Analyticity of the SRB measure for holomorphic families of quadratic-like Collet-Eckmann maps

0.17 MB·English

by  Viviane Baladi

ANALYTICITY OF THE SRB MEASURE FOR HOLOMORPHIC
FAMILIES OF QUADRATIC-LIKE COLLET-ECKMANN MAPS
8
0 VIVIANEBALADIANDDANIELSMANIA
0
2
Abstract. Weshowthatifft isaholomorphicfamilyofquadratic-likemaps

n with all periodic orbits repelling so that for each real t the map ft is a real
a Collet-Eckmann S-unimodal map then, writing µt for the unique absolutely
J continuous invariantprobabilitymeasureofft,themap

2
2 t7→Z ψdµt
isrealanalyticforanyrealanalyticfunction ψ.

]
S
D
h. 1. Introduction and statement of the theorem
t
a If t f is a smooth one-parameter family of dynamics f so that f admits
t t 0
m 7→
a unique SRB measure µ , it is natural to ask whether the map t µ , where t
0 t
7→
[ rangesoverasetΛofparameterssuchthatf has(atleast)oneSRBmeasureµ ,is
t t
differentiable at 0 (in the sense of Whitney if Λ does not contain a neighbourhood
1
v of 0, as suggested by Ruelle [13]). Katok, Knieper, Pollicott, and Weiss [6] gave a
6 positive answer to this question in the setting of C3 families of transitive Anosov
5 flows (here, Λ is a neighbourhood of 0), showing that t ψdµ is differentiable,
t
3 for all smooth ψ. If f is a C3 mixing Axiom A attrac7→torRand the family t f
3 0 7→ t
is C3, Ruelle [12] not only proved that t ψdµ is differentiable, but also gave
1. an explicit formula (the linear response7→forRmula)tfor the derivative. Ruelle [13]
0
suggestedthatthisformula,appropriatelyinterpreted,shouldholdinmuchgreater
8
generality. Indeed,Dolgopyat[5]obtainedthelinearresponseformulaforaclassof
0
: partiallyhyperbolicdiffeomorphisms. Inapreviouswork[3,4],wefoundthatinthe
v
(nonstructurallystable)settingofpiecewiseexpandingunimodalintervalmaps,the
i
X SRBmeasureisdifferentiableifandonlyifthepathf istangenttothetopological
t
r classoff0,thatis,ifandonlyif∂tft t=0 ishorizontal. Whendifferentiabilityholds,
a |
Ruelle’s candidate for the derivative, as interpretedin [2], gives the linear response
formula. (We refer to [2, 3, 4], which also contain conjectures about smooth, not
necessarily analytic, Collet–Eckmann maps, for more information and additional
references.) Then, Ruelle [14] proved the linear response formula for a class of
nonrecurrent 1 analytic unimodal interval maps f , assuming that all f stay in
t t
the topological class of f . In the present work, we consider holomorphic (that is,
0
Date:February2,2008.
1991 Mathematics Subject Classification. 37C4037C3037D2537E05.
V.B. is partially supported by ANR-05-JCJC-0107-01. D.S. is partially supported by CNPq
470957/2006-9 and310964/2006-7, FAPESP2003/03107-9. D.S.thanks theDMAofE´coleNor-
male Sup´erieure for hospitality during a visit where a crucial part of this work was done. V.B.
wrote part of this paper while visiting the Universidad Cat´olica del Norte, Antofagasta, Chile,
whosehospitalityisgratefullyacnowledged. WethankD.Sandsforveryhelpfulcomments.
1I.e.,infkd(ftk(c),c)>0,wherecdenotes thecriticalpoint.
