Analyticity of the Susceptibility Function for Unimodal Markovian Maps of the Interval

0.09 MB·English

by  Yunping Jiang

ANALYTICITY OF THE SUSCEPTIBILITY FUNCTION
FOR UNIMODAL MARKOVIAN MAPS OF THE INTERVAL.
by Yunping Jiang* and David Ruelle**.
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Abstract. We study the expression (susceptibility)

n
a ∞
J d
Ψ(λ) = λn ρ(dx)X(x) A(fnx)
1 Z dx
1 nX=0 I

]
S where f is a unimodal Markovian map of the interval I, and ρ =
D ρ is thecorresponding absolutelycontinuous invariant measure.
f
. WeshowthatΨ(λ)isanalyticnearλ = 1, whereΨ(1)isformally
h
t the derivative of ρ(dx)A(x) with respect to f in the direction
a I
m of the vector fieldRX.
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* Math. Dept., Queens College of CUNY, Flushing, NY 11367 and Math. Dept.,
Graduate Center of CUNY and Math. Inst., AMSS, CAS. email: [email protected]
** Math. Dept., Rutgers University, and IHES. 91440 Bures sur Yvette, France. email:
[email protected]
1
In a previous note [Ru] the susceptibility function was analyzed for some examples of
maps of the interval. The purpose of the present note is to give a concise treatment of the
general unimodal Markovian case (assuming f real analytic). We hope that it will simi-
larly be possible to analyze maps satisfying the Collet-Eckmann condition. Eventually, as
explained in [Ru], application of a theorem of Whitney [Wh] should prove differentiability
of the map f 7→ ρ restricted to a suitable set.
f
Setup
Let I be a compact interval of R and f : I → I be real analytic. We assume that
′ ′ ′
there is c in the interior of I such that f (0) = 0, f (x) > 0 for x < c, f (x) < 0 for x > c,
′′
and f (c) < 0. Replacing I by a possibly smaller interval, we assume that I = [a,b] where
a = f2(c), b = f(c). We assume that the postcritical orbit P = {fnc : n ≥ 1} is finite:
P = {p ,...,p }; in particular, f is Markovian. We shall assume that f is analytically
1 m
expanding in the sense of Assumption A below; in particular the periodic orbits of f are
assumed to be repelling, and therefore c cannot be periodic. We also assume that f is
topologically mixing [this can always be achieved by replacing I by a smaller interval and
f by some iterate fN].
Theorem.
Under the above conditions, and Assumption A stated later, there is a unique f-
invariant probability measure ρ absolutely continuous with respect to Lebesgue on I. If X
is real analytic on I, and A ∈ C1(I), then
∞
d
Ψ(λ) = λn ρ(dx)X(x) A(fnx)
Z dx
nX=0 I
extends to a meromorphic function in C, without pole on {λ : |λ| = 1}.
Change of variable
The finite set {c} ∪ P decomposes I into m subintervals I , with 2m endpoints (we
j
“double” the endpoints ofconsecutive subintervals, distinguishing between a − endpoint at
the right of an interval, and a + endpoint at the left). Note that η = {I : j = 1,...,m} is
j
a Markov partition for the map f. Consider the critical values of fn. Then for large n > 0,
the set of critical values will be stabilized and is always P. We define polar endpoints as
follows:
(1) p ∈ P is a polar −endpoint of an interval in η if p is local maximum value of fn
for n large.
(2) v ∈ P is a polar +endpoint of an interval in η if p is local minimum value of fn
for n large.
Every p ∈ P is a polar − or +endpoint and may be both, c is a nonpolar endpoint on
both sides.
2
We define now an increasing continuous map ̟ : I → R so that J = ̟I is a compact
interval. We write ̟I = J for 1 ≤ j ≤ m. Denote by ω the inverse of ̟. We assume that
j j
ω|J extends to a holomorphic function in a complex neighborhood of J for 1 ≤ j ≤ m
j j
and that for q ∈ {c}∪P, ω has the property
ξ2
ω(̟q±ξ) = ω(̟q)± +O(ξ4)
2
if q is a ±polar endpoint, and
ω(̟q±ξ) = ω(̟q)±ξ +O(ξ2)
if q is a nonpolar endpoint. [We should really consider disjoint copies of the I and J ,
j j
and disjoint neighborhoods of these in C or in a Riemann surface two-sheeted near polar
endpoints. This would lead to notational complications that we prefer to omit].
