Fixed points of diffeomorphism class of Smale and exponential functio

Meyer, K. R.

English

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Per iodic  Points of Diffeomorphisms 
K. R. Meyert 
Center for Dynamical Systems 
Brown University 
Providence, Rhode Island 
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This Research was supported i n  p a r t  by the  National Aeronautics 
and Space Administration under Grant No. NGR SO-OQ?-Ol? and i n  p a r t  
by t h e  National Aeronautics and Space Adm?’.ni,ytr&ion, Humtsville, 
under Contract N e .  NAs8-11264. 
t 
https://ntrs.nasa.gov/search.jsp?R=19670022161 2020-03-12T10:58:32+00:00Z
. 
c' 
I * .  
. Periodic Points of Diffeomorphisms -- 
I. Introduction: 
I n  [l] Art in  and Mazur prove t h a t  t he re  i s  a dense s e t  
i n  t h e  space of Ck mappings of a compact manifold i n t o  i t s e l f  
such t h a t  f o r  each member of t h i s  s e t  t h e  number of f ixed  po in t s  
under i t e r a t i o n  grows a t  most exponentially.  This estimate allows 
one t o  define an ana ly t i c  
morphism t h a t  measures t h e  number of f ixed  po in t s  of t h e  d i f f eo -  
morphism under i t e r a t i o n .  
(-function assoc ia ted  t o  t h e  d i f f eo -  
The theorem of Artin and Mazur gives no ind ica t ion  as t o  
whether o r  no t  a s p e c i f i c  diffeomorphism s a t i s f i e s  such an es t imate .  
I n  t h i s  note we announce (Theorem 1) t h a t  t he  number of 
f ixed  po in t s  of t h e  genera l  c l a s s  of diffeomorphisms r ecen t ly  in -  
troduced by Smale [ 2 , 3 ]  grows a t  most exponentially under i t e r a t i o n .  
It should be noted t h a t  t h i s  new theorem i s  ne i the r  contained i n  
nor  contains the  theorem of Ar t in  and Mazur. 
The method of proof i s  qu i t e  simple. One need only show 
t h a t  t h e  s i z e  of t h e  domain where the re  i s  a unique f i x e d  po in t  of 
t h e  diffeomorphism decreases a t  most exponentially by using an 
estimate on the  domain of v a l i d i t y  of t h e  imp l i c i t  func t ion  theorem. 
The complexity a r i s e s  only from t h e  necess i ty  of checking uniformity 
a t  each s tep .  
11. Notation and Theorem. 
2 
Let M be a compact C -Riemannian manifold and suppose 
t h a t  f :  M -+ M i s  a diffeomorphism of PI. A closed inva r i an t  s e t  
. 
I 
-2- 
A C  M 
T M of M r e s t r i c t e d  t o  A has a continuous inva r i an t  s p l i t t i n g  
T M = EU + ES under df such t h a t  
i s  sa id  t o  have a hyperbolic s t r u c t u r e  if the  tangent bundle 
A 
A 
df :  EU + EU ; df :  ES -+ES 
I I  dfn (4 (4 I1 < chnll UII 
IldfYX) (v>II > c-l~-nllvll 
S for  some f ixed constants  
U E E :  and n E Z .  
C > 0, 0 < h < 1, where x E A, v E E 
X J  
+ 
If f i s  a difreomorphism of M and x E M then x i s  
c a l l e d  a wandering poin t  if the re  e x i s t s  a neighborhood U of x 
such t h a t  fl f n ( U )  = cp.  A poin t  x of M i s  c a l l e d  a non- 
wandering poin t  if it i s  not  a wandering poin t .  Clear ly  the  s e t  of 
nonwandering points of f forms a compact i nva r i an t  subset of M. 
-
n EZ 
The c l a s s  of diffeomorphisms introduced by %ale i n  [2,3] 
i s  the  c l a s s  of diffeomorphisrns of M with a hyperbolic s t r u c t u r e  
on the  s e t  of nonwandering points  of f .  This c l a s s  of d i f f eo -  
morphisms i s  s u f f i c i e n t l y  general  t o  include a l l  known examples of 
diffeornorphisms with g lob ia l  s t a b i l i t y  proper t ies  (see [3] f o r  a 
d e t a i l e d  d iscuss ion) .  
Let N,(f)  be t h e  number of f ixed  poin ts  of fn .  Then 
our main r e s u l t  i s  
2 
