Curtis T. McMullen

The Hausdorff dimension of general Sierpiński carpets

In this note we determine the Hausdorff dimension of a family of planar sets which are generalizations of the classical Cantor set.

Renormalization and 3-manifolds which fiber over the circle

Many parallels between complex dynamics and hyperbolic geometry have emerged in the past decade. Building on work of Sullivan and Thurston, this book gives a unified treatment of the construction of fixed-points for renormalization and the construction of hyperbolic 3- manifolds fibering over the circle.Both subjects are studied via geometric limits and rigidity. This approach shows open hyperbolic manifolds are inflexible, and yields quantitative counterparts to Mostow rigidity. In complex dynamics, it motivates the construction of towers of quadratic-like maps, and leads to a quantitative proof of convergence of renormalization.

Billiards and Teichmüller curves on Hilbert modular surfaces

This paper exhibits an infinite collection of algebraic curves isometrically embedded in the moduli space of Riemann surfaces of genus two. These Teichmuller curves lie on Hilbert modular surfaces parameterizing Abelian varieties with real multiplication. Explicit examples, constructed from L-shaped polygons, give billiard tables with optimal dynamical properties.

Area and Hausdorff dimension of Julia sets of entire functions

We show the Julia set of A sin(z) has positive area and the action of A sin(z) on its Julia set is not ergodic; the Julia set of A exp(z) has Hausdorff dimension two but in the presence of an attracting periodic cycle its area is zero.

Automorphisms of rational maps

Let f(z) be a rational map, Aut(f) the finite group of Mobius transformations commuting with f. We study the question: when can two kinds of more flexible automorphisms of the dynamics of f be realized in Aut(g) for some deformation g of f?

The alexander polynomial of a 3-manifold and the thurston norm on cohomology

Let M be a connected, compact, orientable 3-manifold with b1(M) > 1, whose boundary (if any) is a union of tori. Our main result is the inequality kk A ≤ kk T between the Alexander norm on H 1 (M, Z), defined in terms of the Alexan- der polynomial, and the Thurston norm, defined in terms of the Eu- ler characteristic of embedded surfaces. (A similar result holds when b1(M) = 1.) Using this inequality we determine the Thurston norm for most links with 9 or fewer crossings.

Families of Rational Maps and Iterative Root-Finding Algorithms

In this paper we develop a rigidity theorem for algebraic families of rational maps and apply it to the study of iterative root-finding algorithms. We answer a question of Smales by showing there is no generally convergent algorithm for finding the roots of a polynomial of degree 4 or more. We settle the case of degree 3 by exhibiting a generally convergent algorithm for cubics; and we give a classification of all such algorithms. In the context of conformal dynamics, our main result is the following: a stable algebraic family of rational maps is either trivial (all its members are conjugate by Mobius transformations), or affine (its members are obtained as quotients of iterated addition on a family of complex tori). Our classification of generally convergent algorithms follows easily from this result. As another consequence of rigidity, we observe that the eigenvalues of a nonaffine rational map at its periodic points determine the map up to finitely many choices. This implies that bounded analytic functions nearly separate points on the moduli space of a rational map.

Complex earthquakes and Teichmüller theory

It is known that any two points in Teichmuller space are joined by an earthquake path. In this paper we show any earthquake path R → T (S) extends to a proper holomorphic mapping of a simplyconnected domain D into Teichmuller space, where R ⊂ D ⊂ C. These complex earthquakes relate Weil-Petersson geometry, projective structures, pleated surfaces and quasifuchsian groups. Using complex earthquakes, we prove grafting is a homeomorphism for all 1-dimensional Teichmuller spaces, and we construct bending coordinates on Bers slices and their generalizations. In the appendix we use projective surfaces to show the closure of quasifuchsian space is not a topological manifold.

Self-similarity of Siegel disks and Hausdorff dimension of Julia sets

Let f(z) = ez+ z, where θ is an irrational number of bounded type. According to Siegel, f is linearizable on a disk containing the origin. In this paper we show: • the Hausdorff dimension of the Julia set J(f) is strictly less than two; and • if θ is a quadratic irrational (such as the golden mean), then the Siegel disk for f is self-similar about the critical point. In the latter case, we also show the rescaled first-return maps converge exponentially fast to a system of commuting branched coverings of the complex plane.

Prym Varieties and Teichmüller Curves

This paper gives a uniform construction of infinitely many primitive Teichmuller curves V ⊂ Mg for g = 2, 3 and 4.