C1−generic Pesin`s entropy formula - ICMC

C. R. Acad. Sci. Paris, t. 332, Série I, p. 1–??, 2001
Systèmes dynamiques/Dynamical Systems
- PXMA????.TEX -
C 1−generic Pesin’s entropy formula
Ali Tahzibi
Departamento de Matemática, Instituto de Ciências Matemáticas e de Computação, Universidade de São
Paulo - Campus de São Carlos, Caixa Postal 668, 13560-970 São Carlos SP, Brazil
E-mail: [email protected]
(Reçu le 09 September 2002, accepté après révision le 14 Octobre 2002)
Abstract
The metric entropy of a C 2 −diffeomorphism with respect to an invariant smooth measure µ is equal
to the average of theRsumPof the positive Lyapunov exponents of µ. This is the celebrated Pesin’s entropy
formula, hµ (f ) = M λi >0 λi . The C 2 regularity (or C 1+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
a C 1 −generic diffeomorphism and in particular we obtain a set of volume preserving diffeomorphisms
strictly larger than those which are C 1+α where Pesin’s formula holds.
La formule d’entropie de Pesin C 1 −générique
Résumé.
L’entropie métrique d’un difféomorphisme C 2 , par rapport à une mesure invariante est
égale à la moyenne de la somme des exposants de Lyapunov positifs. Ceci est la célèbre
formule d’entropie de Pesin. La régularité du difféomorphisme est essentielle pour la preuve
de cette égalité. Nous montrons que en dimension deux, cette égalité est satisfaite pour un
difféomorphisme C 1 − générique et montrons qu’en particulier nous obtenons un ensemble
de difféomorphismes conservatifs contenant strictement ceux qui sont C 1+α , où la formule
c 2001 Académie des sciences/Éditions scientifiques et médicales
de Pesin est satisfaite. Elsevier SAS
Version française abrégée
Les exposants de Lyapunov d’une application différentiable de M (une variété compacte) dans M sont
définis par le théorème d’Oseledets. Soit µ une mesure de probabilité invariante pour f ; pour presque tout
point x il existe des nombres λ1 (x) > λ2 (x) > · · · > λk(x) (x) (les exposants) et une unique décomposition
Tx M = E1 (x) ⊕ · · · Ek(x) (x) tels que
lim
n→∞
1
log kDfxn (v)k = λi (x)
n
pour tout 0 6= v ∈ Ei (x), 1 6 i 6 l. (dim(M ) = l) Les exposants caractéristiques définis comme ci1
dessus sont
P en relation avec l’entropie de f . Par exemple pour une mesure invariante ν et f ∈ C , soit
χ(x) := λi >0 λi ; alors par un résultat de Ruelle :
Z
χdm
hν (f ) 6
M
Note présentée par Jean Christophe YOCCOZ
S0764-4442(00)0????-?/FLA
c 2001 Académie des sciences/Éditions scientifiques et médicales Elsevier SAS. Tous droits réservés.
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Ali Tahzibi
Le résultat est en général une inégalité stricte. Mais si m est absolument continue por rapport à la mesure
de Lebesgue sur M , et f ∈ Diff 1+α
m (M ), α > 0, alors
Z
χdm
hm (f ) =
M
En fait, on peut obtenir cette formule pour une plus grande classe de mesures.(voir [3]) Mais la régularité
de f est une condition nécessaire pour la preuve d’une telle égalité.
Nous allons montrer que si dim(M ) = 2, il existe un sous-ensemble générique dans Diff 1m (M ) où la
formule d’entropie de Pesin est satisfaite.
T H ÉOR ÈME 0.1. – Il existe un sous-ensemble générique G ∈ Diff 1m (M ) , tel que toute f ∈ G satisfait
la formule d’entropie de Pesin et G contient strictement ∪α>0 Diff 1+α
m .
L’étape clef de la démonstration sera de prouver que les points de continuité des deux fonctions suivantes
hm (.), L(.) forment une partie résiduelle dans la C 1 topologie.