Theorem 1. I f  f i s  a C -diffeomorphism of M i n t o  i t s e l f  with a 
hyperbolic s t ruc tu re  on the  s e t  of nomrandering pa in ts  of f then 
the re  e x i s t s  a constant  k > 0 such t h a t  
-- - -   -- - ----_ 
- ---- --- -
-- -- --- 
-3 - 
+ N n ( f )  5 k" f o r  n E Z . -
111. Outline of t he  Proof: 
I n  what follows I .I w i l l  denote the  usua l  Euclidean norm 
i n  E" with respect  t o  a f ixed basis.  The following lemma follows 
e a s i l y  from t h e  impl i c i t  funct ion theorem given i n  [4], page 12.  
+ Lemma 1: Let cp 
of rad ius  
Let t he  supremum of t h e  modulus of t he  second p a r t i a l s  of cp be 
l e s s  than bn on Ba. Let I (dcpn(0)-I)I 5 c 
c . Then the re  e x i s t s  a constant d = d(a,b,c) such t h a t  cp has 
a unique poin t  i n  the  sphere of rad ius  
n E Z , be a C2 map from t he  closed b a l l  B n' a 
a about t he  or ig in  i n  E" i n t o  i t s e l f  with cpn(0) = 0. 
n 
and I (d%(O)-I)- 'I 5 
n 
dn about t he  o r ig in .  
Let (Vi,yi) and (U.,".), i = 1 ,... r be a f i n i t e  number 
1 1  
r 
1 
of coordinate systems for M such t h a t  Vi 3 E  uui 3 M  and 
x = yilUi. Consider t he  s e t s  yi(Vi) and xi(Ui) i n  E". There ex- 
i s t s  a constant  a > 0 such t h a t  each poin t  of x.(v.) i s  con- 
t a ined  i n  a sphere of radius  a completely contained i n  y . (V. ) .  
We s h a l l  count t he  number of f ixed  poln ts  of f" i n  each xi(ui). 
i' 
i 
1 1  
1 1  
Let I I*I I  denote the norm induced i n  yi(Vi) by t h e  
metr ic  on M. 
Lemma 2: Let x be a fixed poin t  of fn,x.Of"oxil, and l e t  A 
0 1 
be t h e  Jacobian matr ix  of f" evaluated a t  x, then the re  e x i s t  
constants  N and c > 0 such t h a t  
n -1 I ( A - I ) ]  5 C and I ( A - I )  I 5 Cn 
- 4- 
f o r  n 5 N. 
Comments on the  Proof of Lemma 2. 
A t  t h i s  po in t  the  strong uniformity of ---e hy-perbo i c  
s t r u c t u r e  on the  s e t  of nonwandering poin ts  i s  used. Because the  
s e t  of nonwandering poin ts  i s  compact, and the  s p l i t t i n g  i s  con- 
t inuous the re  e x i s t s  a constant e 2 1 such t h a t  the  norm 1 1 . I I  
and t h e  n o m  I I I ' * I  I I defined by the  coordinates such t h a t  
t h e  form A = 1: 12] where A j = l J 2 ,  i s  t h e  representa t ion  
of t h e  mapptng df on E: and ES respec t ive ly  s a t i s f i e s  t he  
A has 
j J  
condi t ion 
e-? I I * I  I I 
With t h i s  uniformity a t  
A 
0 
(= II I I  
and t e r e s t  of t he  ,emma folloim by 
standard matr ix  methods. 
Since t h e  t o t a l  volume of x.(E.) i s  f i n i t e  and f ixed  
1 1  
poin ts  of f" i n  x.  (E.) can be covered by d i s j o i n t  balls of 
dn rad ius  - 3 
1 1  
t h e  required estimate follows from the  above two lemmas. 
It seems l i k e l y  t h a t  t he  general  ou t l i ne  given above can 
be used t o  give a s imi la r  estimate f o r  the  number of per iodic  o r b i t s  
f o r  a flow on M having a hyperbolic s t ruc tu re  on t h e  s e t  of non- 
wandering poin ts .  The author i s  p re sen t ly  working on t h i s  problem. 
\ 
\ 
References 
b 
1 
[ 1 3  Ar t in ,  M.,  and B. Mazur, On Per iodic  Poin ts ,  Ann. - -- of Math. 
81 (1955), PP- 82-99. 
[2] Snale, S., Dynamical Systems on n-Dimensional Manifolds, 
Proceedings of' t he  In t e rna t ion  Symposium on D i f f e r e n t i a l  -- -
Equations and Dynamical Systems, Academic Press,  to appear.  
[3] Smale, S., D i f f e r e n t i a l  Dynamical Systerris, to appear. 
[ 41 Hartman, P., Ordinary D i f f e r e n t i a l  Equations, John Wiley, 
New York (1964). 


Periodic points of diffeomorphisms

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