SMOOTH

TOPOLOGICALLY

TRANSITIVE

DYNAMIC

SYSTEMS E.

UDC 517.9

A. Sidorov

The paper proves the existence of smooth topologically transitive dynamic s y s t e m s - f l o w s and cascades for any connected n-dimensional region (n -- 3), and cascades for t w o - d i m e n sional regions diffeomorphic to the unit c i r c l e .

Among dynamic s y s t e m s , flows and c a s c a d e s (in the t e r m i n o l o g y adopted in [2], a c a s c a d e is a dynamic s y s t e m with d i s c r e t e time), a p a r t i c u l a r place is a s s u m e d by s y s t e m s which a r e topologically t r a n s i tive and m e t r i c a l l y t r a n s i t i v e , or ergodic. Such s y s t e m s , sufficiently smooth and even analytic, naturally a r i s e on manifolds of a definite type, i.e., t o r u s e s , spaces of linear elements of manifolds of negative c u r v a t u r e (see, for example, [1], Chapter 6, 85, and [2]). F o r the case of Euclidean regions, there a r e c e r t a i n r e s u l t s on the existence of t r a n s i t i v e s y s t e m s . J. Oxtoby [3, 4] put forth a method of constructing topologically t r a n s i t i v e c a s c a d e s in n-dimensional r e gions (n --- 2). Constructions of topologically ergodic s y s t e m s for n-dimensional polyhedra (flows for n -> 3 and c a s c a d e s for n -< 2) w e r e p e r f o r m e d by J. Oxtoby and S. Ulam [5]. In addition, t h e r e a r e a number of e a r l i e r or l a t e r p a p e r s containing constructions of topologically t r a n s i t i v e c a s c a d e s in Euclidean regions [6, 7, 8]. It is n e c e s s a r y to mention that the aforementioned methods of c o n s t r u c t i n g transitive s y s t e m s do not, as a rule, give any b a s i s for drawing conclusions as to the smoothness of these s y s t e m s . An exception is the topologically t r a n s i t i v e c a s c a d e in the $2,plane, given in polar coordinates (p, ~) by the t r a n s f o r m a t i o n Pl = P ef(~~ r =~ + 0 with a p p r o p r i a t e conditions on the continuous f u n c t i o n f ( ~ ) , with period 2~, and on the incommensurability with 2~ of the n u m b e r 0; an example of such a cascade was f i r s t c o n s t r u c t e d by L. S h n i r e l ' m a n [6], while a m o r e general r e s u l t was obtained by A. Besicovitch [7]. Such a c a s c a d e will henceforth be called a S h n i r e l ' m a n - B e s i c o v i t c h cascade. Although the S h n i r e l ' m a n - B e s i c o v i t c h cascade is not differentiable at a fixed point p = 0, it may be analytic at all the remaining points of the ,$z plane (see the L e m m a below). In T h e o r e m s 1 and 2 of this paper we prove for any connected n-dimensional region of space $~ (n -> 3) the existence of topologically transitive flows and cascades of c l a s s C~ . T h e o r e m 3 establishes for any region of the $~,plane diffeomorphic to the unit c i r c l e the existence of topologically t r a n s i t i v e c a s c a d e s of c l a s s C ~. The construction in T h e o r e m 3 is b a s e d on smooth r e p r e s e n t a t i o n s of the S h n i r e l ' m a n - B e s i c o v i t c h t r a n s f o r m a t i o n by a method given by D. V. Anosov. We now adduce the auxiliary definitions and f a c t s used in T h e o r e m s 1 and 2. In space Sn, let t h e r e be given a simple c u r v e T of c l a s s C~~ with length S and equation x = ~(s), ~(s) E C ~ [0, S], where s is a r c length on 7 m e a s u r e d f r o m one of the ends of 7 . The union of the points x E Sn, each of which s a t i s f i e s , for s o m e s E [0, S], the two conditions and

Ix -- ~ (s)l < 6

(1)

(x -- ~ (s))cp' (s) = 0,

(2)

will be called a tubular 6-neighborhood ~(7, 5) of curve 7 if 6 > 0 is so small that for x E 9 ( 7 , 6) Conditions (1) and (2) s i n g l e - v a l u e d l y define the function s = s(x) of class C~ ([10], page 37). It is meaningful to consider the b a s e s of ~ ( 7 , 6 ) to be the two ( n - D - d i m e n s i o n a l s p h e r e s of radius 6, s-i(0) and s-l(S), and N. I. Lobachevskii Gor'kovskii State University. T r a n s l a t e d f r o m Matematicheskie Zametki, Vol. 4, No. 6, pp. 751-759, D e c e m b e r , 1968. Original article submitted F e b r u a r y 29, 1968.

