Corrigendum on Generic fmily with robustly infinitely mny sinks Pierre Berger ll the min results of [Ber16b] re correct, but this would need substntil vrition of the proof s done in [Ber16]. In this corrigendum we prefer to chnge the topologies considered in ll the sttements of [Ber16b]. lso we generlize nd correct the fundmentl property of prblender. Correction of the sttements For d, r, k 0, nd M, N Riemninn mnifolds, two different spces of C d -fmilies (f ) of C r -mps f C r (M, N) prmetred by R k, cn be defined s follows: C d,r (Rk, M, N) = {(f ) : i j zf (z) exists continuously for ll i d, i+j r nd (, z) R k M} C d,r (Rk, M, N) = {(f ) : i j zf (z) exists continuously for ll j r, i d nd (, z) R k M} We endow these spces with the compct open topology w.r.t. the considered derivtives. We notice tht in the importnt cse r = these two spces re equl. In generl, it holds: C d,d+r C d,r Cd,r. The previous rticle delt with the spce C d,r ; the spce Cd,r should be considered insted of C d,r. Then the whole rticle is correct but the sttement of the fundmentl property of the prblender which needs 1 d < r (the cse d = r does not seem to work). Therefore, the topology involved in the sttement of Theorems, C (nd fcts 4.2, 4.3, 4.4 of its proof) must be corrected to C d,r for ny 1 d < r. Likewise, the topology involved in the sttement of Theorems B must be corrected to C d,r for ny 1 d < r <. This correction removes the cse d = r 2. However the vrition [Ber16] of [Ber16b] gives the cse d = r 1 nd lso d r for the topologies C d,r nd Cd,r. In prticulr the sttements of the min theorems of the rticle under correction re correct. Here is the mistke I mde. The spce C d,r is ctully not stble by composition. For instnce if M = M, there exists (f ) C d,r so tht (f f ) does not belong to C d,r. Tht is why we correct it by the spce C d,r which is stble by composition. I m grteful to S. Crovisier for vluble suggestions on the presenttion of the following section. Correction nd generlizton of the fundmentl property Let us fix k 0, 0 d < r. Given Riemnnin mnifold M, nd C d -fmilies of points (z ) R k, its C d -jet t = 0 is denoted by J0 d(z ) = d j=0 j z j! j. Let J0 d M be the spce of C d -jets of C d -fmilies of points in M t = 0. CNRS-LG, Université Pris 13, USPC. 1
We notice tht ny C d,r -fmily (f ) of C r -mps f of M cts cnoniclly J0 d M s the mp: J d 0 (f ) : J d 0 (z ) J d 0 M J d 0 (f (z )) J d 0 M Let us define ctegory of C d -prblenders contining those of [Ber16b, BCP16]. Definition 1. n ffine C d -pr-ifs of R 2 is the dt of finite set of symbols δ, nd polynomils λ δ, Λ δ, p δ, q δ R[X 1,..., X k ] of degrees d, so tht Λ δ (0) < 1/2 < λ δ (0) < 1 nd with () g δ : (x, y) R 2 (Λ δ () x + p δ (), y q δ() ) R 2. λ δ () I e := [ 1, 1], I δ := [p δ Λ δ, p δ + Λ δ ], I δ := [q δ λ δ, q δ + λ δ ] Y e = I e I e Y δ = I δ I e Y δ = I e I δ u Y δ := I δ I e nd s Y δ := I e I δ there exist compct neighborhoods J d 0 R2 of 0 stisfying: (i) for every J d 0 (z ), there exists δ so tht J d 0 ( gδ (z )) is in. (ii) For every smll nd δ, the subsets Y δ () nd Y δ () re included in Y e, with u