Uniform structures and the equivalence of diffeomorphisms

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A new equivalence relation between diffeomorphisms of a compact manifold, viz., 6 -equiv- alence, is defined on the basis of concepts in uniform topology.
LITERATURE 1. 2.

CITED

J. Kushner, "On the stability of p r o c e s s e s defined by s t o c h a s t i c d i f f e r e n c e - d i f f e r e n t i a l equations," J. Diff. E q t s . , 4, No. 3, 424-443 (1969). R. Z. K h a s ' m i n s k i i , Stability of Differential Equation S y s t e m s under Random P e r t u r b a t i o n s of Their P a r a m e t e r s [in Russian], Nauka, Moscow (1969).

UNIFORM

STRUCTURES

A. G. Vainshtein

AND

and

L.

THE

EQUIVALENCE

OF DIFFEOMORPHISMS

M. L e r m a n

UDC513

A new equivalence r e l a t i o n between d i f f e o m o r p h i s m s of a c o m p a c t manifold, viz., 6 - e q u i v a l e n c e , is defined on the b a s i s of concepts in u n i f o r m topology. The 6 - e q u i v a l e n c e c l a s s e s of the identity m a p , the Y - d i f f e o m o r p h i s m s of i n f r a - n u l l m a n i f o l d s , and the connection b e tween 6 - e q u i v a l e n c e and topological e n t r o p y a r e studied. The proofs m a k e use of an e f f e c tive d e s c r i p t i o n of the u n i f o r m - h o m o t o p y type of the " n o , a u t o n o m o u s s u s p e n s i o n s o v e r diff e o m o r p h i s m s w d e s c r i b e d in the p a p e r . The connection between d i f f e o m o r p h i s m s and nonautonomous flows is c o n s i d e r e d ; m o r e o v e r , the nonhomotopy of the Y - d i f f e o m 0 r p h i s m of the identity m a p is proved.

w 0. Introduction. The connections between uniform topology and the qualitative t h e o r y of dynamic s y s t e m s f i r s t a r o s e in the a t t e m p t to c o n s t r u c t a global qualitative t h e o r y of n o , a u t o n o m o u s flows on a c o m p a c t manifold M. With such a flow t h e r e is connected a n a t u r a l g e o m e t r i c object, viz., a 1-foliation in the extended phase s p a c e M x R into integral c u r v e s , i.e., the " i n t e g r a l p o r t r a i t Wof the flow; h o w e v e r , f r o m the point of view of topology, this object is t r i v i a l : we can find a h o m e o m o r p h i s m of M x R into itself, leading the integral p o r t r a i t of a given n o , a u t o n o m o u s flow into a " v e r t i c a l foliation" {x} x R , x E M. But if, following [1], we r i g M x R with the u n i f o r m s t r u c t u r e of d i r e c t product (see [2]), then the object c o n s t r u c t e d by us, in g e n e r a l , will not be t r i v i a l : as a rule a u n i f o r m i s o m o r p h i s m or an e q u i m o r p h i s m (see [3]) does not exist which would take the i n t e g r a l p o r t r a i t of a flow into a v e r t i c a l foliation. On this b a s i s fundamental notions of the qualitative t h e o r y w e r e defined in [1] and the s t r u c t u r a l stability of one c l a s s of n o , a u t o n o m o u s flows on closed s u r f a c e s was proved. We shall examine autonomous dynamic s y s t e m s f r o m an analogous point of view. More p r e c i s e l y , with e v e r y d i f f e o m o r p h i s m f of a c o m p a c t smooth manifold M we a s s o c i a t e the following g e o m e t r i c object: a topological s p a c e M x R rigged with a c e r t a i n u n i f o r m s t r u c t u r e , v i z . , a Wnonautenomous s u s p e n s i o n o v e r f" and a 1-foliation in this s p a c e , topologically equivalent to the v e r t i c a l foliation. By the s a m e token a new equivalence r e l a t i o n is defined between d i f f e o m o r p h i s m s , viz., 5 -equivalence (see Definition 2.1). The following p r o p e r t i e s of 5 -equivalence have been proved: 1) the 5 - e q u i v a l e n c e c l a s s of the identity m a p c o n s i s t s of all those, and only those diffeomorphi': .,~ of M that a d m i t of an invariant m e t r i c compatible with the topology on M ( T h e o r e m 2.6); 2) the 5 - e q u i v a l e n c e of two a l g e b r a i c d i f f e o m o r p h i s m s of a n i n f r a - n u l l m a n i f o l d M is equivalent to the conjugacy of c e r t a i n nonzero p o w e r s of t h e i r actions in group ~lM (Corollary 4.7); 3) the 6 - e q u i v a l e n c e c l a s s of a q u a s i r a n d o m (see [4, 5]) d i f f e o m o r p h i s m contains only quasirandom diff e o m o r p h i s m s ( C o r o l l a r y 6.4). S c i e n t i f i c - R e s e a r c h Institute of M a t h e m a t i c s and C y b e r n e t i c s , Gorki State University. T r a n s l a t e d f r o m M a t e m a t i c h e s k i e Zametki, Vol. 23, No. 5, pp. 739-752, May, 1978. Original a r t i c l e submitted July 7, 1976.

