Resonances in Loewner equations

Theorem 1.2. Let (ϕs,t) be a dilation evolution family such that the eigenvalues of Λ satisfy 2Re α1 < Re αN. (1.5) Then there exists a normal Loewner chain (fs) associated to (ϕs,t), such that S s fs(B) = CN, hence Lr (ϕs,t) = CN. This chain is given by fs = lim t→+∞e−Λtϕs,t, (1.6) where the limit is taken in the topology of uniform convergence on compacta, and it is the unique normal Loewner chain associated to (ϕs,t). A family of univalent mappings (gs) is a Loewner chain associated to (ϕs,t)

Theorem 8.6. Let (ϕs,t) be a dilation evolution family. Then there exists a Loewner chain (fs) associated to (ϕs,t), such that S s fs(B) = CN, hence Lr (ϕs,t) = CN. If no real resonances occur among the eigenvalues of Λ, then (fs) is a normal chain, not necessarily unique. A family of univalent mappings (gs) is a Loewner chain associated to (ϕs,t) if and only if there exists an entire univalent mapping Ψ on CN such that gs = Ψ ◦fs. Notice that (1.5) is a classical condition which ensures the exi

Lemma 2.1. Let M > 0 and f : B →CN be a holomorphic mapping fixing the origin and bounded by M. Then for z in the ball, |f(z)| ≤M|z|. If there is a point z0 ∈B \ {0} such that |f(z0)| = M|z0|, then |f(ζz0)| = M|ζz0| for all |ζ| < 1/|z0|. Moreover, if f(z) = O(|z|k), k ≥2, then for z in the ball, |f(z)| ≤M|z|k. Let Fr,M,A be the family of holomorphic mappings f : rB →CN, bounded by M, fixing the origin and with common differential Az at the origin satisfying ∥A∥< 1.

Lemma 2.2. For each f ∈Fr,M,A, we have |f(z) −Az| ≤C|z|2, where C = C(r, M, A). If moreover f(z) −Az = O(|z|k) for k ≥3, then |f(z) −Az| ≤Ck|z|k, where Ck = Ck(r, M, A).

Lemma 2.3. For each f ∈Fr,M,A we have the following estimate: to each ∥A∥< α < 1 there corresponds s > 0, s = s(r, M, A) such that |f(z)| ≤α|z|, if |z| ≤s.

Lemma 2.4. For each f ∈Fr,r,A we have the following estimate: to each s < r there corresponds K < 1, K = K(r, A), such that |f(z)| ≤K|z|, if |z| ≤s.

Lemma 2.5. Suppose that D is an open set in CN containing the origin. Suppose we have an uniformly bounded family H of holomorphic mappings h: D →CN in Tang1(CN, 0). Then there exist a ball rB ⊂D such that every h ∈H is univalent on rB, and a ball sB such that sB ⊂h(rB) for all h ∈H.

Lemma 4.2. Assume that supn deg Tn,n+1 < ∞, then supn deg T0,n < ∞.

Lemma 4.3. Let (Tn,m) be a triangular evolution family of bounded degree and bounded coefficients. Let ∆be the unit polydisc. Then there exists a constant γ > 0 such that Tk,0(∆) ⊂γk∆, k ≥1.

Corollary 4.4. Let (Tn,m) be a triangular evolution family of bounded degree and bounded coefficients. Let 1 2∆be the polydisc of radius (1/2). Then there exists β ≥0 such that for all k ≥1 and all z, z′ ∈1 2∆, |Tk,0(z) −Tk,0(z′)| ≤βk|z −z′|.

Lemma 4.5. Let (Tn,m) be a triangular evolution family, with bounded degree and bounded coefficients. Then T0,n(z) →0 uniformly on compacta. Hence for each neighborhood V of 0 we have ∞ [ n=1 Tn,0(V ) = CN.

Lemma 5.2. Let 0 < r < t. Let (ϕn,m; tB) be a discrete dilation evolution family. Suppose there exists a Loewner chain (fn) associated to the evolution family (ϕn,m; rB). Then there exists a Loewner chain (f e n) associated to (ϕn,m; tB) which extends (fn) in the following sense: f e n(z) = fn(z), z ∈rB, n ≥0.

Proposition 5.4. A discrete dilation evolution family (ϕn,m) admits a normal Loewner chain if and only if there exists a normal family (hn) of univalent mappings in Tang1(CN, 0) which conjugates it to its linear part: hm ◦ϕn,m = Am−nhn, 0 ≤n ≤m. Next we show how to find conjugations, provided we start with a discrete dilation evolution family close enough to a triangular evolution family.

Proposition 5.5. Suppose that (ϕn,m; tB) is a discrete dilation evolution family, and that (Tn,m) is a triangular evolution family with bounded degree and bounded coefficients. Let β be the constant given by Corollary 4.4 for (Tn,m), and let k be an integer such that |λ1|k < 1 β. If for each n ≥0 we have ϕn,n+1(z) −Tn,n+1(z) = O(|z|k), then (ϕn,m) is conjugate to (Tn,m).

