!

!
!

!

!
!

!


k
x " Rk.
t = 0
Ą : [0, ") � Rk Rk,
Ą(t, x) t, x
0. Ą
Ą(0, x) = x x
Ą(t, Ą(s, x)) = Ą(t + s, x) t, s e" 0 x " Rk
0

Ą.
Ą,

Ą.







t [0, ")
N 0).
f(x) := Ą(1, x), Ą(t, x) = ft(x) t = 1, 2, . . .

Rk f : Rk Rk. Rk f
X �" Rk

f
f-1
0
fn : X X
n " N, n " Z.


f : X X x0 " X.
X Rk. x
ł(x) = {xm : m " Z}
x0 = x
fn(xm) = xn+m, n " N, m " Z.
x
ł+(x) = {xm : m e" 0}
xm = fm(x),
{xm : m d" 0} x
ł(x)
m < 0 xm = (f-1)-m(x).
f : X X
x0 " f(X)
/
ł(x) = {x}.
f(x) = x.
ł(x) = {xm : m " Z}, xm = z m e" m0 > 0.
f(z) = z, z


f : R R, f(x) = x(x2 - 1), x0 = -1 : x-1 =
f-1(-1), x-2 = f-1(x-1), . . . , x-m-1 = f-1(x-m), f(x) = y
"
y < -2 3/9 x1 = 0 = xm m > 1

{xm : m " Z},
n0 > 1 xm+n = xm m. m0
0


xm+n = xm
0
m e" m0 > 0.
xm
0


f : R R,
ńł
x+3
x < -1,
�ł
2
f(x) = -x |x| d" 1,
ół
x-3
x > +1.
2
x = 5 x0 = 5, xm = (-1)m+1 m e" 1
x-m-1 = 2x-m + 3 m e" 0
ą1 2.
xm
m
f : R R, f(x) = x/2. 1 xm = 2-m m " Z

�- x " X f : X
X fn(x), n " N
�(x). �(x) X
x. ą- ą(x) :
x-m
k
x. ą(x)
f-n(x), n " N.
x �(x) = {x}.
ą(x) = {x},
�

ą(5) = ".
5


f : R R
f y = x.
x0 " R. (x0, 0)

 x = x0 (x0, f(x0)).
y = f(x0)
y = x. x1 = f(x0).

y = x.
x2 = f(x1)

f y = x).
y = x,
2.


f
[a, b] x0
x[N].


!
!
!

! !

! !!

! ! ! !!
! ! ! !!

! ! !

f : [0, 1]
[0, 1], f(x) = �x(1 - x), � = 2, 7 0, 3.

1
0.8
0.6
0.4
0.2
0
0.2 0.4 0.6 0.8 1
� = 3, 4
1
0.8
0.6
0.4
0.2
0
0.2 0.4 0.6 0.8 1
� = 4

1
0.8
0.6
0.4
0.2
0
0.2 0.4 0.6 0.8 1



A �" X f(A) =
A = f-1(A).



A �" X,
f(A) = A.
�(x)
f-1(�(x)) �" �(x),
�(x) (yk)k"N �" �(x)
y " X. y " �(x). yk

(k)
m
fn (x), m " N.
(1)
1
z1 := fn (x), z2 x f
(k)
m
n(2) > n(1), zk := fn (x), n(k)
m m
1
zk-1. (zk)
y.
m
f(�(x)) �" �(x). y " �(x), y = lim fn (x) f(y) =
m
lim fn +1(x) f), f(y) " �(x).


f �(x) = f(�(x))
f-1 f-1(�(x)) = f-1 ć% f(�(x)) = �(x).
A �" X
f|A : A A.
X


f : X X
f|A.
A �" X
f : X X, B �" X
� > 0 n0 " N n e" n0 y " fn(B)
z " A |z - y| < �.

