Chung, Yong Moo

Osaka City University Repository

Chung, Y-.M.
Osaka J. Math.
38 (2001), 1–12
TOPOLOGICAL ENTROPY FOR DIFFERENTIABLE MAPS
OF INTERVALS
YONG MOO CHUNG
(Received February 9, 1998)
Let I be a compact interval of the real line. For a continuous map f : I ! I by
Misiurewicz et al. ([1, 12, 13]) the following relation between the topological entropy
h(f ) and the growth rate of the number of periodic points is known:
h(f )  lim sup
n!1
1
n
log ℄Per(f; n)()
where Per(f; n) denotes the set of all fixed points of f n for n  1, and ℄A the number
of elements of a set A. (The equality of the expression () does not hold in general.
For instance, the topological entropy of the identity map is zero, nevertheless all of
points of the interval are fixed by this map.)
For a periodic point p of f with period n we put
O+
f
(p) = fp; f (p);    ; f n 1(p)g:
Then we say that q is a homoclinic point of p if q =2 O+
f
(p) and there are a positive
integer m with f m(q) = p and a sequence q0, q1, : : :, qk , : : : 2 I with q0 = q such
that
f (q
k
) = q
k 1 (k  1); lim
k!1
jq
k
 O+
f
(p)j = 0
where jx   Aj = inffjx   yj : y 2 Ag for x 2 I , A  I . It is known by Block ([2, 3])
that h(f ) is positive if and only if f has a homoclinic point of a periodic point.
In this paper we shall establish more results (Theorems 1 and 2) for differentiable
maps of intervals. To describe them we need some notations.
Let f : I ! I be a C1+ map ( > 0). A periodic point p of f with period n is
a source if
(p) = j(f n)0(p)j1=n > 1:
For n  1,  > 1 and Æ > 0 we define an f -invariant set by
1991 Mathematics Subject Classification. 28D20, 58F11, 58F15
2 Y-.M. CHUNG
Per(f; n; ; Æ) = fp 2 Per(f; n) : (p)  ; jf 0(f i(p))j  Æ for all 0  i  n  1g:
Then we have
Per(f; n; 1; Æ1)  Per(f; n; 2; Æ2) if 1  2; Æ1  Æ2;
and
fp : source of f g =
[
>1
[
Æ>0
1
[
n=1
Per(f; n; ; Æ):
One of our results is the following:
Theorem 1.
h(f ) = max

0; lim
!1
lim
Æ!0
lim sup
n!1
1
n
log ℄Per(f; n; ; Æ)

:
By Theorem 1 it is clear that for a C1+ map of a compact interval if the topo-
logical entropy is positive then the map has infinitely many sources. However, the con-
verse is not true in general. In fact, for any r  1 it is easy to constract a Cr diffeo-
morphism of a compact interval having infinitely many source fixed points. But every
diffeomorphism of an interval has zero entropy.
REMARK. It is known that if f is a C2 map with non-flat critical points, then any
periodic point of f with sufficiently large period is a source ([10]). In Theorem 1 we
do not assume any conditition concerned with critical points. Then the map f may
have flat critical points.
For a source p of f with period n we denote by W uloc(p) the maximal interval J
of I containing p such that
j(f n)0(x)j  f(1 + (p))=2gn for all x 2 J:
We say that a homoclinic point q of p is transversal if there are non-negative integers
m1, m2 and a point q 0 2 W uloc(p) such that
f
m1 (q 0) = q; f m1+m2 (q 0) = f m2 (q) = p and (f m1+m2 )0(q 0) 6= 0:
If f has a transversal homoclinic point of a source, then there is a C1 neighborhood
U of f such that every map g belonging to U has a transversal homoclinic point of
a source. We denote the set of transversal homoclinic points of a source p of f by
TH(p), and its closure by TH(p): We call TH(p) the transversal homolinic closure of
TOPOLOGICAL ENTROPY 3
p. It is easy to see that p 2 TH(p), and TH(p) is f -invariant. For m  1 and Æ > 0
define
H (p;m; Æ) = fq 2 W uloc(p) : f m(q) = p; jf 0(f i(q))j  Æ for all 0  i  m  1g:
Then we have
H (p;m; Æ1)  H (p;m; Æ2) if Æ1  Æ2
and
TH(p) =
[
Æ>0
1
[
m=1
m 1
[
i=0
f
i
H (p;m; Æ) nO+
f
(p):
The second result of this paper is the following:
Theorem 2. If h(f ) > 0 then
h(f ) = supfh(f jTH(p)) : p is a source of f g;
and for a source p of f we have
h(f jTH(p)) = max