0001-4346/78/2356-0407507.50 9 1978 Plenum Publishing C o r p o r a t i o n

407

We delineate as well the c l a s s of nonautonomous flows on M, which r e p r o d u c e the s t r u c t u r e of the diff e o m o r p h i s m s of M (a natural g e n e r a l i z a t i o n of periodic flows; see Definition 2.2). In p a r t i c u l a r , we p r o v e that no Y - d i f f e o m o r p h i s m [6] is homotop with the identity one ( T h e o r e m 4.4), and, t h e r e f o r e , its s t r u c t u r e is not r e p r o d u c e d by any nonautonomous flow ( C o r o l l a r i e s 4.1 and 4.3). The m a j o r i t y of these r e s u l t s a r e b a s e d on our d e s c r i p t i o n of the u n i f o r m - h o m o t o p y type of nonautonomous suspensions o v e r d i f f e o m o r p h i s m s ( T h e o r e m 3.4). A p a r t of the r e s u l t s w e r e announced by us in [7]. w1. Uniform S t r u c t u r e s and Coverings. In this section we cite one s i m p l e construction of u n i f o r m t o p o l ogy, which we r e q u i r e subsequently. All the definitions connected with the concept of a uniform s t r u c t u r e a r e a s s u m e d known (e.g., see Chaps. II and X in [2]). Let X be a s e p a r a b l e uniform space. We shall call X an a d m i s s i b l e s p a c e if we can find an e n c i r c l e m e n t of the diagonal V* of the uniform s t r u c t u r e on X such that for all xEX the neighborhood V* (x) is s i m p l y connected. In what follows all the uniform s p a c e s a r e a s s u m e d a d m i s s i b l e . Let e: X ~ X be a covering. We fix s o m e b a s e {V~ [ a ~ A, V~ C V* C X x X}, of the ira|form s t r u c t a r e on X a n d w e a s s o c i a t e w i t h i t t h e s y s t e m { ~ ] a ~ A, ~r= C X~• X} of e n c i r e l e m e n t s of the d i ~ o n a l in the following m a n n e r : f o r all ~ EA and all xEX we r e q u i r e that V~(x) be l i n e a r l y connected and 0(V~(x))= P r o p o s i t i o n 1.1. The diagonal e n c i r c l e m e n t s y s t e m defined above is the b a s e of s o m e s e p a r a b l e u n i f o r m s t r u c t u r e on X, which is independent of the choice of b a s e ~ V~} and is a m i n i m a l s e p a r a b l e s t r u c t u r e r e l a t i v e to which the covering 0 : X -* X is a u n i f o r m l y continuous m a p . Definition 1.2. The uniform s t r u c t u r e c o n s t r u c t e d on X will be called a lifting of the u n i f o r m s t r u c t u r e on X with r e s p e c t to covering e. Example 1.3. Let M be a R i e m a n n manifold with a nonzero radius of injectivity and 0: M ~ M be a smooth covering. Then the lifting onto M of the R t e m a n n m e t r i c (relative to which 0 is a local i s o m e t r y ) defines on M a u n i f o r m s t r u c t u r e lifted f r o m M with r e s p e c t to 0. P r o p e s i t i o n 1.4. Let X 1 and X~ be a d m i s s i b l e uniform s p a c e s , 0i: Xi ~ Xi (i = 1, 2) be c o v e r i n g s , f: X I --* X~ be a u n i f o r m continuous m a p , and {fv I v ~ r, fv: X~ -+ X2} be an a r b i t r a r y f a m i l y of continuous m a p s c o v e r i n g f. Then ~ ~ } is a u n i f o r m l y equicontinuous family of (uniformly continuous) m a p s . COROLLARY 1.5__:. If 0 : X ~ X i s a r e g u l a r covering, then the m o n o d r o m y group is a u n i f o r m l y equicontinuous group of e q u i m o r p h i s m s of X. Example 1.6. Under the a s s u m p t i o n s in Example 1.3, if covering 0 is r e g u l a r , then the m o n o d r o m y group acts on ~I by i s o m e t r i c s . All the a s s e r t i o n s in this section a r e c o m p l e t e l y obvious. w2. 6 -Equivalence of D f f f e o m o r p h i s m s : Definition and E l e m e n t a r y P r o p e r t i e s . L e t M be a closed smooth (of c l a s s C~) manifold and let f E Diff(M). (Here and l a t e r Diff(M) is the group of C i - d i f f e o m o r p h i s m s of M in the Cl-topology and Top(M) is the group of h o m e o m o r p h i s m s of M in the C~ Following Smale [8], we define a d i f f e o m o r p h i s m ~ on the manifold M • R by the f o r m u l a f ( ~ , t) = (1 (~), t -

1),

9 ~ .~r

t ~ n.

(~.l)

Since f acts on M x R without fixed points, by identifying in M • R the points y and f(y) we obtain a c o m p a c t C l manifold Mf; m o r e o v e r , the natural p r o j e c t i o n ex~'p:M • R --- Mf will be a smooth covering. (The choice of the notation ~ b e c o m e s c l e a r below; see P r o p o s i t i o n 2.4.) By lifting with r e s p e c t to this c o v e r i n g the unique uniform s t r u c t u r e Mf, compatible with the t r a j e c t o r y , w e o b t a i n a new~unIform s p a c e which we shall call a nonautonomous s u s p e n s i o n over f and denote by the s y m b o l Mr. In s p a c e Mf we shall examine a v e r t i c a l 1 - f o l i a tion which we denote by the symbol Lf. Definition 2.1. D i f f e o m o r p h i s m s f,g E Diff(M) a r e said to be ~ -equivalent (denoted f ~ g) if we can find an e q u i m o r p h i s m 9 : Mr--- Mg such that 9 (Lf) = Lg; 9 will be called the 5-equivalence between f and g. Definition 2.2. We shall s a y that a nonautenomous flow ~ ~ ~'~ on manifold M r e p r o d u c e s the s t r u c t u r e of d i f f e o m o r p h i s m f if we can find an e q u i m o r p h i s m 9 : M • R ~ ~If taking the integral p o r t r a i t of flow ~ ~ " into foliation Lf. (M • R can be examined with the uniform s t r u c t u r e of d i r e c t product.) 408

