Entropy in Smooth Dynamical Systems 1. Introduction /7,(a,/) = lim

Entropy in Smooth Dynamical Systems
Sheldon
E.Newhouse
Department of Mathematics, University of North Carolina
Chapel Hill, NC 27599-3250, USA
1. Introduction
The topological entropy of a continuous dynamical system is now well established
as an important invariant of the system. It was first defined by Adler, Konheim,
and McAndrew [AKM] in 1965 using open coverings *. Nowadays it is convenient
to think of the topological entropy as a limit of the number of the distinct orbits
of a given length which can be obtained with a fixed small precision. Thus, it
measures the orbit growth of the system.
More precisely, suppose that / : M —• M is a continuous self-mapping of
the compact metric space M with metric d. Given a positive integer n, and a
small real number ö > 0, we say that a set E is an (7i,<5)-separated set if, for any
x ^ y e E there is a j e [0,n) such that d(fJx,fjy)
> ö. Let r(n,ö,f) be the
maximum cardinality of an (n, ö)-separated set in M. Let
h(ö,f) = lim sup - log r(n, ö9f)
n—>oo
n
and
h(f) = limh(ö,f) =
<5-»0
suph(ö,f).
^>o
This is the topological entropy of / . The most interesting properties of h(f) arise
from its relation to set of invariant probability measures of / . Let us denote this
set by Jt(f), and recall that it is a compact metrizable set. Given p e J4(f),
the measure-theoretic or metric entropy, hfi(f)9 is defined as follows. For a finite
Borei measurable partition a, let
Given two finite partitions a, ß, set a V ß = {A f] B : A e a, B e ß}. Then, set
1
n_1
/7,(a,/) = lim-H,(Vr/(a))
i=0
1
Adler recently pointed out to us that topological entropy generalizes Shannon's notion
of channel capacity (see [SW, p.7])
Proceedings of the International Congress
„ r Aifn)i,„ m n < : n ;n« n v,,~+~ T„„n„ l o o n
1286
Sheldon E. Newhouse
and
hfl(f) = sup^(a,/) =
lim
Ma,/).
diam a->0
a
The following properties hold:
1. h(fn) = nh(f), hfl(fn) = nh,(f) for n e Z+
2. h(fl) = | 11/iC/1) if {f}teR is a continuous flow.
3. h(f) = s u p ^ ^ h„(f)
4. fr(f) = M#) if/ is topologically conjugate to g; i.e., there is a homeomorphism
cj) with 0 / ^ _ 1 = g.
5. /i(/) can be computed using (n, ö)— separated subsets of sets of large measure
for various p e J?(f)(sQG [Nl]). That is,
h(f)=
sup lim inf ]imr(ö,A)
where
r(ö, A) = Hm sup - log f (ö, n, A)
n—»oo
n
and r(ö,n,A) is the maximal cardinality of an (n,<5)-separated subset of A.
The relationship between topological and metric entropies (statement 3 above)
is a combination of the work of Goodman, Goodwyn, and Dinaburg. It is referred
to as the Variational Principle for Topological Entropy. An elegant proof has
been given by Misiurewicz (see [DGS], pp. 140-146 for a more general result).
Statement 4 above states that h(f) is a topological conjugacy invariant of / .
It is generally not a complete invariant of topological conjugacy. However, for
certain important systems it is close to being complete. For instance, Adler and
Marcus [AM] have shown that two mixing subshifts of finite type with the same
topological entropy are almost topologically conjugate in the sense that each is
a boundedly finite to one factor of a third mixing subshift of finite type. Also,
Milnor and Thurston [MT] have shown that a piecewise monotone continuous
map / of an interval with positive topological entropy is semi-conjugate in a
simple way to a piecewise linear map g with the same number of turning points
as / and such that the slope of each monotone piece of g has absolute value
equal to eh^K
It was known quite early that some form of regularity affected the finiteness of the topological entropy. Embedding a sequence of larger and larger
shift automorphims topologically in a homeomorphism of the two-sphere shows
that homeomorphisms need not have finite topological entropy. However, Kushnirenko proved that every C1 self-map of a compact manifold has finite topological entropy. A natural question arises: When are there measures p for which
hpif) = h(f)l. Misiurewicz was the first to construct examples of Ck diffeomorphisms of compact manifolds with no measure of maximal entropy. His first
examples were on manifolds of dimension greater than three, but now it is known
that such examples arise in small perturbations of diffeomorphisms having a degenerate homoclinic orbit (an intersection of stable and unstable manifolds of a
Entropy in Smooth Dynamical Systems
1287
hyperbolic saddle point with infinite order contact) even on the two dimensional
sphere. His construction worked for everyfinitek, but it failed in the C00 case. The
question of the existence of measures of maximal entropy was quite important.