A m +".
n0
A
f|A :
�
A. d(y, A) := inf{|y - z| : z " A}
� A K(A, �) := {y " X : d(y, A) <
�}. A
"�>0"B - "n "ne"n fn(B) �" K(A, �).
0 0

x, fn(x) A,
f : R R
ńł
2x+1
x e" 0,
�ł
3
11x+2
f(x) = -1 < x < 0,
6
ół
x-2
x d" -1.
2
p1 = 1, p2 = -2/5 p3 = -2.
{p1} {p3}.
> p2, < p2.
p2

X


A


 A = {p1, p3}


A f : X X)
� > 0 � > 0, d(x, A) < �, d(fn(x), A) < �
n " N.
�0 > 0 x " K(A, �0)
limn" d(fn(x), A) = 0. A A

A = {p}, p
A = {p1, p2, . . . , pr} f(pi) = pi+1 i = 1, 2, . . . , r - 1, f(pr) = p1
f C1
X f
X, C1).
p ||f2 (p)|| < 1, p

f = (f1, f2, . . . , fk), fi


"fi
f2 (p) = (p) ,
"xj i,jd"k

||f2 (p)|| := sup |f2 (p) � x|.
x"Rk,|x|d"1
|f2 (p) � x| d" q|x| q < 1
x " Rk. |f(p + h) - p| = |f(p + h) - f(p)| d" q|h|
|fn(p + h) - p| d" qn|h| h < �,
C1 f K(p, �). fn(p + h) p.

f2 (p) < 1.
A = {p1, p2, . . . , pr} f(pi) = pi+1 i = 1, 2, . . . , r - 1,
f(pr) = p1 r.
||f2 (pr) . . . f2 (p2)f2 (p1)|| < 1,

d(fn(pi + h), A) d"
|fn(pi + h) - pj| j fn(pi) = pj (fn)2 (p1) =
f2 (pr) . . . f2 (p2)f2 (p1).
k = 1, f2 (p) " R ||f2 (p)||


 fn(x), n " N,
x.
n-1

1
(x) := lim sup ln |f2 (fi(x))|.
n
n"
i=0

f2 (x)
x.

x.
A
(x) < 0, x " A,
A
x " A. -ą < 0 n0 " N
n e" n0
n-1

1 1
n
-ą e" ln |f2 (fi(x))| = ln |(fn)2 (x)| .
n
i=0

n -ą
|(fn)2 (x)| d" q := < 1
|(fn)2 (x)| d" qn 0.

d(fn(x + h), A) d" |fn(x + h) - fn(x)| d" qn|h| 0.

f : [0, 1] [0, 1], f(x) = x,
f2 (x) = 1, g : [0, ") [0, "), g(x) = x(x + 1),
x0 = 0, g(x) = x + x2 > x x > 0.


A f :
Rk Rk. V : Rk [0, ")
V (x) = 0 �! x " A,
lim|x|" V (x) = ",
V (f(x)) d" V (x) x " Rk.
A
� > 0 � := inf{V (x) : x " K(A, �)}. � = 0,
/
(xn) �" Rk \ K(A, �) V (xn) 0.
xn x.
m

 V, V (x) = 0, x " A, Rk \ K(A, �)
� > 0.
{x : V (x) < �} A.
A � > K(A, �) �" G.
n " N xn d(xn, A) < 1/n xn " G,
/
V (xn) e" �.
xn x. d(x, A) = 0, x " A,
m
V (x) e" �.
d(x, A) < �, V (x) < �,
V (fn(x)) d" V (fn-1(x)) d" . . . V (x) < �
� d(fn(x), A) < �.

