Fractional Susceptibility Functions for the Quadratic Family: Misiurewicz–Thurston Parameters

Abstract

For $f_t(x)= t-x^2$ the quadratic family, we define the fractional susceptibility function $\Psi ^\Omega _{\phi ,t_0}(\eta , z)$ of $f_t$, associated to a $C^1$ observable $\phi$ at a stochastic parameter ${t_0}$. We also define an approximate, “frozen,” fractional susceptibility function $\Psi ^\mathrm {fr}_{\phi ,t_0}(\eta , z)$ such that $\lim _{\eta \rightarrow 1} \Psi ^{\mathrm {fr}}_{\phi ,t_0}(\eta , z)$ is the susceptibility function $\Psi _{\phi ,t_0}(z)$ studied by Ruelle. If ${t_0}$ is Misiurewicz–Thurston, we show that $\Psi ^\mathrm {fr}_{\phi ,t_0}(1/2, z)$ has a pole at $z=1$ for generic $\phi$ if ${\mathcal {J}}_{1/2}(t_0)\ne 0$, where ${\mathcal {J}}_\eta (t)=\sum _{k=0}^\infty \mathrm {sgn}(Df^k_{t}(c_1))|D f^k_{t}(c_1)|^{-\eta }$, with $c_1=t$ the critical value of $f_{t}$. We introduce “Whitney” fractional integrals $I^{\eta ,\Omega }$ and derivatives $M^{\eta ,\Omega }$ on suitable sets $\Omega$. We formulate conjectures on $\Psi ^\Omega _{\phi ,t_0}(\eta , z)$ and ${\mathcal {J}}_\eta (t)$, supported by our results on $M^{\eta ,\Omega }$ and $\Psi ^{\mathrm {fr}}_{\phi ,t_0}(1/2, z)$, for the former, and numerical experiments, for the latter. In particular, we expect that $\Psi ^\Omega _{\phi ,t_0}(1/2, z)$ is singular at $z=1$ for Collet–Eckmann ${t_0}$ and generic $\phi$. We view this work as a step towards the resolution of the paradox that $\Psi _{\phi ,t_0}(z)$ is holomorphic at $z=1$ for Misiurewicz–Thurston $f_{t_0}$ (Jiang and Ruelle in Nonlinearity 18:2447–2453, 2005, Ruelle in Commun Math Phys 258:445–453, 2005), despite lack of linear response (Baladi et al. in Invent Math 201:773–844, 2015).

Appendices

Appendix A. Proof of Lemma 3.2. (Abel’s remark, two-sided)

Recall that if $|x|<1$ then

$$\begin{aligned} \sqrt{1+x}=1+\sum _{n=1}^\infty b_n x^n \text{ with } b_n=\frac{(1/2)(-1/2)\cdots (1/2-n+1) }{n!}. \end{aligned}$$

(68)

(In particular $b_1=1/2$ and $b_2=-1/8$.) We shall also use the fact [17, 195.01] that for any real numbers $a>\tau$ and $b>\tau$

$$\begin{aligned} \partial _\tau \log \bigl ( |\sqrt{a-\tau } -\sqrt{b-\tau }|\bigr ) =\frac{1}{2} \frac{1}{\sqrt{a-\tau }\sqrt{b-\tau }}. \end{aligned}$$

(69)

Proof of Lemma 3.2

Set $x_0=x-c_k$. To prove the claim on $I^{1/2}_{+,t} (\phi _{c_k,+,{\mathcal {Z}}})(x,t)$, it is enough to show that, for any $c_k+t-{\mathcal {Z}}<x<c_k+t+{\mathcal {Z}}$,

$$\begin{aligned} \sqrt{\pi }\cdot I^{1/2}_{+,t }(\phi _{c_k,+,{\mathcal {Z}}})(x,t)= - \log |x_0-t| + \log {\mathcal {Z}}+G_{\mathcal {Z}}(t-x_0)\, , \end{aligned}$$

(70)

where, for $y \in (-{\mathcal {Z}}/2,{\mathcal {Z}}/2)$,

$$\begin{aligned} G_{\mathcal {Z}}(y)=-2\log H_{\mathcal {Z}}(y), \quad H_{\mathcal {Z}}(y)=\frac{\sqrt{1+\frac{y}{{\mathcal {Z}}}}-1}{y/{{\mathcal {Z}}}}>0\, . \end{aligned}$$