1
2 VIVIANEBALADIANDDANIELSMANIA
complex analytic)families f ofquadratic-likeholomorphic Collet–Eckmannmaps.
t
Our assumptions imply (using classical holomorphic motions) that all f lie in the
t
sameconjugacyclass. Generalisingoneoftheargumentsin[6],weareabletoshow
that t ψdµ is real analytic for any real analytic function ψ.
t
7→
Let usRnow state our resultmoreprecisely. Let I =[ 1,1]. AC3 mapf :I I
− →
isanS-unimodalmapifithasc=0asuniquecriticalpoint,andf hasnonpositive
Schwarzianderivative,thatis f′′′ 3 f′′ 2 0exceptatc. AnS-unimodalmapis
f′ −2 f′ ≤
called Collet-Eckmann if there exist C(cid:0) >(cid:1)0 and λ >1 so that (fn)′(f(c)) Cλn
c | |≥ c
foralln 1. Inthispaper,weshallonlyconsiderS-unimodalmapswithf′′(c)=0.
≥ 6
In Section 2 we shall define precisely the notion of a holomorphic (complex
analytic)family of quadratic-like maps in a neighbourhood of I and provethe main
result of this work:
Theorem 1.1. Let t f be a holomorphic family of quadratic-like maps in a
t
7→
neighbourhood of I, with all periodic orbits repelling. Assume in addition that for
each small real t the map f restricted to I is a (real) Collet-Eckmann S-unimodal
t
map. Then there exists ǫ>0 so that for each real analytic ψ :I C, the map
→
t ψρ dx,
7→Z t
where ρ is the invariant density of f , is real analytic on ( ǫ,ǫ).
t t
−
The quadratic-like assumption implies that f′′(c) < 0. The fact that periodic
t
orbits are repelling implies that f is topologically conjugated with f (see our use
t 0
ofMan˜´e-Sad-Sullivan[8]inthebeginning oftheproofofthe theoreminSection2).
BesidesMan˜´e-Sad-Sullivan[8]theothermainingredientofourproofaretheresults
andconstructionsofKellerandNowicki[7] whichallowustoexploitdynamicalzeta
functions, following the argument in the work of Katok–Knieper–Pollicott–Weiss
[6, First proof of Theorem 1].
The extension from quadratic-like to polynomial-like is straightforward,and we
stick to the nondegenerate case f′′(c) = 0 for the sake of simplicity of exposition.
6
As the proof uses only real-analyticity of the holomorphic motions t h , it is
t
7→
conceivable that the conclusion of the theorem holds if f is a real analytic family
t
of quadratic-like maps, using ideas of [1], but this generalisation appears to be
nontrivial.
2. Proof of the Theorem
Beforewe provethe theorem,letus define preciselythe objectswe arestudying:
Definition. We say that f is a holomorphic family of quadratic-like maps in a
t
neighbourhood of I ifthereexistsacomplexneigbourhoodU ofI sothatt f isa
t
7→
holomorphicmapfromacomplexneighbourhoodofzerototheBanachspaceB(U)
of holomorphic functions on U extending continously to U (with the supremum
norm), such that:
Forrealt,themapf isrealon U,withf (I) I andf ( 1)=f (1)= 1.
t t t t
• ℜ ⊂ − −
There existsimply connectedcomplex domainsW andV, whoseboundaries
•
areanalyticJordancurves,withI W,I V,V U,V W,andsothat
⊂ ⊂ ⊂ ⊂
f : V W is a double-branched ramified covering, with c = 0 as a unique
0
7→
critical point. (That is, f :V W is a quadratic-like restriction of f .)
0 0
7→
ANALYTICITY OF THE SRB MEASURE FOR FAMILIES OF COLLET-ECKMANN MAPS 3
Iff isaholomorphicfamilyofquadratic-likemapsinaneighbourhoodofI then
t
itiseasytosee2thatforsmallcomplext,denotingbyV theconnectedcomponent
t
of f−1(W) containing 0, then f :V W is a quadratic-like restriction of f . We
t t t 7→ t
may then give another definition:
Definition. We say that f is a holomorphic family of quadratic-like maps in a
t
neighbourhood of I with all periodic orbits repelling, if f is a holomorphic family
t
of quadratic-like maps in a neighbourhood of I so that, for each small complex t,
the map f only has repelling periodic orbits in V .