939

t h e r e m a i n i n g p o r t i o n o f t h e b o u n d a r y of fi(T, 6 ) , a l s o of d i m e n s i o n a l i t y n - l , to b e a c o n c a v e s u r f a c e . d e n o t e b y D a c o n n e c t e d r e g i o n in $~.

We

W i t h the c o n d i t i o n t h a t ~(T, 5) c D, we c o n s i d e r t h e flow in D of c l a s s C r176d e f i n e d b y the s y s t e m (3)

dx

e-/-= X (x),

w h e r e X(x) = go, (s(x)) 9 h(x), x E D, a n d h(x) is a f u n c t i o n of c l a s s C ~ in D, w i t h

h(x)>0,

x E I n t ~ (~?, 6 ) , h ( x ) = 0 ,

x ED~(~,

6).

T h e b o u n d a r y of ~(T, 6) c o n s i s t s of t h e f i x e d p o i n t s of t h e flow of (3). S i n c e

W

9 ~

= h> 0

in

t h e i n t e r i o r o f ~ (T, ~), a n d d

/ dz

ds (x)

( s e e , (2) a n d (3)), t h e n the c l o s u r e of a t r a j e c t o r y of t h e flow o f (3) p a s s i n g t h r o u g h t h e i n t e r i o r of ~)(T, 6) i s a s i m p l e a r c w i t h e n d s on the b a s e s o f ~ ( T , 6). O b v i o u s l y , c u r v e 3~ w i l l b e a t r a j e c t o r y of flow (3). Note 1. To a r e c t i n l i n e a r p o r t i o n of c u r v e T c o r r e s p o n d t h e s a m e p o r t i o n s i n t e r n a l to ~ ( T , 6) of a t r a j e c t o r y of flow (3). T H E O R E M 1. F o r a n y c o n n e c t e d r e g i o n D c $ n ( n ~ 3 ) l o g i c a l l y t r a n s i t i v e in D.

t h e r e e x i s t s a flow o f c l a s s C r162w h i c h is t o p o -

P r o o f . We i n t r o d u c e a s e q u e n c e of s p h e r i c a l n e i g h b o r h o o d s o k ~ D (k = 0, 1, 2 . . . . r k ~ 0 a s k ~ ~ a n d a s e q u e n c e , e v e r y w h e r e d e n s e in D, o f t h e i r c e n t e r s .

) with radii

T o p r o v e t h e t h e o r e m , it s u f f i c e s to c o n s t r u c t in r e g i o n D a flow of c l a s s C ~ f o r w h i c h t h e r e e x i s t s a s e m i t r a j e c t o r y i n t e r s e c t i n g a l l the o k . We c o n s t r u c t s u c h a flow b y i n d u c t i o n on k. On t h e f i r s t s t e p , w e c h o o s e t h e c u r v e T1 ~ Int D w i t h r e c t i n l i n e a r b o u n d a r y s e g m e n t s p a s s i n g t h r o u g h the c e n t e r s ql and q l . L e t 6~ > 0 b e l e s s t h a n r0 a n d r I and s o s m a l l t h a t 9 ( % 6 t) ~ Int D, w h i l e D\9(~/1, 61) is a connected n-dimensional region. F o r c u r v e ~/1, w e d e f i n e a flow in D o f c l a s s C ~ b y t h e s y s t e m = siX1 (x),