The s e n s e of Definitions 2.1 and 2.2 is c l a r i f i e d b y the following P r o p o s i t i o n 2.3. Let x, y E M. Then the f i b e r s {x} x R and {y} x R of the foliation Lf a r e p r o x i m a t e in Mf if and only if the o r b i t s of x and y r e l a t i v e to f a r e p r o x i m a t e in M. (We r e c a l l t h a t two s u b s e t s of a uniform s p a c e a r e said to be p r o x i m a t e (see [3]) if it is i m p o s s i b l e to separate them by uniform neighborhood.} The following s i m p l e s t a t e m e n t shows how to c o n s t r u c t the u n i f o r m s p a c e Mr. P r o p o s i t i o n 2.4. a) T h e r e e x i s t s the c o m m u t a t i v e d i a g r a m exp: ~7] -~ M 1 Ip I1 exp: l l - - + S w h e r e exp : R -~ S t is the s t a n d a r d u n i v e r s a l covering, p: ~If - - R is the p r o j e c t i o n of M x R onto the second f a c t o r , and p: M f - - 81 is a smooth bundle with f i b e r M and a " c h a r a c t e r i s t i c d i f f e o m o r p h i s m " f; b) ex~p: ~ I f - - Mf is a r e g u l a r c o v e r i n g c o r r e s p o n d i n g to the n o r m a l d i v i s o r k e r ( p # : 7r 1 Mf -+ Z); m o r e o v e r , f is one of the g e n e r a t o r s of the m o n o d r o m y group of this covering; c} the m a p ~: ~If - - R is u n i f o r m l y continuous; d) Mf is a c o m p l e t e u n i f o r m space. A s s e r t i o n a} is obvious, b) follows instantly f r o m a) if we c o n s i d e r the e x a c t sequence of bundle p: M f - - $1, 9 c) follows f r o m P r o p o s i t i o n 1.4, and d) follows f r o m e) and the c o m p l e t e n e s s of R. Let us now p r e s e n t s e v e r a l e l e m e n t a r y s t a t e m e n t s on the 5-equivalence c l a s s e s of c e r t a i n diffeomorphismE P r o p o s i t i o n 2.5. The following d i f f e o m o r p h i s m s a r e always 6-equivalent: a} f and fk, k ~ 0 is an integer; b) @l and ~0t, t ~ 0, {~ot } is an autonomous flow; c) topologically conjugate d i f f e o m o r p h i s m s ; d) flow-equivalent (see [9]) d i f f e o m o r p h i s m s , if ~ 1M = 0. To p r o v e a) it is sufficient to note that by virtue of P r o p o s i t i o n 2.4 the homothety (.l/k): R - - R can be continued up to an e q u i m o r p h l s m between Mf and Mg, p r e s e r v i n g the v e r t i c a l foliation, b} is proved in exactly the s a m e way, and c) is c o m p l e t e l y obvious. To p r o v e d) we note that the flow-equivalence of two d i f f e o m o r p h i s m s f and g of a s i m p l y connected manifold M can be lifted up to 5-equivalence between f and g (it is enough to apply P r o p o s i t i o n s 2.3 and 1.4 and to note as well that by definition ~xp (Lf) is a foliation into t r a j e c t o r i e s of the flow of the suspension o v e r f on Mr}. Finally, let us d e s c r i b e the 5-equivalence c l a s s of the identity map. THEOREM 2.6. In o r d e r that d i f f e o m o r p h i s m s f: M ~ M be equivalent to idM, it is n e c e s s a r y and sufficient t h a t on M t h e r e e x i s t an f - i n v a r i a n t m e t r i c compatible with its topology. Proof. a} Sufficiency. Let a n f - i n v a r i a n t m e t r i c e x i s t on M. Then f g e n e r a t e s a u n i f o r m l y equicontinuous group of e q u i m o r p h i s m s both of Mf [by v i r t u e of P r o p o s i t i o n 2.4.b) and C o r o l l a r y 1.5] as well as of M x R , Using the c o m p a c t n e s s of M and P r o p o s i t i o n 2.4.c), f r o m this it is e a s y to deduce that idMxR:M x R ~f[s the 6-equivalence between id M and f. b) N e c e s s i t y . It is evident that if f ~ idM, then the map r : ~ [ f - - M (the projection of M x R onto the f i r s t factor} is uniformly continuous. By the s a m e token, the f a m i l y of m a p s {~ o'~: +lTi-~ M I k ~ Z} is unif o r m l y equicontinuous and, so, f g e n e r a t e s a p r e c o m p a c t subgroup in Top(M}. The f - i n v a r i a n t m e t r i c on M can now be obtained, e.g., by a v e r a g i n g an a r b i t r a r y m e t r i c , compatible with its topology, o v e r a c o m p a c t group {]~lk~Z}~ Top (M). R e m a r k 2.7. N e c e s s a r y and sufficient conditions a r e given in [10] for the t r a n s f o r m a t i o n group of a unif o r m s p a c e to admit of an invariant m e t r i c compatible with the uniform s t r u c t u r e . w3. Uniform Classification and U n i f o r m - H o m o t o p y Type of Nonautonomous Suspensions o v e r Diffeom o r p h i s m s . We c o n s i d e r the following p r o b l e m : when a r e nonautonomous s u r g e r i e s o v e r f, g E Diff (M)

409

equimorphic (denoted ~If ~ ~Ig)?