Applications of this singular change of coordinate have been used in [Ji1], [BJR],
and [Ru]; the reference [Ji2] contains some more relevant study regarding the method
of singular change of coordinates in one-dimensional dynamical systems. The reader is
encouraged to compare this method with orbifold metrics in [Th, Chapter 13]. Another
relevant application of this method in complex dynamical systems can be found in [DH].
From now on we shall say that ̟q is a ±polar (nonpolar) endpoint if q is ±polar
(nonpolar).
The dynamical system viewed after the change of variable.
For any two intervals I ,I ∈ η with fI ⊃ I , we define
j k j k
ψ = ̟◦(f|I )−1 ◦(ω|J )
jk j k
Note that the ψ are restrictions of inverse branches of g = ̟◦f ◦ω : J → J to intervals
jk
in η. The function ψ : J → J extends holomorphically to a complex neighborhood of
jk k j
J . Indeed, note that (f|I )−1 is holomorphic except if I is one of the two intervals around
k j j
c, in which case the singularity is corrected by ω|I , where I is the rightmost interval in
n n
η. In other cases ω|I cancels the singularity of ̟|I by our definition of ω. [Note that
k j
′
ψ (x) ≥ 0 or ≤ 0 on J and may vanish only at an interval endpoint].
jk k
Assumption A.
Each J , for k = 1,...,m, has a bounded open connected neighborhood U in C such
k k
that ψ : J → J extends to a continuous function ψ : U¯ → C holomorphic in U , and
jk k j jk k k
with ψ U¯ ⊂ U .
jk k j
One checks that the sets U can be assumed to be in ǫ-neighborhoods of the J . Also,
k k
Assumption A implies that periodic points for g are strictly repelling. The smoothness of
ω, ̟ in the interior of subintervals shows that the same property holds for f, apart from
interval endpoints where we however also assume the property to hold:
The periodic orbits of f are strictly repelling.
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Markovian graph.
ConsidertheMarkovpartitionη = {I }. Letuswritej ≻ k (j coversk)iffI ⊃ I (we
j j k
allow j ≻ j). This defines a directed graph with vertex set {1,...,m} and oriented edges
(j,k) for j ≻ k. Since we have assumed our dynamical system f to be topological mixing,
our graph is also mixing in the sense that there is N ≥ 1 such that for all j,k ∈ {1,...,m}
we have j ≻ ... ≻ k (N edges) corresponding to fNI ⊃ I .
j k
Transfer operators.
For a function Φ = (Φ ) defined on ⊔J , write
j j
′
(LΦ) (z) = sgn(j)ψ (z)Φ(ψ z)
k jk jk
X
j:j≻k
(L Φ) (z) = sgn(j)Φ(ψ z)
0 k jk
X
j:j≻k
where sgn(j) is +1 if ψ is increasing on J , and −1 if ψ is decreasing on J . If H
jk k jk k
is the Hilbert space of functions on ⊔ U¯ which are square integrable with respect to
j∈L j
Lebesgue, and have holomorphic restrictions to the U , then L and L acting on H are
j 0
holomorphy improving, hence compact and of trace class.
Properties of L.
For x ∈ J we have
k
′
(LΦ) (x) = |ψ (x)|Φ (ψ x)
k jk j jk
X
j≻k
hence Φ ≥ 0 implies LΦ ≥ 0 (L preserves positivity) and
dx(LΦ)(x) = dx(LΦ) (x) = dxΦ (x) = dxΦ(x)
Z Z k Z j Z
J Xk Jk Xj Jj J
(L preserves mass). Using mixing one finds that L has a simple eigenvalue µ = 1
0
corresponding to an eigenfunction σ > 0. The other eigenvalues µ satisfy |µ | < 1,
0 ℓ ℓ
and their (generalized) eigenfunctions σ satisfy dxσ (x) = 0. If we normalize σ by
ℓ J ℓ 0
dxσ (x) = 1, then σ (dx) = σ (x)dx is the Runique g-invariant probability measure
J 0 0 0
aRbsolutely continuous with respect to Lebesgue on J. In particular, σ (x)dx is ergodic.