0; lim
Æ!0
lim sup
m!1
1
m
log ℄H (p;m; Æ)

:
A result corresponding to Theorem 2 is known for surface diffeomorphisms by
Mendoza ([11]). As an easy corollary of Theorem 2 we have:
Corollary 3. The following statements are equivalent:
(i) h(f ) > 0;
(ii) f has a transversal homoclinic point of a source;
(iii) f has a homoclinic point of a periodic point.
1. Proofs of Theorems
Let f : I ! I be a continuous map. For integers k; l  1 we say that a closed f -
invariant set 0 is a (k; l)-horseshoe of f if there are subsets 00, : : :, 0k 1 of I such
that
0 = 0
0
[    [ 0
k 1
; f (0j ) = 0j+1 (mod k)
and f k j
0
0 : 00 ! 00 is topologically conjugate to a one-sided full shift in l-symbols.
If 0 is a (k; l)-horseshoe, then it is clear that
h(f j
0
) = 1
k
log l and ln  ℄[Per(f; kn) \ 0]  kln
4 Y-.M. CHUNG
for all n  1. It was proved by Misiurewicz et al. ([1, 12, 13]) that if the topological
entropy of f is positive then there are sequences k
j
, l
j
of positive integers with a (k
j
,
l
j
)-horseshoe 0
j
of f (j  1) such that
h(f ) = lim
j!1
h(f j
0
j
) = lim
j!1
1
k
j
log l
j
:
Then the formula () follows from this fact.
In order to prove our results, we need the notion of hyperbolic horseshoe and
ideas of the theory of hyperbolic measures ([14, 15]). Katok ([9]) has proved that if a
C
1+ diffeomorphism of a manifold has a hyperbolic measure then its metric entropy
is approximated by the entropy of a hyperbolic horseshoe. The author has shown in
[5] that the result of Katok is also valid for C1+ (non-invertible) maps.
Let f : I ! I be a differentiable map. For integers k; l  1, numbers  > 1
and Æ > 0 we say that 0 is a (k; l; ; Æ)-hyperbolic horseshoe of f if 0 is a (k; l)-
horseshoe and
j(f k)0(x)j  k; jf 0(x)j  Æ (x 2 0):
The following lemma plays an important role for the proofs of Theorems 1 and 2.
Lemma 4. Let f : I ! I be a C1+ map. If h(f ) > 0, then for a number 0
with 1 < 0 < expfh(f )g there exist sequences kj ; lj of positive integers and Æj > 0
(j  1) such that for j  1 there is a (k
j
; l
j
; 0; Æj )-hyperbolic horseshoe 0j of f so
that
h(f ) = lim
j!1
h(f j
0
j
) = lim
j!1
1
k
j
log l
j
:
This is corresponding to the result obtained by Katok for surface diffeomorphisms
([9]). For the proof we use the result stated in [5].
Proof of Lemma 4. For a number 0 with 1 < 0 < expfh(f )g we take a se-
quence 
j
of positive numbers (j  1) such that expfh(f )  3
j
g > 0 and j ! 0 as
j ! 1. By the variational principle for the topological entropy ([6, 7, 8]), we have
an f -invariant ergodic Borel probability measure 
j
on I such that
h
j
 h(f )  
j
> 0
where h
j
denotes the metric entropy of 
j
with respect to f . If 
j
denotes the Lya-
punov exponent of 
j
, that is,

j
=
Z
log jf 0(x)jd
j
(x);
TOPOLOGICAL ENTROPY 5
then by the Ruelle inequality ([17]) we have

j
 h
j
> 0;
and so 
j
is a hyperbolic measure of f . Then by Theorem C (and its proof) of [5],
we can construct sequences of integers k
j
, l
j
 1 with (1=k
j
)log l
j
 h
j
 
j
, numbers

j
 1 and closed sets 3
j
 I (j  1) such that:
(1) f kj (3
j
) = 3
j
;
(2) j(f kj i)0(x)j  
j
 1
 expfk
j
i(
j
  