The s i m p l e s t sufficient condition of e q u l m o r p h i c i t y is given by

P r o p o s i t i o n 3 . 1 . If i n t e g e r s m , n ~ 0 and ~ E Top(M) exist such that f - m o ~-1 o gn o ~ E Top0(M ), then ~If and Mg a r e equimorphic. (Here and below GO is a component of the linear connectivity of unity in a topological s p a c e G.) We o m i t the p r o o f since it is obvious. Now it is c l e a r that the p r o b l e m of the uniform c l a s s i f i c a t i o n of nonautonomous s u s p e n s i o n s o v e r diff e o m o r p h i s m s is s i m p l e r than the p r o b l e m of d e s c r i b i n g the 6-equivalence c l a s s e s : It is sufficient to c o n s i d e r E x a m p l e 3.2. If Sn is an n-dimensional s p h e r e , then Top(S n) c o n s i s t s of two components; t h e r e f o r e , S~ ~ Sn x R for all f E Diff(sn). F u r t h e r m o r e , since Diff (Sn) c o n s i s t s of only a finite n u m b e r of components, the e q u i m o r p h i s m between S~ and S x R can be chosen as smooth, satisfying a t w o - s i d e d Lipscbitz condition, and fibered; this p e r m i t s us to c o n s t r u c t on Sn a nonautonomous flow r e p r o d u c i n g the s t r u c t u r e of f (cf. [11]). The nontriviality of the p r o b l e m being c o n s i d e r e d is i l l u s t r a t e d by P r o p o s i t i o n 3.3. The nonautonomous suspension o v e r a h y p e r b o l i c a u t o m o r p h i s m of an n - d i m e n s i o n a l t o r u s is not e q u i m o r p h i c with T n x R. P r o o f . L e t f E GL(n, Z) be a h y p e r b o l i c a u t o m o r p h i s m of T n. As shown in [12], the u n i v e r s a l covering o v e r T~ cannot be equimorphic with R n+! since its volume invariant is the o r d e r of growth of the exnpn+nential c u r v e ; see [3]. But f r o m the equimorphicity of 2"~ and T n x R , would follow the e q u i m o r p h i c i t y of R n 1 and the u n i v e r s a l c o v e r i n g o v e r T ~ , which s i m u l t a n e o u s l y is a u n i v e r s a l c o v e r i n g o v e r T n. This r e a s o n i n g e m e r g e d as an a n s w e r to a question a s k e d one of the authors by E f r e m o v i c h when d i s c u s s i n g C o r o l l a r y 4.1 (see below). Subsequent p r o g r e s s in the p r o b l e m being examined is connected with the study of the u n i f o r m - h o m o t o p y (u.h.) type of nonautonomous s u s p e n s i o n s o v e r d i f f e o m o r p h i s m s . We r e c a l l that the c o n cept of a u.h. type w a s introduced and studied in [13]: a uniform homotopy of continuous m a p s is a homotopy u n i f o r m l y continuous "with r e s p e c t to the a g g r e g a t e of v a r i a b l e s , ~ and on this b a s i s we can define such c o n cepts as u.h. equivalence and u.h. type just as this is done in homotopy topology on the b a s i s of the concept of homotopy; see [14], f o r instance. We f o r m u l a t e the c e n t r a l r e s u l t of the p r e s e n t paper: THEOREM 3.4 (Classification T h e o r e m ) . In o r d e r that the s p a c e s ~If and ~Ig be u.h. equivalent (denoted ~If ~ ~Ig) it is n e c e s s a r y and sufficient that i n t e g e r s m , n ~ 0 and a homotopy equivalence ~: M ~ M be found such that 9

~ g~ 9 %

(3.1)

w h e r e ~ denotes the homotopy of the m a p s . The c l a s s i f i c a t i o n t h e o r e m is proved in Sec. 5. w4. C o r o l l a r i e s of the Classification T h e o r e m . In this section we show how T h e o r e m 3.4 can be used to solve p r o b l e m s on the uniform c l a s s i f i c a t i o n of nonautonomous suspensions o v e r c e r t a i n d i f f e o m o r p h i s m s and, in a n u m b e r of c a s e s , also f o r solving p r o b l e m s on the d e s c r i p t i o n of 6-equivalence c l a s s e s . COROLLARY 4.1. If Mf ~ M x R , then for s o m e integer k # 0 we have fk ~ idM. P r o o f . We set g = id M in (3.1). R e m a r k 4.2. In [15] i t w a s p r o v e d that if M is a closed s u r f a c e , ~ E Top (M), ~ ~ id M, then ~ E ToP0(M ). T h e r e f o r e , P r o p o s i t i o n 3.1 and C o r o l l a r y 4.1 in a g g r e g a t e d e s c r i b e the n e c e s s a r y and sufficient conditions for the existence of an e q u i m o r p h i s m between a nonautonomous suspension o v e r a d i f f e o m o r p h i s m of a closed s u r face and the d i r e c t product of this s u r f a c e by R. COROLLARY 4.3. If f:M ~ M is a Y - d i f f e o m o r p h i s m , then ~If ~ M x R. Since an a r b i t r a r y nonzero power of a y - d i f f e o m o r p h i s m is a y - d i f f e o m o r p h i s m , the a s s e r t i o n of Coroll a r y 4.3 follows i m m e d i a t e l y f r o m C o r o l l a r y 4.1 and the next t h e o r e m . THEOREM 4.4. If f:M ~ M is a Y - d i f f e o m o r p h i s m , then f ~ id M. Proof. We c o n s i d e r two c a s e s : 410

