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Karshiboev Kh.K., candidate of physical and mathematical sciences associate professor Head of Department of Higher Mathematics
SamIES
ON A FAMILY OF CIRCLE HOMEOMOHPHISMS WITH ONE BREAK
POINT
Abstract. In this article, we study a one-parameter family of circle homeomorphisms with one break point. It is proved that in the case of a rational rotation number the number of periodic trajectories does not exceed two.
Key words: circle homeomorphism, renormalization, rotation number.
Consider a one-parameter family of mappings of the unit circle [1]: Tqx = {f (x, Q)}, x e S1 = [0,1), Q e [0; 1]
where the bracket {•} - denotes the fractional part of the number, and f (x, Q)- satisfies the following conditions:
a) at a fixed Q, f (x; Q) - continuous monotonically increasing function;
b) f (0; 0) = 0, f (x +1; Q) = f (x; Q) +1, for anyone x e R1;
, f (x; Q)
c)JK > const > 0; dQ
d) t0 : [0;1] ^ [0;1] continuous curve;
e) for every fixed Qe [0;1], f (x; Q) > const > 0; for
dx
Vx e S1 \ {t0(Q)}, f (x; Q) e C2+e(Sl \ {t0(Q)}), at some e > 0 and
¿MQhQ) = c(Q)* J. /+(t0(Q), Q)
Let us Pq denote the number of rotations corresponding to, responsible f [2]:
r f (n)(x, Q)
Pq = lim ^-^-
n ^(X n
From the d) - e) conditions it follows that TQ x for each fixed value of the parameter has only one break point t0 (Q). The number c(Q) is called the break
point Tq . Everywhere below we will denote by the f(n) - n st superposition of the function f. It is easy to see that Pq monotonically (not strictly) depends on
p
the parameter Q. Note that each rational p = — corresponds to a non-degenerate
q
p
segment (values Q such that pQ= —, while irrational p corresponds only to Q
q
Let A =
r \
Pi P2
с (0,1) - be the Faria interval of the n - nd level [1]:
qi
1) P2 qi - pq =1
2) All rational numbers inside the interval A have the form kPi + P .
kqi + q
Rational number with minimum denominator is Pi + P2 .
q + ^2
We choose an arbitrary point x0 on the circle and a segment of the
trajectory of this point {xi = TqXq,0 < i < qi + q2). Denote A^ and A(02) segments [x0, xq ] and [xq , x0], respectively. We also denote the images of
these segments under the action of Tq by A(1) and [1]:
a9 = Г A(1)
0
a(72) = T A(2)
The following assertion was proved in [1] and works without any changes
f \
in our situation. Лемма 1. Suppose p(T) e —,P2-
l qi 42
Trajectory segment
ixi = Tqxo,0<i<qi + 42} divides the circle into non-intersecting segments
e 0 < i < 42 and A\-), 0 < j < q1. Denote the constructed partition Ç( A, xo).
v(2)
Let's
put
v = vars 1 ln f ' < да, v = v+1 ln f '(x0 - 0) + ln f '(x0 + 0) | ,
q = max(qi,42), p = max(pi, pj). Consider an arbitrary trajectory yi = TQyo, У0 e ^ such that yi * xo = 0, 0 <i < q2-
Лемма 2. Suppose p(T ) e
or p(T) =
f , ^ Pi + P2 P
qi + q2, q.
- q-i -
^ "v<n f '( уг ) < ev. i=0
f \ v q j
Then
Let An =
f \ Pi P2
be a Farey interval of rank n - [1], and Am,m < n
qi qi
be some Farey interval of rank m containing An. Let p(T) e An. Let's choose An -arbitrary element of partition £(Am, t0) containing An . Let's denote by | A |.
