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Correction to: Generic family with robustly infinitely many sinks

  • Pierre BergerEmail author
Correction
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1 Correction to: Invent. Math.  https://doi.org/10.1007/s00222-015-0632-6

All the main results of [2] are correct, but this would need a substantial variation of the proof as done in [3]. In this corrigendum, we prefer to change the topologies considered in all the statements of [2]. Also we generalize and correct the fundamental property of parablender.

Correction of the statements For \(d,r ,k\ge 0\), and MN Riemaniann manifolds, two different spaces of \(C^d\)-families \((f_a)_a\) of \(C^r\)-maps \(f_a\in C^r(M,N)\) parametred by \(\mathbb {R}^k\), can be defined as follows:
$$\begin{aligned} C^{d,r}_A(\mathbb {R}^k ,M,N)= & {} \{(f_a)_a: \partial _a^i\partial _z^j f_a (z)\text { exists continuously for all } \\&\quad i\le d\text { , }i+j\le r \text { and } (a,z)\in \mathbb {R}^k\times M\}\\ C^{d,r}_{PS}(\mathbb {R}^k ,M,N)= & {} \{(f_a)_a: \partial _a^i\partial _z^j f_a (z)\text { exists continuously for all } \\&\quad j\le r\text { , }i\le d \text { and } (a,z)\in \mathbb {R}^k\times M\} \end{aligned}$$
We endow these spaces with the compact open topology w.r.t. the considered derivatives.
We notice that in the important case \(r=\infty \) these two spaces are equal. In general, it holds:
$$\begin{aligned} C^{d,d+r}_A\subset C^{d,r}_{PS}\subset C^{d,r}_A. \end{aligned}$$
The previous article dealt with the space \(C^{d,r}_{PS}\); the space \(C^{d,r}_A\) should be considered instead of \(C^{d,r}_{PS}\). Then the whole article is correct, but the statement of the fundamental property of the parablender which needs \(1\le d <r\le \infty \) (the case \(d=r\) does not seem to work).

Therefore, the topology involved in the statement of Theorems A, C (and facts 4.2, 4.3, 4.4 of its proof) must be corrected to \(C^{d,r}_A\) for any \(1\le d <r\le \infty \). Likewise, the topology involved in the statement of Theorems B must be corrected to \(C^{d,r}_{A}\) for any \(1\le d<r< \infty \).

This correction removes the case \(d=r\ge 2\). However, the variation [3] of [2] gives the case \(d=r\ge 1\) and also \(d\ge r\) for the topologies \(C^{d,r}_A\)and\(C^{d,r}_{PS}\). In particular the statements of the main theorems of the article under correction are correct.

Here is the mistake I made. The space \(C^{d,r}_{PS}\) is actually not stable by composition. For instance, if \(N=M\), there exists \((f_a)_a\in C^{d,r}_{PS}\) such that \((f_a\circ f_a)_a\) does not belong to \(C^{d,r}_{PS}\). That is why we correct it by the space \(C^{d,r}_A\) which is stable by composition.

I am grateful to S. Crovisier for valuable suggestions on the presentation of the following section.

2 Correction and generalization of the fundamental property

Let us fix \(k\ge 0\), \(0\le d<r\le \infty \). Given a Riemannian manifold M, and \(C^d\)-families of points \((z_a)_{a\in \mathbb {R}^k }\), its \(C^d\)-jet at \(a=0\) is denoted by \(J^d_0 (z_a)_a= \sum _{j=0}^d \frac{\partial ^j_a z_a}{j!} a^{\otimes j}\). Let \(J^d_0M\) be the space of \(C^d\)-jets of \(C^d\)-families of points in M at \(a=0\).

We notice that any \(C^{d,r}_A\)-family \((f_a)_a\) of \(C^r\)-maps \(f_a\) of M acts canonically \(J^d_0M\) as the map:
$$\begin{aligned} J^d_0 (f_a)_a:J^d_0 (z_a)_a\in J^d_0M\mapsto J^d_0 (f_a(z_a))_a\in J^d_0M \end{aligned}$$
Let us define a category of \(C^d\)-parablenders containing those of [1, 2]. To this end, put \(I_e=[-1,1]\), \(Y_e= I_e\times I_e \), \(\partial ^ s Y_e= \partial I_e\times I_e \) and \(\partial ^u Y_e= I_e\times \partial I_e \).