a) The expanding bundle E u of Y - d i f f e o m o r p h i s m f is orientable. Then t h e indices of all fixed points of fn (n > 0) a r e equal to e a c h other; since f has infinitely m a n y p e r i o d i c points (see [6]), by v i r t u e of L e f s c h e t z ' f o r m u l a f . : H . {lVI)~ H . (M) h a s eigenvalues g r e a t e r than unity in absolute value. b) The bundle E u is nonorientable. is o r i e n t a b l e .

Then a t w o - s h e e t e d c o v e r i n g 0 :~I - - M e x i s t s such that d0-1(E u)

It is advisable to isolate the next stage of the p r o o f of T h e o r e m 4.4 as an independent s t a t e m e n t . LEMMA 4.5. Under the hypothesis of c a s e b) f can be c o v e r e d by the Y - d i f f e o m o r p h i s m f :M -* M f o r which d0-1(E u) is the expanding bundle. P r o o f . The c o v e r i n g 0:~I -~ M c o r r e s p o n d s to the n o r m a l d i v i s o r N(E u) C ~ 1M g e n e r a t e d by those loops with b a s e point x E M, the t r a v e r s e along which p r e s e r v e s the o r i e n t a t i o n of the fiber E xu of bundle EU. Since df p r e s e r v e s EU,_f# p r e s e r v e s N(EU). (Here and below f# is the action of f in v i M . ) T h e r e f o r e , f can be c o v e r e d b y the m a p f : M ~ M , a n d , m o r e o v e r , it is obvious that f" is a d i f f e o m o r p h i s m of the smooth manifold M. F u r t h e r , the expansion T . M = d 0-1(T, M) = d 0 - I (E u $ E s) = d0-1(EU)$ d0-1(ES), i n v a r i a n t r e l a t i v e to d r , holds. Lifting the R i e m a n n m e t r i c f r o m M onto M, convinces us that f is a Y - d i f f e o m o r p h i s m for which the expanding bundle is d0-1(EU). R e m a r k 4.6. This v e r s i o n of the l e m m a w a s pointed out to us by Anosov. We continue with the proof of T h e o r e m 4.4: let f ~ id M. Then f o 0 ~ 0 ; by c o v e r i n g this homotopy, we get that f is homotop with the m a p g : M - * M, which is included in the c o m m u t a t i v e d i a g r a m

o'~ ~o M and, consequently, g~ = idl~, since 0 is a t w o - s h e e t e d covering. But then (f)~ ~ idi~i, which c o n t r a d i c t s the r e suit of case a). By the s a m e token, T h e o r e m 4.4 is proved. COROLLARY 4.7. If M is an i n f r a - n u l l m a n i f o l d (see [16, pp. 111-112]) and f and g a r e a l g e b r a i c autom o r p h i s m s of it, then f ~ g only if we can find i n t e g e r s m , n ~ 0 such that f~ and g~ a r e conjugate in group Aut 0r 1M).

fif

P r o o f . The n e c e s s i t y follows f r o m T h e o r e m 3.4: o t h e r w i s e , ~Ig, In [17] it w a s proved that e v e r y a u t o m o r p h i s m of the fundamental group of an i n f r a - n u l l m a n l f o l d can be uniquely continued up to an a l g e b r a i c a u t o m o r p h i s m of this manifold; it is c l e a r that the sufficiency follows i m m e d i a t e l y f r o m this. COROLLARY 4.8. If M is an i n f r a - n u l l m a n i f o l d and f and g a r e Y - d i f f e o m o r p h i s m s of it, than f ~ g if and only if we can find i n t e g e r s m, n ~ 0 such that ia~#and g~ a r e conjugate in Aut (TrlM). P r o o f . In [18] it w a s proved that e v e r y Y - d i f f e o m o r p h i s m of an i n f r a - n u l l m a n i f o l d is topologically c o n jugate with an a l g e b r a i c one. It now r e m a i n s to apply C o r o l l a r y 4.7. w5. P r o o f of the Classification T h e o r e m . a) Sufficiency. F r o m P r o p o s i t i o n 1.4 it is e a s y to d e r i v e the following P r o p o s i t i o n 5.1. Let ~ and ~:~be a d m i s s i b l e uniform s p a c e s , 0i:.~i - - Xi, be coverings c o r r e s p o n d i n g to c l a s s e s of conjugate subgroups ~ ~ : h X s = t, 2), q): X~--~ X~ be a u.h. equivalence, @,(~) = ~. Then (~ is c o v e r e d by the u.h. equivalence (~ : X 1 - - X 2. The sufficiency of the conditions of T h e o r e m 3.4 now follow f r o m this proposition, Whitehead's th [13, Sec. 10], and the exact sequence of bundle p:Mf--* S 1.

em

b) N e c e s s i t y . We p r e f a c e the p r o o f of the n e c e s s i t y of the conditions of T h e o r e m 3.4 by two l e m m a s . LEMMA 5.2. Let K be a c o m p a c t u m , N be a Riemann manifold, and {h~: K -~ N I a ~ A, [ A ] ~ N0} be a f a m i l y of m a p s , p r e c o m p a c t in the C~ Then f o r some o~ E A we can find infinitely m a n y ~ ' E A such that h~ ~ l ~ , . P r o o f . As is w e l l known, two sufficiently C ~ a r e homotop.

m a p s of a c o m p a c t u m into a R i e m a n n manifold

5

Let ], g ~ Diff (M), ~ I ~ ~Tg and let (I): ~ff] -~ ~ff,z be a u.h. equivalence.