Y δ () u Y e nd s Y δ () s Y e, moreover they stisfy g δ (Y δ ()) = Y δ (). (iii) for every Z = d i=0 z i i, the vlue z 0 belongs to the interior of Y δ (0) for every δ. Definition 2. Let r d. fmily (f ) of locl diffeomorphisms of R 2 defines nerly ffine C d -prblender if the fmily of its inverse brnches ((g δ ) ) δ is C d,r -perturbtion of (( gδ ) ) δ. Then for smll R k nd δ, with Îδ smll neighborhood of I δ, the imge by g δ of [ 1, 1] Îδ intersects Y e t set Y δ (f ) close to Y δ (). The set Y δ (f ) is bounded by two segments u Y δ (f ) of u Y e, nd two curves s Y δ (f ) close to s Y δ (). The imge by f of Y δ (f ) is denoted by Y δ (f ). It is filled squre close to Y δ (). The set Y δ (f ) is bounded by two segments s Y δ (f ) of s Y e nd two segments u Y δ (f ) close to u Y δ (). We notice tht (i)-(ii)-(iii) re still stisfied by (g δ ) insted of ( g δ ) nd Y δ (f ) insted of Y δ (). Then, for every δ = (δ i ) i 1 Z we define the following locl unstble mnifold: W u loc (δ; f ) := n 1 f n (Y δ n (f )). Exmple 3. In [BCP16], we showed exmple of nerly ffine C d -prblender with Crd = 2. It is precisely for this exmple tht we consider the topology on the inverse brnches rther thn on the dynmics, since the degenerte cse Λ δ = 0 does occur in the limit of renormliztion process. Exmple 4. In [Ber16b], we defined in 2.2, the fmily of mps (f ɛ ) for f U 0 nd ɛ > smll enough. The covering property (i) is shown in section 2.3.2 with with = {P J0 2R2 : P (0) [ 1, 1] [ 2/3, 2/3] i P (0) [ 1, 1] [ 2ɛ, 2ɛ]} nd neighborhood of {P J0 2R2 : P (0) [ 1/2, 1/2] [ 1/2, 1/2] i P (0) [ 1/2, 1/2] [ 3 2 ɛ, 3 2 ɛ]}. Let us fix n ffine C d -pr-ifs (( g ) δ R k) δ. Let γ : x [ 1, 1] (x, x 2 ). 2
Fundmentl property of the prblender If r > d 1 nd Λ δ (0) < λ δ d (0) for every δ, there exist C d,r -neighborhood V γ of the constnt fmily of functions ( γ) nd C d,r - neighborhood V g of (( g ) δ ) δ so tht for every nerly ffine prblender (f ) with inverse brnches ((g δ ) ) δ V g, every (γ ) V γ hs its imge (Γ = γ ([ 1, 1])) which is C d -prtngent t = 0 to locl unstble mnifold of (f ). We recll tht (Γ ) is C d -prtngent to locl unstble mnifold (W u loc (δ; f )) if there re C d -fmilies of points (C ) in (Γ ) nd (Q ) in (W u loc (δ; f )) so tht: J d 0 (C ) = J d 0 (Q ) nd J d 0 (T C Γ ) = J d 0 (T Q W u loc (δ; f )). To prove the fundmentl property we re going to define sequence of symbols (δ k ) k 1 nd C d -fmily of points (C ) of (Γ ) stisfying the following property: (H 1 ) for every k 1, J d 0 (Gk C ) is in, with G k = g δ k (H 2 ) for every k 1, J d 0 (DGk T C Γ ) is smll. g δ 1, Proof tht (H 1 ) (H 2 ) implies the fundmentl property. By proceeding s in the proof of Thm B. [BCP16], (H 1 ) implies the existence of C d -curve of points (Q ) in (W u loc (δ; f )) so tht: J d 0 (C ) = J d 0 (Q ). Note tht J0 d(gk Q ) is equl to J0 d(gk C ) for every k nd so it is in, for every k 1. s (g ) is C d,d+1 -perturbtion of ( g ), (Wloc u (δ; f )) is C d,d+1 -close to be horizontl, by Prop. 1.6 [Ber16b]. The sme holds for (Wloc u (σk (δ); f )), with σ k (δ) = (δ i+k ) i 1. Hence, with L PR 1 the line