t t
Proof. Sinceweassumedthatallperiodicpointsoff arerepelling,[8,TheoremB]
t
(theresultthereisquotedforpolynomialmaps,buttheproofimmediately extends
to polynomial-like) implies that there exists a holomorphic motion of the Julia set
K(f ) of f , that is, a map h : D K(f ) C where D = z C z < ǫ for
0 0 0 0
× → { ∈ | | | }
some ǫ >0, such that for each x K(f ) the map t h (x) is holomorphic, and
0 0 t
∈ 7→
for every t D the function x h (x) is continuous and injective on K(f ), with
t 0
∈ 7→
h f =f h .
t 0 t t
◦ ◦
(In particular, h is a homeomorphism from K(f ) to K(f ).) Our assumptions
t 0 t
implythat[f2(0),f (0)]=K(f ) Randh (K(f ) R)=K(f ) R=[f2(0),f (0)].
0 0 0 ∩ t 0 ∩ t ∩ t t
From now on, we only use real analyticity of t f (x) and t h (x) for x
t t
7→ 7→ ∈
[f2(0),f(0)].
Wenextclaimthatourassumptionsguaranteethateachf satisfiesthetechnical
t
requirement needed by Keller and Nowicki [7, (1.2)]. Denoting by var φ the total
J
variation of a function φ on an interval J, and writing f = f , we need to check
t
that there is that a constant M >0 such that:
a. M−1 <sup |x−c| +var |x−c| <M,
I |f′(x)| I|f′(x)|
b. var |f(x)−f(u)| <M where J =[ 1,u] if u<c and =[u,1] if u>c.
Ju|x−u||f′(x)| u −
Let δ > 0 be so that f′′(y) > f′′(c)/2 if y c < δ . It suffices to prove (a.)
1 1
| | | | | − |
and (b.) for x c < δ and u c < δ , and we restrict to such points. Noting
1 1
| − | | − |
that for every such x=c there exist y , z , and z˜ , between x and c, so that
x x x
6
x c x c 1
| − | = − = ,
f′(x) −f′(x) f′(c) −f′′(y )
x
| | −
and(use f′′(x)=f′′(c)+f(3)(z )(x c)andf′(x)=f′′(c)(x c)+f(3)(z˜ )(x−c)2)
x − − x 2
x c f′(x)+(x c)f′′(x) (x c)2 f(3)(z˜ )
∂ | − | = − − = − f(3)(z ) x ,
x f′(x) (f′(x))2 (f′(x))2 x − 2
| | (cid:0) (cid:1)
the first two conditions hold because f is C3. For the third condition, consider
x u>c (the other case is symmetric). Since
≥
f(x) f(u) x uf′′(z ) x u f′′(z )
x x
− =1+ − =1+ − ,
(x u)f′(x) f′(x) 2 f′(x) 2f′′(y )
x
−
2Indeed, ∂W is an analytic Jordan curve, and f0 has no critical point on ∂V. If ft ∈ B(U)
is close to f0, there is a simplyconnected domain Vt close to V such that ft(Vt)=W, and the
boundary of ∂Vt is a Jordan curve, by the implicit function theorem. Then ft : Vt → W is a
quadratic-likeextension.
4 VIVIANEBALADIANDDANIELSMANIA
and 0 < x−u < x−c , we get that f(x)−f(u) is bounded on [u,1], uniformly
−f′(x) −f′(x) (x−u)f′(x)
in u. Finally, since (cid:12) (cid:12)
(cid:12) (cid:12)
x u f′(x) (x u)f′′(x)
∂ − = − − ,
xf′(x) (f′(x))2
analyticity of f implies that ∂ x−u changes signs finitely many times, uniformly
xf′(x)
in u, proving (b.).
Also, the results of Nowicki–Sands [11] and Nowicki–Przytycki [10] ensure (see
Appendix A)thatthere existλ >1,λ >1,λ >1,andǫ >0sothat,foreach
c per η 1
t <ǫ , there is C >0 with
1 t
| |
(1) (fn)′(f (0) C λn, n 1,
| t t |≥ t c ∀ ≥
and so that for each x I so that fp(x)=x for some p 1, we have
∈ t ≥
(2) (fp)′(x) C λp ,
| t |≥ t per
and, finally, setting
λ (t):=liminf η −1/n η I is the biggest monotonicity interval of fn ,
η n→∞ {| | | ⊂ t }
(3) inf λ (t)>λ .