j
)g for all x 2 3
j
and i  1;
(3) f kj j
3
j
: 3
j
! 3
j
is topologically conjugate to a one-sided full shift in l
j
-
symbols.
For j  1 we set
0
j
= 3
j
[ f3
j
   [ f
k
j
 1
3
j
:
Then 0
j
is f -invariant. Moreover we put
Æ
j
= minfjf 0(x)j : x 2 0
j
g > 0;
e
j
= max

jf
0(x)j
jf
0(y)j : x; y 2 0j

2 [1;1)
and take an integer n
j
 1 large enough so that
expfk
j
n
j

j
g  
j
e
j
k
j
:
Then we have
j(f kjnj )0(x)j  0kjnj (x 2 0j ):
This follows from the fact that for 0  i  k
j
  1 and x 2 f i3
j
j(f kjnj )0(x)j = j(f kjnj )0(f kj i(x))j  j(f kj i)0(x)j  j(f kj i)0(f kjnj (x))j 1
 
j
 1
 expf(k
j
n
j
)(
j
  
j
)g  e
j
 k
j
+i
 expfk
j
n
j
(
j
  2
j
)g
 expfk
j
n
j
(h(f )  3
j
)g
 0
k
j
n
j
:
It is easy to see that f kjnj j
3
j
: 3
j
! 3
j
is topologically conjugate to a one-sided full
shift in l
j
n
j
-symbols. Thus 0
j
is a (k
j
n
j
; l
j
n
j
; 0; Æj )-hyperbolic horseshoe, and from
which
h(f j
0
j
) = 1
k
j
n
j
log l
j
n
j
6 Y-.M. CHUNG
=
1
k
j
log l
j
 h
j
  
j
 h(f )  2
j
:
Since 
j
! 0 as j !1, we have
h(f ) = lim
j!1
h(f j
0
j
):
Lemma 4 was proved.
Proof of Theorem 1. For  > 1 and Æ > 0 we want to find 0 = 0(; Æ) > 0 such
that Per(f; n; ; Æ) is an (n; 0)-separated set of f for all n  1. Take 1 = 1(Æ) > 0
so small that if x; y 2 I satisfy jx   yj  1 then
jf
0(x)  f 0(y)j  Æ
2
:
We put
I
Æ
=

x 2 I : jf 0(x)j  Æ
2

:
Obviously, I
Æ
is closed. For n  1 and x 2 I ,
jx   Per(f; n; ; Æ)j  1 implies that x 2 IÆ:
Since a function x 7! log jf 0(x)j is bounded and varies continuously on I
Æ
, there is
2 = 2(; Æ) > 0 such that if x; y 2 IÆ satisfy jx   yj  2 then

 log jf 0(x)j   log jf 0(y)j  1
2
log :
We put 0 = minf1; 2g. Then it is checked that Per(f; n; ; Æ) is an (n; 0)-separated
set of f for n  1. Indeed, if a pair p; p0 2 Per(f; n; ; Æ) with p  p0 satisfies
maxfjf i(p)  f i(p0)j : 0  i  n  1g  0;
then we see that for x 2 [p; p0] and 0  i  n  1,
jf
i(x)  f i(p)j  0; f i(x) 2 IÆ:
On the other hand, by the mean value theorem there is a point  2 [p; p0] such that
jf
n(p)  f n(p0)j = j(f n)0( )j  jp   p0j:
TOPOLOGICAL ENTROPY 7
Since f i( ), f i(p) 2 I
Æ
and jf i( )  f i(p)j  0 for 0  i  n  1, we have

 log j(f n)0( )j   log j(f n)0(p)j 
n 1
X
i=0

 log jf 0(f i( ))j   log jf 0(f i(p))j

n
2
log ;
and so
j(f n)0( )j
j(f n)0(p)j  exp

 
n
2
log 

= 
 n=2
:
Since p; p0 2 Per(f; n; ; Æ), we have
jp   p
0
j = jf
n(p)  f n(p0)j
= j(f n)0( )j  jp   p0j
=
j(f n)0( )j
j(f n)0(p)j  j(f
n)0(p)j  jp   p0j
 