tngent to Wloc u (δ; f ) t Q, it holds tht J0 d(l ) is smll, nd J0 d(d Q G k L ) s well. Consequently, by (H 2 ), J d 0 (DGk T C Γ ) is close to J d 0 (D C G k L ) for every k 1. Let us notice tht the ction of T Q G k of DG k on PR 1 is exponentilly expnding t the neighborhood of L. The sme holds for J d 0 (T Q G k ) : it is exponentilly expnding t bll centered t J d 0 (L ) nd which contins J d 0 (T C Γ ). Thus they re equl. The proof of (H 1 ) (H 2 ) is done by constructing by induction on n 0 sequence of symbols δ n 1,..., δ 1 so tht there exists C d -curve (C n ) (Γ ) stisfying: () C n 0 is in the interior of the domin of Gn 0. (b) G n (C n ) is the point of G n (Γ ) with the miniml y-coordinte. (c) J d 0 (Gk (C n )) is in for every n k 0 nd J d 0 (Gn 1 (C n )) is in. We observe tht ny C d -curve (C ) in (Γ ), so tht J d 0 (C ) is cluster vlue of (J d 0 (Cn ) ) n stisfies (H 1 ) by (c). To see tht it stisfies lso (H 2 ), let us bring some mterils. Let κ < 1 nd let η > 0 be smll nd such tht: ( ) 1 > κ exp(η), η + η/(1 κexp(η)) < d(, c ) nd 1 > κ > mx δ Λ δ λ d δ (0)exp(η). 3
Let p y : R 2 R be the 2 sd coordinte projection. For ny n 1, nd δ n,..., δ 1, we define the line field: L(δ n δ 1, f ) := ker D(p y g δ n g δ 1 ) nd L(, f ) := ker p y = R {0}. We will define norm on the spce of C d,d -line field fmilies nd show the following below: Lemm 5. For V g sufficiently smll, there exists smll neighborhood V of 0 R k such tht for ll n < 0 nd δ n 1,..., δ 1 d, the C d,d -distnce between the fmilies (L(δ n 1 δ 1, f )) V nd (L(δ n δ 1, f )) V (restricted to the intersection of their definition domins) is t most ηκ n (λ δ n λ δ 1 ). In prticulr (L(δ n δ 1, f )) V is η(1 κ) 1 -C d,d -close to the horizontl line field (L(, f )) V. Proof tht ( b c) implies (H 2 ). By (b), the curve Γ is tngent to L(δ n δ 1, f ) t C n, for every smll. Note lso tht DG k T Γ is equl to L(δ n δ k 1, f ) G k t C n for every n k 1. Thus J d 0 (DGk T Γ ) is equl to J d 0 (L(δ n δ k 1, f ) G k ) t J d 0 (Cn ). By (c), J d 0 (Gk C ) is in the compct set for every k 1. lso J d 0 (L(δ n δ k 1, f )) is η(1 κ) 1 -smll by Lemm 5. Thus J d 0 (DGk T Γ ) is bounded by η(1 κ) 1 dim t J d 0 (Cn ), for every n k 1. Hence (H 2 ) holds true t the cluster vlue J d 0 (C ) of (J d 0 (Cn ) ) n. Proof of the induction hypothesis (-b-c). Let n = 0. Let C 0 be the point which relizes the y- minimum of Γ. s it is C d -close to 0 for V γ smll, its C d -jet J d 0 (C0 ) is in. Hence by (ii), there exists symbol δ 1 so tht G 1 C 0 = g δ 1 C 0 hs its C d -jets t = 0 in. Let n 1. Let us ssume δ n 1,..., δ 1 constructed so tht (C m ) stisfies ( b c) for every n m 0. We put L m := L(δ m δ 1, f ) for every n 1 m 0. For every n 1 m 0, we cn extend (L m ) on Y e so tht nerby = 0, the line fields (L m ) nd (L n 1 ) re η n j=m κ j λ δj λ δ 1 -C d,d -close by Lemm 5. Therefore there is unique point C n 1 () t which L n 1 nd T Γ re equl, this proves (b). Moreover, J0 d(cn 1 ) nd J0 d(cm ) re η n j=m κ j λ δj λ δ 1 -C d,d -close. We will define the norm involved nd we will prove the following below: Lemm 6. For V g sufficiently smll, for every ((g δ ) ) δ V g, for every δ, the following mp is exp(η)/λ δ (0) Lipschitz: Hence J d 0 (Gm 1 (C n 1 J d (g δ ) : J d 0 (z ) J d 0 (g δ z ) J d 0 R 2. )) nd J0 d(gm 1 C m ) is less thn: ηκ m λ δm 1 exp( m 1 η) + η n j=m 1 κ j λ δj λ δm 2 exp( m 1 η) By the first inequlity of ( ) nd since 1 > λ δm 1 (0) 1/2, the bove sum is t most η + η/(1 κexp(η)). 4
