C1−generic Pesin`s entropy formula - ICMC

C. R. Acad. Sci. Paris, t. 332, S´ erie I, p. 1–??, 2001 - PXMA????.TEX -
Syst` emes dynamiques/Dynamical Systems
C1−generic Pesin’s entropy formula
Ali Tahzibi
Departamento de Matem´ atica, Instituto de Ciˆ encias Matem´ aticas e de Computac¸˜ ao, Universidade de S˜ ao
Paulo - Campus de S˜ ao Carlos, Caixa Postal 668, 13560-970 S˜ ao Carlos SP, Brazil
E-mail: [email protected]r
(Rec¸u le 09 September 2002, accept´ e apr` es r´ evision le 14 Octobre 2002)
Abstract
The metric entropy of a C2−diffeomorphism with respect to an invariant smooth measure µis equal
to the average of the sum of the positive Lyapunov exponents of µ. This is the celebrated Pesin’s entropy
formula, hµ(f) = RMPλi>0λi. The C2regularity (or C1+α) of diffeomorphism is essential to the
proof of this equality. We show that at least in the two dimensional case this equality is satisfied for
aC1−generic diffeomorphism and in particular we obtain a set of volume preserving diffeomorphisms
strictly larger than those which are C1+αwhere Pesin’s formula holds.
La formule d’entropie de Pesin C1−g´ en´ erique
R´ esum´ e. L’entropie m´etrique d’un diff´eomorphisme C2, par rapport `a une mesure invariante est
´egale `a la moyenne de la somme des exposants de Lyapunov positifs. Ceci est la c´el`ebre
formule d’entropie de Pesin. La r´egularit´e du diff´eomorphisme est essentielle pour la preuve
de cette ´egalit´e. Nous montrons que en dimension deux, cette ´egalit´e est satisfaite pour un
diff´eomorphisme C1−g´en´erique et montrons qu’en particulier nous obtenons un ensemble
de diff´eomorphismes conservatifs contenant strictement ceux qui sont C1+α, o`u la formule
de Pesin est satisfaite. c 2001 Acad´emie des sciences/´ Editions scientifiques et m´edicales
Elsevier SAS
Version franc¸aise abr´ eg´ ee
Les exposants de Lyapunov d’une application diff´erentiable de M(une vari´et´e compacte) dans Msont
d´efinis par le th´eor`eme d’Oseledets. Soit µune mesure de probabilit´e invariante pour f; pour presque tout
point xil existe des nombres λ1(x)> λ2(x)>···> λk(x)(x)(les exposants) et une unique d´ecomposition
TxM=E1(x)⊕ · · · Ek(x)(x)tels que
lim
n→∞
1
nlog kDf n
x(v)k=λi(x)
pour tout 06=v∈Ei(x),16i6l. (dim(M) = l) Les exposants caract´eristiques d´efinis comme ci-
dessus sont en relation avec l’entropie de f. Par exemple pour une mesure invariante νet f∈C1, soit
χ(x) := Pλi>0λi; alors par un r´esultat de Ruelle :
hν(f)6ZM
χdm
Note pr´ esent´ ee par Jean Christophe YOCCOZ
S0764-4442(00)0????-?/FLA
c
2001 Acad´emie des sciences/´ Editions scientifiques et m´edicales Elsevier SAS. Tous droits r´eserv´es. 1

Ali Tahzibi
Le r´esultat est en g´en´eral une in´egalit´e stricte. Mais si mest absolument continue por rapport `a la mesure
de Lebesgue sur M, et f∈Diff1+α
m(M), α > 0, alors
hm(f) = ZM
χdm
En fait, on peut obtenir cette formule pour une plus grande classe de mesures.(voir [3]) Mais la r´egularit´e
de fest une condition n´ecessaire pour la preuve d’une telle ´egalit´e.
Nous allons montrer que si dim(M) = 2, il existe un sous-ensemble g´en´erique dans Diff1
m(M)o`u la
formule d’entropie de Pesin est satisfaite.
TH´ EOR` EME 0.1. – Il existe un sous-ensemble g´ en´ erique G∈Diff1
m(M), tel que toute f∈Gsatisfait
la formule d’entropie de Pesin et Gcontient strictement ∪α>0Diff1+α
m.