Definition 1

An affine \(C^d\)-para-IFS of \(\mathbb {R}^2\) is a finite set of family of maps \(\{(\mathring{g}_a^\delta )_{a\in \mathbb {R}^k}: \delta \in \Delta \}\) from planar subsets \(Y^\delta (a)\) onto \(Y_\delta (a)\) of the form:
where \(I_\delta := [ p_\delta -\Lambda _\delta , p_\delta +\Lambda _\delta ]\), \(I^\delta := [ q_\delta -\lambda _\delta , q_\delta +\lambda _\delta ]\), the functions \(\lambda _\delta ,\Lambda _\delta , p_\delta , q_\delta \) are polynomial such that \( |\Lambda _\delta (0)|<1/2<|\lambda _\delta (0)|<1\) and such that, with \(\partial ^u Y_\delta := I_\delta \times \partial I_e\) and \(\partial ^s Y^\delta := \partial I_e \times I^\delta \), there exist compact neighborhoods \( A'\Subset A\subset J^d_0 \mathbb {R}^2\) of 0 satisfying:
  1. (i)

    for every \(J_0^d(z_a)_a\in A\), there exists \(\delta \in \Delta \) such that \(J^d_0(\mathring{g}^\delta _a (z_a))_a\) is in \(A'\).

     
  2. (ii)

    For every \(\delta \in \Delta \), for every a small, the subsets \(Y_\delta (a)\) and \(Y^\delta (a)\) are included in \(Y_e\), with \(\partial ^u Y_\delta (a)\Subset \partial ^u Y_e\) and \(\partial ^s Y^\delta (a)\Subset \partial ^s Y_e\).

     
  3. (ii)

    for every \(Z=\sum _{i=0}^d z_i a^{\otimes i}\in A\), the value \(z_0\) belongs to the interior of \(Y^\delta (0)\) for every \(\delta \in \Delta \).

     

Definition 2

Let \(r\ge d\). A family \((f_{a})_a\) of local diffeomorphisms of \(\mathbb {R}^2\) defines an affine-like\(C^d\)-parablender if a finite set of its inverse branches \(\{(g^\delta _a)_a : \delta \in \Delta \}\) is a \(C^{d,r}_A\)-perturbation of an affine \(C^d\)-para-IFS \(\{(\mathring{g}^\delta _a)_a: \delta \in \Delta \}\).

Then, for small \(a\in \mathbb {R}^k \) and \(\delta \in \Delta \), with \(\hat{I}^\delta \) a small neighborhood of \(I^\delta \), the image by \(g_a^\delta \) of \([-1,1]\times \hat{I}^\delta \) intersects \(Y_e\) at a set \(Y_\delta (f_a)\) close to \(Y_\delta (a)\). The set \(Y_\delta (f_a)\) is bounded by two segments \(\partial ^u Y_\delta (f_a)\) of \(\partial ^u Y_e\), and two curves \(\partial ^s Y_\delta (f_a)\) close to \(\partial ^s Y_\delta (a)\). The image by \(f_a\) of \(Y_\delta (f_a)\) is denoted by \(Y^\delta (f_a)\). It is a filled square close to \(Y^\delta (a)\). The set \(Y^\delta (f_a)\) is bounded by two segments \(\partial ^s Y^\delta (f_a)\) of \(\partial ^s Y_e\) and two segments \(\partial ^u Y^\delta (f_a)\) close to \(\partial ^u Y^\delta (a)\).

We notice that (i)-(ii)-(iii) are still satisfied by \((g_a^\delta )_a\) instead of \((\mathring{g}_a^\delta )_a\) and \(Y_\delta (f_a)\) instead of \(Y_\delta (a)\).

