Correction to: Generic family with robustly infinitely many sinks
 28 Downloads
1 Correction to: Invent. Math. https://doi.org/10.1007/s0022201506326
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.
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 \(a=0\).
Definition 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 \(A'\).
 (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 \(\partial ^s Y^\delta (a)\Subset \partial ^s Y_e\).
 (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 affinelike\(C^d\)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 \(C^d\)paraIFS \(\{(\mathring{g}^\delta _a)_a: \delta \in \Delta \}\).
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 \(Y_\delta (a)\).
Example 3
In [1], we showed an example of affinelike \(C^d\)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 \(C^d\)paraIFS \(((\mathring{g}_a^\delta )_{ a\in \mathbb {R}^k})_{\delta \in \Delta }\). 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 affinelike 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\((f_a)_a\).
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 \(C^d\).
 \((H_1)\)

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.
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. \(\square \)
 (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 ycoordinate.
 (c)

\(J^d_0 (G_a^k(C^n_a))_a\) is in A for every \(n\le k\le 0\) and \(J^d_0 (G_a^{n1}(C^n_a))_a\) is in \(A'\).
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 \(\delta '_{n1},\dots , \delta '_{1}\in \Delta _d\), the \(C^{d,d}_A\)distance between the families \((L(\delta '_{n1} \cdots \delta '_{1}, f_a))_{a\in V}\) 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\((abc)\)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 \(L(\delta _{n}\cdots \delta _{k1}, f_a) \circ G_a^k\) 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 \(J^d_0(L(\delta _{n}\cdots \delta _{k1}, f_a) \circ G_a^k)_a\) 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 \(J^d_0 (L(\delta _{n}\cdots \delta _{k1}, f_a))_a\) 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 (abc)
\(\underline{\hbox {Let } n=0.}\) Let \(C_a^0 \) be the point which realizes the yminimum 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 \(\delta _{n1},\dots , \delta _{1}\in \Delta \) constructed such that \((C^m_a)_a\) satisfies \((abc)\) for every \(n\le m\le 0\). We put \(L_{m\, a} := L(\delta _{m}\cdots \delta _{1}, f_a)\) for every \(n1\le m\le 0\).
For every \(n1< m\le 0\), we can extend \((L_{m\, a})_a\) on \(Y_e\) such that nearby \(a=0\), the line fields \((L_{m\, a})_{a}\) and \((L_{n1\, a})_{a}\) 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 \(C_{n1}(a)\) at which \(L_{n1 a}\) and \(T\Gamma _a\) are equal, this proves (b). Moreover, \(J^d_0(C^{n1}_a)_a\) 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
By the second inequality of \((\star )\) and the second part of (c) at step m, \(J^d_0(G^{m1}_a(C^{n1}_a))_a\) is in A for every \(n\le m\le 0\). This proves the first part of (c) at step \(n1\).
Note that for \(m=n\), we obtained that \(J^d_0(G^{n1}_a(C^{n1}_a))_a\) is in A and such that \(G^{n1}_0(C^{n1}_0)\) belongs to the domain of any \(g_0^\delta \), hence (a) is satisfied.
Let \(\delta _{n2}\in \Delta \) be such that \(J^d_0 (g_a^{\delta _{n2}})_a\) sends \(J^d_0(G^{n2}_a(C^{n1}_a))_a\) 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.
Proof of Lemma 6
Notes
References
 1.Berger, P., Crovisier, S., Pujals, E.: Iterated functions systems, blenders, and parablenders. In: Conference of Fractals and Related Fields, pp. 57–70. Springer, (2015)Google Scholar
 2.Berger, P.: Generic family with robustly infinitely many sinks. Invent. Math. 205(1), 121–172 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
 3.Berger, P.: Emergence and nontypicality of the finiteness of the attractors in many topologies. Proc. Steklov Inst. Math. 297(1), 1–27 (2017)MathSciNetCrossRefzbMATHGoogle Scholar