(4)

w h e r e Xl(x) is g i v e n in the f o r m of (3) w h i l e s 1 > 0 i s s t i l l a r b i t r a r y . T h e n , t h e t r a j e c t o r i e s of flow (4) p a s s i n g t h r o u g h the i n t e r i o r of ~ (,/~, 6I) f o r m a n open s h e a f of t r a j e c t o r i e s fI i ( a m o n g t h e m i s t r a j e c t o r y l 1 w i t h c a r r i e r 71), i n t e r s e c t i n g ~0 a n d a 1, w h i l e t h e i r i n i t i a l a n d final p o r t i o n s w i l l b e r e c t i l i n e a r s e g m e n t s of c o n s t a n t l e n g t h s ( s e e , Note 1). W e now a s s u m e t h a t a f t e r k s t e p s w e s h a l l h a v e c o n s t r u c t e d t h e c u r v e s "Yi c Int D (i = 1 . . . . . with rectilinear

b o u n d a r y p o r t i o n s a n d t u b u l a r n e i g h b o r h o o d s ~(~'i, 6i) C Int D s u c h t h a t D \

k)

h

U ~ ('~,, ~t~)

i s a c o n n e c t e d n - d i m e n s i o n a l r e g i o n , and a l s o t h e flows in D o f c l a s s C ~ g i v e n b y the s y s t e m s dt

~-

8,X~ (z)

(i = t ,

"~"

k)

(5)

i n a f o r m a n a l o g o u s to (4), w i t h s t i l l a r b i t r a r y e i > 0. T h e i n i t i a l (final) s e g m e n t o f ~ ( T k , 6k) l i e s o n l y in t h e i n i t i a l (final) s e g m e n t of ~(~'l, ~l), w h i l e t h e r e m a i n i n g p o r t i o n of ~ ( T k , 5k) h a s no p o i n t s in c o m m o n h--l,

with

U ~(~,@. i=1

F o r t h e k - t h flow in D of c l a s s C ~~, g i v e n b y t h e s y s t e m h dx

d"T = ~ e~Xt(x),

940

(6)

k

the b o u n d a r y of r e g i o n

k

(3 ~ (7~, ~*) c o n s i s t s of the fixed p o i n t s , while the t r a j e c t o r y i n t e r n a l to U ~ (7,, 8t) ~=I

{=I

begins with the trajectory of flow (4) and has as carrier the union of a number,bounded above, of trajectories of the flows of (5). F r o m the trajectories of flow (6) intersecting neighborhoods a 0, ~I .... , ak, it is possible to single out an open sheaf of trajectories IIk (among them the trajectory Ik with initial segment/k-l) having no c o m m o n points with the concave boundaries of ~7(Ti, 6i). O n the (k + l)st step, after selecting from the Ok+ 1 the smallest neighborhood of the s a m e form and the s a m e notation, w e arrive at one of two cases _

f o r s o m e i <--k, with lk e i t h e r i n t e r s e c t i n g ak+ ~ o r h a v i n g no c o m m o n points with ~ ' 1 t i o n is the s a m e in both e a s e s 1) and 2), we shall c o n s i d e r in detail only c a s e 2).

Since the c o n s t r u c -

If Ik i n t e r s e c t s ak+l, we single out the open s h e a f IIk+ 1 r II k containing Ik, and we s e t Y,+I = r

X,+l(x)

-- 0, x 6 D ; l ~ 1 = l ~ , If lk N ~'k+l = ~b, we shall then a s s u m e a l s o t h a t IIk N ~--/1 = ~b, which can e a s i l y be a c h i e v e d b y identifying in IIk the p a r t i a l l y open s h e a f ( c o n t a i n i n g / k ) with the s a m e d e s i g n a t i o n . By IIk+ l we denote the open s h e a f of t r a j e c t o r i e s of flow (6) which i n t e r s e c t Crk+1 and have no c o m m o n points with the c o n c a v e b o u n d a r i e s of ~(Yi, 6i) (i = 1 . . . . . k) ( a m o n g t h e m we note the t r a j e c t o r y / ~ + l ) " A f t e r t h i s , we c o n s t r u c t c u r v e Yk+l c Int D, whose initial (final) r e c t i l i n e a r s e g m e n t c o i n c i d e s with the final (initial) s e g m e n t of h lk (/1~+I), with Yk+i h a v i n g no o t h e r c o m m o n points with U ~2 (~i, ~). We then c h o o s e / } k + l > 0 so s m a l l t h a t the initial (final) s e g m e n t of ~ ( Y k + l , 6k+l) lies in the final (initial) s e g m e n t of IIk0rk+ l) while k

~(Yk+l, 6k+i) c D and has no o t h e r points in c o m m o n with

k+,

U f~ (Yo 6i), with D ~ i=1

connected n-dimensional region.

