 1. (3.2)
The structure of the dynamical partitions implies that xC1r>(n1) = f-l(xCr>) G [xq ,xq 1 ], where l G (0,qm-1) if x^r1^) G [xqni ,xo], and l G (0,qri1) if xjlJ n) G [xo,xqn—1 ]. Since h conjugation between f1 and f2, we get
f2 (h(xj;1))) = f2-1(f2(h(xC1)))) = f2-1(h(f1 (x£))) = ■■■ = h(fl (x£)) = h(x£) = x™.
Hence xj?r>(n1) = f2 l(xj2>) G [tqni ,tq 1 ]. The points x(1^(n1) and xj2>(n1) are called the qni -pre-images of the critical points xC2 and x(:2\ respectively.
Next we introduce the concept of a "regular" cover of the critical point. Let zi G S1, i = 1,4, z1 ~< z2 ~< zs ~< z4 ~< z1. Define for each j, 0 < j < qn
f If (Z2 ),fj (zs)]) .. If (Z2),fj (zs)])
^ l([ft(z1 ),fj(z2)]Y l(№(zs),fj(z4)])'
Definition 3.1. Let M > 1, Z G (0,1), S > 0 be constant numbers, n is a positive integer and xo G S1. We say that a triple of intervals ([z1,z2], [z2,zs], [zs,z4]), zi G S1, i = 1, 2, 3,4, covers the critical point of xi1 "(M, Z, 0, S; xo)-regularly", if the following conditions hold:
1) [z1, z4] C (xo - S,xo + S), and the system of intervals {fj ([z1, z4]), 0 ^ j ^ qn - 1} cover critical point only once;
2) z2 = f-l(x(2>) for some l, 0 < l < qn;
3) f (l) < Z and f (l) > M.
Denote
L = min{2m 1 + 1, 2m2 + 1, 2\m 1 - m2\}.
Lemma 3.2. Suppose that the homeomorphisms fi, i = 1,2 satisfy the conditions of Theorem 1.4. Then for any xo G S1 and S > 0 there exist constant Mo > 1 and Zo G (0,1), such that for all triples of intervals [zs, zs+1] C (xo - S, xo + S), s = 1, 2, 3, and [h(zs), h(zs+1)], s = 1, 2, 3, covering the critical points xi1 and xC221 regularly with constants Mo and Zo the following in-
equalities hold:
e2m1 vjT1 (l) + e2m1-1V fo1-1 (l) + ■■■ + 1
1 - f (l)+■■+fm1 (l) fm1 (l)+cm f1-1(l)+■■■+1
- (2m1 + 1)
L
< 16,
1
1 - f (l) +
+f2 (i)
e2m fm2 (l) + e2m2-mIim2-L(l) + ■■■ + 1
fm2 (i)+eu n)m2-l(i)+■■■+1
- (2m2 + 1)
where mi and m2 are orders of critical points x^ and x^ respectively.
L
< Y6 '
Assume that the homeomorphism fi satisfies the conditions of Theorem 1.4. Let ^„(xiï1) be a dynamical partition of the circle by fi. We take a natural number r, such that A^^ (xiï1) U
A(r i)(xCi)) c UU1 (xCr). Suppose that h'(x0)
p0 > 0 for some x0 G 5. Consider the dynamical partition £n(x0) of the point x0 under fi. Suppose that n > r an odd natural
number. Let xiï1
f-l(xir ) G [xq
-1 ].
Let [Cn+k(xcr^)}kL0 be a sequence of dynamical partitions of the point xcr. We define the points zi, i = 1, 2, 3, 4 as follows
zi = fqn+k0 (x^), z2 = x(cr\ z3 = fqn+fc0+fcl (x^l), z4 = fqn+fc0+fcl +qn+k2 (x^l).