0
′
Let now H ⊂ H consist of those Φ = (Φ ) such that the derivative Φ vanishes
1 k
at the (polar) endpoints ̟a ,̟b of J, and such that at the common endpoint ̟q (q ∈
{c}∪P\{a,b})) of two subintervals we have equality on both sides of a quantity which is
either
• the value of Φ for a nonpolar endpoint, or
′
• the value of ±Φ for a polar ± endpoint.
4
WenotethatLH ⊂ H [thisrequiresacaseby casediscussion]. Furthermoreσ ∈ H
1 1 0 1
′
[take φ ∈ H such that φ ≥ 0, dyφ(y) = 1, and φ,φ vanish at subinterval endpoints;
J
then φ ∈ H and σ = lim RLnφ ∈ H ].
1 0 n→∞ 1
Evaluating Ψ(λ).
The image ρ(dx) = ρ(x)dx of σ (y)dy by ω is the unique f-invariant probability
0
measure absolutely continuous with respect to Lebesgue on I. We have
′
ρ(x) = σ (̟x)̟ (x)
0
Consider now the expression
∞
d
Ψ(λ) = λn ρ(dx)X(x) A(fnx)
Z dx
X I
n=0
where we assume that X extends to a holomorphic function in a neighborhood of each I
k
and takes the same value at both sides of common endpoints of intervals in η (continuity).
Also assume that A ∈ C1(I). For sufficiently small |λ|, the series defining Ψ(λ) converges.
Writing B = A◦ω (B has piecewise continuous derivative) and x = ωy we have
d 1 d
X(x) A(fnx) = X(ωy) B(gny)
dx ω′(y)dy
hence
∞
X(ωy) d
Ψ(λ) = λn dyσ (y) B(gny)
Z 0 ω′(y) dy
nX=0 J
′
Defining Y(y) = σ (y)X(ωy)/ω (y), we see that Y extends to a holomorphic function
0
in a neighborhood of each J , which we may take to be U , except for a simple pole at
k k
each polar endpoint of J . Since σ ∈ H , the properties assumed for ω imply that also
k 0 1
(X ◦ω)×σ ∈ H . Note that near a nonpolar subinterval endpoint ̟q
0 1
′
ω (̟q±ξ) = 1+O(ξ)
and near a ± polar endpoint
ω′(̟q ±ξ) = ξ +O(ξ3)
Therefore
1
± ±
Y(̟q ±ξ) = A +B +O(ξ)
ξ
where B+ = B− for the two sides of ̟q, and B+ = 0 at the left endpoint ̟a of J, B− = 0
at the right endpoint ̟b of J. We may write
X(ωy) d
dyσ (y) B(gny) = dyY(y)g′(y)···g′(gn−1y)B′(gny)
Z 0 ω′(y) dy Z
J J
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= ds(LnY)(s)B′(s)
Z 0
J
where L has been defined above, and we have thus
0
∞
Ψ(λ) = λn ds(LnY)(s)B′(s)
Z 0
X J
n=0
Properties of L .
0
We let now H ⊂ H be the space of functions Φ = (Φ ) vanishing at the endpoints
0 k
̟a, ̟b of J, and such that the values of Φ on both sides of common endpoints of intervals
J coincide (continuity). Therefore L H ⊂ H .
j 0 0 0
There is a periodic orbit γ ,...,γ (with gγ = γ ) of polar endpoints where
1 p j j+1(modp)
γ is the ± endpoint of some subinterval J . Choose P to be 0 on subintervals different
α k(α) α
from J , and to be holomorphic on a complex neighborhood of J except at γ . Also
k(α) k(α) α
assume that
1
P (γ ±ξ) = +O(ξ)
α α
ξ
and that P vanishes at the endpoint of J different from γ . Then
α k(α) α
L P −|f′(γ(α)|1/2P ∈ H
0 α α+1(modp) 0
Therefore LpP −ΛP = u ∈ H where Λ = p |f′(γ(α)|1/2 > 1. Since the spectrum of
0 1 1 0 α=1
′
L acting on H is contained in the closed uniQt disk, and since the derivative u is in H, we
may define v = (Lp −Λ)−1u′ ∈ H. Since dyu′(y) = 0 we also have dyv(y) = 0 and
J J
we can take w ∈ H such that w′ = v. WeRhave thus R
0
((Lp −Λ)w)′ = (Lp −Λ)w′ = (Lp −Λ)v = u′
0
p p
so that (L −Λ)w = u [there is no additive constant of integration since (L −Λ)w and u
0 0
are in H ]. Finally
0
p
(L −Λ)(P −w) = 0
0 1
There is thus a L -invariant p-dimensional vector space spanned by vectors P −w with
0 α α
w ∈ H , such that the spectrum of L restricted to this space consists of eigenvalues ω
α 0 0 ℓ
with
p
ω = Λ1/pe2πℓi/p = | f′(γ )|1/2pe2πℓi/p
ℓ α
Y
α=1
for ℓ = 0,...,p−1.