By the second inequlity of ( ) nd the second prt of (c) t step m, J0 d(gm 1 C n 1 ) is in for every n m 0. This proves the first prt of (c) t step n 1. Note tht for m = n, this proves tht J0 d(gn 1 to the domin of ny g0 δ, hence () is stisfied. Let δ n 2 so tht J d 0 (gδ n 2 is stisfied. ) sends J d 0 (Gn 2 (C n 1 C n 1 )) is in nd so tht G n 1 0 (C n 1 0 ) belongs ) into. Note tht the second prt of (c) Proof of Lemm 6. We only need to prove this proposition for ( g ) since the Lipschitz constnt depends continuously on the C d;d -perturbtion. With J0 d(z ) = d j=0 z j j, nd since z g j 0 δ = 0 for every j 2, it holds: J d ( g δ (z )) = d n=0 i+k 1 + +k j =n ( i j z g δ 0 ) i!j! ( l z kl ) n = d n=0 i+k=n ( i z g δ 0 ) i! z k n Hence J d ( g δ ) is liner mp with upper tringulr mtrix in the bse (z n ) 1 n d nd with digonl equl to z g 0 δ id which is λ 1 δ (0) -Lipschitz. Hence there exists (c i ) i d with c i smll w.r.t. c j whenever i < j, so tht for the norm d j=0 z j j J0 dr2 j d c j z j, the mp J d ( g ) δ is (0)exp(η/2)-Lipschitz. By tking V g smll in function of η, we get the lemm. λ 1 δ To prove Lemm 5, let us order the set {(i, j) : i + j d} by: (i, j) (i, j ) iff i > i or i = i nd j > j ). Proof of Lemm 5. Let δ, nd let L(Ω) be the spce of C d,d -fmilies of line fields (L ) V over set Ω R 2. We notice tht the following mp fix the horizontl line field H : (, z) R {0}. φ: (L ) L(Y δ ) ((D g δ ) 1 L g δ ) L(Y δ ). In the identifiction of PR 1 which ssocites to line its slope, H is the zero section nd the mp (D g δ ) 1 is the multipliction by Λ δ ()λ δ (). Thus the mp φ is liner: φ(l ) = (Λ δ ()λ δ () L g δ ). Note tht Lemm 5 is proved if we show tht φ is exp(3η/4)λ δ (0)λ 1 d δ (0)-contrcting for C d,d - equivlent norm, independent of δ. Then, there exist continuous functions (C i,j i,j ) (i,j )<(i,j) so tht, for every smll (L ) L(Y δ ) : i j zφ(l ) = Λ δ ()λ δ () i j z(l ) g δ ( z g δ ) i + (i,j ) (i,j) C i,j i,j (z, ) i j z (L ) g δ Thus the derivtives [ i j z] i+j d re bounded from bove by n upper tringulr mtrix with digonl coefficients t most Λ δ λ δ ( z g δ ) d. For V sufficiently smll, the ltter is t most exp(η/2) Λ δ (0) λ δ (0) d 1. Hence there exists (c i,j ) i+j d with c i,j smll w.r.t. c i,j whenever (i, j) (i, j ), so tht the mp φ is exp(3η/4) Λ δ (0) λ δ (0) d 1 -contrcting for the norm (L ) = i+j d c i,j i j zl C 0. 5
References [BCP16] P. Berger, S. Crovisier, nd P. Pujls. Iterted functions systems, blenders nd prblenders. rxiv:1603.01241, to pper Proc. FRFIII, 2016. [Ber16] P. Berger. Emergence nd non-typiclity of the finiteness of the ttrctors in mny topologies. rxiv, 2016. [Ber16b] P. Berger. Generic fmily with robustly infinitely mny sinks. Invent. Mth., 205(1):121 172, 2016. 6