Lemma 3.3. Suppose that the homeomorphisms fi and f2 satisfies the conditions of Theorem 1.4. Let h'(x0) = p0 > 0 for some x0 € S1, 5 € (0,1) and k0 € N. Then there exist natural numbers k1, k2 such that for sufficiently large n, the triple of intervals [zs, zs+1] C (x0 - 5, x0 + 5), s = 1, 2, 3 satisfies the following properties:
(1) the intervals {[fj(z1),fj(z4)\, 0 ^ j ^ qn} cover each point at most once;
(2) the intervals [zs, zs+1\ and [fqn( of Lemma 3.1 with some constant R1 > 1 depending on k0, k1, k2;
(3) the triples of intervals ([zs,zs+1], s = 1, 2, 3) and ([h(zs), h(z.
zs),fqn (zs+i)], s = 1, 2, 3 satisfy conditions (a) and (b)
critical points x
(i)
x(2)
xcr ,
_i)], s = 1, 2, 3) cover the "(Mo, Co, S; x0 )-regularly" and "(Mo, (0, S; h(x0))-regularly" , respectively.
Lemma 3.4. Suppose the circle homeomorphisms fi and f2 satisfy the conditions of Theorem 1.4. Then there exists natural number k0 such that for intervals [zs,zs+i], s = 1, 2, 3 satisfying conditions (1)-(3) of Lemma 3.3, and for sufficiently large n the following inequality holds
Dist(zi, Z2, Z3, z4; fqn)
Dist(h(zi),h(z2),h(z3),h(z4); ft ) where the constant R2 depends only on fi and f2.
1
> R2 > 0,
(3.3)
Proof of Theorem 1.4. Let f1 and f2 be circle homeomorphisms satisfying the conditions of Theorem 1.4. The lift H(x) of the conjugating map h(x) is a continuous and monotone increasing function on R1. Hence H(x) has a finite derivative H'(x) for almost all x with respect to Lebesgue measure. We claim that h'(x) = 0 at all points x where the finite derivative exists. Suppose h'(x0) > 0 for some point x0 € S1. Fix e > 0. We take a triple of intervals [zs,zs+1\ C (x0 - 5, x0 + 5), s = 1,2,3, satisfying the conditions of Lemma 3.4.
Using the assertion of Lemma 3.1 we obtain
Hence
Dist(zi, z2, z3 ,z4; h) — 1 ^ C3£, Dist(fin (zi),fln (z2),ffn (z3),fin (z4); h) — 1 Dist(zi, z2, z3, z4; h)
Dist(f1n (z 1), f1n(z2), f1n(zs), f1n(z4); h) where the constant C4 > 0 does not depend on e and n.
1
< C3£.
< C4e,
(3.4)
(3.5)
(3.6)
1
x
cr
Since h is conjugating f1 and f2 we can readily see that
Cr(h(ft (z1)),h(ftn (z2)),h(ffn (zs)),h(f? (z4 ))) =
= Cr(ft (h(z1)),fqn (h(z2)),ft (h(zs)),ft (h(z4))).
Disf (z1),ft (z2),ffn (zs),ft (z4); h) _
Hence we obtain
Dist(z1, z2, zs, z4; h) Cr(h(f1n (z1)),h(f1n (z*)),h(f? (zs)),h(f1n (z4)))
Cr(f1n (z1),f1n (z2),f1n (zs),f1n (z4)) Cr(z1,z2,zs,z4) _ Cf (h(z1)),f2jn (h(z2)),ft (h(zs)),ft (h(z4))) .
Cr(h(z1), h(z2), h(zs), h(zA)) Cr(h(z1), h(z2), h(zs), h(z4))
. Cr(ft (z1), ft (z2),f?n (zs), ft (z4)) = Dist(h(z1),h(z2),h(zs),h(z4); ft)
' Cr(z1,z2,zs,z4) Dist(z1,z2,zs,z4; ft) '
This, together with (3.6) obviously implies that
Dist(z1,z2,zs,z4; fqn)
Dist(h(z1),h(z2),h(zs),h(z4); ft)
1
< C5£,
where the constant C5 > 0 does not depend on e and n. This contradicts equation (3.3). Therefore Theorem 1.4 is completely proved. □
4. The proofs of Lemmas 3.2-3.4
Proof of Lemma 3.2. Denote
M(f1 m=1 - u (')+'■■ + tf1 (l) •
and
.. e2m1 f (l) + e2m1-1V2fm1-1(l) + ■■■ + 1
(l)) = fm)+Cm^m-+1 ■
It is easy to check that for f (l) > 0 the function 42(nf1 (l)) is monotone increasing and 1 < 42(rlf1 (l)) < 2m1 + 1. Obviously
e l™ nM(h (l)) = 1, 1™ 42 f (l)) = 2m 1 + 1.