Comme nous considérons des difféomorphismes conservatifs en dimension deux, il existe tout au plus
un exposant de
R Lyapunov positif. Définissons
– L(f ) = M λ1 dm pour f ∈ Diff 1m (M ) et
– hm (f ) = l’entropie métrique de f pour f ∈ Diff 1m (M ).
Maintenant nous procédons en utilisant la formule d’entropie pour les difféomorphismes dans ∪ Diff 1+α
m (M ).
1
Soit f un point de continuité pour L(.) et hm (.). Par la densité de Diff 1+α
(M
)
dans
Diff
(M
)
prouvée
m
m
1
dans [4], il y a une suite fn ∈ Diff 1+α
m (M ) telle que fn converge vers f dans la C topologie. Par la formule
de Pesin, hm (fn ) = L(fn ) et par la continuité en f , hm (f ) = L(f ).
The Lyapunov exponents of a diffeomorphism f of a compact manifold M are defined by Oseledets
theorem which states that, for any invariant probability measure µ, for almost all points x ∈ M there
exist numbers λ1 (x) > λ2 (x) > · · · > λk(x) (x) (Lyapunov exponents) and a unique splitting Tx M =
E1 (x) ⊕ · · · ⊕ Ek(x) (x) such that
1
log kDf n (x)vk = λi (x)
n→∞ n
lim
for all 0 6= v ∈ Ei (x), 1 6 i 6 m. The characteristic exponents defined as above
P are related to the entropy
of f . For example for any invariant measure ν and f ∈ C 1 , let χ(x) := λi >0 λi , then by a result of
Ruelle:
Z
χdν.
hν (f ) 6
M
An estimation from below in terms of positive Lyapunov exponents is not true for general invariant measures, but if the measure m is absolutely continuous with respect to the Lebesgue measure of M , Pesin’s
formula states that for m-preserving diffeomorphisms with Hölder continuous derivative, f ∈ Diff 1+α
m (M )
Z
χdm.
hm (f ) =
M
In fact this entropy formula holds for a larger class of measures [3]. But the regularity of f is 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 Diff 1m (M ) such that the
diffeomorphisms in this subset satisfy the Pesin’s entropy formula.
T HEOREM 0.1. – There exists a C 1 -residual subset G ⊂ Diff 1m (M ) such that any f ∈ G satisfy the
Pesin’s entropy formula and G strictly contains ∪α>0 Diff 1+α
m .
2
C 1 generic 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 C 1 topology. As we are considering volume preserving diffeomorhisms in dimension
two, there exists
most one positive Lyapunov exponent. Define
R atP
• L(f ) = M λi >0 λi (x)dm for f ∈ Diff 1m (M ) and
• hm (f ) = the metric entropy of f for any f ∈ Diff 1m (M ).
Now we proceed by using the entropy formula for diffeomorphisms in ∪ Diff 1+α
m (M ). Let f be a conti1
nuity point for L(.) and hm (.). By density of Diff 1+α
(M
)
in
Diff
(M
)
proved
in [4], there is a sequence
m
m
1
fn ∈ Diff 1+α
(M
)
such
that
f
converges
to
f
in
C
topology.
By
Pesin’s
formula,
hm (fn ) = L(fn ) and
n
m
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 f and Λi (f ) =
R Pi
j=1 λj (average of sum of the i-greatest exponents) then it is well known that f →Λi (f ) is an upM
per semi-continuous function.
L EMMA 1.1. – The application f →Λi (f ) is upper semi-continuous for f ∈ Diff 1m (M )
Proof . – By an standard argument we see that
Z
1
Λi (f ) = inf
log k ∧i (Df n (x))kdm(x).
n>1 n M
In fact to see why the limit is substituted by infimum, observe that the sequence
Z
log k ∧i (Df n (x))kdm(x)
an =
M
is subadditive, i.e (an+m 6 an + am ) and consequently lim ann = inf ann . Now, as an (f ) varies continuously with f in C 1 topology 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 Diff 1m (M ) independent of the dimension of
M.
R
P
L EMMA 1.2. – Let L(x, f ) =
λi >0 λi (x, f ) then f →L(f ) = M L(x, f )dm(x) is upper semicontinuous.