L’´etape clef de la d´emonstration sera de prouver que les points de continuit´e des deux fonctions suivantes
hm(.), L(.)forment une partie r´esiduelle dans la C1topologie.
Comme nous consid´erons des diff´eomorphismes conservatifs en dimension deux, il existe tout au plus
un exposant de Lyapunov positif. D´efinissons
–L(f) = RMλ1dm pour f∈Diff1
m(M)et
–hm(f) = l’entropie m´etrique de fpour f∈Diff1
m(M).
Maintenant nous proc´edonsen utilisant la formule d’entropie pour les diff´eomorphismesdans ∪Diff1+α
m(M).
Soit fun point de continuit´e pour L(.)et hm(.). Par la densit´e de Diff1+α
m(M)dans Diff1
m(M)prouv´ee
dans [4], il y a une suite fn∈Diff1+α
m(M)telle que fnconvergevers fdans la C1topologie. Par la formule
de Pesin, hm(fn) = L(fn)et par la continuit´e en f,hm(f) = L(f).
The Lyapunov exponents of a diffeomorphism fof a compact manifold Mare defined by Oseledets
theorem which states that, for any invariant probability measure µ, for almost all points x∈Mthere
exist numbers λ1(x)> λ2(x)>··· > λk(x)(x)(Lyapunov exponents) and a unique splitting TxM=
E1(x)⊕ · · · ⊕ Ek(x)(x)such that
lim
n→∞
1
nlog kDf n(x)vk=λi(x)
for all 06=v∈Ei(x),16i6m. The characteristic exponents defined as above are related to the entropy
of f. For example for any invariant measure νand f∈C1, let χ(x) := Pλi>0λi, then by a result of
Ruelle:
hν(f)6ZM
χdν.
An estimation from below in terms of positive Lyapunov exponents is not true for general invariant mea-
sures, but if the measure mis absolutely continuous with respect to the Lebesgue measure of M, Pesin’s
formula states that for m-preserving diffeomorphisms with H¨older continuous derivative, f∈Diff1+α
m(M)
hm(f) = ZM
χdm.
In fact this entropy formula holds for a larger class of measures [3]. But the regularity of fis always
used to get results of lower bounds for entropy.
We are going to show that if dim(M) = 2 then there exists a residual subset in Diff1
m(M)such that the
diffeomorphisms in this subset satisfy the Pesin’s entropy formula.
THEOREM 0.1. – There exists a C1-residual subset G⊂Diff1
m(M)such that any f∈Gsatisfy the
Pesin’s entropy formula and Gstrictly contains ∪α>0Diff1+α
m.
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C1generic Pesin’s entropy formula
The key idea is to prove that the set of the continuity points of the following two functions, L(.)and
hm(.)is residual in C1topology. As we are considering volume preserving diffeomorhisms in dimension
two, there exists at most one positive Lyapunov exponent. Define
•L(f) = RMPλi>0λi(x)dm for f∈Diff1
m(M)and
•hm(f) = the metric entropy of ffor any f∈Diff1
m(M).
Now we proceed by using the entropy formula for diffeomorphisms in ∪Diff1+α
m(M). Let fbe a conti-
nuity point for L(.)and hm(.). By density of Diff1+α
m(M)in Diff1 m(M)proved in [4], there is a sequence
fn∈Diff1+α
m(M)such that fnconverges to fin C1topology. By Pesin’s formula, hm(fn) = L(fn)and
by continuity at f,hm(f) = L(f).
1. Continuity points of L(f)and hm(f)
The continuous dependence of Lyapunov exponents on diffeomorphism is an important problem. In
fact let λ1(x, f)>λ2(x, f)>··· >λd(x, f)denotes all Lyapunov exponents of fand Λi(f) =
RMPi
j=1 λj(average of sum of the i-greatest exponents) then it is well known that f→Λi(f)is an up-
per semi-continuous function.
LEMMA 1.1. – The application f→Λi(f)is upper semi-continuous for f∈Diff1
m(M)
Proof. – By an standard argument we see that
Λi(f) = inf
n>1
1
nZM
log k ∧i(Df n(x))kdm(x).