Taking these remarks into account and using the explicit form of the functions 41 ((f1 (l)) and 42(rlf1 (l)) we can now estimate \ 41 ■ 42 - (2m 1 + 1) \. Firstly, we estimate 42 for large value of nf1 (l). Using the explicit form of the function 42(rlf1 (l)), we see that the inequality
\42 - (2m 1 + 1)\ = o(—^ < Rs(—^ , (4.1)
\Vf1 (l)J \Vf1 (l)J
R2(—w) < L, then
2 Vnf1(l)) 32
where the constant Rs > 0 depends only on f1. If we choose nf1 (l) satisfying the inequality
1_ ^ < L
32'
\42(Vf1 (l)) - (2m1 + 1)\ < L, - 295 -
f m ^ 32R3 for f (I) > ■
We next estimate — 1| for small value of £f1 (l). Using the explicit form of the function
$1(£f1 (l)), we see that \$1(£f1 (l)) — 1\ = O(£f1 (l)) < R4£f1 (l)■ It follows from this together with
(4.1) that \• $2 — (2mi + 1)\ < \$2 — (2mi + 1)\ + • — 1\ < + (2mi + 1)Ri£f1 (l). If we take
where R5 = max{R3, R4}, then for all f (l) < ^ and f (l) > Mi the following inequality holds
• $2 — (2mi + 1)\ < L 16
Similarly it can be shown that with
C2 := min ( —-L-—, 1), M2 :=max ( , 1), (4/2)
C2 132(2m2 + 1)R6, J' 2 I L , J' ( )
and £f2 (l) < C2 and f (l) > M2, the second assertion of Lemma 3.2 holds. In (4.2) the constants R6 > 0 depends only on f2. Finally, if we set Co := min{C1,C2} and M0 := max{M1 ,M2}, then Lemma 3.2 holds for f (l), f2 (l) G [0, Co) and nf1 (l), Vf2 (l) ^ M0. Lemma 3.2 is proved. □
Proof of Lemma 3.3. Firstly, we prove the third assertion of the lemma. By the construction of the points Zi, i = 1, 2, 3, 4, it implies that the intervals [zs,zs+1] and [h(zs),h(zs+1)\, s = 1, 2, 3 satisfy the 1) and 2) conditions of definition of "regularly" covering. We consider dynamical partition £n(x'ycr). According to Lemma 2.2 the intervals A0"\xi11)) and AO" 1)(xi^) are K-comparable, i.e. there exist constant K > 1 such that K-1l(A0" ^(xCT1)) < £(A0"^ (xCr1)) < xCr1)). Thus it follows that there exists k2^ G N such that the following inequality
holds
f1
Indeed, it is clear that
Kxttjin+k0+k1) (xCr>)p If0 (xi'Axtt])
< Co■ (4.3)
^A0qn+ko+3)(xil))) 1 1 K
<
¿A^o^X^)) e№n+k0+1\x<$)) ^ 1 + K K +1
A^ n+ko+3 UV))
Hence £(A0qn+ko+3)(x2Jr))) < K+ie(A<0qn+ko+1)(x^)). Using the last inequality we obtain that for any k
£(A~qn+ko+k)(X£)) < (K^) kl(A0qn+ko+1)(xCl))).