 n=2
 
n
 jp   p
0
j
= 
n=2
 jp   p
0
j;
and so p = p0 because of  > 1. Thus Per(f; n; ; Æ) is an (n; 0)-separated set of f ,
and then
lim sup
n!1
1
n
log ℄Per(f; n; ; Æ)  lim sup
n!1
1
n
log s(f; n; 0)(1.1)
 lim
!0
lim sup
n!1
1
n
log s(f; n;  )
= h(f )
for  > 1 and Æ > 0, where s(f; n;  ) denotes the maximal cardinality of (n;  )-
separated sets for f . Therefore we have the conclusion of Theorem 1 when h(f ) = 0.
Thus it remains to give the proof for the case when h(f ) > 0. Fix 1 < 0 <
expfh(f )g. Take sequences k
j
, l
j
, Æ
j
and 0
j
(j  1) as in Lemma 4. Since
l
j
n
 ℄[Per(f; nk
j
; 0; Æj ) \ 0j ]  kj lj n
for all n  1, we have
lim
Æ!0
lim sup
n!1
1
n
log ℄Per(f; n; 0; Æ)  lim
n!1
1
nk
j
log ℄[Per(f; nk
j
; 0; Æj ) \ 0j ]
=
1
k
j
log l
j
:
8 Y-.M. CHUNG
If j !1, then
lim
Æ!0
lim sup
n!1
1
n
log ℄Per(f; n; 0; Æ)  h(f ):(1.2)
Combining (1.1) and (1.2) we have
h(f )  lim
Æ!0
lim sup
n!1
1
n
log ℄Per(f; n; 0; Æ)
 lim
!1
lim
Æ!0
lim sup
n!1
1
n
log ℄Per(f; n; ; Æ)
 h(f ):
Theorem 1 was proved.
REMARK. In fact, from the proof of Theorem 1 it follows that
h(f ) = lim
Æ!0
lim sup
n!1
1
n
log ℄Per(f; n; 0; Æ)
if 1 < 0 < expfh(f )g.
Proof of Theorem 2.
Proof of the first statement. Under the assumption of Theorem 2 we fix a num-
ber 0 with 1 < 0 < expfh(f )g. By Lemma 4, for j  1 there are kj , lj  1 and
Æ
j
> 0 with a (k
j
; l
j
; 0; Æj )-hyperbolic horseshoe 0j = 00
j
[    [ 0
k
j
 1
j
such that
h(f j
0
j
) = 1
k
j
log l
j
! h(f )
as j !1. For j  1 define a product space
6
j
=
1
Y
m=1
f1; : : : ; l
j
g
with the product topology and a shift 
j
: 6
j
! 6
j
by

j
((a
m
)
m1) = (am+1)m1 ((am)m1 2 6j ):
From the definition of hyperbolic horseshoe, there is a homeomorphism '
j
: 6
j
! 0
0
j
such that '
j
Æ 
j
= (f kj j
0
0
j
) Æ '
j
. Then p
j
= '
j
(1; 1; : : :) is a source of f . For m  1
and a1; : : : ; am 2 f1; : : : ; lj g with ai 6= 1 for some 1  i  m, 'j (a1; : : : ; am; 1; 1; : : :)
is a transversal homoclinic point of p
j
. Thus, TH(p
j
)  0
j
, from which
TOPOLOGICAL ENTROPY 9
h(f ) = lim
j!1
h(f j
0
j
)
 lim
j!1
h(f j
TH (p
j
))
 supfh(f j
TH (p)) : p is a source of f g
 h(f ):
The first statement was proved.
Proof of the second statement. Let p be a source of f . Without loss of general-
ity we may assume that p is a fixed point, i.e., f (p) = p. To show that for Æ > 0
h(f jTH(p))  lim sup
m!1
1
m
log ℄H (p;m; Æ);
take 0 = 0(Æ) > 0 so small that if x; y 2 I satisfy jx   yj  0 then
jf
0(x)  f 0(y)j  Æ
2
:
Then, for m  1 and a pair q; q 0 2 H (p;m; Æ) satisfying
maxfjf i(q)  f i(q 0)j : 0  i  m  1g  0;
we can find a sequence 0, : : :, m 1 2 I such that
j
i
  f
i(q)j  0
and
jf
i+1(q)  f i+1(q 0)j = jf 0(
i
)j  jf i(q)  f i(q 0)j (0  i  m  1):
Since f m(q) = f m(q 0) = p, we have
0 = jf m(q)  f m(q 0)j = jf 0(
m 1)j  jf m 1(q)  f m 1(q 0)j
=    =
m 1
Y
i=0
jf
0(
i
)j  jq   q 0j