Then t h e r e holds

LEMMA 5.3. A m a p l: Z ~ Z e x i s t s such that: 411

a) the f a m i l y of m a p s {h~ = ~7(~) o (I) o ]~ IM.: M --* .g/g I k ~ Z}

(5.1)

I kl I L g I Z (k) l g L Ik I"

(5.2)

is p r e c o m p a c t in the C~ b) for s o m e L - 1,

(Here and below g : ~ I g - - ~Ig is defined analogously to ~ and M 0 = ~ - l ( 0 ) C MI is identified in s t a n d a r d fashion with M.) P r o o f . By v i r t u e of the A s c o l i - A r z e l a t h e o r e m is is sufficient to produce a m a p /:Z - - Z such that the f a m i l y {hk} be u n i f o r m l y bounded and equicontinuous. Since ~ and g g e n e r a t e u n i f o r m l y equicontinuous equim o r p h i s m groups, however we m a y choose the m a p l the f a m i l y ~hk} will always be equicontinuous by virtue of the uniform continuity of r Under our a s s u m p t i o n s the u n i f o r m s t r u c t u r e s Mf and ~Ig can be given by c o m plete R i e m a n n m e t r i c s of c l a s s C ~ (see [6, p. l i D , i.e., Mf and Mg a r e geodesic s p a c e s (see [3]). In [13] it was p r o v e d that if X and Y a r e geodesic s p a c e s and ~ : X--- Y is a u.h. equivalence, then we can find c m 0 and L > 1 such that for all ~, ~ E X, f r o m p(~, ~) > c it follows that

L-' ~ p ((:I) (~j), (I) (~I))IP (~, ~1) ~ L. T h e r e f o r e it is sufficient to s e t l (k).=-- inmgez(inf {t I p-i (t) ~ qb o]f" (Mo) =/= ~}),

(5.3)

which s a t i s f i e s all the r e q u i r e m e n t s of L e m m a 5.3. We now derive the n e c e s s i t y of the condition in T h e o r e m 3.4 f r o m L e m m a s 5.2 and 5.3: having applied L e m m a 5.2 to the f a m i l y {hk} constructed in L e m m a 5.3, we find k, k' E Z such that h k ~ hk,, k ~ k ' , l (k) l(k') (the l a s t by virtue of (5.2)). Since the projection ~r : M x R - - M is homotop with idMx R , by substituting (5.1) we obtain _

We now set m = k' - k, n = l (k) - l (k') and define the homotope equivalence r ] M0. Then (5.4) turns into (3.1) if we take into account that

(5.4)

-* M by the equality r = v o

T h e o r e m 3.4 h a s been proved. w 6. T h e o r e m on a P r e c 0 m p a c t F a m i l y f o r 5 - E q u i v a l e n t D i f f e o m o r p h i s m s and Its C o r o l l a r i e s . In this section we consider two p r o p e r t i e s of c l a s s e s of 5-equivalence of d i f f e o m o r p h i s m s , which do not follow f r o m T h e o r e m 3.4. The r e s u l t s obtained a r e b a s e d on the following g e n e r a l i z a t i o n of L e m m a 5.3: THEOREM 6.1 ( P r e c o m p a c t FamUy). Let f , g E Diff (M), f ~ g. Then we can find ~ E Top (M) and a m a p / : Z ~ Z satisfying (5.2), f o r which the f a m i l y of m a p s {h k = fk o r o g/(k} [ k E Z} is p r e c o m p a c t in the C~ topology. .Proof. L e t 9 : ~Ig --- ~If be a (~ -equivalence between g and f. We define the h o m e o m o r p h i s m r : M --- M by the f o r m u l a r = 7to ~I'l M 0 and we give m a p 1 by a f o r m u l a analogous to (5.3): 1 (k) = --.integ-er(inf {tl P-' (t) (~ W-~ o f (M0) =/= ~}). 2he r e s t is c o m p l e t e l y analogous to the p r o o f of L e m m a 5.3. COROLLARY 6.2. If f, g E Diff(M), f ~ g, then we can find i n t e g e r s m,n ~ 0 and ~p E Top(M) such that ]-~ o q~-t o g~ o q~ ~ Topo (M). P r o o f . This follows i m m e d i a t e l y f r o m T h e o r e m 6.1 if we use the local linear connectivity of group Top(M), p r o v e d in [19]. COROLLARY 6.3. Let f , g E Diff (M), f ~ g, dim M = 4s - 1. Then we can find i n t e g e r s m , n ~ 0 such that f o r all expansions w of n u m b e r s into a s u m of positive i n t e g e r s u m m a n d s the relation m 9 pw(Mf) = n 9 pco(Mg) is fulfilled between the P o n t r y a g i n n u m b e r s of manifolds Mf and Mg.