_

_

U f~ (Yi, 6t)

being a

i=l

If the flow in D of c l a s s Coo is given b y the s y s t e m dz

(7)

d-T = 8k +lXk § (z),

c o n s t r u c t e d in a n a l o g y to (4), the (k + D - s t flow in D of c l a s s Coo, defined by the s y s t e m dx

k+t

(8)

then has an open s h e a f of t r a j e c t o r i e s IIk+ 1 i n t e r s e c t i n g a0, al . . . . . Ok, Crk+ 1, containing t r a j e c t o r y / k + l with c a r r i e r l k U Yk+l U/l~+l. It is e a s y to c o n v i n c e o n e s e l f that t h e ( k + l ) - s t flow of (8) s a t i s f i e s c o n d i tions of the s a m e f o r m as the k - t h flow of (6), i.e., the inductive s t e p f r o m k to k + 1 is justified. As the r e s u l t , we obtain a flow in D, defined by the s y s t e m dx

oo

-d? = Y'k=l s

in which t r a j e c t o r y l, with c a r r i e r

~ lk

h =I

(X),

will i n t e r s e c t all the n e i g h b o r h o o d s ~k (k = O, 1, 2 . . . .

(9)

) and,

s i n c e its initial s e g m e n t c o i n c i d e s with Yt, the s a m e p r o p e r t y is p o s s e s s e d b y the p o s i t i v e s e m i t r a j e c t o r y / . To c o m p l e t e the p r o o f of T h e o r e m 1, we note that, with the a p p r o p r i a t e c h o i c e o f values of ek > 0, the flow of (9) will be of c l a s s Coo in r e g i o n D. T H E O R E M 2. F o r any c o n n e c t e d r e g i o n D c Sn(n~3) t o p o l o g i c a l l y t r a n s i t i v e in D.

t h e r e e x i s t s a c a s c a d e of c l a s s C ~ w h i c h is

P r o o f . The t r a n s f o r m a t i o n 8-, shifting the t r a j e c t o r i e s of flow (9) by t i m e T, will lie in c l a s s C oo a c c o r d i n g to T h e o r e m 1 while, a c c o r d i n g to T h e o r e m 6 of [5], f o r all T E (--oo, + oo) b e s i d e s a s e t of c a t e g o r y I, the c a s c a d e ~ } is t o p o l o g i c a l l y t r a n s i t i v e in r e g i o n D. T u r n i n g now to the c o n s i d e r a t i o n of the t w o - d i m e n s i o n a l c a s e , we f i r s t obtain an a u x i l i a r y r e s u l t w h i c h is of i n t e r e s t in its own right.

941

L E M M A . T h e r e exists a t o p o l o g i c a l l y t r a n s i t i v e S h n i r e l ' m a n - B e s i c o v i t c h c a s c a d e in plane in p o l a r c o o r d i n a t e s (p, r b y t r a n s f o r m a t i o n U:

g2, given

Pl = Pet(~), ch ------~ q-0,

(10)

analytically, except for fixed point p = 0, at all other points of plane $2. Proof. As proved in [7], for any continuous function f(~o) withperiod 2v andhavingthetwoproperties: 1) SI~ 1(,~) dqo = O, 2) for a n y 6 and n, the h m e t i o n