One consequence of the results described here is that for every C°° self-map / of
a compact smooth manifold, the function p -» h^if) is uppersemicontinuous on
J4(f). Hence, / does indeed possess measures of maximal entropy.
In this article we shall survey several recent results about topological and
metric entropy, particularly as they relate to smooth systems. We view the results
described here as part of the natural evolution of certain topological aspects of
the qualitative theory of dynamical systems following the rieh development in the
sixties due, in large part, to Anosov, Sinai, and Smale. The recent developments
on quadratic mappings due to Carleson and Benedicts may be viewed as part of
the evolution of certain quantitative aspects of this theory. In the case of uniformly
hyperbolic dynamics, these two types of aspects come together beautifully in the
theory of equilibrium states as described, for example, in [B2]. It is natural to
search for a generalization of the Equilibrium State Theory which encompasses
all of these results.
2. Entropy and Volume
To motivate the results, we first consider the case of mappings of the interval.
Let M be the unit interval and let / : M —> M be a continuous map with finitely
many turning points. Let log+(x) = max(logx,0).
Let *f(/) = length of image of/ with multiplicities:
m = [ I fix) I dx
Theorem 1 (Misiurewicz-Szlenk [MS]). With f as above, the following results hold.
1. /iCf)= lim,^ œ ilog + £{f>).
2. p —> h^(f) is uppersemicontinuous on M(f).
3. / —• h(f) is uppersemicontinuous for f C1 where one perturbs in the C1 topology keeping the same number of turning points.
4. / —> h(f) is lowersemicontinuous for certain f.
Theorem 2 (Misiurewicz [M2]). The map f -> h(f) is lowersemicontinuous for all
c°f.
Corollary 3. The map f —• h(f) is continuous for f in the set of C1 maps with a
uniformly bounded number of turning points.
This last corollary was also proved by Milnor and Thurston [MT] .
We consider a direct generalization of the preceding results. A clue as to how
to proceed comes from work of Margulis in the sixties on the geodesic flow on
compact negatively curved manifolds. Following his work, it became known that
1288
Sheldon E. Newhouse
the topological entropy of the time-one map equals the maximum volume growth
rate of compact disks in the unstable manifolds.
Let Dk be the unit fc-disk in Rfe, and let M be a C00 manifold. Let Cr(M,M)
be the space of Cr self maps of M , and let <®r(M,M) be the space of Cr
diffeomorphisms of M where r is an integer greater than 1.
A Cr fc-disk in M is a Cr map y : Dk —> M.
Define the fc-volume of y by
| y \k = /
k
| AkTy \dX
JD
where dX is Lebesgue measure on Dk, and ^ T y is the fc—th exterior power of
the derivative Ty
For / G Cr(M, M), let A be a compact /—invariant set, and let U be a compact
neighborhood of A Given a positive integer n, set Ws(n, U) = C\o<j<nf~j(U).
This is just the set of points whose iterates from time 0 through n — 1 remain in
U. For a fc-disk y in 17, set
Gk(y,f, U) = lim sup - log + (| f" 1 o y | r W f a £/)) | fc ).
H->00
W
This is the volume growth of the / n _ 1 -st iterate of the part of y which remains
in U from time 0 through time n — 1.
A collection sé of fe-disks in U with 1 < k < dim M, is called ample for yl if
it contains a subcollection J / I for which 3K > 0 such that
1. K" 1 < | Dxy(v) | < X and | D2xy(v,v) \ < K
Vx G domain y, | u | = 1 and y e i i
2. For x e A, and for each fc-dimensional subspace H of TXM, there exists a
sequence of fe-disks yi,y2,-..,€ sé\ whose tangent spaces at y,(0) approach H in
the Grassmann sense as i -» oo.