Then, for every \(\underline{\delta }= (\delta _i)_{i\le -1}\in \Delta ^{\mathbb {Z}^-}\), we define the following local unstable manifold:
$$\begin{aligned} W^u_{loc}(\underline{\delta }; f_a):= \bigcap _{n\ge 1} f^n_a(Y_{\delta _{-n}}(f_a)). \end{aligned}$$

Example 3

In [1], we showed an example of affine-like \(C^d\)-parablender with \(\text {Card}\, \Delta =2\). It is precisely for this example that we consider the topology on the inverse branches rather than on the dynamics, since the degenerate case \(\Lambda _\delta = 0\) does occur in the limit of a renormalization process.

Example 4

In [2], we defined in §2.2, the family of maps \((f_{a\, \epsilon })_a\) for \(f\in U_0\) and \(\epsilon>\) small enough. The covering property (i) is shown in section §2.3.2 \(A=\{P\in J^2_0\mathbb {R}^2: P(0)\in [-1,1]\times [-2/3,2/3]\, \; \, \partial ^i P(0)\in [-1,1] \times [-2\epsilon , 2\epsilon ]\}\) and \(A'\) a neighborhood of \(\{P\in J^2_0\mathbb {R}^2: P(0)\in [-1/2,1/2]\times [-1/2,1/2]\, \; \, \partial ^i P(0)\in [-1/2,1/2] \times [-\frac{3}{2} \epsilon , \frac{3}{2} \epsilon ]\}\).

Let us fix an affine \(C^d\)-para-IFS \(((\mathring{g}_a^\delta )_{ a\in \mathbb {R}^k})_{\delta \in \Delta }\). Let \(\mathring{\gamma }: x\in [-1,1]\mapsto (x,x^2)\).

Fundamental property of the parablenderIf\(\infty \ge r>d\ge 1\)and\(|\Lambda _\delta |(0)<|\lambda _\delta |^{d}(0)\)for every\(\delta \in \Delta \), there exist a\(C^{d,r}_A\)-neighborhood\(V_\gamma \)of the constant family of functions\((\mathring{\gamma })_a\)and a\(C^{d,r}_A\)-neighborhood\(V_g\)of\(((\mathring{g}_a^\delta )_{ a})_\delta \)such that for every affine-like parablender\((f_a)_a\)with inverse branches\(((g_a^\delta )_{ a})_\delta \in V_g\), every\((\gamma _a)_a\in V_\gamma \)has its image\((\Gamma _a=\gamma _a([-1,1]) )_a\)which is\(C^d\)-paratangent at\(a=0\)to a local unstable manifold of\((f_a)_a\).

We recall that \((\Gamma _a)_a\) is \(C^d\)-paratangent to a local unstable manifold \((W^u_{loc}(\underline{\delta }; f_a))_a\) if there are \(C^d\)-families of points \((C_a)_a\) in \((\Gamma _a)_a\) and \((Q_a)_a\) in \((W^u_{loc}(\underline{\delta }; f_a))_a\) such that:
$$\begin{aligned} J^d_0(C_a)_a = J^d_0(Q_a)_a \quad \text {and}\quad J^d_0 (T_{C_a} \Gamma _a)_a =J^d_0 (T_{Q_a}W^u_{loc} (\underline{\delta }; f_a))_a. \end{aligned}$$

Remark 5

The fundamental property of parablender cannot be satified in the topology \(C^{d,d}_A\) since the map \(a\mapsto T_{Q_a}W^u_{loc} (\underline{\delta }; f_a)\) is in general not of class \(C^d\).

To prove the fundamental property, we are going to define a sequence of symbols \((\delta _{k})_{k\le -1}\) and a \(C^d\)-family of points \((C_a)_a\) of \((\Gamma _a)_a\) satisfying the following property:
\((H_1)\)

for every \(k\le -1\), \(J^d_0( G^{k}_a\circ C_a)_a\) is in A, with \(G^k_a= g_a^{\delta _k}\circ \cdots \circ g_a^{\delta _{-1}}\),

\((H_2)\)

for every \(k\le -1\), \(J^d_0( DG^{k}_a\circ T_{C_a} \Gamma _a)_a\) is small.