412

Proof.

By v i r t u e of C o r o l l a r y 6.2 the manifolds Mfm and Mgn a r e h o m e o m 0 r p h i c f o r s o m e i n t e g e r s

m , n ~ 0. Since Mfm is an m - s h e e t e d c o v e r i n g o v e r Mf, pco(Mfm) = m 9 p~0(Mf). It now r e m a i n s to apply Novik o v ' s t h e o r e m on the topologic i n v a r i a n c e of rational P o n t r y a g i n c l a s s e s . 6 COROLLARY 6.4. If f, g E Diff(M), f ~ g, then the p o s i t i v e n e s s of the topological e n t r o p y of g follows f r o m the p o s i t i v e n e s s of the topological e n t r o p y of f. R e m a r k 6.5. The definition of topological entropy, convenient for us, can be found in [4]; t h e r e i n it was p r o v e d that d i f f e o m o r p h i s m s with a p o s i t i v e topological e n t r o p y a r e p r e c i s e l y the q u a s i r a n d o m d i f f e o m o r p h i s m s in the s e n s e of [5]. P r o o f of C o r o l l a r y 6.4. We denote the topological e n t r o p y of f by H(f); let H(f) > 0. Let @ and l be the s a m e as in T h e o r e m 6.1 and let the constant L f r o m (5.2) be an integer. We s e t f = ~ - 1 o f o ~ ; as i s w e l l known, H(f) = H(f). L e t Sn = {xl . . . . . X#(n) I x ~ E M} be an (n, ~)-distinguishable s e t for f and let ~ > 0 be fixed (i.e. , for all ~ , fl, l _ o ~ , fl 0 such that Sn is an (L 9 n, O - d i s t i n g u i s h a b l e s e t for g. As a m a t t e r of f a c t , f o r e v e r y e > 0 we can find 5 > 0 such that for a l l ~ , ~ E M f r o m p ( ~ , ~ ) > e , it follows that p ( h ~ l o r 1~162 > e f o r all k E Z. But then H(g) >-- H(f)/L > 0. COROLLARY 6.6. If f, g E Dill(M), f ~ g, f i s a Morse-Smale diffeomorphism, and g satisfies Smale's Axiom A (see [8]), then g too is a Morse-Smale diffeomorphism. Proof. In [20] it was proved that Morse-Smale diffeomorphisms are precisely those diffeomorphisms that satisfy Axiom A and have a zero topological entropy. Now it is enough to refer to Corollary 6.4. The concept of a n o , autonomous s u s p e n s i o n o v e r a d i f f e o m o r p h i s m was introduced b y L e r m a n ; he posed the p r o b l e m on the connection of this concept with the qualitative t h e o r y of dynamic s y s t e m s and f o r m u l a t e d the a s s e r t i o n s of C o r o l l a r i e s 4.1, 4.3, and 6.4 as c o n j e c t u r e s . T h e o r e m s 3.4 and 6.1, C o r o l l a r i e s 4.7, 6.2, and 6.3, and the p r o o f of T h e o r e m 4.4 w e r e obtained by Vainshtein. The r e m a i n i n g r e s u l t s a r e o u r s jointly. The a u t h o r s thank V. M. Alekseev, D. V. Anosov, V. A. E f r e m o v i c h , and A. G. Kushnirenko for attention to the w o r k and valuable d i s c u s s i o n s . LITERATURE 1.

2.

3.

4. 5. 6. 7. 8. 9. 10. 11. 12. 13.

14.

CITED

L. M. L e r m a n and L. P. Shil'nikov, "On the c l a s s i f i c a t i o n of s t r u c t u r a l l y stable nonantonomous s e c o n d o r d e r dynamic s y s t e m s with a finite n u m b e r of cells, w Dokl. Akad. Nauk SSSR, 209, No. 3, 544-547 (1973). N. Bourbaki, G e n e r a l Topology. Fundamental S t r u c t u r e s [Russian t r a n s l a t i o n ] , F i z m a t g i z , Moscow (1968); The Use of Real N u m b e r s in G e n e r a l Topology. F u n c t i o n a l Spaces. S u m m a r y of R e s u l t s [Russian t r a n s l a tion], F i z m a t g i z , Moscow (1971). V. A. E f r e m o v i c h , " P r o x i m i t y i n v a r i a n t s , n Uch. Zap. Ivanovsk. Gos. Ped. Inst., 31, 71-84 (1963). E. I. Dinaburg, ~Connection between the v a r i o u s e n t r o p y c h a r a c t e r i s t i c s of dynamic s y s t e m s , n Izv. Akad. Nauk SSSR, Ser. Mat., 3._.55,No. 2 , 3 2 4 - 3 6 6 (1971). V. M. A l e k s e e v , " Q u a s i r a n d o m dynamic s y s t e m s . I. Q u a s i r a n d o m d i f f e o m o r p h i s m s , ~ Mat. Sb., 7_.6.6,NO. 1, 72-134 (1968). D. Vo Anosov, ~Geodesic flows on c o m p a c t R i e m a n n manifolds of negative c u r v a t u r e , ~ T r . Mat. Inst., Akad. Nauk SSSR, 9_O0(1967). A. G. Vainshtein and L. M. L e r m a n , " P r o x i m i t y g e o m e t r y and n o , autonomous s u s p e n s i o n s o v e r diffeom o r p h i s m s , ~ Usp. Mat. Nauk, 3_1, No. 5 , 2 3 1 - 2 3 2 (1976). S. Smale, ~Differentiable dynamic s y s t e m s , w Usp. Mat. Nauk, 2_~5,No. 1, 113-185 (1970). A. V e r j o v s k y , "Flows with c r o s s s e c t i o n s , " P r o c . Nat. Acad. Sci. U.S.A., 66_6,No. 4, 1154-1156 ( 1 ~ 0 ) . F. Rhodes, ,, G e n e r a l i z a t i o n of i s o m e t r i c s in u n i f o r m topology, n P r o c . Camb. Phil. Soc., 5_22,No. 3 , 3 9 9 - 4 0 5 (1950. L. M. L e r m a n , "Nonautonomous dynamic s y s t e m s of M o r s e - S m a l e type, n Usp. Mat. Nauk, 30, No. 5, 195 (1975). A. S. S h v a r t s , "Volume invariant of c o v e r i n g s , " Dokl. Akad. Nauk SSSR, 105, No. 1, 32-34 (1955). A. G. Vainshtein, "Uniform homotopy and p r o x i m i t y i n v a r i a n t s , " Vestn~ Mosk. Gos. Univ., Ser. Mat. Mekh., 1, 17-21 (1970); " E q u i m o r p h i s m s of geodesic s p a c e s and uniform homotopy, n Vestn. Mosk. Gos. Univ., Ser. Mat. Mekh., 3, 66-69 (1973). D. B. Fuks, A. T. F o m e n k o , and V. L. G u t e u m a k h e r , Homotopy Topology [in Russian], Moscow State Univ., Moscow (1970). 413