If M is complex analytic with a hermitian metric, A and 1/ are as above, and
sé is a collection of holomorphic disks in U, then J?/ is holomorphically ample or
h-ample if, in condition 2 above, if is assumed to be a complex subspace of the
holomorphic tangent space of TXM.
Clearly the family of all disks through points in A is ample for A. If M = R N ,
with the usual metric, then the collection sé of affine disks through points in A
is ample.
Theorem 4 gives an upper bound for entropy in terms of volume growth rates
of smooth disks. An earlier upper bound in terms of the average (relative to
Lebesgue measure) of the maximum growth of the norms of the exterior powers
of the derivative had been obtained by Przytycki [P] for diffeomorphisms.
Entropy in Smooth Dynamical Systems
1289
Theorem4 [Nl]. Suppose f e Cr(M,M), A is a compact invariant set, U is a
compact neighborhood of A, and sé is an ample family of Cr disks for A, Iff and
M are complex analytic assume that sé is h-ample. Then,
h(f\A)<^suipG(y,f,U).
yes/
r
Iffe@> {M,M),then
h(f\A)<
sup
G(y, f,U).
yes/
dim y < dim M
In particular, for f e @r(M2), h(f | A) < supcurves y G(y,f, U).
Using Theorem 4, one can give simple proofs of the following results.
Theorem 5. 1. Let f : RN -* RN be a polynomial map with coordinate functions
of degree < d; i.e., for x G RN,f(x) = ( / I ( X ) , / 2 ( X ) , . . . , / J V ( X ) ) with //(x) a
polynomial in x of degree < d. If A is a compact invariant set for f, then
h(f \A)<Nlogd
2
(Gromov).
2
2. If f : S -* S is a rational map of degree d, then
h(f) < log d (Gromov — Ljubich).
3. If f : 0>N(C) - • &N(C) is globally defined and holomorphic, then h(f) <
log(topological degree) (Gromov)
S. Friedland [F] has obtained results on volume growth in quasi-projective
varieties which generalize Theorem 5.1 .
It is well-known (Misiurewicz-Przytycki [MP] ) that if M is compact and
/ : M —> M is C1, then h(f) > log(topological degree).
So, Theorems 5.2 and 5.3 above are equalities.
From now on, we assume that M is a compact C00 manifold.
The above theorems give an upper estimate of h(f) in terms of volume growth
rates of disks. For the lower estimate we have
Theorem 6 (Yomdin [Yl]). For f e ^(M,]^),
and any C00 disk y in M,
h(f)>G(y,f).
Yomdin's results can be used to give a proof of a generalization of the
Shub entropy conjecture in the C00 case. To recall this conjecture, first define
the homology growth of a map / , HG(f), to be lim sup,,.^ \ log | / ï | where
/* : if*(M,R) —• if*(M,R) is the induced map on the direct sum of the real
homology groups of/ (given any norm). The entropy conjecture states that for
a C 1 diffeomorphism / of the compact manifold M, one has h(f) > HG(f). Of
course, HG(f) is the same as the maximum logarithm of the absolute values of
the eigenvalues of/* : #*(M,R) -> H*(M,R). Yomdin's results show that this
holds for arbitrary C00 maps.
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Sheldon E. Newhouse
Corollary 7 (Yomdin) (C00 Entropy Conjecture). Iffe
HG(f).
C°°(M, Af), then h(f) >
We note that the Entropy Conjecture fails in general for piece-wise linear
homeomorphisms although it is true for "typical" piecewise linear maps. There
is a large literature on various cases of the entropy conjecture (see [FS]). The
general conjecture is still unproved for / G &'(M,M) with 1 < r < oo.
The next result states that, for positive h(f), there always exist disks y for
which G(y,f) assumes the maximum value. In addition, it can be shown that
there are such disks for which G(y,f) is actually a limit and not just a lim sup.