^eMMa 3. Let's put X = (1 + e °) 2 < 1
| An |< constX1 m | Am | An |< constX1. Let the continued fraction expansion of p be of the form
p(f (x, Q)> = p = [kl9k2,...,kn],kn > 2.
q
p
Let's designate I(—) the segment of the value of the parameter Q such that
q
p p
p(Q) = . Fix some Q e I(—) and denote f = fQ, Tf = Tf . For a rational
q q j j q
p
rotation number of p(Q) = —, there always exists at least one periodic trajectory
q
of period q. Let {y(i),0 < i < q -1} be an arbitrary periodic trajectory. Let
[yi,y2] denote the segment formed by the trajectory {y(i),0 < i < q -1} and containing the singular point to. Let's move on to renormalized coordinates:
x = y2 + (yi - y2)*
and define the function corresponding to T q in the renormalized coordinate
system:
f ( z) =
1
yi - y2
[Tj (y2 + (yi - y2z)) - y2], z E[0,1]
Denote by d the renormalized coordinate of the point to :
d = (to - y2)/(y1 -y2)
and define the function Fd (z), z e [0,1] :
Fd (z) =
zc
d (1 - c2) + c2 + z (c -1):
z e [0, d]
d (1 - c2) + zc 2
2 . d(1 - c ) + c + zc(c -1)
-, z e[d,1]
Theorem 1. There is a constant c3 > 0 such that
f (z) - Fd (z)
< c3^. (1)
C 2([0,]\{d})
Proof. Consider the partition of the circle generated by the trajectory (y(z) ,0 < i < q -1). Denote A0 = [y1,y2], At = Te A0,1 < i < q -1. Obviously
Tq A 0 = A 0. It is not difficult to show [1] that |A^ < const X1,1 < i < q -1. Function f (z) can be represented as a superposition of two functions f and f2
l
<
, corresponding to mappings Te : A0 ^ A1? Tqj 1: A1 ^ Aq = A0. Let us
determine the relative coordinates inside the segments Ai:
* = fi + f - T)yi)z .
Then the functions fi and fi can be written as:
fi( *o) =
1
(ТвУ1 - Тв У 2)
[Тв (У2 + (У1 - У2 )z0 ) - ТвУ2 ]
1
f2(Z1) = -1(ТвУ2 + (ТвУ1 - ТвУ2)z1) - У2]
У1 - У2
Wherein
In [1], it was proved f2( ^1)
f (z) = f2(f1(z)). (2) Mz-
1
1 + z1(M -1)
< constAns (3)
С 2([0,1])
where
q-1 f" (v)
ln M = Z j ^У: /■=1A 2f (У)
(4)
q-1 f " (У) Z j ^У
,=0 A,2 f (У , A 0
j ™ ^У = ln c -j i™ ^У д. 2f '(У) У A. 2f '(У) У
Insofar as
f2( Z1) -
J f™ dy
L 2f'(У) У
< const An, we get
CZ1
1 + z1(c -1)
< const A (5)
С 2([0,1])
It is easy to see that function fi (z0) is close to piecewise linear function fd(z0), where
fd (z0)=
z,
~2-0-, z0 e [0, d]
c 2(1 - d) + d
d(1 - c ) + z0c2
(6)
c 2(1 - d) + d
z0 e [d, 1]
Since |A0 < const A is valid estimate: Using (2)-(6) we obtain (1).
Theorem 1 implies that f (z) is convex at 0 < c < 1 and concave at c > 1
Indeed, by direct calculation it is easy to verify that
d2
Fd(z) > 2c2(1 - c), z ф dпри 0 < c < 1
dz
2
<
Ä.
d2 2
(z) <—-(c -1), z * d npn c > 1.
dz c3
Let's put
N =
11 1 9
ln(-| c - 1|min(- c2))
Ä c3 c3
p
Denote the interval I (—) =
q
^1(P), Q2(P) q q _
Let's put J = [0,1] \ U I( ) • Denote the Lebesgue measure on [0,1] by
p. q
o< P <1 q
We now formulate the main results of our work.
Theorem 2. For all n > N, the following statements are true: p P
(a) if Q = Q1^) or Q = ), then Tq has a unique periodic trajectory q q
of period q;
(c) at Q e
^Q1(P), Q2(P
q
q
there are equal to two periodic trajectories of
period q.
Theorem 3. The Lebesgue measure of the set J is equal to zero, i.e. A( J) = 0.
References:
1. K. M. Khanin and E. B. Vul. Circle Homeomorphisms with weak Discontinuities. Advances in Soviet Mathematics, v. 3, 1991, p. 57-98.
2. I.P. Kornfeld, Ya.G. Sinai, S.V. Fomin. Ergodic theory. -M. Science, 1980.
3. H.K. Karshiboev. Behavior of renormalizations of ergodic mappings of a circle with a break// Uzbek mathematical journal. - Tashkent, 2009. - No. 4. -p.82-95.