η η
|t|<ǫ1
In other words, the hyperbolicity constants are uniform in t, guaranteeing unifor-
mity when applying the results of Keller and Nowicki [7]. (We choose ǫ <ǫ .)
1 0
We now adapt the strategy used in the first proof of [6, Theorem 1]. Fix ψ and,
for x I so that fp(x)=x for p 1, and for small real s and t, consider
∈ 0 ≥
esψ(ht(x))
(4) g (x)= .
s,t f′(h (x))
| t t |
Since ψ is real analytic, the analyticity of t h and of t f together with (2)
t t
7→ 7→
imply that there is ǫ > 0 so that, for every periodic point x I of period p 1
2
∈ ≥
for f, the function
(t,s) g(p)(x):= esPpk−=01ψ(ht(fk(x))
7→ s,t (fp)′(h (x))
| t t |
is real analytic in s <ǫ and t <ǫ , uniformly in x. We take ǫ <ǫ .
2 2 2 1
| | | |
Therefore, the dynamical zeta function defined by
∞ zp
(5) ζ(s,t,z):=exp g(p)(x)
p s,t
Xp=1 x∈I:Xf0p(x)=x
hasthefollowingproperty: Thereexistsδ >0sothatforeach z <δ thefunction
2 2
| |
ζ(s,t,z)is realanalyticin t <ǫ , s <ǫ , andsothatforeach(s,t)with t <ǫ ,
2 2 2
| | | | | |
s <ǫ the map ζ(s,t,z) is holomorphic and nonvanishing in z <δ .
2 2
| | | |
Now, h f =f h immediately implies
t 0 t t
◦ ◦
∞ zp esPpk−=01ψ(ftk(y))
(6) ζ(s,t,z)=exp .
p (fp)′(y)
Xp=1 y∈I:Xfp(y)=y | t |
t
Recall (1, 2, 3) and take Θ (0,1) with
∈
Θ−1 <min λ , min(λ ,λ ) .
η c per
{ q }
ANALYTICITY OF THE SRB MEASURE FOR FAMILIES OF COLLET-ECKMANN MAPS 5
KellerandNowicki[7,Theorem2.1]provethat,ifǫ (0,ǫ )issmallenough,then
3 2
∈
for s <ǫ and t <ǫ the transfer operator 3
3 3
| | | |
ω (y)exp(sψ(y))
t
ϕ(x)= ϕ(y),
Ls,t fˆtX(y)=xωt(x) |fˆt′(y)|
acting on functions of bounded variation on a suitable Hofbauer tower extension
fˆ : Iˆ Iˆ of f [7, Section 3], endowed with an appropriate [7, 6.2] cocycle ω
t t t
→ §
(which embodies the singularities along the postcritical orbit of f ), is a bounded
t
operator. If s = 0 then the spectral radius λ of is equal to 1, it is a simple
0,t s,t
L
eigenvalue (whose eigenvector gives the invariant density ρ of f ), and the rest
t t
of the spectrum is contained in a disc of strictly smaller radius. In addition, the
essential spectral radius θ of satisfies sup θ < Θ, and for each
s,t Ls,t |t|<ǫ3,|s|<ǫ3 s,t
t <ǫ the spectral radius 4 λ > Θ of is an analytic function [7, Prop. 4.2]
3 s,t s,t
| | L
of s. Also, perturbation theory gives (see [7, (5.2)])
(7) ∂ logλ = ψρ dx.