Indeed, using (68), we have (the power series below are absolutely convergent)

$$\begin{aligned}&H_{\mathcal {Z}}(y)=1/2+ \sum _{j=1}^\infty b_{j+1} \left( \frac{y}{{\mathcal {Z}}}\right) ^{j} , \,\, H_{\mathcal {Z}}(0)=1/2,\\&\partial _y G_{\mathcal {Z}}(y)=-\frac{2}{H_{\mathcal {Z}}(y)}\cdot \sum _{j=1}^\infty j \cdot b_{j+1} \frac{y^{j-1}}{{\mathcal {Z}}^{j}} , \\&\partial ^2_y G_{\mathcal {Z}}(y)=\frac{2}{(H_{\mathcal {Z}}(y))^2}\cdot \sum _{j=1}^\infty j \cdot b_{j+1} \frac{y^{j-1}}{{\mathcal {Z}}^{j}} -\frac{2}{H_{\mathcal {Z}}(y)}\cdot \sum _{j=2}^\infty j (j-1) b_{j+1} \frac{y^{j-2}}{{\mathcal {Z}}^{j}}. \end{aligned}$$

In particular, $\lim _{{\mathcal {Z}}\rightarrow \infty }\sup _{ y\in (-{\mathcal {Z}}/2,{\mathcal {Z}}/2)} | \partial _y G_{\mathcal {Z}}(y)|=0$, and

$$\begin{aligned} \sup _{{\mathcal {Z}}} \,\, \sup _{ y\in (-{\mathcal {Z}}/2,{\mathcal {Z}}/2)} \max \{| G_{\mathcal {Z}}(y)|, | \partial _y G_{\mathcal {Z}}(y)|, | \partial ^2_y G_{\mathcal {Z}}(y)|\} < \infty \, . \end{aligned}$$

We proceed to show (70). From the definition (26) of $I^{1/2}_{+,t}$, we get

$$\begin{aligned} \sqrt{\pi } I^{1/2}_{+,t} \phi _{c_k,+,{\mathcal {Z}}} (x,t)&=\int _{-\infty }^t\frac{\phi _{c_k,+,{\mathcal {Z}}}(x,\tau )}{\sqrt{t-\tau }} \, \mathrm {d}\tau \\&= \int _{x-c_k-{\mathcal {Z}}}^{\min (t,x-c_k)} \frac{1}{\sqrt{x-c_k-\tau } }\frac{1}{\sqrt{t-\tau }}\, \mathrm {d}\tau . \end{aligned}$$

Recalling that $x_0=x-c_k$, if $x< c_k+t$, then we find

$$\begin{aligned} \sqrt{\pi } I^{1/2}_{+,t} \phi _{c_k,+,{\mathcal {Z}}} (x,t)=\int _{x_0-{\mathcal {Z}}}^{x_0} \frac{1}{\sqrt{x_0-\tau }} \frac{1}{\sqrt{t-\tau }}\, \mathrm {d}\tau . \end{aligned}$$

(This term did not appear in Lemma 3.1.) Using (69), we find

$$\begin{aligned} \int _{x_0-{\mathcal {Z}}}^{x_0}&\frac{1}{\sqrt{x_0-\tau }\sqrt{t-\tau }}\, \mathrm {d}\tau =2\log ( |\sqrt{x_0-\tau }-\sqrt{t-\tau }|) \Bigr |^{x_0}_{x_0-{\mathcal {Z}}}\\&= \log (t-x_0) -2 \log (\sqrt{t-x_0+{\mathcal {Z}}}-\sqrt{\mathcal {Z}}). \end{aligned}$$