F (~, 8, n) = ~ = 0

(~ + v6)

h a s no i n t e r v a l s of c o n s t a n c y in (P, t h e r e e x i s t s a n u m b e r O, i n c o m m e n s u r a b l e with 2% s u c h that c a s c a d e {ITn} will be t o p o l o g i c a l l y t r a n s i t i v e in plane $~. In o r d e r to e s t a b l i s h the L e m m a ' s v a l i d i t y , it t h e r e f o r e s u f f i c e s to find an a n a l y t i c function f ( q ) with p e r i o d 2% which has p r o p e r t i e s 1) and 2). Such a function will be the a n a l y t i c function/(~P), with p r o p e r t y 1), which f o r any n a t u r a l m has a n o n z e r o F o u r i e r s e r i e s c o e f f i c i e n t w h o s e o r d i n a l n u m b e r is a multiple of m . Indeed, it would follow f r o m the e x i s t e n c e of an i n t e r v a l of c o n s t a n c y of r of function F ( q , 6, n) t h a t F(~P, 5, n) would v a n i s h for all 9 . The value o f the k - t h F o u r i e r s e r i e s c o e f f i c i e n t o f function F ( 9 , 5, n) is g i v e n in c o m p l e x f o r m b y the formula C "k-'an--1 e ivh~

~=o

'

(11)

w h e r e Ck is the c o r r e s p o n d i n g c o e f f i c i e n t of function f ( ~ ) . The c o e f f i c i e n t s of (11) c a n n o t equal z e r o f o r all k, s i n c e when 6 is i n c o m m e n s u r a b l e with 2v, (11) is n o n z e r o t o g e t h e r with Ck, while w h e n 5 is c o m m e n s u r a b l e with 21r, we can find, in a c c o r d a n c e with the condition on f(q~), an o r d i n a l n u m b e r k such that k6 is a multiple of 2~r while c k ~ 0. T h u s , the a n a l y t i c function f ( 9 ) ity of o u r L e m m a follows.

s a t i s f i e s conditions 1),and 2),whence, as m e n t i o n e d above, the v a l i d -

T H E O R E M 3. F o r any r e g i o n D c $2, d i f f e o m o r p h i c to the unit c i r c l e , t h e r e e x i s t s a c a s c a d e {~-~} of c l a s s C ~ which is t o p o l o g i c a l l y t r a n s i t i v e in D. P r o o f . O b v i o u s l y , it s u f f i c e s to c o n s i d e r the c a s e when D is the unit c i r c l e , x 2 +y2 _ 1, with x and y b e i n g r e c t a n g u l a r c o o r d i n a t e s in g2. We denote b y r , q~ the p o l a r c o o r d i n a t e s c o r r e s p o n d i n g to t h e m . We s h a l l obtain our t r a n s f o r m $- f r o m the t r a n s f o r m a t i o n U c o n s t r u c t e d in the L e m m a , b y r e p l a c i n g c o o r d i n a t e p. L e t the function r -= g (z), defined for z E (-~o, + ~) be s u c h that: 1)g ( - - o o ) = 0 , g (q-oo)=t; g' ( z ) > 0,z E (--o~,-boo).

2) Functions [g~ (z)l~ g(~ (z + a) Ig' (z)l1"~-~' [g (z)l-~ --~ 0 when z --.-oo (j, k , l = 1 , 2 . . . . r e s p e c t i v e l y , inf and s u p f ( q ) ,

), and a r e b o u n d e d in the r e g i o n z ~ 0, a E Ira, M], w h e r e m and M a r e , cp E [0, 21r).

3) Functions

re" (z)l' g(k~ (z + ~) [g' (z)l -~-~' (k, l = t, 2 . . . . ) a r e bound in the r e g i o n z >-- 0, a E [m, M]. T h e n , if $-, obtained f r o m U by the r e p l a c e m e n t r ---g(ln p), is the t r a n s f o r m a t i o n of unit c i r c l e D, xl = x, (x, y) = g [ g - ' (r) + f (~)lr -i (z cos0 -- y sin0),~ Yl Yl (x, y) = g [g-' (r) -4- l (~)]r-' (x sin0 -4- y cos0), ]

942

(12)

where, when r =0 and r =1, the right side of (12) is understood in the sense of the limit, then {S"n} will be a topologically transitive c a s c a d e in D of c l a s s C r176 The topological transitivity in D of c a s c a d e {~-n} as well as its m e m b e r s h i p in c l a s s C ~, when 0 < r < 1, follows f r o m condition 1) on g(z); it remains to v e r i f y that S'is in c l a s s C ~~ when r = 0 and r = 1. We p e r f o r m this verification only for x 1 = xl(x, y) at point (0, 0). We have xl = {g[g-t (r) + ] ( r