s s,t|s=0 Z t
Keller and Nowicki also show [7, Theorem 2.2] that for t <ǫ and s <ǫ the
3 3
| | | |
power series ζ(s,t,z) defined by (6) extends meromorphically to the disc of radius
Θ−1 (where it does not vanish, by [7, Prop. 4.3 and Lemma 4.5]), and its poles
z in this disc are in bijection with the eigenvalues λ of , via λ =z−1. (The
k k Ls,t k k
order of the zero coincides with the algebraic multiplicity of the eigenvalue.) It
follows that z ζ(s,t,z)−1 is holomorphic in the disc of radius Θ−1. This disc
7→
contains λ−1, which is a simple zero.
s,t
Toendtheproof,recalling(7),itsufficestoseethat(s,t) λ isrealanalytic,
s,t
7→
but this easily follows from Shiffman’s [15] real analytic Hartogs’ theorem (see
Appendix B or [6, Thm p. 589]) applied to d(s,t,z) = ζ(s,t,z)−1, which implies
that for each (s,t) ( ǫ ,ǫ ) ( ǫ ,ǫ ) the map z d(s,t,z) is holomorphic in
3 3 3 3
∈ − × − 7→
z < Θ−1. Indeed, by the implicit function theorem, the simple zeroes of d(s,t, )
| | ·
depend real analytically on s and t. (We used the same ǫ discs for the s and t
i
variable, but a more careful analysis shows that ǫ in the statement of the theorem
may be selected independently of ψ.) (cid:3)
Appendix A. Uniformity of the hyperbolicity constants
We start with a preliminary observation5: Let g be an S-unimodal Collet–
Eckman map (with g′′(0) < 0, say). Denote by λ (g), λ (g), and λ (g) the
c per η
constants defined by (1, 2, 3) (replacing f by g). Nowicki and Sands [11] proved
t
that if g is an S-unimodal map and λ (g)>1 then λ (g)>1. A careful study of
per c
theirproofshowsthatλ (g)>λ (g)α,wheretheexponentα>0onlydependson
c per
the maximum length N(g) of “almost-parabolicfunnels” of g (see [11, Lemma 6.6]
for a definition of N(g), which can be bounded by a function of 1/log(λ (g))
per
and sup g′ ). Since N(g) is in fact invariant under topological conjugacy and f is
t
| |
3Our parameter s is called t in [7], the parameter β in [7] is β = 1, and our parameter t
correspondstochangingthedynamics.
4Note that λs,t isthe exponential of the topological pressure of sψ−log|ft′| for ft, and that
ρtdxistheequilibriumstateforft and−log|ft′|.
5WethankDuncanSands forhisexplanations.
6 VIVIANEBALADIANDDANIELSMANIA
topologically conjugated to f , we conclude that λ (f ) > λ (f )α, with α > 0
0 c t per t
uniform in small t.
Next, recall that Nowicki and Przytycki [10] proved that if g and g˜ are S-
unimodal maps (with g′′(c) = 0 and g˜′′(c) = 0, say) conjugated by a homeomor-
6 6
phism of the interval and g is Collet–Eckmann, then g˜ is Collet–Eckmann. Take
g =f and g˜=f (in particular, f is C2 close to f and t h is smooth). Then
0 t t 0 t
7→
it is not very difficult to see that the constants M = M(f )> 0, P = P (f ) > 0,
t 4 4 t
and δ = δ (f ) > 0 from the topological characterisation (“finite criticality”) of
4 4 t
Collet–Eckmannin [10, (4) p. 35]) are uniform in small t.
Recallthatourassumptionsimply f′′(c)=0forallsmallt,sothatthe constant
t 6
denoted l in [10] is l = 2. Section 2 of [10], and in particular the use of the
c c
Koebe principle there, implies that there exists a (universal) function q : R+
∗ ×
(0,1) (0,1) with q(M,1/4)<1/2 for any M (see [10, Lemma 2.2]), and so that
→
−1
λ (f ) > 1 2q(M(f ),1/4) . Therefore, λ (f ) > 1 is uniformly bounded
per t t per t
−
away from (cid:0)1 for small t. The p(cid:1)reliminary observation then implies that λ (f ) is
c t
also uniformly bounded in t. By [9, Proposition 3.2] (see also [10, p. 35]), this
implies a uniform lower bound for λ (f ). (Indeed, in the notations of [9, 3], we
η t
§
have λ =λ =λ λ =λ √λ .)