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→∞ n1 log k ∧p Df n (x)k exists and is equal to λ1 + λ2 (x) + · · · +
λp (x). This functions
P varies upper semi-continuously with respect to f. From this we claim that for any x
the function f → λi >0 λi (f, x) is upper semi-continuous. Because let f ∈ Diff 1m (M ) and for x ∈ M ,
λ1 (x) > · · · > λp (x) > 0 > λp+1 (x) > · · · > λd (x). Take Uǫ a neighborhood of f such that for all
1 6 k 6 d and any g ∈ Uǫ
k
k
X
X
λi (x, g) + ǫ
(1)
λi (x, g) 6
i=1
i=1
This is possible by means of Lemma 1.1. Now take any such g and let λ1 (x) > · · · > λ′p (x) > 0 >
P
P ′
λp′ +1 > · · · > λd (x) for some 1 < p′ < d. Using (1) we see that pi=1 λi (x, g) 6 pi=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) 6 C for some uniform C in a neighborhood of f .
Define
1
ǫ
An = {x ∈ M ; d(f, g) 6 ⇒ L(x, g) − L(x, f ) 6 }
n
2
3
Ali Tahzibi
ǫ
As m(∪An ) = 1 then for some large n we have m(An ) > 1 − 4C
. So,
Z
Z
Z
L(x, g) − L(x, f )dm +
L(x, g) − L(x, f )dm =
6
L(x, g) − L(x, f )dm
Acn
An
M
ǫ
ǫ
+ 2C
=ǫ
2
4C
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 f varying in Diff 1m (M ) is not known. In fact using Ruelles
inequality and Pesin’s equality we can show upper semi-continuity of hm (f ) in the C 2 topology. (In this
paper all C 2 statements can be replaced by C 1+α ). Let g ∈ Diff 2m (M ) be near enough to f , by semi
continuity of L(.) and Pesin’s equality in C 2 topology:
hm (g) 6 L(g) 6 L(f ) + ǫ = hm (f ) + ǫ
So, we pose the following question:
Q UESTION 1.3. – Is it true that hm (f ) is an upper semi-continuous function with C 1 volume 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 C 1 topology.
P ROPOSITION 1.1. – The continuity points of the map hm : Diff 1m (M )→R is a residual set.
Proof . – We use the result of Bochi [1] which gives a C 1 generic subset G′ = A ∪ Z such that any
g ∈ A is Anosov and for g ∈ Z both Lyapunov exponents vanish almost everywhere. We show that G′
contains a generic subset G and each diffeomorphism in G is a continuity point of hm . Firstly we state the
following lemma:
L EMMA 1.4. – Any f ∈ Z is a continuity point of hm .
Proof . – Let g be near enough to f by Ruelle’s inequality and upper semi-continuity of L(.) we get
hm (g) 6 L(g) 6 L(f ) + ǫ = ǫ
Now we prove that hm : A→R is upper semi-continuous. As A ⊂ Diff 1m (M ) is open we conclude that the
continuity points of hm |A is generic inside A.
P ROPOSITION 1.5. – hm restricted to C 1 Anosov diffeomorphisms is upper semi-continuous.
Proof . – To prove the upper semi-continuity of hm |A we recall the definition of hm (f ). By a theorem
of Sinai we know that that if P is a generating partition then
hm (f ) = hm (f, P) = lim
n→∞
1
Hm (P ∨ f −1 (P) · · · ∨ f −n+1 (P))
n
(2)
If f is Anosov then there is ǫ > 0 such that any g in a C 1 neighborhood of f is expansive with ǫ as
expansivity constant. By the definition of generating partition any partition with diameter less than ǫ is
generating and so we can choose a unique generating partition for a neighborhood of f . As m is a smooth
measure we see that the function
1
1X
f → Hm (P ∨ f −1 (P) · · · ∨ f −n+1 (P)) =
m(P ) log(m(P ))
n
n
P
4
C 1 generic 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′ ⊂ A such that the diffeomorphisms in
G = A′ ∪ Z are continuity point of hm . Now we claim that G is C 1 generic in Diff 1m (M ). To proof the
above claim we show a general fact about generic subsets.
