Abstract
We study explicit examples of Loewner chains generated by ab-
solutely continuous driving measures, and discuss how properties of driving
measures are reflected in the shapes of the growing Loewner hulls.
Results & Lemmas (6)
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Lemma 3.1.
Lemma 3.1. The solution to (3.4) is given by the conformal maps (3.7) φm(z, t) = et(zm −1 + e−mt)1/m, z ∈∆. Applying the inversion map z…
Lemma 3.1. The solution to (3.4) is given by the conformal maps (3.7) φm(z, t) = et(zm −1 + e−mt)1/m, z ∈∆. Applying the inversion map z 7→1/z, we recover the maps of [26, Example 4, Table 1]. We note that the angular limits of φm fix the points on the unit circle corresponding to the m-th roots of unity, that is, the m zeros of the density ρm, and that the growing hulls are smooth. In the case m = 1, the hulls are simply disks tangent to T at ζ = 1.
Proposition 3.2.
Proposition 3.2. Suppose that two functions p and ˜p appearing in the right-hand side of the Loewner equation (2.2) satisfy (3.8) ˜p(z) =…
Proposition 3.2. Suppose that two functions p and ˜p appearing in the right-hand side of the Loewner equation (2.2) satisfy (3.8) ˜p(z) = p(zm), z ∈∆. Then the corresponding conformal mappings satisfy ˜f(z, t)=[f(zm, mt)]1/m. A sufficient condition on the two families of driving measures for this to occur to hold is provided by the next lemma.
Lemma 3.3.
Lemma 3.3. Let µ t≥0 be a family of absolutely continuous probability measures with densities ρt t≥0, and let µm t≥0 be the family of…
Lemma 3.3. Let {µ}t≥0 be a family of absolutely continuous probability measures with densities {ρt}t≥0, and let {µm}t≥0 be the family of measures whose densities are given by ρm t (e2πix) = ρt(e2πimx). Then, if S[µt](z) = p(z, t), we have S[µm t ](z) = p(zm, t).
Lemma 3.4.
Lemma 3.4. The solutions to (3.15) are given by the conformal mappings σr(z, t) = et 2z h r2z2+2(1 −e−t)rz+1 + (rz + 1) p r2z2+2(1…
Lemma 3.4. The solutions to (3.15) are given by the conformal mappings σr(z, t) = et 2z h r2z2+2(1 −e−t)rz+1 + (rz + 1) p r2z2+2(1 −2e−t)rz+1 i To see what the dilation of the function p = S[µt] means in terms of general driving measures we look at the Fourier representation pr(z, t) = p(rz, t) = 1 + 2 ∞ X j=1 ˆµt(−j)(rz)−j = 1 + 2
Proposition 3.5.
Proposition 3.5. Let f(·, t) t≥0 denote the conformal mappings that solve the Loewner equation driven by the family of probability measures…
Proposition 3.5. Let {f(·, t)}t≥0 denote the conformal mappings that solve the Loewner equation driven by the family of probability measures {µt}t≥0. Then the solution to the Loewner equation driven by the convolution mea- sures with (3.18) dµr t = d(µt ∗νr), r > 1, are given by {fr(·, t)}t≥0, where fr(z, t) = r−1f(rz, t). This means that solving the Loewner equation for a measure smoothed by convolution with the Poisson kernel amounts to considering the dilated versions of the conformal mapping
Lemma 4.1.
Lemma 4.1. The solutions to the Loewner equation driven by (4.3) and (4.4) are given by Φm(z, t) = et zm −t + 1 m(1 −e−mt) 1/m, z ∈∆, t…
Lemma 4.1. The solutions to the Loewner equation driven by (4.3) and (4.4) are given by Φm(z, t) = et zm −t + 1 m(1 −e−mt) 1/m , z ∈∆, t ∈[0, 1], and Φ∗ m(z, t) = et zm + t −m + 1 m