Proof that\((H_1)-(H_2)\)implies the fundamental property  By proceeding as in the proof of Thm B. [1], \((H_1)\) implies the existence of a \(C^d\)-curve of points \((Q_a)_a\) in \((W^u_{loc}(\underline{\delta }; f_a))_a\) such that:
$$\begin{aligned} J^d_0(C_a)_a = J^d_0(Q_a)_a. \end{aligned}$$
Note that \(J^d_0 (G_a^k\circ Q_a)_a\) is equal to \(J^d_0 (G_a^k\circ C_a)_a\) for every k and so it is in A, for every \(k\le -1\).

As \((g_a)_a\) is a \(C^{d,d+1}_A\)-perturbation of \((\mathring{g}_a)_a\), \((W^u_{loc}(\underline{\delta }; f_a))_a\) is \(C^{d,d+1}_A\)-close to a be horizontal, by Prop. 1.6 [2]. The same holds for \((W^u_{loc}(\sigma ^k (\underline{\delta }); f_a))_a\), with \(\sigma ^k (\underline{\delta })=(\delta _{i+k})_{i\le -1}\). Hence, with \(L_a\in \mathbb {P}\mathbb {R}^1\) the line tangent to \(W^u_{loc}(\underline{\delta }; f_a)\) at \(Q_a\), it holds that \(J ^d_0 (L_a)_a\) is small, and \(J ^d_0 (D_{Q_a}G ^k_a\circ L_a)_a\) as well. Consequently, by \((H_2)\), \(J^d_0 (D G_a^k\circ T_{C_a}\Gamma _a)_a\) is close to \(J ^d_0 (D_{C_a}G ^k_a\circ L_a)_a\) for every \(k\le -1\).

Let us notice that the action of \(T_{Q_a}G^k_a\) of \(DG^k_a\) on \(\mathbb {P}\mathbb {R}^1\) is exponentially expanding at the neighborhood of \(L_a\). The same holds for \(J^d_0(T_{Q_a}G^k_a)_a\): it is exponentially expanding at a ball centered at \(J^d_0(L_a)_a\) and which contains \(J^d_0 (T_{C_a}\Gamma _a)_a\). Thus they are equal. \(\square \)

The proof of \((H_1)-(H_2)\) is done by constructing by induction on \(n\le 0\) a sequence of symbols \(\delta _{n-1},\dots , \delta _{-1}\in \Delta \) such that there exists a \(C^d\)-curve \((C^{n}_a)_a\in (\Gamma _a)_a\) satisfying:
(a)

\(C^{n}_0\) is in the interior of the domain of \(G_0^n\).

(b)

\(G_a^n(C^n_a)\) is the point of \(G_a^n(\Gamma _a)\) with the minimal y-coordinate.

(c)

\(J^d_0 (G_a^k(C^n_a))_a\) is in A for every \(n\le k\le 0\) and \(J^d_0 (G_a^{n-1}(C^n_a))_a\) is in \(A'\).

We observe that any \(C^d\)-curve \((C_a)_a\) in \((\Gamma _a)_a\), such that \(J_0^d(C_a)_a\) is a cluster value of \((J^d_0 (C_a^n)_a)_n\) satisfies \((H_1)\) by (c). To see that it satisfies also \((H_2)\), let us bring some materials.
Let \(\kappa <1\) and let \(\eta >0\) be small and such that:
Let \(p_y:\mathbb {R}^2\rightarrow \mathbb {R}\) be the \(2^{sd}\) coordinate projection. For any \(n\le -1\), and \(\delta '_{n},\dots , \delta '_{-1}\in \Delta \), we define the line field:
$$\begin{aligned}&L(\delta '_{n}\cdots \delta '_{-1}, f_a):= \ker \, D(p_y\circ g_a^{\delta '_{n}}\circ \cdots \circ g_a^{\delta '_{-1}})\quad \text {and}\\&L(\varnothing , f_a):= \ker \, p_y=\mathbb {R}\times \{0\}. \end{aligned}$$
We will define a complete distance on the space of \(C^{d,d}_A\)-line field families and show the following below:

Lemma 6

For \(V_g\) sufficiently small, there exists a small neighborhood V of \(0\in \mathbb {R}^k\) such that for all \(n<0\) and \(\delta '_{n-1},\dots , \delta '_{-1}\in \Delta _d\), the \(C^{d,d}_A\)-distance between the families \((L(\delta '_{n-1} \cdots \delta '_{-1}, f_a))_{a\in V}\) and \((L(\delta '_{n}\cdots \delta '_{-1}, f_a))_{a\in V}\) (restricted to the intersection of their definition domains) is at most \(\eta \kappa ^{|n|} ( \lambda _{\delta '_{n}}\cdots \lambda _{\delta '_{-1}})\). In particular, \((L(\delta '_{n} \cdots \delta '_{-1}, f_a))_{a\in V}\) is \(\eta (1-\kappa )^{-1}\)-\(C^{d,d}_A\)-close to the horizontal line field \((L(\varnothing ,f_a))_{a\in V}\).

Proof that\((a-b-c)\)implies\((H_2)\) By (b), the curve \(\Gamma _a\) is tangent to \(L(\delta _{n}\cdots \delta _{-1}, f_a)\) at \(C^n_a\), for every a small. Note also that \(DG_a^k\circ T\Gamma _a\) is equal to \(L(\delta _{n}\cdots \delta _{k-1}, f_a) \circ G_a^k\) at \({C^n_a}\) for every \(n\le k\le -1\). Thus \(J^d_0 (DG_a^k\circ T\Gamma _a)_a\) is equal to \(J^d_0(L(\delta _{n}\cdots \delta _{k-1}, f_a) \circ G_a^k)_a\) at \(J^d_0 (C^n_a)_a \).

By (c), \(J^d_0(G_a^k\circ C_a)_a\) is in the compact set A for every \(k\le -1\). Also \(J^d_0 (L(\delta _{n}\cdots \delta _{k-1}, f_a))_a\) is \(\eta (1-\kappa )^{-1}\)-small by Lemma 6.

Thus, \(J^d_0 (DG_a^k\circ T\Gamma _a)_a\) is uniformly dominated by \(\eta (1-\kappa )^{-1}\text {diam}\,A\) at \(J^d_0 (C^n_a)_a \), among \(n\le k\le -1\). Hence \((H_2)\) holds true at the cluster value \( J^d_0 (C_a)_a\) of \((J^d_0 (C^n_a)_a)_n\).

Proof of the induction hypothesis (a-b-c)

\(\underline{\hbox {Let } n=0.}\) Let \(C_a^0 \) be the point which realizes the y-minimum of \(\Gamma _a\). As it is \(C^d\)-close to 0 for \(V_\gamma \) small, its \(C^d\)-jet \(J^d_0 (C_a^0)_a\) is in A. Hence by (i), there exists a symbol \(\delta _{-1}\) such that \(G^{-1}_a\circ C_a^{0}=g_a^{\delta _{-1}}\circ C_a^0\) has its \(C^d\)-jets at \(a=0\) in \(A'\).

\(\underline{\hbox {Let } n\le -1.}\) Let us assume \(\delta _{n-1},\dots , \delta _{-1}\in \Delta \) constructed such that \((C^m_a)_a\) satisfies \((a-b-c)\) for every \(n\le m\le 0\). We put \(L_{m\, a} := L(\delta _{m}\cdots \delta _{-1}, f_a)\) for every \(n-1\le m\le 0\).

For every \(n-1< m\le 0\), we can extend \((L_{m\, a})_a\) on \(Y_e\) such that nearby \(a=0\), the line fields \((L_{m\, a})_{a}\) and \((L_{n-1\, a})_{a}\) are \(\eta \sum _{j=m}^{n} \kappa ^{|j|} \lambda _{\delta _{j}}\cdots \lambda _{\delta _{-1}}\)-\(C^{d,d}\)-close by Lemma 6.