A c < 1
V (f(x)) d" cV (x) x " Rk.


f : M �X X X
Rk), M
R R2. f� f(�, �),
� " M
X. f (�, x),
� f�

f : R � R R,
f(�, x) = � - x2.
f�(x) = x � < -1/4,
� = -1/4 � > -1/4 :
"
-1 ą 1 + 4�
x1,2 = .
2

"
2
f�(x1,2) = 1 " 1 + 4�, < 1,
> 1.
� = -1/4

m
2
1.5
1
0.5
0
 2  1 1 2
x
 0.5
 1
2
� = -1/4 f-1/4(x = -1/2) = 1

f : R � R R,
f(�, x) = (� + 1)x - x2.
� " R x0 = 0
� x1 = � � = 0
2 2
f� = � + 1 - 2x, f�(x0) = 1 + �
2 2
f0(x0) = 1 f�(x0) < 1 � < 0 > 1 � > 0.
2
f�(x1) = 1-�,
� = 0
� = 0


m
2
1
 2  1 1 2
x
 1
 2

� = �0, � < �0 f�
� > 0)
2.
p
�0 2p,
p.
f�(x) = �x(1 - x), x " [0, 1]; �
[0, 4], [0, 1]
�-1
x0 = 0 x1 = ,
�
� > 1.
2
f�(x) = �2x(1 - (� + 1)x - 2�x2 - �x3)
2
x0 x1 f�,


� + 1 ą �2 - 2� - 3
x2,3 = .
2�
2;
f�(x2) = x3 f�(x3) = x2,
� > 3. � = 3
2
x2 = x3 = x1. � = 3 f�(x1) = -1,
� > 3 < -1 x1

2 2
f�(x3) � f�(x2) = �2(1 - 2x3)(1 - 2x2)

= (-1 + �2 - 2� - 3)(-1 - �2 - 2� - 3) = -�2 + 2� + 4.

 � < 3 1 � > 3,
{x2, x3}
�,
-1 2
"
� = 1+ 6 H" 3, 449489743.
4.
2
f�
ą1.
2
f�(x0).
x0 f : X X, C1,
 f2 (x0) || =

1.
f : R �R R
C1 x. f� := f(�, �).
2
f� (x0) = x0 f� (x0) = 1, V �0 W

0
0
x0 g : V W
f� (g(�)) = g(�),
g(�0) = x0
g(�) f� W.
F (�, x) = f�(x) - x



f : R � R R
C1 C2 x.
2
f� (0) = 0, f� (0) = 1,
0
0
"f�
2 2
f� (0) = 0, |�=� (0) = 0,

0
0
"�
W 0 h : W R C2
fh(x)(x) = x, h(0) = �0 h2 (0) = 0 h2 2 (0) = 0.

F (�, x) = f�(x) - x.
F (�0, 0) = 0
"F "f�
(�0, 0) = (0) = 0.

"� "�
h x F (h(x), x) = 0.
h G x C2. x

"F "F
0 = (h(x), x) � h2 (x) + (h(x), x)
"� "x

 x = 0, � = �0
-1 -1

"F "F "F
2
h2 (0) = - (�0, 0) (�0, 0) = - f� (0) - 1 (�0, 0) = 0.
0
"x "� "�
h2 2 (0) :
"2F "F
h2 2 (0) = (�0, 0) (�0, 0)-1.
"x2 "�
f : R �
R R C1 C3
x.
f�(0) = 0, � ,
2 2 2 2
f� (0) = -1, f� (0) = 0,

0 0
2
("f�)2
(0) |�=� = 0,

0
"�
W h : W R
fh(x)(x) = x, h(0) = �0,

2
fh(x)(x) = x.
2
F (�, x) := f�(x) - x F (�, �)
G : R2 R

F (�,x)
x = 0,

x
G(�, x) :=
"F
(�, 0) x = 0,
"x
C1 x C2),
"G "2F
(�0, 0) = (�0, 0)
"x "x2
"2G "3F
(�0, 0) = (�0, 0).
"x2 "x3
G :
2 2 2
G(�0, 0) = (f� )2 (0) - 1 = f� (0) � f� (0) - 1 = 0,
0 0 0

"G "
2
(�0, 0) = |�=� (f�)2 (0) - 1 = 0.

0
"� "�
h F (h(x), x) = 0.
"G "G
h2 (0) = - (�0, 0) (�0, 0)-1
"x "�


2 2 2 2 2 2 2 2 2 2 2 2 2
(f� )2 2 (x) |x=0 = f� (f� (x))f� (x)2+f� (fmu (x))f� (x) |x=0 = f� (0)f� (0)2+f� (0)f� (0) = 0
0 0
0 0 0 0 0 0 0 0 0

-1
"
2 2 2 2
h2 2 (0) = -f� (0) |�=� (f�)2 (0) = 0.