If in addition $c_k+t-{\mathcal {Z}}<x$, then, by (68) we find

$$\begin{aligned} -2&\log (\sqrt{t-x_0+{\mathcal {Z}}}-\sqrt{\mathcal {Z}}) = - 2 \log \bigl (\sqrt{\mathcal {Z}}\biggl (\sqrt{1+\frac{t-x_0}{{\mathcal {Z}}}}-1 \biggr ) \bigr ) \nonumber \\&= - 2 \log \biggl ( \frac{t-x_0}{\sqrt{\mathcal {Z}}} \sum _{n=1}^\infty b_n \left( \frac{t-x_0}{{\mathcal {Z}}}\right) ^{n-1}\biggr )\nonumber \\&= +\log {\mathcal {Z}}- 2 \log (t-x_0)-2\log \biggl (1/2+ \sum _{j=1}^\infty b_{j+1} \left( \frac{t-x_0}{{\mathcal {Z}}}\right) ^{j}\biggr ), \end{aligned}$$

(71)

and we have shown that if $c_k+t-{\mathcal {Z}}<x<c_k+t$ then

$$\begin{aligned}&\sqrt{\pi } I^{1/2}_{+, t} \phi _{c_k,+,{\mathcal {Z}}} (x,t)\nonumber \\&=-\log (t-x_0) +\log {\mathcal {Z}}-2\log \biggl (\frac{1}{2}+ \sum _{j=1}^\infty b_{j+1} \left( \frac{t-x_0}{{\mathcal {Z}}}\right) ^{j}\biggr ) \, . \end{aligned}$$

(72)

We now consider the case $c_k+t<x<c_k+t+{\mathcal {Z}}$. Then

$$\begin{aligned} \sqrt{\pi } I^{1/2}_{+,t} \phi _{c_k,+,{\mathcal {Z}}} (x,t)= \int _{x_0-{\mathcal {Z}}}^{t} \frac{1}{\sqrt{x_0-\tau }} \frac{1}{\sqrt{t-\tau }}\, \mathrm {d}\tau . \end{aligned}$$

Using again (69), and we find

$$\begin{aligned} \int _{x_0-{\mathcal {Z}}}^{t} \frac{1}{\sqrt{x_0-\tau }} \frac{1}{\sqrt{t-\tau }}\, \mathrm {d}\tau&= \log (x_0-t) -2 \log (\sqrt{\mathcal {Z}}-\sqrt{t-x_0+{\mathcal {Z}}}). \end{aligned}$$

Similarly as for (71), we find, using (68),

$$\begin{aligned}&-2 \log (\sqrt{\mathcal {Z}}-\sqrt{t-x_0+{\mathcal {Z}}})= - 2 \log \bigl (\sqrt{\mathcal {Z}}\biggl (1-\sqrt{1+\frac{t-x_0}{{\mathcal {Z}}}} \biggr ) \bigr )\\&\quad = - 2 \log \bigl (\frac{x_0-t}{\sqrt{\mathcal {Z}}} \sum _{n=1}^\infty b_n \biggl (\frac{t-x_0}{{\mathcal {Z}}} \biggr )^{n-1} \bigr ) \\&\quad =+\log {\mathcal {Z}}- 2 \log (x_0-t)- 2 \log \biggl (\frac{1}{2}-\sum _{j=1}^\infty b_{j+1} \left( \frac{t-x_0}{{\mathcal {Z}}}\right) ^j\biggr )\, . \end{aligned}$$

We have thus shown that if $c_k+t-{\mathcal {Z}}<x<c_k+t+{\mathcal {Z}}$ then

$$\begin{aligned} \sqrt{\pi }\cdot&I^{1/2}_{+,t} \phi _{c_k,+,{\mathcal {Z}}} (x,t)= -\log |x_0-t| +\log {\mathcal {Z}}- 2 \log \biggl ( \frac{1}{2}+\sum _{j=1}^\infty b_{j+1} \left( \frac{t-x_0}{{\mathcal {Z}}}\right) ^j\biggr )\, . \end{aligned}$$

With (72), the above identity shows the claim on the right-handed spike ($\sigma =+$).