Therefore, there is a unique point \(C_{n-1}(a)\) at which \(L_{n-1 a}\) and \(T\Gamma _a\) are equal, this proves (b). Moreover, \(J^d_0(C^{n-1}_a)_a\) and \(J^d_0(C^{m}_a)_a\) are \(\eta \sum _{j=m}^{n} \kappa ^{|j|} \lambda _{\delta _{j}}\cdots \lambda _{\delta _{-1}} \hbox {-}C^{d,d}\)-close.

We will define the norm involved, and we will prove the following below:

Lemma 7

For \(V_g\) sufficiently small, for every \(((g_a^\delta )_a)_\delta \in V_g\), and for every \(\delta \in \Delta \), the following map is \(\text {exp}(\eta ) /\lambda _\delta (0)\) Lipschitz:
$$\begin{aligned} J^d(g_a^\delta )_a:J^d_0(z_a)_a\in A\mapsto J^d_0(g_a^\delta \circ z_a)_a \in J^d_0\mathbb {R}^2. \end{aligned}$$
Hence, \(J^d_0(G^{m-1}_a(C^{n-1}_a))_a\) and \(J^d_0(G^{m-1}_a(C^{m}_a))_a\) is less than:
$$\begin{aligned} \frac{\eta \kappa ^{|m|}}{\lambda _{\delta _{m-1}}} \text {exp}(|m-1| \eta ) + \eta \sum _{j=m-1}^{n} \kappa ^{|j|} \lambda _{\delta _{j}}\cdots \lambda _{\delta _{m-2}}\text {exp}(|m-1| \eta ) \end{aligned}$$
By the first inequality of \((\star )\) and since \(1>|\lambda _{\delta _{m-1}}(0)|\ge 1/2\), the above sum is at most \(2\eta \text {exp}(\eta )+ \eta /(1-\kappa \text {exp}(\eta ))\).

By the second inequality of \((\star )\) and the second part of (c) at step m, \(J^d_0(G^{m-1}_a(C^{n-1}_a))_a\) is in A for every \(n\le m\le 0\). This proves the first part of (c) at step \(n-1\).

Note that for \(m=n\), we obtained that \(J^d_0(G^{n-1}_a(C^{n-1}_a))_a\) is in A and such that \(G^{n-1}_0(C^{n-1}_0)\) belongs to the domain of any \(g_0^\delta \), hence (a) is satisfied.

Let \(\delta _{n-2}\in \Delta \) be such that \(J^d_0 (g_a^{\delta _{n-2}})_a\) sends \(J^d_0(G^{n-2}_a(C^{n-1}_a))_a\) into \(A'\). Note that the second part of (c) is satisfied. \(\square \)

Proof of Lemma 7

We only need to prove this proposition for \((\mathring{g}_a)\) since the Lipschitz constant depends continuously on the \(C^{d,d}\)-perturbation.

With \(J^d_0(z_a)_a = \sum _{j=0}^d z_j a^{\otimes j}\), and since \(\partial _z^j \mathring{g}_0^\delta =0\) for every \(j\ge 2\), it holds:
$$\begin{aligned} J^d(\mathring{g}_a^\delta ( z_a) )_a= & {} \sum _{n=0}^d \sum _{i+k_1+\cdots +k_j=n} \frac{(\partial _a^i \partial _z^j \mathring{g}_0^\delta ) }{i! j!}\cdot (\otimes _l z_{k_l})\otimes a^{\otimes n} \\= & {} \sum _{n=0}^d \sum _{i+k=n} \frac{(\partial _a^i \partial _z\mathring{g}_0^\delta ) }{i! }\cdot z_{k}\otimes a^{\otimes n} \end{aligned}$$
Hence \(J^d(\mathring{g}_a^\delta )_a \) is a linear map with upper triangular matrix in the base \((z_n)_{1\le n\le d}\) and with diagonal equal to \(\partial _z\mathring{g}_0^\delta \cdot id\) which is \(|\lambda _\delta ^{-1} (0) |\)-Lipschitz. Hence there exists \((c_i)_{i\le d}\) with \(c_i\) small w.r.t. \(c_{j}\) whenever \(i<j\), such that for the norm \( \sum _{j=0}^d z_j a^{\otimes j}\in J^d_0\mathbb {R}^2\mapsto \sum _{j\le d} c_j \Vert z_j\Vert \), the map \(J^d(\mathring{g}_a^\delta )_a\) is \(\lambda _\delta ^{-1} (0)\text {exp}(\eta /2)\)-Lipschitz. By taking \(V_g\) small in function of \(\eta \), we get the lemma. \(\square \)
To prove Lemma 6, let us order the set \(\{(i,j):i+j\le d\}\) by:
$$\begin{aligned} (i,j)\succ (i',j')\text { iff }i>i'\text { or }i=i'\text { and }j>j'). \end{aligned}$$