15. 16. 17, 18. 19. 20.

D . B . A . Epstein, " C u r v e s on 2 - m a n i f o l d s and isotopy, ~ Acta Math., 1.15, No. 1-2, 83-107 (1966). Z. Nitetskii, Introduction to Differential Dynamics [Russian translation], M i r , Moscow (1975). L. A u s l a n d e r , " B i e b e r b a c h ' s conjecture on s p a c e groups and d i s c r e t e subgroups of Lie g r o u p s , " Ann. Math., 71,579-590 (1960). A. Manning, WThere a r e no new Anosov d i f f e o m o r p h i s m s on t o r i , w Am. J. Math., 96, No. 3 , 4 2 2 - 4 2 9 (1974). A . V . Chernavskii, "Local c o n t r a c t a b i l i t y of the h o m e o m o r p h i s m g r o u p of a manifold," Dokl. Akad. Nauk SSSR, 182, No. 3, 510-513 (1968). M. Shub and D. Sullivan, "Homology t h e o r y and dynamical s y s t e m s , " Topology, 14, No. 2 , 1 0 9 - 1 3 2 (1975).

REMARK

ON THE

STANDARD

IDENTITY

A. R. Kemer

UDC 519.4

It is p r o v e d that the standard identity of d e g r e e n implies all Capelli identities of s o m e o r d e r m depending on n.

Let F(X) be a f r e e a s s o c i a t i v e a l g e b r a (without unity) o v e r a f i e l d of c h a r a c t e r i s t i c 0, with countable s e t of f r e e g e n e r a t o r s X. Let I n denote the ideal of identities generated by the s t a n d a r d identity Sn(x I . . . . , Xn) = 0, where

S(n} being the s y m m e t r i c group of d e g r e e n, and let V m denote the ideal of Capelli identities of o r d e r m (see [1]) g e n e r a t e d b y all identities of the f o r m ~~

( - - i )axa(i)ylxo(~)Y2' "" "' y,~-ix~(~) = O,

w h e r e s o m e of the v a r i a b l e s Yi m a y be absent. H e r e we p r o v e the following THEOREM 1. The standard identity of d e g r e e n implies all Capelli identities of s o m e o r d e r m depending on n, i.e., In D Vm for s o m e m . F r o m this t h e o r e m and f r o m [1, T h e o r e m 3] we obtain the following COROLLARY. A finitely g e n e r a t e d P I - a l g e b r a of c h a r a c t e r i s t i c 0 h a s nllpotent Jacobson r a d i c a l if and only if the a l g e b r a s a t i s f i e s a standard identity of s o m e d e g r e e . In this note we also d e s c r i b e the identities which i m p l y all CapelU identities of s o m e o r d e r , n a m e l y , we have THEOREM 2. An identity f = 0 i m p l i e s all Capelli identities of s o m e o r d e r if and only if the identity f = 0 is not satisfied in the G r a s s m a n n a l g e b r a Goo of countable rank. Suppose G is an a s s o c i a t i v e a l g e b r a o v e r F, p r e s e n t e d by g e n e r a t o r s Yl, ..., Yn; el, -.., en, *.. and defining relations eluej ~ --ejue~,

i, ] ~ 4,2, ...,

w h e r e u is any word (possibly empty) in the g e n e r a t o r s . We divide the generating set X of the a l g e b r a F ( X ) into two disjoint countable s u b s e t s Y = ~Yl . . . . . Yn, ... } and T = ~t 1. . . . , t n, ...}. We will identify Y with the K o m s o m o l s k - o n - A m u r Polytechnic Institute. T r a n s l a t e d f r o m M a t e m a t i c h e s k i e Zametki, Vol. 23, No. 5, pp. 753-757, May, 1978. Original article submitted July 23, 1976.

414

0001-4346/78/2356- 0414507.50

9 1978 Plenum Publishing C o r p o r a t i o n