Proposition 6.2. Let (ϕn,m; tB) be a discrete dilation evolution family. For each i ≥2 there exist (1) an evolution family (ϕi n,m) defined on a ball Bi, (2) a uniformly bounded family (ki n) of univalent maps defined on a ball B′ i ⊂tB which conjugates (ϕn,m) to (ϕi n,m), (3) a triangular evolution family (T i n,m) with deg T i n,n+1 ≤i −1 for all n ≥0 and bounded coefficients such that for all n ≥0, ϕi n,n+1 = T i

Lemma 2.5 yields a ball rB ⊂D such that every kn is univalent on rB, and a ball sB such that sB ⊂kn(rB) for all n ≥0. On sB we can define a family of holomorphic mappings ϕi+1 n,n+1 = kn+1 ◦ϕi n,n+1 ◦k−1 n . By Lemma 2.3 there exists a ball Bi+1 invariant for each ϕi+1 n,n+1. Hence (ϕi+1 n,n+1; Bi+1) is a discrete evolution family. Since (kn) is an equicontinuous family, there exists a ball uB such that kn(uB) ⊂Bi+1 for all n ≥0, so that (kn) conjugates (ϕi n,n+1; Bi) to (ϕi+1 n,n+1; Bi+1): kn+1

Proposition 6.4. A discrete dilation evolution family (ϕn,m; tB) admits an associated normalized Loewner chain (fn) such that S n fn(tB) = CN. If no real resonances occur (fn) is a normal chain.

Proposition 5.5 obtaining a uniformly bounded family (hn) given by hn = lim m→∞Tm,n ◦ϕl n,m, defined on a ball rB ⊂Bl, which conjugates (ϕl n,m; Bl) to (Tn,m). Thus a Loewner chain associated to (ϕl n,m; Bl) is given by the mappings (Tn,0 ◦hn)e = lim m→∞Tm,0 ◦ϕl n,m. Since (kl n) conjugates (ϕn,m; tB) to (ϕl n,n+1; Bl), a Loewner chain associated to (ϕn,m; tB) is given by

Proposition 7.1. Let (ϕn,m; tB) be a discrete dilation evolution family, and let (fn) be the Loewner chain given by Proposition 6.4. A family of holomorphic mappings (gn) is a subordination chain associated to (ϕn,m) if and only if there exists an entire mapping Ψ on CN such that gn = Ψ ◦fn.

Lemma 8.4. Let kB be the Kobayashi metric of B. Given a dilation evolution family (ϕs,t), a dilation Loewner chain (fs) and a dilation Herglotz vector field H(z, t) the following hold: to each T > 0 and to any compact set K ⊂B there correspond positive constants cT,K, CT,K and kT,K such that for all z ∈K and 0 ≤s ≤t′ ≤t ≤T, (1) kB(ϕs,t(z), ϕs,t′(z)) ≤cT,K(t −t′), (2) |fs(z) −ft(z)| ≤kK,T(t −s), (3) |H(z, t)| ≤CK,T. Therefore (ϕs,t) is an L∞-evolution family, (fs) is an L∞-Loewner chain, and H(z,

Theorem 2.8]. □ Recall if (ϕs,t) is an L∞-evolution family, each mapping ϕs,t is univalent [4, Proposition 5.1]. If we restrict the time to integer values in a dilation evolution family (ϕs,t) we obtain the discretized dilation evolution family (ϕn,m). We have Az = d0ϕn,n+1(z) = (eα1z1, . . . , eαNzN) = eΛz. In the continuous framework a real resonance is an identity Re ( N X j=1 kjαj) = Re αl, where kj ≥0 and P j kj ≥2. It is easy to see that a continuous real resonance corresponds to a real re

Lemma 8.5. Let (ϕs,t) be a dilation evolution family, and let (ϕn,m) be its discretized evolution family. Assume there exists a discrete Loewner chain (fn) associated to (ϕn,m). Then we can extend it in a unique way to a dilation Loewner chain associated to (ϕs,t). If (fn) is a normal Loewner chain, then also (fs) is normal.

Theorem 8.6. Let (ϕs,t) be a dilation evolution family. Then there exists a dilation Loewner chain (fs) associated to (ϕs,t), such that S s fs(B) = CN. If no real resonances occur then (fs) is a normal chain. A family of holomorphic mappings (gs) is a subordina- tion chain associated to (ϕs,t) if and only if there exists an entire mapping Ψ on CN such that gs = Ψ ◦fs.

Theorem 8.9. Let H be a dilation Herglotz vector field, and let t 7→ϕs,t be the solution of the associated Loewner ODE. Then if (fs) is the Loewner chain associated to the dilation L∞-evolution family (ϕs,t) given by Theorem 8.6, the mapping t 7→ft is a solution for the Loewner PDE ∂ft(z) ∂t = −dzftH(z, t). Moreover, a family (gs) of holomorphic mappings on the ball satisfies (1) the mapping t 7→gt is locally absolutely continuous in t, uniformly on compacta with respect to z ∈B, (2) the mapping t