0
0
"�
h h(x) = x.


f : X X g : Y Y
X, Y
f, g
h : X Y

g ć% h = h ć% f.
h f
g



h f g,
h(F ix(f)) = F ix(g)
f g.
h(P ern(f)) = P ern(g) n


h(�(x)) = �(h(x)) gn ć% h = h ć% fn.

x f
h(x) g.

A f, h(A) g

f : [0, 1] [0, 1]

1
2x x " [0, ],
2
f(x) =
2 - 2x x " [1, 1]
2
g : [0, 1] [0, 1], g(x) = 4x(1 - x). f g
h : [0, 1] [0, 1], h(x) = sin2 Ą x.
2

Ą Ą Ą Ą
g(h(x)) = 4 sin2 x 1 - sin2 x = 4 sin2 x cos2 x = (sin Ąx)2 .
2 2 2 2

 2h(x) x " [0, 1/2], sin Ąx = sin Ą(1-x),
2 - 2h(x) x " [1/2, 1].



X
0 1 :
Ł2 := {(sn)n"N : sn = 0 lub sn = 1 dla n " N}

"

d((sn)n, (tn)n) := 2-n|sn - tn|.
n=0

Ł2
(s(k))n - (sn)n przy k " �!�! lim s(k) = sn dla n = 0, 1, 2, . . . .
n n
k"

k
sn = tn n = 0, 1, . . . , m, d((sn), (tn)) d" 2-m

d((sn), (tn)) < 2-m sn = tn n = 0, 1, . . . , m.


�(s0s1s2s3 . . .) := (s1s2s3 . . .)
�((sn)n) = (tn)n, tn = sn+1. �
Ł2.

N :
N.
(000 . . .) (111 . . .), 2
(101010 . . .) (010101 . . .).
Ł2.
Ł2.

0, 1,
00, 01, 10, 11,
000, 001, 010, 100, 011, 101, 110, 111,
4


 f : X X
U V �" X n " N
fn(U) )" V = ".


F : X X
M x " X
xk x N "" nk "
k k
d(fn (xk), fn (x)) e" M
d X). M = 2
(sn)n (t(k))n, k " N, t(k) = sn n d" k t(k)
n n n
sn n > k 1 sn = 0 0
sn = 1. �k(t(k)) �k(s)
"
2-n = 2.
n=0




f : X X
f

f : X X

dC0(f1, f2) := sup |f1(x) - f2(x)|
x"X
X Rk.
f : X Rl Cp p " N *" {"}), f
X Rk Cp.

X f(p)(x) x " X.
f2 (x) k � k

"fi
f2 (x) = (x) ,
"xj i,jd"k
f2 2 (x)

"fi2
f2 2 (x) =
"xj "xj i,j1,j2d"k
1 2


 Cp f : X X
p

(r) (r)
p
dC (f1, f2) := sup |f1 (x) - f2 (x)|,
x"X
r=0
r = 0 f(0)(x) = f(x) | � | Rk, r = 1
2 3
| � | Rk , r = 2 Rk
k = 1 :
2 2
dC1(f1, f2) = sup |f1(x) - f2(x)| + sup |f1(x) - f2(x)|,
x"X x"X
2 2 2 2 2 2
2
dC (f1, f2) = sup |f1(x) - f2(x)| + sup |f1(x) - f2(x)| + sup |f1 (x) - f2 (x)|,
x"X x"X x"X
p

(r) (r)
p
dC (f1, f2) = sup |f1(x) - f2(x)| + sup |f1 (x) - f2 (x)|,
x"X x"X
r=1
| � |
fn f Cp
(p)
2
fn �! f, fn �! f2 , . . . , fn �! f(p).
f : X X
Cp Cp � > 0
p
g : X X dC (g, f) < �
f.
f
C0
[a, b] C1


f : M � X
X, M M X
C1 x " X. � = �0 " M
�n �0 f(�n, �)
f(�0, �) n.
f(�0, �) C1
!
xn n
n + 1
xn.
xn+1 = f(xn)
f : [0, ") [0, "). f(N) �" N,

106

 f(0) = 0.
f(x) = rx, r > 0
xn, x0
xn = x0rn.

x n
u > 0

xn+1 = rxn - uxn = (r - u)xn,

xn = x0(r - u)n.

















n,
n xn.
n
r-u
r.



