L EMMA 1.6. – Let A ∪ Z be a generic subset of a topological space T where A is an open subset. If
A′ ⊂ A is generic inside A then A′ ∪ Z is 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 = ∩n Cn where Cn are open and dense. So, we have
A′ ∪ Z = A′ ∪ (∩Cn ∩ Ac ) = ∩n (A′ ∪ Cn ) ∩ (A′ ∪ Ac )
(3)
First observe that each A′ ∪ Cn is an open and dense subset and their intersection is generic. To complete
the proof it is enough to show that A′ ∪ (Ac )◦ ⊆ A′ ∪ Ac is 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
Diff 1m (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>0 Diff 1+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>0 Diff 1+α
m (M ).
S
1+α
1
L EMMA 1.7. – Diff 1+
n (M ) :=
α>0 Diff m (M ) is not generic with C topology.
Proof . – We show that the complement of Diff 1+
m (M ) is a generic subset and this implies that
Diff 1+
m (M ) can not be generic.
As in what follows we are working locally, one may suppose that M = R2 . Let’s define
kf kα = Supx6=y∈M
d(Df (x), Df (y))
d(x, y)α
and denote
Hn,k = {f ∈ Diff 1m (M ), kf k n1 > k}.
T
By the above definition we get Diff 1m (M ) \ Diff 1+
m (M ) =
n,m∈N Hn,m . To prove the lemma We claim
that for any n, each Hn,k is an open dense subset.
1. Openness
d(Df (x), Df (y))
Let f ∈ Hn,k , by definition there exist x, y and η > 0 such that
> k + η. Take any g,
d(x, y)α
ǫ-near to f in C 1 topology by the triangular inequality we get:
kgk n1 >
d(Dg(x), Dg(y))
d(x, y)
1
n
>
d(Df (x), Df (y))
d(x, y)
1
n
−
2ǫ
1
(d(x, y)) n
Taking ǫ small enough the above inequality shows that kgk n1 > k and the openness is proved.
2. Density
Let f ∈ Diff 1m (M ) we are going to find g ∈ Diff 1m (M ) \ Diff 1+
m (M ) such that g is near enough to f . For
this purpose we construct h ∈ Diff 1m (M ) \ Diff 1+
m (M ) near enough to identity and then put g = h ◦ f.
Considering local charts, it is enough to construct a C 1 volume preserving diffeomorphism I˜ from R2 to
R2 such that:
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Ali Tahzibi
1. I˜ is C 1 near to identity inside B(0, ǫ) for small ǫ > 0
2. I˜ is identity outside the ball B(0, 2ǫ)
2
3. I˜ ∈ Diff 1m (R2 ) \ Diff 1+
m (R )
Let us parameterize R2 with polar (r, θ) coordinates and ξ : R2 →R be a C 1 bump function which is equal
to one inside the ball {r < ǫ} and vanishes outside the ball of radius 2ǫ. Consider the following C 1 but not
C 1+α (for any α) real diffeomorphism:
(
r + logr 1 if r > 0
r
η(r) =
r
if r 6 0
˜ θ) = (r, θ + ξ(r)η(r)θ0 ) for small θ0 . The jacobian matrix of I˜ is
and define I(r,
1
0
DI˜ =
θ0 (ξ(r)η(r))′ 1
and it is obvious that I˜ is volume preserving and taking θ0 and ǫ, small enough it is near enough to identity
in C 1 topology.
Acknowledgements. This work was supported by FAPESP, Brazil. I am grateful also to the discussions of C.
Guttierez and A. Lopez at USP, São Carlos and Vanderlei Horita at Unesp de São Jose do Rio Preto.
References
[1] J. Bochi. Genericity of zero lyapunov exponents. Ergodic Theory and Dynamical Systems, pages 1667–1696,
2002.
[2] K Kuratowski. Topology, Vol. 1. Academic Press, 1966.
[3] F. Ledrappier and L.-S. Young. The metric entropy of diffeomorphisms i. characterization of measures satisfying
pesin’s entropy formula. Ann. of Math, 122:509–539, 1985.
[4] E. Zehnder. A note on smoothing symplectic and volume preserving diffeomorphims. volume 597 of Lecture notes
in Math, pages 828–854. Springer Verlag, 1977.
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