Theorem 8. For f G Cœ(M,M),
sé, an ample family of C°° disks, we have
h(f) = sup G(y,f)= max G(y,f).
ye*/
y^
In general, the disks y for which G(y,f) = h(f) are not easily identifiable.
However, for an area decreasing self-embedding of a surface with boundary, the
entropy is just the growth rate of the length of the boundary.
Theorem 9 [N2]. For f G Dœ(M2),dM2
^ 0 , / area decreasing, we have
h(f) = G(ÔM2,f).
3. Continuity Properties of Entropy
The methods used in the proofs of the above results concerning volume growth
and entropy have local analogs by which we mean that one considers the growth
rates of the cardinalities of (n,ô)—separated sets or of the volumes of disks
which remain in small neighborhoods of the orbits of given points. These can
be combined with general results estimating the defect in uppersemicontinuity of
both topological and metric entropy to obtain various continuity properties of
entropy for C00 systems.
We begin with a description of the so-called local entropy of a dynamical
system / : M -* M.
Given A a M,x e A,ne Z+,s > 0, set
Wx(n,E,A) = {yeA:
dtfy,fx)
< E VJ G [0,n)},
r(n,ö,E,A) = supmax{card E : E cz Wx(n,s,A),E is (n,8) — separated},
xeA
r(E, A) = lim lim sup - log r(n, ö, E, A).
<5-»o
n
n
Ifp€ Ji(f), let
hfAOC(s,f) = lim inf r(£,A),
ff->l fl{A)>G
and set,
h\oc(E,f) = sup/^ioc(e,/).
Entropy in Smooth Dynamical Systems
1291
The quantity h\oc(E,f) is called the E— local entropy of / , and h^\0C(E,f) is
called the c—local entropy of / relative to p.
The next theorem states that h\oc(E,f) gives an upper bound for the difference
h(f) — h(E,f) while h^ocfaf) gives and upper bound for the difference h^(f) —
hfl(ß,f) for any partition ß with diameter less than e. Earlier estimates of these
differences were given by Bowen in [B].
Theorem 10 [N2]. For any continuous self map f of the compact metric space M,
and e > 0,
1. h(f)<h(E,f) + hÌ0C(E,f)
2. If p G M(f) and ß is a finite Borei partition with diam ß < E, then
h,(f)<h(i(ß,f)
+ h,loc(E,f)
Next we consider local volume growth.
Let Wx(n,E) = Wx(n,E,M).
For a disk y, set
ewe, y) =lim
SU
H->OO
P - l o g + S ^P I /"_1 ° y I 7"1 (Wx (w, e)) I
n
Xç-M
and
G\0C(EJ)
= supG]oc(ß,y)
y
Theorem 11 [N2]. For f G Cr(M,M), r > 1,
hoc(s,f) < Gioc(2E,f)
Theorem 12 (Yomdin[Yl]). For f G
^(M,^!),
lim G\0C(E,f) =0
From Theorems 10,11,12 with some elementary arguments (see [N2]) we have
Theorem 13. 1. For f G ^(M,!^)^
—> h^ff) is uppersemicontinuous
2. / —• h(f) is uppersemicontinuous on C™(M,M).
Yomdin [Yl] has an independent proof of Theorem 13.2.
In general, / -> h(f) is not lowersemicontinuous, but Katok has proved that
this does hold on surfaces.
Theorem 14 (Katok). / —• h(f) is lowersemicontinuous on @2(M2)
Corollary 15. / -> h(f) is continuous on ^°°(M 2 ).
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Sheldon E. Newhouse
Question 16. Is the preceding map Holder continuous?
Answer: No
Yomdin [Y2] has examples of curves {ft} of real analytic maps on S2, t G [—e, e]
with h(ft) = 0 for t G [-s, 0], and for t G (0, e),
h(ft)~h(fo)>C-
,log| logt I
logt\
In view of Corollary 15, we wish to point out some analogies between the
entropy map / -> h(f) on ^(M2)
and the rotation number map / —> g(f) G R / Z
on Homeo+(Sx), the set of orientation preserving homeomorphisms of the circle.
Both h(f) and g(f) are topological invariants which depend continuously on / .