For the left-handed spike, we have, recalling $x_0=x-c_k$ and (27),

$$\begin{aligned} \phi _{c_k,-,{\mathcal {Z}}} (x,t)= \phi _{c_k,+,{\mathcal {Z}}} (x,2x_0-t)= Q \circ T_{2 x_0} (\phi _{c_k,+,{\mathcal {Z}}} ) (x,t)\, . \end{aligned}$$

Thus, we find

$$\begin{aligned} I^{1/2}_{-,t} \phi _{c_k,-,{\mathcal {Z}}} (x,t)&= I^{1/2}_{-,t} \circ Q \circ T_{2 x_0} (\phi _{c_k,+,{\mathcal {Z}}} ) (x,t)\\&= I^{1/2}_{+,t} \circ T_{2 x_0}(\phi _{c_k,+,{\mathcal {Z}}}) (x,-t) = I^{1/2}_{+,t} \phi _{c_k,+,{\mathcal {Z}}} (x,-t+2x_0). \end{aligned}$$

Finally, note that $-(x-c_k-t)=x-c_k-(2x_0-t)$. $\quad \square$

Appendix B. Vanishing of $X_t$ at the image of endpoints

It is sometimes convenient to assume that $X_t$ vanishes at the endpoints $\pm 1$. This can be achieved in several ways, as we explain now. For $t\in (1,2)$, setting

$$\begin{aligned} {\tilde{f}}_t(y)=\frac{f_t(|a_t|y)}{|a_t|}=\frac{t}{|a_t|}-|a_t| y^2= a_t y^2-\frac{t}{a_t}, \end{aligned}$$

gives a family of maps ${\tilde{f}}_t$ preserving $[-1,1]$, with $c_{0,t}=0$, and such that

$$\begin{aligned} {\tilde{f}}_t(-1)=-1={\tilde{f}}_t(1), \,\, \forall \, t\in (-1,2), \end{aligned}$$

so that $\partial _t {\tilde{f}}_t$ vanishes at the endpoints $-1$ and 1. This is a transversal family of quadratic maps in the sense of Tsujii [41], or [5, 6]. The formula for $\partial {\tilde{f}}_t$ being unwieldy, it is convenient to work with the family ${\tilde{h}}_t:[0,1]\rightarrow [0,1]$ defined by ${\tilde{h}}_t(x)=t x(1-x)$, $t \in (1,4]$. The critical point of each ${\tilde{h}}_t$ is 1/2, and $X_t(x)=\partial _t {\tilde{h}}_t \circ {\tilde{h}}_t^{-1}=tx$ (there is a typo in [8, eq. (2)] where it is stated incorrectly that $X^{{\tilde{h}}_t}_t(x)\equiv t$). Then ${\tilde{h}}_t(0)={\tilde{h}}_t(1)=0$ for all t, so that $\partial _t {\tilde{h}}_t$ vanishes at the endpoints 0 and 1. A variant of ${\tilde{h}}_t$ is $h_t(x)=t(1-x^2)-1$ for $t\in (1,2]$ on $[-1, 1]$ (there, $c=0$ and $h_t(-1)=h_t(1)=-1$). However, the formulas for $f_t$ are easier to manipulate than those of ${\tilde{f}}_t$, $h_t$, or ${\tilde{h}}_t$, compensating for the non vanishing of the vector field at the endpoints of a common invariant interval. In addition,³⁸ for any fixed $t_0 \in (1,2)$ and all t close enough to $t_0$, we may extend $f_t$ on $[-2, 2]$ to a $C^4$ map, also called $f_t$, with negative Schwarzian derivative, such that $Df_t$ is positive on $[-2, t-t^2]$ and negative on [t, 2], with $f_t(-2)=f_t(2)=-2$ and $f_t - f_{t_0}= O(|t-t_0|)$. (The extended family $f_t$ is not needed in the present paper, but we expect it should be useful to prove equality [ii] in Conjecture A in future works.) Finally, using that $c_{2,t}=t-t^2>-|a_t|$, one can easily show that for any $t_0\in (1,2)$ there are $\epsilon >0$ and an interval $I'_{t_0}\subset (-2,2)$ such that $f^k_t (I'_{t_0})\subset I'_{t_0}$ for all $t \in (t_0-\epsilon , t_0+\epsilon )$. In other words, $f_t$ for $t \in (t_0-\epsilon , t_0+\epsilon )$ is a transversal family of unimodal maps on $I'_{t_0}$ in the sense of Tsujii [41] since, recalling (10), if ${\mathcal {J}}({t_1})$ is absolutely convergent then ${\mathcal {J}}({t_1})\ne 0$.