2(1662+4000)/64 H" 2, 5 � 1026
2
.



r x x > A, r(x) < 1, x < A, r(x) > 1,
A
|r - 1| r(x) =
r max(1 - x/A, 0). n

+
xn
xn+1 = rxn 1 - ,
A
z+ := max(z, 0).
x A xn
n /A),
[0, 1] :
f�(x) = �x(1 - x)
[0, 1] � " (0, 4].

� �
�.
� 4 xn
(� - 1)/�.

x " R x = (a1, a2, a3) "
R3, a1 a2
a3 f : R3 R3
�ł łł
a1-a2-a3
ra2 1 -
A
�ł �ł
f(a1, a2, a3) = a1 .
a2











x y



+
xn yn
xn+1 = r1xn 1 - - ,
A1 B1
+
xn yn
yn+1 = r2yn 1 + - .
A2 B2
B1 A2,
1/B1 1/A2
x
y,
R2,


r1, r2, A1, A2, B1, B2 r1 + r2 d" 1
(x, y)
x y
+ d" 1,
A1 B1
x y
- + d" 1,
A2 B2
f
�ł łł
y
x
r1x 1 - -
A1 B1
�ł �ł
f(x, y) :=
y
x
r2y 1 + -
A2 B2

x y
x e" 0, + d" 1,
A1 B1
x y
y e" 0, - + d" 1
A2 B2
A1 = 1, B1 = 3, A2 = B2 = 2.

2
1.5
1
0.5
0
 0.2 0.2 0.4 0.6 0.8 1 1.2

f
�ł + łł
x y
r1x 1 - -
A1 B1
�ł
+ �ł
f(x, y) :=
x y
r2y 1 - - .
A2 B2


!


g : R R C1. x0

g(xn)
xn+1 = xn - .
g2 (xn)
g
x g
�
g C2
[a, b], g(a) � g(b) < 0 g2 (x)
max |g2 2 |
|xn+1 - x| d" |xn - x|2.
� �
2 min |g2 |


 xn
x
�
g(x)
f(x) = x - .
g2 (x)
p |f2 (p)| < 1

f2 (p) = 0

g2 (x)2 - g(x)g2 2 (x) g2 2 (x)g(x)
f2 (x) = 1 - =
g2 (x)2 g2 (x)2
x
�



x2 = g(t, x), x(t0) = x0,
g : R2 R. h > 0.
� t0 + nh
�(t0 +nh) xn, xn x0
t0
xn+1 = xn + h � g(t0 + nh, xn).
g
t f : R R

fh(x) = x + h � g(x).


x2 = g(x), x(t0) =
x0 xn h N



!

! ! !


x2 = x2 + 1, x(0) = 0,
�(t) = t h = 0, 01
155
�(0, 10) = 0, 1003346721, x10 = 0, 1002858314],

�(0, 20) = 0, 2027100355, x20 = 0, 2025044393,
�(0, 30) = .3093362496, x30 = 0, 3088431654,
�(0, 50) = 0, 5463024898, x50 = 0, 5446309253,
�(1, 00) = 1, 557407725, x100 = 1, 536977456,
�(1, 40) = 5, 797883715, x140 = 5, 273821003,
�(1, 50) = 14, 10141995, x150 = 10.60617297,
�(1, 55) = 48, 07848248, x155 = 20, 01089898.