Moreover, g(f) is rational iff* / has periodic points for / G Homeo+(S{) while
(as proved by Katok) h(f) is positive iff / has transverse homoclinic points for
/ G ^°°(M 2 ).
Problems
1. (Monotonicity of entropy) For fr(x) = r — x2, the function r -> h(fr) is
monotone increasing (Douady and Hubbard). What about r -> /i(/?>&) for fixed b
with frtb(x, y) = (r - x2 + fcy,x) ?.
2. Let ^max(/) denote the set of measures of maximal entropy for a mapping
/ . As a consequence of Theorem 13, for any C00 / , J?max(f) Ì 0- For / € ^°°(M 2 )
with h(f) > 0, is ^maxt/) a finite dimensional simplex?
Related to this problem, Hofbauer ([H]) has developed an interesting theory concerning piecewise monotone mappings of an interval with finitely many
monotone continuous pieces. His theory can be described by introducing the
notion of a zero-entropy set (0-entropy set). Hofbauer calls these small sets.
Let / : X —> X be a Borei automorphism of a standard Borei space. A
0-entropy set is an f-invariant subset I j c l such that for any ergodic p G J£(f),
with p(X\) = 1, we have h^(f) = 0. By convention, if Jt(f) = 0, then, every
invariant subset of X is a 0-entropy set. Periodic orbits are simple 0-entropy
sets as are stable manifolds of hyperbolic periodic orbits in the smooth setting.
There is a natural notion of isomorphism mod 0-entropy: Borei automorphisms
(f,X), (g, Y) are isomorphic mod 0-entropy if there are 0-entropy sets X\ <= X,
Y\ c Y, and a Borei isomorphism cj) : X \ X\ -> Y \ Yi, with gej) = cj)f. We say
two Borei endomorphisms / : X -> X, g : Y —> Y are isomorphic mod 0-entropy
if their natural extensions are isomorphic mod 0-entropy. Finally, we say that
the Borei endomorphism (f, X) is Markov mod 0-entropy if it is isomorphic mod
0-entropy to a finite or countable state topological Markov chain (cr, IA). We will
say that a measure preserving endomorphism (X,f,p) is essentially Markov if its
natural extension (X,f,p) is isomorphic to a Markov process. In this case we
also say that p is essentially Markov.
Theorem 17 (Hofbauer). Let f : I —> J be a piecewise monotone map of the interval. Then, (f, I) is Markov mod 0-entropy. Moreover, there are only finitely many
ergodic measures of maximal entropy and each is essentially Markov.
Entropy in Smooth Dynamical Systems
1293
Moving to general C°° maps of an interval with positive topological entropy,
we can prove that there are at most a countable number of ergodic measures of
maximal entropy, and that each is essentially Markov.
In dimension greater than 1, it is not always true that positive entropy implies
that Jtmax(f) is a finite dimensional simplex: take the direct product of the
identity transformation and any transformation with positive entropy. However,
it is possible that, generically, i.e. for elements of a residual set of diffeomorphisms,
^max(f) is a finite dimensional simplex.
3. For / G @r(M2),h(f) = 0, can / be C perturbed to be Morse-Smale?
This is true if the limit set of / is finite and hyperbolic [MaP], but not even
known if the non-wandering set is finite.
4. For / G 3e0(M2), let %+(x) denote the positive characteristic exponent of x
(defined for a total probability set of x). Let c/)(x) = — x+ (x). Then, c6 is bounded
and Borei measurable. Define
P(cj))= sup hfi(f) + Icj>dp.
Is there always a po such that
P{<t>) = hlia(S) +
jwn«.
This is true for continuous ç6. Such pf0s are called co-equilibrium states. For
a hyperbolic attractor A and any p supported in the basin of A and absolutely
continuous with respect to Lebesgue measure, it is known that
1 "- 1
lim-Yfk(p)
k=0
is the unique ^-equilibrium state on A (Ruelle). In general, one would expect
^-equilibrium states to be related to weak limits of the measures {^ Xl/c^o/*^)}In this connection, Pesin and Sinai have shown that the weak limits of the iterates
of Lebesgue measure have absolutely continuous densities along unstable leaves
for partially hyperbolic attractors [PS].
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