Abstract. The paper concerns numerical algorithms for solving the Beltrami equation fz(z) = µ(z)fz (z) for a compactly supported µ.First, we study an efficient algorithm that has been proposed in the literature, and present its rigorous justification. We then propose a different scheme for solving the Beltrami equation which has a comparable speed and accuracy, but has the virtue of a greater simplicity of implementation.

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We study one of the central open questions in one-dimensional renormalization theory -the conjectural universality of golden-mean Siegel disks. We present an approach to the problem based on cylinder renormalization proposed by the second author. Numerical implementation of this approach relies on the Constructive Measurable Riemann Mapping Theorem proved by the first author. Our numerical study yields a convincing evidence to support the Hyperbolicity Conjecture in this setting.

It is known that the famous Feigenbaum-Coullet-Tresser period doubling universality has a counterpart for area-preserving maps of R 2 . A renormalization approach has been used in (Eckmann et al 1982) and (Eckmann et al 1984) in a computer-assisted proof of existence of a "universal" area-preserving map F * -a map with orbits of all binary periods 2 k , k ∈ N. In this paper, we consider maps in some neighbourhood of F * and study their dynamics.We first demonstrate that the map F * admits a "bi-infinite heteroclinic tangle": a sequence of periodic points {z k }, k ∈ Z,(1) whose stable and unstable manifolds intersect transversally; and, for any N ∈ N, a compact invariant set on which F * is homeomorphic to a topological Markov chain on the space of all two-sided sequences composed of N symbols. A corollary of these results is the existence of unbounded and oscillating orbits.We also show that the third iterate for all maps close to F * admits a horseshoe. We use distortion tools to provide rigorous bounds on the Hausdorff dimension of the associated locally maximal invariant hyperbolic set: 0.7673 ≥ dim H (C F ) ≥ ε ≈ 0.00013 e −7499 .

The period doubling Cantor sets of strongly dissipative Henon-like maps with different average Jacobian are not smoothly conjugated. The Jacobian Rigidity Conjecture says that the period doubling Cantor sets of two-dimensional Henon-like maps with the same average Jacobian are smoothly conjugated. This conjecture is true for average Jacobian zero, e.g. the one-dimensional case. The other extreme case is when the maps preserve area, e.g. the average Jacobian is one. Indeed, the period doubling Cantor set of area-preserving maps in the universality class of the Eckmann-Koch-Wittwer renormalization fixed point are smoothly conjugated.Comment: 55 pages, incl. references; 2 figure