In fact to see why the limit is substituted by infimum, observe that the sequence
an=ZM
log k ∧i(Df n(x))kdm(x)
is subadditive, i.e (an+m6an+am)and consequently lim an
n= inf an n.Now, as an(f)varies continu-
ously with fin C1topology and the infimum of continuous functions is upper semi-continuous, the proof
of the lemma is complete. 
Let us show that L(f)is an upper semi-continuous function in Diff1
m(M)independent of the dimension of
M.
LEMMA 1.2. – Let L(x, f) = Pλi>0λi(x, f)then f→L(f) = RML(x, f)dm(x)is upper semi-
continuous.
Proof. – Observe that the proof of this lemma for two dimensional case is the direct consequence of the
Lemma 1.1. In fact for any x∈M,limn→∞ 1
nlog k ∧pDfn(x)kexists and is equal to λ1+λ2(x) + ···+
λp(x).This functions varies upper semi-continuously with respect to f. From this we claim that for any x
the function f→Pλi>0λi(f, x)is upper semi-continuous. Because let f∈Diff1
m(M)and for x∈M,
λ1(x)>··· >λp(x)>0>λp+1(x)>··· >λd(x). Take Uǫa neighborhood of fsuch that for all
16k6dand any g∈Uǫ
k
X
i=1
λi(x, g)6
k
X
i=1
λi(x, g) + ǫ(1)
This is possible by means of Lemma 1.1. Now take any such gand let λ1(x)>··· >λ′
p(x)>0>
λp′+1 >··· >λd(x)for some 1< p′< d. Using (1) we see that Pp′
i=1 λi(x, g)6Pp i=1 λi(x, f) + ǫ
(consider the three cases p′< p, p =p′, p < p′) and the claim is proved.
Now we prove the lemma. By definition L(x, g)6Cfor some uniform Cin a neighborhood of f.
Define
An={x∈M;d(f, g)61
n⇒L(x, g)−L(x, f)6ǫ 2}
3

Ali Tahzibi
As m(∪An) = 1 then for some large nwe have m(An)>1−ǫ
4C.So,
ZM
L(x, g)−L(x, f)dm =ZAn
L(x, g)−L(x, f)dm +ZAc
n
L(x, g)−L(x, f)dm
6ǫ
2+ 2Cǫ 4C=ǫ
and the proof of the lemma is complete. 
The upper semi-continuity is the key for the proof of our main theorem, because by a classical theorem in
Analysis we know that the continuity points of a semi-continuous function on a Baire space is always a
residual subset of the space. (see e.g [2].)
The upper semi-continuity of hm(f)for fvarying in Diff1
m(M)is not known. In fact using Ruelles
inequality and Pesin’s equality we can show upper semi-continuity of hm(f)in the C2topology. (In this
paper all C2statements can be replaced by C1+α). Let g∈Diff2
m(M)be near enough to f, by semi
continuity of L(.)and Pesin’s equality in C2topology:
hm(g)6L(g)6L(f) + ǫ=hm(f) + ǫ
So, we pose the following question:
QUESTION 1.3. – Is it true that hm(f)is an upper semi-continuous function with C1volume preserving
diffeomorphisms as its domain.
However we are able to show that at least in two dimensional case the continuity points of hm(f)is generic
in C1topology.
PROPOSITION 1.1. – The continuity points of the map hm: Diff1
m(M)→Ris a residual set.
Proof. – We use the result of Bochi [1] which gives a C1generic subset G′=A∪Zsuch that any
g∈Ais Anosov and for g∈Zboth Lyapunov exponents vanish almost everywhere. We show that G′
contains a generic subset Gand each diffeomorphism in Gis a continuity point of hm. Firstly we state the
following lemma:
LEMMA 1.4. – Any f∈Zis a continuity point of hm.
Proof. – Let gbe near enough to fby Ruelle’s inequality and upper semi-continuity of L(.)we get
hm(g)6L(g)6L(f) + ǫ=ǫ
Now we prove that hm:A→Ris upper semi-continuous. As A⊂Diff1
m(M)is open we conclude that the
continuity points of hm|Ais generic inside A.