t = Ą/2 H" 1, 57.
x2 = rx(1 - x)
xn h

hr
xn+1 = (1 + hr)xn 1 - xn ,
1 + hr
� =
1 + hr, [0, 1 + (hr)-1].
�


x0 exp(rt)
�(t) = .
x0(exp(rt) - 1) + 1
Yn
n. Cn
In Gn

Cn = ąYn-1


In = �(Cn - Cn-1)

Gn = 1



Yn = ą(1 + �)Yn-1 - ą�Yn-2 + 1,
R2 :

ą(1 + �)X - ą�Y + 1
F (X, Y ) := .
X

 F (Yn, Yn-1) = (Yn+1, Yn). F
((1 - ą)-1, (1 - ą)-1);
ą� < 1.

n

Sn, In.

Sn + łIn + A
Sn+1 = ,
1 + �hIn + dh
In + �SnIn
In+1 = .
1 + d + ł + ą
ą, �, ł, A, d h

P1 = (A/(dh), 0)

d + ł + ą dh(d + ł + ą) - �A
P2 = , .
� �ł - �h(d + ł + ą)



P1
�A < dh(d + ł + ą).














































Y X

Y = aX + b,
a b.

(xj, yj) " R2, j = 1, 2, . . . , N.

 a b,
N

R(a, b) := (yj - axj - b)2
j=1


N

"R
= -2 (yj - axj - b) � xj = 0,
"a
j=1
N

"R
= -2 (yj - axj - b) = 0.
"b
j=1

N N

1 1
x := xj, y := yj,
N N
j=1 j=1

b = y - ax,

N
xj(yj - y)
j=1
a =
N
xj(xj - x)
j=1

N
xj ((xj - x)y - (yj - y)x)
j=1
b = .
N
xj(xj - x)
j=1
R +", (a, b) ", R R2


a b


a = (a1, a2, . . . , am) " Rm.
f(�, a) : Rk Rk.


x0, x1, x2, . . . , xN " Rk.
a,
N

R(a) := |xj - f(xj-1, a)|2
j=1

 Rk)
a " Rm,

"R "R "R
R2 (a) = , , . . . , = 0,
"a1 "a2 "am
m m
R



x0 = 0, 1, x1 = 0, 3 x2 = 0, 67 x3 = 0, 59 x4 = 0, 71 x5 = 0, 52
x6 = 0, 78 x7 = 0, 43 x8 = 0, 76 x9 = 0, 42 x10 = 0, 78;

r A

x
rx 1 - .
A
B := A-1.
10

R(r, B) := (xj - f(xj-1, r, B))2,
j=1
f(x, r, B) := rx(1 - Bx).
"f "f
= x(1 - bx), = -rx2
"r "B
10

"R
= 2 (xj - rxj-1(1 - Bxj-1)) xj-1(1 - Bxj-1),
"r
j=1
10

"R
= -2r (xj - rxj-1(1 - Bxj-1)) x2
j-1
"B
j=1


3.218800000 r+1.830666000 B-4.199568000 rB+1.427195080 rB2-3.129000000 = 0

r(2.099784000 r - 1.427195080 rB - 1.830666000) = 0.

r = 0, B = 1.709214024, r = 3.364804190, B = 1.090054795.


R(0, 1.709214024) = 3.8172, R(3.364804190, 1.090054795) = 0.003282780347.


r = 3.364804190, B = 1.090054795.





x1 = 0, 43, x2 = 0, 78.
2
fr,B(x) = r2x - r2B(1 + r)x2 + 2r3B2x3 - r3B3x4.
x = 0 x = (r - 1)/(rB),
2
fr,B(x) - x x(x - (r - 1)/(rB))
x1
x2.
r2B + rB
2 = 0, 43 + 0, 78,
r2B2
r + 1
= 0, 43 � 0, 48.
r2B2

r = 3.365235540, B = 1.072029769.


























f :
[0, 1] [0, 1], f(x) = 4x(1 - x) (j, xj), j = 0, . . . , 100,
(j, yj), j = 0, . . . , 100,
yj [0, 1]
1
0.8
0.6
0.4
0.2
0
20 40 60 80 100

1
0.8
0.6
0.4
0.2
0
20 40 60 80 100



xj xj+1 f.