˜
For the postcritical but nonperiodic polar points γ˜ ,...,γ˜ define P like P above,
1 q β α
with γ replaced by γ˜ . For each β there is α = α(β) with
α β
Lq(P˜ −Λ˜ P ) ∈ H
0 β β α 0
6
˜
with some Λ 6= 0, hence
β
Lq(P˜ −Λ˜ (P −w )) = Y˜ ∈ H
0 β β α α β 0
Poles of Ψ(λ).
We may now write
Y = Y +Y +Y
0 1 2
where
Y ∈ H
0 0
p
Y = c (P −w )
1 α α α
X
α=1
q
Y = c˜ (P˜ −Λ˜ (P −w ))
2 β β β α(β) α(β)
X
β=1
and there is a corresponding decomposition Ψ(λ) = Ψ (λ)+Ψ (λ)+Ψ (λ). Here Ψ (λ) is
0 1 2 1
a sum of terms C /(λ−ω ) where ω = Λ1/p ×p-th root of unity; Ψ (λ) = polynomial of
ℓ ℓ ℓ 2
degree q−1 in λ plus λq q c˜ Ψ˜ (λ) where Ψ˜ is obtained if we replace Y by Y˜ in the
β=1 β β β β
definition of Ψ. The polePs of Ψ(λ) are thus those of Ψ (λ) at the ω and those of Ψ (λ)
1 ℓ 0
and Ψ˜ (λ). The discussion is the same for Ψ and the Ψ˜ , we shall thus only consider Ψ .
β 0 β 0
Since Y ∈ H and L H ⊂ H we have
0 0 0 0 0
∞ ∞
Ψ (λ) = λn ds(LnY )(s)B′(s) = − λn ds(LnY )′(s)B(s)
0 Z 0 0 Z 0 0
nX=0 J nX=0 J
∞
= − λn ds(LnY′)(s)B(s)
Z 0
nX=0 J
It follows that Ψ (λ) extends meromorphically to C with poles at the µ−1. We want to
0 k
show that the residue of the pole at µ−1 = 1 vanishes. Since dyσ (y) = 0 for k ≥ 1, the
0 J k
coefficient of σ in the expansion of Y′ is proportional to R
0 0
′
dyY (y) = Y(̟b)−Y(̟a) = 0
Z 0
J
because Y ∈ H . Therefore Ψ (λ) is holomorphic for |λ| = 1, and the same holds for the
0 0 0
Ψ˜ (λ), concluding the proof of the theorem. In fact we know that the poles of Ψ(λ) are
β
located at µ−1 for k ≥ 1, and at ω−1 for ℓ = 0,...,p−1, so that |µ−1| > 1,|ω−1| < 1.
k ℓ k ℓ
Acknowledgments.
One of us (Y.J.) is partially supported by grants from the NSF and the PSC-CUNY
Program, and by the Hundred Talents Program of the CAS.
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References.
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conjugacies for postcritically finite polynomials.” J. Statist. Phys. 108, 973-993(2002).
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tion of rational functions.” Acta Math., Vol. 171, 263-297(1993).
[Ji1] Y. Jiang, “On Ulam-von Neumann transformations.” Commun. Math. Phys.,
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[Ji2] Y. Jiang, “Renormalization and Geometry in One-Dimensional and Complex
Dynamics.” Advanced Series in Nonlinear Dynamics, Vol. 10 (1996), World Scientific
Publishing Co. Pte. Ltd., River Edge, NJ.
[Ru] D. Ruelle. “Differentiating the absolutely continuous invariant measure of an
interval map f with respect to f.” Commun. Math. Phys. to appear.
[Th] W. Thurston, “The Geometry and Topology of Three-Manifolds”, Electronic
version 1.1-March 2002, http://www.msri.org/publications/books/gt3m/
[Wh] H. Whitney. “Analytic expansions of differentiable functions defined in closed
sets.” Trans. Amer. Math. Soc. 36,63-89(1934).
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