Since Agqn+ko+1)(xCr^) and Apq"+ko) (x^) are K-comparable, there exists a k^ G N such that the inequality (4.3) is true. Similarly, we can show that there exists a k2^ G N such that the following inequality holds
K [x2Jr\fqin+k0 + k1 (xCr})])
£/ ffqn + ko+k1 X^)) fln + k0+k11) +qn+k21 X^r)]
> Mo ■
(2) (2)
Similarly, it can be shown that with natural numbers k1 and k1 the inequalities
e(\x2\fq2n+k0+k12) (xg)]) Z i([x(3\ f92n+k0+k12) (x$)]) M
< —q-^-^^-T-Tq-^-- > Mo
[ft+ ^ (xcr), ]) ^ {xw)t ^ ^
hold. If we take k1 = maxf^1-1, k^} and k2 = maxlk^, k^} then the third assertion of Lemma 3.3 holds for k1 and k2. By the definition of the points zi, i = i, 2, 3 it implies the first assertion of the lemma.
Let XcV) be a dynamical partition of the point x^. According to Lemma 2.2 the intervals A0n)(x(^) and A0n 1)(x{cr) are K-comparable. Hence, it implies that the intervals [zs,zs+1], s = i, 2,3 are pairwise Kkl+k2- comparable. It is easy to see that the intervals [ft(zs),fqn(zs+1)j, s = 1,2, 3 are pairwise Kkl+k2-comparable. Obviously,
i < e(Ar1)(xi1r))) < Kk0+1 i < i(Ar1)(xíl))) < Kk0,
< nA0 (xcr )) < Kk0 + 1 1 < nA0 (xcr )) < Kk0 + 1
Kkl([z1,z20 " ' Kk0+^ l([ft (z1),ft (z2)])
Since the intervals A^ ^(x^) and A^ 1\f- qn-1 (xCV)) are K-comparable and xo G A^if-"-1 (x^)) U Aon-1)(x(2)) we get
max {l([fqn(zi),xo]), l([zi,xo])} < (K + 1)Kk0+1l([z1,z2]).
If we take R1 = (K +1)Kko+k1+k2, then we obtain the proof of the second assertion of Lemma 3.3 with constant R1. Lemma 3.3 is proved. □
Proof of Lemma 3.4. Suppose, the triples of intervals ([zs,zs+1], s = 1, 2, 3) and ([h(zs), h(zs+1)], s = 1,2,3) satisfy the conditions of Lemma 3.3. We want to compare the distortion Dist(z1, z2,zs, z4; ft) and Dist(h(z1),h(z2), h(zs),h(z4); ft). We estimate only the first distortion, the second one can be estimated analogously. Obviously
qn-1
Dist(z1,z2,zs,z4; ft) = It Dist(fi(z1), f1 (z2),f1(zs),fi(z4); f1).
i=o
We denote
Jr(x(2) = Ao\x£) U At1)(x(c2), A = {i : (f^), f{(z4)) n J(xC2}) = %},
B = {i : (f1 (z1),fi(z4)) n Jr (x(1)) = %}.
It is clear that A U B = {0,1,..., qn}.
Next we rewrite Dist(z1, z2, zs, z4; ft) in the form
Dist(z1, z2,zs, z4; ft) = n Dist(f1(z1), f1 (z2), f1(zs), flM; h)x
ieA
x n Dist(fi(z1), f1 (z2), fi(zs),fi(z4); f1). (4.4)
ieB
We estimate the first factor in (4.4). Using the Lemmas 2.4 we obtain I n Dist(f1 (z1), fi(z2), f1 (zs), f1(z4); f1) - 1 = I n (1 + °(e([f1 (z1), fi(z4)]))1+") - 1
ieA
ieA
max (l([f{ (z1),f1(z4)]))v O^K, [fÍ(z1),fí z)])) = Of),
ieA
where v > 0 and 0 < Xf1 < 1. We fix e > 0. There exists N0 = N0(e) > 1 such that for any n > N0 the estimate
| H Dist(fi(zi), fi(z2), fi(zs),fi(z4); fi) — 1
ieA
< C6e
(4.5)
holds. We now estimate the second factor in (4.4). We rewrite the second factor in the following form