Appendix C. Averaging

For regular parameters t (also called “hyperbolic”), the physical measure $\mu _t^{sink}= P^{-1} \sum _{j=1}^P \delta _{x_{t,j}}$ is atomic, supported on an attracting periodic orbit $f^{P} (x_{1,t})=x_{1,t}$ with $P=P(t)$, and can be obtained as

$$\begin{aligned} \lim _{k \rightarrow \infty } \sum _{j=0}^{P-1}\int {\mathcal {L}}_t^{k+j}(\psi ) \phi \, \mathrm {d}m= \lim _{k \rightarrow \infty } \sum _{j=0}^{P-1}\int \psi (\phi \circ f_t^{k+j}) \, \mathrm {d}m = \int \psi \, \mathrm {d}m \cdot \frac{1}{P} \sum _{j=1}^P \phi ({x_{j,t}}) \end{aligned}$$

The convergence is however not uniform in any interval of regular parameters so one cannot a priori sum over k even if $\int _{I_t}\psi \, \mathrm {d}m =0$.

Since almost every parameter is either regular or stochastic [24] it is natural to consider, for a $C^1$ observable, say, a Collet–Eckmann parameter t, and $\epsilon >0$, the double Lebesgue integral

$$\begin{aligned} {\mathcal {A}}_\epsilon (t):=\int _{[-\epsilon , \epsilon ]} \int \phi (x) \mathrm {d}\mu _{t+\delta }(x)\, \, \mathrm {d}\delta , \end{aligned}$$

where $\mu _{t+\delta }=\mu _{t+\delta }^{sink}$ for regular parameters, and $\mu _{t+\delta }=\rho _{t+\delta } dm$ is the SRB measure for stochastic parameters. Then it is not hard to see, using Lebesgue differentiation, that the (ordinary) t-derivative ${\mathcal {A}}_\epsilon '(t)$ exists for almost every $\epsilon >0$ and coincides with $\int \phi (x) \mathrm {d}\mu _{t+\epsilon }(x)$ (see also [44, §3] and [45, (16)]). This does not resolve the paradox described in the introduction, since the derivative depends on $\epsilon$ and does not coincide with $\Psi _\phi (1)$ in general. (Note that the “weakening of the linear response problem” in the introduction of [8]—existence and value of the limit as $\epsilon \rightarrow 0$ of the derivative ${\mathcal {A}}'_\epsilon$—does not explain the paradox either.)

Appendix D. Complements to the proof of Theorem C

We record here interesting facts which are not needed for our proofs.

Remark D.1

(Spectrum on a pole-extended Banach space) For $t \in \mathrm {MT}$, let $\Lambda =\Lambda _t:[-a_t,a_t]\rightarrow [0,1]$ be the absolutely continuous bijection defined by $\Lambda _t(x)=\int _{-a_t}^x \rho _t(u) \mathrm {d}u$. Then (see [30, 38]) the map $F_t:[0,1]\rightarrow [0,1]$ defined by $F_t (\Lambda _t(x))=\Lambda _t (f_t(x))$ is Markov (for the partition $J_\ell$ defined by the endpoints $\Lambda _t(c_k)$, $k=0, \ldots L+P-1$), and $F_t$ is $C^1$ on the interior of each interval of monotonicity $J_\ell$, with $\inf | F_t'|>1$. At the endpoints, we have³⁹

$$\begin{aligned} (D_t F_t)^k(\Lambda _t(c_{1,t})_\pm )=s_k \sqrt{ |(Df_t^k)(c_{1,t})|} \end{aligned}$$

(73)