m 1
Y
i=0

jf
0(f i(q))j   Æ
2

 jq   q
0
j


Æ
2

m
 jq   q
0
j;
and so q = q 0. Thus H (p;m; Æ) is an (m; 0)-separated set of f jTH(p), from which it
follows that
10 Y-.M. CHUNG
lim sup
m!1
1
m
log ℄H (p;m; Æ)  lim sup
m!1
1
m
log s(f jTH(p);m; 0)(1.3)
 h(f jTH(p)):
If h(f jTH(p)) = 0, then nothing to prove for the second statement. Thus we must check
the conclusion for the case when h(f jTH(p)) > 0. To do so fix a number 0 with
1 < 0 < minf(p); exp h(f jTH(p))g. By the same way as in the proof of Lemma 4,
we can take sequences of integers k
j
, l
j
 1, numbers Æ
j
> 0 with (k
j
; l
j
; 0; Æj )-
hyperbolic horseshoes 0
j
= 0
0
j
[    [ 0
k
j
 1
j
containing p (j  1) such that
h(f j
0
j
) = (1=k
j
)  log l
j
! h(f jTH(p)) as j !1:
Then there is a homeomorphism '
j
: 6
j
! 0
0
j
such that '
j
Æ
j
= (f kj j
0
0
j
)Æ'
j
, where

j
: 6
j
! 6
j
is the shift defined as in the proof of the first statement. Without loss
of generality we may assume that '
j
(1; 1; : : :) = p. By taking an integer n
j
 1 large
enough we have
'
j
([1; : : : ; 1]
n
j
)  W uloc(p)
where
[1; : : : ; 1]
n
j
= f(b
m
)
m1 2 6j : bm = 1 for all 1  m  nj g:
Since
'
j
(
n
j
times
z }| {
1; : : : ; 1; a1; : : : ; am n
j
; 1; 1; : : :) 2 H (p;mk
j
; Æ
j
)
holds for all m  n
j
+ 1 and a1, : : :, am n
j
2 f1; : : : ; l
j
g, we have
℄H (p;mk
j
; Æ
j
)  lm nj
j
:
Thus,
lim
Æ!0
lim sup
m!1
1
m
log ℄H (p;m; Æ)  lim sup
m!1
1
mk
j
log ℄H (p;mk
j
; Æ
j
)
 lim
m!1
m  n
j
mk
j
log l
j
=
1
k
j
log l
j
for j  1. If j !1, then we have
lim
Æ!0
lim sup
m!1
1
m
log ℄H (p;m; Æ)  h(f jTH(p)):(1.4)
TOPOLOGICAL ENTROPY 11
Combining (1.3) and (1.4),
h(f jTH(p))  lim
Æ!0
lim sup
m!1
1
m
log ℄H (p;m; Æ)
 h(f jTH(p)):
The second statement was proved. This completes the proof of Theorem 2.
2. Circle Maps
In the same way as above, it can be checked that our results (Theorems 1 and 2)
are also valid for C1+ maps ( > 0) of the circle S1. However, the existence of a
homoclinic point does not imply that the topological entropy is positive. In fact, we
know an example of a C1 map g : S1 ! S1 such that g has a homoclinic point of
a source fixed point, nevertheless h(g) = 0 ([16]). It is known that the topological en-
tropy of a continuous circle map is positive if and only if the map has a nonwandering
homocinic point of a periodic point ([4]). Since any transversal homoclinic point of a
source is nonwandering, we have:
Corollary 5. For a C1+ map f : S1 ! S1 ( > 0) the following statements are
equivalent:
(i) h(f ) > 0;
(ii) f has a transversal homoclinic point of a source;
(iii) f has a nonwandering homoclinic point of a periodic point.
Added in proof. After this manuscript was completed the author learned from
A. Katok that he and A. Mezhirov had obtained a result that overlaps with Theorem
1 for C1 maps with finitely many critical points ([18]).
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Department of Applied Mathematics
Faculty of Engineering
Hiroshima University
Higashi-Hiroshima 739-8527, Japan
e-mail: chung@amath.hiroshima-u.ac.jp


Topological entropy for differentiable maps of intervals

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Topological entropy for differentiable maps of intervals

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