n Distf i(z 1), f i(z2),fi(z3), fi(z4); fi) =
ieB
= n Dist(fi (zi ),n(z2),fi(z3),fi (z4); fi )X Distfl (z 1 ),fl (z2),fl (z3),fl (z4); fi)■ (4.6)
ieB,i=i
By applying Lemmas 2.5 and 3.2 we obtain
\Dist(f{(zi),fl(z2),f[(z3), f l(z4); f i) — (2mi + 1)\ <
(4.7)
Using Lemma 2.6 for the first factor in (4.6), we get n Dist(fi(zi),fi(z2),fi(z3),fi(z4); fi) — 1 = n (1 +
ieB,i=i
m (z 1 ),fi(z4)])\2 di
1
expj ^ log[1 + O
ieB,i=i
l([fi(z iUl (z4)])\2 di
1
const ^^
E
q=0 i:[ft2z1),ft2z4)]c2Jn-q 2xi1J)\Jn-q + 12xiJr) )),i=l
< const £ (e([fi (zif(z4)])) (l([fi(z 1 ),fi(z4)])x2
V di
Obviously,
E
i:[f!2z1),f12z4)]C2Jn-q 2xi1J)\Jn-q+12xir')),i = l
(l([f i(z 1 ),fi(z4)])
V di
= const
and it follows from Lemma 2.3 that (^fi( 1 ( 4)]) ^ const 1+q. Consequently
n Dist(fi (z 1 ),fi(z2),fi (z3),fi(z4); fi) — 1 ieB,i=l
< C7X'
f1
(4.8)
where C7 > 0 depends only on f 1.
Similarly one can show that for the triple of intervals ([h(zs), h(zs+1)], s = 1, 2, 3) the following inequality
n Dist(f2(h(z 1 )),fi(h(z2)),f2(h(z3)),f2(h(z4)); f2) — 1
ieB,i=l
< C8,
(4.9)
where C8 > 0 depends only on f2 and 0 < Xf2 < 1 is defined in Lemma 2.3. If we choose
k0 = max
logA
L
f1 (16m 1 + 8 + L)C7_
+ 1,
lo§A
L
f2 (16m2 + 8 + L)C8_
+ 1 ,
k0
where constants 0 < Xf1 ,Xf2 < 1 are defined in Lemma 2.3, then from the relations (4.4)-(4.8) it implies that for sufficiently large n
\Dist(z1,z2,zs,z4; ft) - (2m 1 + 1)| < Ll. (4.10)
Similarly
\Dist(h(z1),h(z2),h(zs),h(z4); ft) - (2m2 + 1)\ < L. (4.11) The inequalities (4.10) and (4.11) implies
Dist(z1,z2 ,zs,z4; ft)__. > 8(m1 - m2) - 2L > 0 (412)
Dist(h(z1),h(z2),h(zs),h(z4); ft) > 8m2 + L + 4 , ( . )
if m1 > m2, and
Dist(z1,z2 ,zs,z4; ft)__. < 8(m1 - m2) +2L < 0 (413)
Dist(h(z1), h(z2), h(zs), h(z4); ft) < 8m2 - L + 4 < , (. )
if m1 < m2. If we set
. (\8(m1 - m2) - 2L\ \8(m1 - m2) + 2L\
R2 := mm < ---, ---
2 \ 8m2 - L + 4 ' 8m2 + L + 4
then it follows from (4.12)-(4.14) that the assertion of the lemma holds.
(4.14)
The author would like to thank Professors A. A. Dzhalilov, K. M. Khanin and A. Davydov for useful discussion.
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О сопряжение между двумя критическими отображениями окружности
Уткир А. Сафаров
Туринский политехнический университет Ташкент, Узбекистан Ташкентский государственный экономический университет
Ташкент, Узбекистан
Аннотация. В статье изучается сопряжение между двумя критическими гомеоморфизмами окружности с иррациональным числом вращения. Пусть fi, i = 1, 2 являются C3-гомеоморфизмы окружности с критической точкой хСГ порядка 2mi + 1. Доказано, что если 2m1 + 1 = 2m2 + 1, то сопряжение между f1 и f2 — сингулярная функция.
Ключевые слова: гомеоморфизм окружности, критическая точка, сопрягащий гомеоморфизм, число вращения, сингулярная функция.