(taking right or left-sided limits in the left-hand side according to the dynamical orbit). In fact, $G_t:=1/F_t'$ extends to a $C^1$ map on the closure of each $J_\ell$, with $\sup G''_t<\infty$. On the Banach space ${\mathcal {B}}_{\Lambda _t}$ of bounded functions $\phi$ on [0, 1] such that each $\phi |_{\mathrm {int}J_\ell }$ is $C^1$ and admits a $C^1$ extension to the closure of $J_\ell$, the transfer operator ${\mathcal {L}}^\Lambda _t \phi (y)= \sum _{F_t(z)=y} \phi (z)/|F'_t(z)|$ thus has spectral radius equal to one, with a simple eigenvalue at 1, for the eigenvector $\rho ^\Lambda _t(y):=\rho _t(\Lambda _t^{-1}(y))$, and the rest of the spectrum is contained in a disc of radius $\kappa$ strictly smaller than 1. Then ${\mathcal {B}}_t=\{ \phi \circ \Lambda _t , \, \phi \in {\mathcal {B}}_{\Lambda _t}\}$ is a Banach space for the norm induced by ${\mathcal {B}}_{t,\Lambda }$ and the operator ${\mathcal {L}}_t$ on ${\mathcal {B}}_t$ inherits the spectral properties of ${\mathcal {L}}^\Lambda _t$ on ${\mathcal {B}}_{\Lambda _t}$. Any element of ${\mathcal {B}}_t$ belongs to $L^1(dm)$, with $\int _{I_t} |\varphi |\, \mathrm {d}m \le \Vert \varphi \Vert _{{\mathcal {B}}_t}$. Recall the notations ${\mathcal {Y}}_t$, $\chi _k$, $\chi (\mathbf {Y})$, and ${\mathcal {M}}_t$ from Remark 5.7. Then we claim that we may extend ${\mathcal {L}}_t:{\mathcal {B}}_t\rightarrow {\mathcal {B}}_t$ to a bounded operator ${\mathbb {L}}_t$ on the Banach space ${\mathbb {B}}_t:={\mathcal {B}}_t\oplus {\mathcal {Y}}_t$, whose nonzero spectrum is the union of the Pth roots of $\mathrm {sgn}( Df^P_t (c_{L}))$ with the nonzero spectrum of ${\mathcal {L}}_t$ on ${\mathcal {B}}_t$. Morever the following holds if $\mathrm {sgn}(Df^p(c_L))=+1$: First, setting ${\mathcal {M}}^0_t(\mathbf {Y})={\mathcal {M}}_t(\mathbf {Y})-\rho _t \int _{I_t} {\mathcal {M}}_t(\mathbf {Y}) \mathrm {d}m$, and letting $\mathbf {S}_t$ be the fixed point of ${\mathbb {S}}_t$,

$$\begin{aligned} \psi ^*_t:=(\mathrm {id}-{\mathcal {L}}_t)^{-1} ( {\mathcal {M}}^0_t(\mathbf {S}_t))\in {\mathcal {B}}_t, \end{aligned}$$

and the (rank-two) spectral projector $\Pi _1$ for the eigenvalue 1 of ${\mathbb {L}}_t$ satisfies

$$\begin{aligned} \Pi _1(\varphi , \chi (\mathbf {Y}))=\int \varphi \mathrm {d}m \cdot \rho _t +\frac{ \langle \mathbf {S}^*_t, \mathbf {Y}\rangle }{\langle \mathbf {S}^*_t, \mathbf {S}_t\rangle } \cdot (\psi ^*_t+\chi (\mathbf {S}_t))\, . \end{aligned}$$

Second, if $\int _{I_t} {\mathcal {M}}_t(\mathbf {S}_t) \, \mathrm {d}m=0$ then $\psi ^*_t+\chi (\mathbf {S}_t) \in {\mathbb {B}}_t$ is a fixed point⁴⁰ of ${\mathbb {L}}_t$, while if $\int _{I_t} {\mathcal {M}}_t(\mathbf {S}_t) \, \mathrm {d}m\ne 0$ then there exists a non zero $\nu ^N_t\in {\mathbb {B}}_t^*$ such that the (rank-one) nilpotent operator for the eigenvalue 1 of ${\mathbb {L}}_t$ satisfies $N_1^2=0$ and $\Pi _{{\mathcal {B}}_t}\circ N_1=\nu ^N_t \cdot \rho _t$.