PROPOSITION 1.5. – hmrestricted to C1Anosov diffeomorphisms is upper semi-continuous.
Proof. – To prove the upper semi-continuity of hm|Awe recall the definition of hm(f). By a theorem
of Sinai we know that that if Pis a generating partition then
hm(f) = hm(f, P) = lim
n→∞
1
nHm(P∨f−1(P)· · · ∨ f−n+1(P)) (2)
If fis Anosov then there is ǫ > 0such that any gin a C1neighborhood of fis expansive with ǫas
expansivity constant. By the definition of generating partition any partition with diameter less than ǫis
generating and so we can choose a unique generating partition for a neighborhood of f. As mis a smooth
measure we see that the function
f→1
nHm(P∨f−1(P)· · · ∨ f−n+1(P)) = 1 nX
P
m(P) log(m(P))
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C1generic Pesin’s entropy formula
is continuous where the sum is over all elements of P∨f−1(P)· · · ∨ f−n+1(P).The limit in (2) can be
replaced by infimum and we know that the infimum of continuous function is upper semi-continuous. 
So, up to know we have proved that there exists a generic subset A′⊂Asuch that the diffeomorphisms in
G=A′∪Zare continuity point of hm. Now we claim that Gis C1generic in Diff1
m(M). To proof the
above claim we show a general fact about generic subsets.
LEMMA 1.6. – Let A∪Zbe a generic subset of a topological space Twhere Ais an open subset. If
A′⊂Ais generic inside Athen A′∪Zis also generic in T.
Proof. – As a countable intersection of generic subsets is also generic, we may suppose that A′is open
and dense in A. By hypothesis, A∪Z=∩nCnwhere Cnare open and dense. So, we have
A′∪Z=A′∪(∩Cn∩Ac) = ∩n(A′∪Cn)∩(A′∪Ac)(3)
First observe that each A′∪Cnis an open and dense subset and their intersection is generic. To complete
the proof it is enough to show that A′∪(Ac)◦⊆A′∪Acis open and dense. Openness is obvious and
denseness is left to reader as an easy exercise of general topology. 
So, as the intersection of generic subsets is again a generic set we conclude that there is generic subset of
Diff1
m(M)where the diffeomorphisms in this generic subset are the continuity point of both L(.)and hm(.)
and so for this generic subset the Pesin’s entropy formula is satisfied.
To finish the proof of the Theorem 0.1 we have to show that ∪α>0Diff1+α
m(M)is not a generic subset
and so the generic subset of Theorem 0.1 gives us some more diffeomorphisms satisfying Pesin’s formula
than ∪α>0Diff1+α
m(M).
LEMMA 1.7. – Diff1+
n(M) := Sα>0Diff1+α
m(M)is not generic with C1topology.
Proof. – We show that the complement of Diff1+
m(M)is a generic subset and this implies that
Diff1+
m(M)can not be generic.
As in what follows we are working locally, one may suppose that M=R2.Let’s define
kfkα=Supx6=y∈M
d(Df (x), Df (y))
d(x, y)α
and denote
Hn,k ={f∈Diff1
m(M),kfk1
n> k}.
By the above definition we get Diff1
m(M)\Diff1+
m(M) = Tn,m∈NHn,m. To prove the lemma We claim
that for any n, each Hn,k is an open dense subset.
1. Openness
Let f∈Hn,k,by definition there exist x, y and η > 0such that d(Df (x), Df (y))
d(x, y)α> k +η. Take any g,
ǫ-near to fin C1topology by the triangular inequality we get:
kgk1
n>d(Dg(x), Dg(y))
d(x, y)1
n
>d(Df (x), Df (y))
d(x, y)1
n
−2ǫ
(d(x, y)) 1
n
Taking ǫsmall enough the above inequality shows that kgk1
n> k and the openness is proved.
2. Density
Let f∈Diff1
m(M)we are going to find g∈Diff1 m(M)\Diff1+ m(M)such that gis near enough to f. For
this purpose we construct h∈Diff1
m(M)\Diff1+
m(M)near enough to identity and then put g=h◦f.
Considering local charts, it is enough to construct a C1volume preserving diffeomorphism ˜ Ifrom R2to
R2such that:
5