η 5 4 3 1 c
≥ ≥
Appendix B. Shiffman’s real analytic Hartogs’ extension theorem
Theorem B.1. [15] Let δ >0 and 0<r <R. Assume that
d:( δ,δ)2 z C z <R C
− ×{ ∈ || | }→
satisfies the following conditions:
For each (s,t) ( δ,δ)2 the map z d(s,t,z) is holomorphic in z <R.
• ∈ − 7→ | |
For each z <r the map (s,t) d(s,t,z) is real analytic in ( δ,δ)2.
• | | 7→ −
Then d(s,t,z) is real analytic on ( δ,δ)2 z <R .
− ×{| | }
NotethattheabovetheoremfailsifrealanalyticityisreplacedbyCk fork .
≤∞
Thetheoremholdsbecause z <r isnotpluripolarin z <R. Shiffman’sresult
| | | |
is based on deep work of Siciak [16]
References
1. A.Avila,M.Lyubich,andW.deMelo,Regularorstochasticdynamicsinrealanalyticfamilies
of unimodal maps,Invent. Math.154(2003)451–550.
2. V.Baladi,Onthesusceptibilityfunctionofpiecewiseexpandingintervalmaps,Comm.Math.
Phys.275(2007) 839–859.
3. V.BaladiandD.Smania,Linearresponseforpiecewiseexpandingunimodalmaps,arxiv.org
preprint,2007.
4. V. Baladi and D. Smania, Smooth deformations of piecewise expanding unimodal maps,
arxiv.org preprint,2007.
5. D. Dolgopyat, On differentiability of SRB states for partially hyperbolic systems, Invent.
Math.155(2004) 389–449.
6. A. Katok, G. Knieper, M. Pollicott, and H. Weiss, Differentiability and analyticity of topo-
logical entropy for Anosov and geodesic flows,Invent. Math.98(1989) 581–597.
7. G. Keller and T. Nowicki, Spectral theory, zeta functions and the distribution of periodic
points forCollet-Eckmann maps,Comm.Math.Phys.149(1992)31–69.
8. R.Man˜´e,P.Sad,andD.Sullivan,Onthe dynamicsof rational maps,Ann.Sci.E´coleNorm.
Sup.16(1983)193–217.
9. T.Nowicki,Somedynamical properties ofS-unimodal maps,Fund.Math.142(1993)45–57.
ANALYTICITY OF THE SRB MEASURE FOR FAMILIES OF COLLET-ECKMANN MAPS 7
10. T. Nowicki and F. Przytycki, Topological invariance of the Collet-Eckmann property for S-
unimodal maps, Fund.Math.155(1998)33–43.
11. T. Nowicki and D. Sands, Nonuniform hyperbolicity and universal bounds for S-unimodal
maps, Invent. Math.132(1998) 633–680.
12. D.Ruelle,DifferentiationofSRBstates,Comm.Math.Phys.187(1997)227–241,Differenti-
ationofSRBstates: Correctionsandcomplements,Comm.Math.Phys.234(2003)185–190.
13. D. Ruelle, Application of hyperbolic dynamics to physics: some problems and conjectures,
Bull.Amer.Math.Soc.41(2004)275–278.
14. D. Ruelle, Structure and f-dependence of the a.c.i.m. for a unimodal map f of Misiurewicz
type,arxiv.org, 2007.
15. B. Shiffman, Separate analyticity and Hartogs theorems, Indiana Univ. Math. J. 38 (1989)
943–957.
16. J. Siciak, Separately analytic functions and envelopes of holomorphy of some lower dimen-
sional subsetsof Cn,Ann.Polon.Math.22(1969) 145–171.
D.M.A., UMR8553,E´coleNormaleSup´erieure,75005Paris, France
E-mail address: [email protected]
Departamento de Matema´tica, ICMC-USP, Caixa Postal 668, Sa˜o Carlos-SP, CEP
13560-970Sa˜oCarlos-SP, Brazil
E-mail address: [email protected]