(xj, xj+1) y.
1
0.8
0.6
0.4
0.2
0
0.2 0.4 0.6 0.8 1

1
0.8
0.6
0.4
0.2
0
0.2 0.4 0.6 0.8 1





(xk-2, xk-1, xk),










f : X X x " X
U " x m

fk�m(x) " U.
AP (f);
P ern(t) n F ix(f)

 P er1(f) = F ix(f)
"

P ern(f) �" AP (f).
n=1
x
U " x n fn(x) " U.
Rec(f).
AP (f) �" Rec(f).
f(x) = 4x(1 - x), x " [0, 1],
[0, 1])

x
U " x n
fn(U) )" U = ".

&!(f). Rec(f) �"
&!(f).
&!(f)
AP (f), Rec(f), &!(f)
X, f.
A �" X f|A
A. x " A A x " &!(f|A), x
X.
&!(f|A) �" &!(f).
&!(f|&!(f)) �" &!(f).
M1 = &!(f), Mn+1 = &!(f|Mn).


"

M� = Mn
n=1



Mn+1 = Mn

X



 Z(f).
Z(f) Z(f)










p f : X X,
X Rk. f(p) = p
f2 (p)

A
k � k
k � k S-1AS


�ł łł �ł łł
 0 0 . . . 0  1 0 . . . 0
�ł śł �ł śł
0  0 . . . 0 0  1 . . . 0
�ł śł �ł śł
, ,
�ł �ł �ł �ł
. . . . . . . . . . . . . . . . . . . . . . . .
0 0 . . .  0 0 . . . 
 A


A Q(z) = det(A - zI) I

S-1AS A.
A Ck Ck;
Ck, S-1AS;
S
Rk,

A

 = � + i�
 = � - i�



 S S-1AS

�ł łł �ł łł
� � 0 0 . . . 0 � � 1 0 . . . 0
�ł śł �ł śł
-� � 0 0 . . . 0 -� � 0 1 . . . 0
�ł śł �ł śł
�ł śł �ł śł
0 0 � � . . . 0 0 0 � � . . . 0
�ł śł �ł śł
�ł śł �ł śł
0 0 -� � . . . 0 , 0 0 -� � . . . 0 ;
�ł śł �ł śł
�ł śł �ł śł
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
�ł śł �ł śł
�ł �ł �ł �ł
0 0 . . . � � 0 0 . . . � �
0 0 . . . -� � 0 0 . . . -� �


� �
-� �
1
2�2.
 = �+i�.

 .
Rk
A
� ą i�
L �" Rk, A
A|L : L L � ą i�.

L-

L+.
Rk = L- �" L+,
A(L-) �" L-, A(L+) �" L+
A|L- < 1, A|L+ > 1.
x " L-
lim An(x) = 0,
n"
x " L+
lim A-n(x) = 0;
n"
A|L+ A|L+
C1 f :
Rk Rk p.
k = 2. f2 (p)

 1,2
� ą i�. < 1
|� + i�| = |� - i�|), p
f, x p
limn" fn(p) = p. > 1,
limn" f-n(p) = p
f p).
|1| < 1, |2| > 1

s
Wloc(p) := {x " U : lim fn(p) = p},
n"
u
Wloc(p) := {x " U : lim f-n(p) = p},
n"
U p.
f : R2 R2 Cr, p
f2 (p)
s u
< 1, > 1, Wloc(p) Wloc(p) Cr-1
s
p, p + L- Wloc(p)
u
p, p + L+ Wloc(p) p.
s
p Wloc(p)
u
n +", Wloc(p) n -".

dim L- dim L+.

!



p
f : Rk Rk C1.
U p f|U
f2 (p).


R




�
�((a, b)) = b - a,




" "

� An = �(An)
n=1 n=1
�(A)
A. �
+".