-i(xcos{} --ysin{}) + (xcos0 - - y s i n 0 ) ,

so that the question reduces to the investigation of the function h = ~g[g-l(r) + f ( ~ ) ] - r } r -1 at the point ix = 0, y =0). F r o m the r e p r e s e n t a t i o n

h -~ r-t~t(~)g['g-l(r)+oo

t]dt

we find that the partial derivative of any o r d e r of h with r e s p e c t to x, y when r ~ 0 equals the sum of f r a c tions or the integral of fractions. In the denominator of such a fraction, one of the f a c t o r s will be g'[g-l(r)] to a power less than, o r equal t o , a - l , w h e r e a is the sum of the o r d e r s of all the derivatives of g entering into the n u m e r a t o r (there exist fractions where equality is attained). M o r e o v e r , powers of r will o c c u r in the denominator, while the n u m e r a t o r can contain monomials in x, y, and derivatives of the c o r r e s p o n d i n g o r d e r off(r with the difference between the degree of r in the denominator and the d e g r e e of the m o n o mial in the numerator, will for some fractions be a r b i t r a r i l y l a r g e , together with the o r d e r of the partial d e r i v a t i v e s of h. It follows f r o m condition 2) on function g that the partial derivative of h with r e s p e c t to x, y has the limit value of z e r o as r --* 0. Thus, t r a n s f o r m a t i o n $- is in c l a s s C r176 in unit c i r c l e D, and T h e o r e m 3 is proven. An example of function g is

g(z)=[la(e+l/'-~+t--z)]

-1,

z E(~oo,+oo).

Note 2. The set of points of region D which a r e transitive for the dynamic s y s t e m s of T h e o r e m s 1-3 will differ f r o m D on a set of c a t e g o r y I (see, [9], Chapter 4, w 8). The author wishes to thank D. V. Anosov for his advice and his i n t e r e s t in this work. LITERATURE 1. 2. 3. 4. 5. 6. 7. 8. 9. 10.

CITED

V . V . Nemytskii and V. V. Stepanov, Qualitative T h e o r y of Differential Equations [in Russian], Moscow, Leningrad (1949). D . V . Anosov, "Geodesic flows on closed Riemann manifolds of negative c u r v a t u r e , " Trudy Matem. In-ta AN SSSR, 90, 3-210 (1967). J . C . Oxtoby, "Note on t r a n s i t i v e t r a n s f o r m a t i o n s , " P r o c . Nat. Acad. Sci. U.S.A., 233, No. 8 , 4 4 3 - 4 4 6 (1937). E. Hopf, "Statistische P r o b l e m e und E r g e b n i s s e inder K l a s i s c h e n Mechanik," Actaulit~s Scientifiques et Industrielles, No. 737, 5-16 (1938). J . C . Oxtoby and S. M. Ulam, "Measure p r e s e r v i n g h o m e o m o r p h i s m s and m e t r i c a l t r a n s i t i v i t y , " Ann. of Math., 4_22,No. 4, 874-920 (1941). L . G . S h n i r e l ' m a n , "Example of a plane t r a n s f o r m a t i o n , " Izv. Donskogo 1Dolitekhnicheskogo In-ta, 1_~4, 67-74 (1930). A . S . Besicovitch, nA p r o b l e m on topological t r a n s f o r m a t i o n of the plane," Fund. Math., 2_~8,61-65 (1937). E . A . Sidorov, "Topologically indecomposable t r a n s f o r m a t i o n s of n - d i m e n s i o n a l s p a c e , " Volzhskii Matem. Sb., No. 5 , 3 2 6 - 3 3 0 (1966). L. A u s l a n d e r , L. Green, and F. Kahn, Flows on Homogeneous Spaces [Russian translation], Moscow (1966). G. De Rham, Differentiable Manifolds [Russian translation], Moscow (1956).

943