Applied Mathematicsematics

Communications In Mathematical Physics - Volume 287 by M. Aizenman (Chief Editor)

By M. Aizenman (Chief Editor)

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Then for t > t0 we get that −at ∈ / Dm+1 , since we are outside m+1 . Similarly, a0 −at ∈ Dm+1 for t < t0 . 1. Let us first handle the harder case, when the critical tableau of f c is recurrent. Extend the holomorphic motion on m 0 at depth m 0 by the λ-Lemma, so that we get a holomorphic motion on m 0 with dilatation K = K (δ(a, ∂ m 0 )), which depends on the conformal distance δ(a, ∂ m 0 ) from a to the boundary of m 0 . Let us call this extended motion h˜ m 0 . m −m Now, lift the motion h˜ m 0 via the unbranched covering maps Ra j 0 for a ∈ m j .

7. We have φ1 (z) = φ2 (w) if and only if z ∼r w. 36 M. Aspenberg, M. Yampolsky Proof. If z ∈ K ◦ ◦ is uni-accessible then let −β be the angle of the external ray landing at z and put ζ = φ1 (z). ,−θq (−β) = σ (ζ ). If ζ = φ2 (w), then w lies in the same puzzle-pieces as the point ζ , by definition. ,θq (γ ) = σ (ζ ). Obviously, one solution is γ = −β, and therefore z ∼r w. Conversely, if z ∼r w, then φ1 (z) = φ2 (w) by construction. 4. We conclude: Main Theorem, the existence part. Suppose c is a non-renormalizable parameter value outside the 1/2-limb of M.

Since H N is injective on the boundary of A N , it follows that H N ◦ φ −1 maps φ(A N ) conformally onto AaN0 . It follows that H N must be a homeomorphism. 6 – we select the initial thickened piece so that it moves holomorphically with a. Consider the thickened annulus A˜ m 0 (−a0 ) for a = a0 . The map H N (N = m j ) defined above then easliy extends to the thickened non-degenerate parameter annulus A˜ N . It follows that 1 1 mod A˜ am0j ≤ mod A˜ m j ≤ mod A˜ am0j . 2 that the parameter pieces N shrink to a single point, which has to be a0 .

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