Proof of Lemma 6

Let \(\delta \in \Delta \), and let \(\mathcal {L}(\Omega )\) be the space of \(C^{d,d}\)-families of line fields \((L_a)_{a\in V}\) over a set \(\Omega \subset \mathbb {R}^2\). We notice that the following map fixes the horizontal line field \(H:(a,z)\mapsto \mathbb {R}\times \{0\}\).
$$\begin{aligned} \phi :(L_a)_a\in \mathcal {L}( Y^\delta ) \mapsto ((D\mathring{g}^\delta _a)^{-1}\circ L_a\circ \mathring{g}^\delta _a)_a\in \mathcal {L}(Y_\delta ). \end{aligned}$$
In the identification of \(\mathbb {P}\mathbb {R}^1\) which associates with a line its slope, H is the zero section and the map \((D\mathring{g}^\delta _a)^{-1}\) is the multiplication by \(\Lambda _\delta (a)\lambda _\delta (a)\). Thus the map \(\phi \) is linear:
$$\begin{aligned} \phi (L_a)_a= (\Lambda _\delta (a)\lambda _\delta (a)\cdot L_a\circ \mathring{g}^\delta _a)_a. \end{aligned}$$
Note that Lemma 6 is proved if we show that \(\phi \) is \(\text {exp}(3\eta /4) \Lambda _\delta (0) \lambda ^{1-d}_\delta (0)\)-contracting for a \(C^{d,d}_A\)-equivalent norm, independent of \(\delta \).
Then, there exist continuous functions \((C^{i,j}_{i',j'})_{(i',j')<(i,j)}\) such that, for every small \((L_a)_a\in \mathcal {L}(Y^\delta )\):
$$\begin{aligned} \partial _a^i\partial _z^j \phi (L_a)_a= & {} \Lambda _\delta (a)\lambda _\delta (a)\cdot \partial _a^i\partial _z^j (L_a) \circ \mathring{g}^\delta _a\cdot (\partial _z\mathring{g}^\delta _a)^i\\&+\sum _{(i',j')\prec (i,j)} C^{i,j}_{i',j'}(z,a) \partial _a^{i'}\partial _z^{j'} (L_a) \circ \mathring{g}^\delta _a \end{aligned}$$
Thus the derivatives \([\partial ^i_a\partial ^j_z]_{i+j\le d}\) are bounded from above by an upper triangular matrix with diagonal coefficients at most \(\Vert \Lambda _\delta \lambda _\delta \cdot (\partial _z\mathring{g}^\delta _a)^d\Vert \). For V sufficiently small, the latter is at most \(\text {exp}(\eta /2) |\Lambda _\delta (0)|\cdot |\lambda _\delta (0)^{d-1}|\). Hence, there exists \((c_{i,j})_{i+j\le d}\) with \(c_{i,j}\) small w.r.t. \(c_{i',j'}\) whenever \((i,j)\prec (i',j')\), such that the map \(\phi \) is \(\text {exp}(3\eta /4) |\Lambda _\delta (0)|\cdot |\lambda _\delta (0)^{d-1}|\)-contracting for the norm \(\Vert (L_a)_a\Vert = \sum _{i+j\le d} c_{i,j} \Vert \partial _a^i\partial _z^j L_a\Vert _{C^0}\). \(\square \)

Notes

References

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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Université Paris 13VilletaneuseFrance

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