�x
x " R

1, x " A,
�x(A) =
0, x " A,
/
x.
[0, 1]
�[0,1](A) = �(A )" [0, 1]).
�x �[0,1] [0, 1],




+"
� R, �
-"


+"
�(x) dx = 1;
-"


b
��((a, b)) = �(x) dx
a

��
�.



f : �"X X. �
�(X) = 1)
�(f-1(A)) = �(A),

 A �" X.
�(f(A)) = �(A).


1
2x x " [0, ],
2
f(x) =
2 - 2x x " [1, 1],
2
(a, b) �" [0, 1/2]
(c, d), (1 -
b, 1 -a) [1/2, 1]. �(f-1(c, d)) = 2�((a, b))
�[0,1]


A X
x0,
N

1
lim �A(fn(x))
N"
N
n=1
�A(y) = 1, y " A �A(y) = 0, y " A,
/
A �(A).
X



R,



f(x) = 4x(1 - x), x " [0, 1],
1
�(x) = .
Ą x(1 - x)

X
m I1, I2, . . . ,
Im. x0

m
N N
m. N Ij; Nj
Ij, j = 1, 2, . . . , m.
pj = Nj/N
Ij.

pj H" �(x) dx,
Ij


pj
�(x) =
�(Ij)
x " Ij �
�.

1
�(x) =
Ą x(1 - x)
4x(1 - x)
(0, 01j, pj)
N = 10000 :
12
10
8
6
4
2
0 0.2 0.4 0.6 0.8 1
x





















v,
d,
�,
�.
F

L,
t,
m.

L
v ,
t
d L,
m
� ,
L3
m
� ,
L�t
m�L
F .
t2


v, d �.
1, 2.
1 1 1
1 = va db �c �,
2 2 2
2 = va db �c F,
aj, bj, cj, j = 1, 2,
L, t, m

 L,
t m 1
ńł
a1 +b1 -3c1 -1 = 0,
�ł
-a1 -1 = 0,
ół
c1 +1 = 0,
2 :
ńł
a2 +b2 -3c2 +1 = 0,
�ł
-a2 -2 = 0,
ół
c2 +1 = 0.
a1 = b1 = c1 = -1 a2 = b2 = -2, c2 = -1.

� F
1 = , 2 = .
�vd �v2d2

1 2,
( 1 2),
2 = A 1 + B.
A B


Ą. m n >
m A1, A2, . . . , An, n - m

1 1 1
1 = Aa Ab . . . Az An-m+1,
1 2 n-m
2 2 2
2 = Aa Ab . . . Az An-m+2,
1 2 n-m
. . . . . . . . . . . . . . . ,
n-m n-m n-m
n-m = Aa Ab . . . Az An.
1 2 n-m




f : X X.


x
y. n xn yn.
xn+1 =
xn + r1xn y ri

y
a1 x; x
b2 y;



> 0,



xn yn
xn+1 = xn 1 + r1 - - ,
a1 b1

xn yn
yn+1 = yn 1 + r2 - - ;
a2 b2
b1 a2
x y


ri, ai, bi, i = 1, 2. x
a1, y b2.
xn yn
un = , vn = .
a1 b2


b1
a1un+1 = a1un 1 + r1 - un - vn ,
b2

a2
b2vn+1 = b2vn 1 + r2 - un - vn .
a1
a2 b2
�
ć = , b =
a1 b1

�
un+1 = un(1 + r1 - un - bvn),
vn+1 = vn(1 + r2 - ćun - vn).

xn yn







x1, x2,

. . . , xk n i xi , i = 1, 2, . . . , k.
n
k

n n + 1 j i pij(n). Xn
x1, x2, . . . , xk, P (n)
n n n
k � k pij(n).
Xn+1 = P (n) � Xn, .


k

Z(n) := xi
n
i=1
n.


k

pij(n) = 1 j = 1, 2, . . . , k, n " N.
i=1


k k k

Z(n + 1) = xi = pij(n)xj
n+1 n
i=1 i=1 j=1

k k k

= pij(n) xj = xj = Z(n).
n n
j=1 i=1 j=1