We justify the claims above: First, there exists $\kappa <1$ such that, on ${\mathcal {B}}_t$,

$$\begin{aligned} {\mathcal {L}}_t^j(\varphi )=\int \varphi \, \mathrm {d}y \cdot \rho _t + {\mathcal {Q}}^j_t(\varphi ), \,\, j \ge 1, \end{aligned}$$

where for any $\epsilon >0$ there exists C such that $\Vert {\mathcal {Q}}_t^j\Vert _{{\mathcal {B}}_t}\le C (\kappa +\epsilon )^j$ for all $j\ge 1$. Note that ${\mathcal {M}}_t(\mathbf {Y})=\Pi _{{\mathcal {B}}_t} \bigl ( {\mathbb {L}}_t(\chi (\mathbf {Y}))\bigr )$. Identifying $\chi (\mathbf {Y})$ and $\mathbf {Y}$, we have

$$\begin{aligned} {\mathbb {L}}_t( \varphi , \chi (\mathbf {Y}))=\left( \begin{array}{ccc} 1&{}0&{} \rho _t\cdot \int {\mathcal {M}}_t \, \mathrm {d}m \\ 0&{}{\mathcal {Q}}_t&{}{\mathcal {M}}^0_t \\ 0&{}0&{}{\mathbb {S}} \end{array} \right) \left( \begin{array}{c} \rho _t \cdot \int \varphi \, \mathrm {d}m \\ \varphi -\rho _t \cdot \int \varphi \,\mathrm {d}m \\ \ \mathbf {Y} \end{array} \right) \, . \end{aligned}$$

In particular if 1/z does not belong to the spectrum of ${\mathcal {L}}_t$ or ${\mathbb {S}}_t$, then

$$\begin{aligned} \bigl (\mathrm {id}-z{\mathbb {L}}_t\bigr )^{-1}=\left( \begin{array}{ccc} \frac{1}{1-z}&{}0&{} -\frac{ \rho _t \int {\mathcal {M}}_t(\mathrm {id}-z{\mathbb {S}} )^{-1} \,\mathrm {d}m}{1-z} \\ 0&{}(\mathrm {id}-z{\mathcal {Q}}_t)^{-1}&{}-(\mathrm {id}-z{\mathcal {Q}}_t)^{-1}{\mathcal {M}}^0_t(\mathrm {id}-z{\mathbb {S}} )^{-1} \\ 0&{}0&{}(\mathrm {id}-z{\mathbb {S}} )^{-1} \end{array} \right) . \end{aligned}$$

If $\int _{I_t} {\mathcal {M}}_t (\mathbf {S}_t) \, \mathrm {d}m=0$ (with $\mathbf {S}_t$ the fixed point of ${\mathbb {S}}_t$) then a direct computation gives that ${\mathbb {L}}_t$ inherits a (second) fixed point $\psi ^*_t+\chi (\mathbf {S}_t) \in {\mathbb {B}}_t$ from the fixed point $\mathbf {S}_t$ of ${\mathbb {S}}_t$. If $\int {\mathcal {M}}_t(\mathbf {S}_t)\, \mathrm {d}m\ne 0$ then the eigenvalue 1 of ${\mathbb {L}}_t$ has algebraic multiplicity two but geometric multiplicity one, and the associated nilpotent $N_1$ satisfies our claim. In both cases, the claim on $\Pi _1$ follows.

Remark D.2

In the Collet–Eckmann case with an infinite postcritical orbit, the finite matrix ${\mathbb {S}}_t$ appearing in the proof of Theorem C will be replaced by a shift to the right, also denoted ${\mathbb {S}}_t$, weighted by $s_{1,k}=\pm 1$, acting on a space of infinite sequences (for example $\ell ^\infty ({\mathbb {Z}}_+)$). Then ${\mathbb {S}}_t$ does not have any eigenvalues, and its spectrum is contained in the closed unit disc. Also, ${\mathbb {M}}_t(z):=(\mathrm {id}-z {\mathbb {S}}_t)^{-1}$ is the infinite matrix with $({\mathbb {M}}_t(z))_{j,j}=1$, $({\mathbb {M}}_t(z))_{\ell ,j}=(-1)^{1+\ell -j} z^{\ell -j} \prod _{k=\ell }^{j-1} s_{1,k} =(-1)^{1+\ell -j} z^{\ell -j} s_{j-1,\ell }$ for $j<\ell$, and $({\mathbb {M}}_t(z))_{\ell ,j}=0$ for other $\ell$, j.