Welcome to the preprint server of the Institute for Mathematical Sciences at Stony Brook University.

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**PREPRINTS IN THIS SERIES, IN PDF FORMAT.**

* Starred papers have appeared in the journal cited.

*A Comparison of Harmonic and Balanced Measures on Cantor Repellors*

Let *J* be a Cantor repellor of a conformal map *f*. Provided *f* is a polynomial-like or *R*-symmetric, we prove that harmonic measure on *J* is equivalent to the measure of maximal entropy if and only if *f* is conformally equivalent to a polynomial. We also show that this is not true for general Cantor repellors: there is a non-polynomial algebraic function generating a Cantor repellor on which above two measures coincide.

*Dynamics of Certain Non-Conformal Degree Two Maps on the Plane*

In this paper we consider maps on the plane which are similar to quadratic maps in that they are degree 2 branched covers of the plane. In fact, consider for $\alpha$ fixed, maps $f_c$ which have the following form (in polar coordinates): $$f_c(r\,e^{i\theta})\;=\;r^{2\alpha}\,e^{2i\theta}\,+\,c$$ When $\alpha=1$, these maps are quadratic ($z \mapsto z^2 + c$), and their dynamics and bifurcation theory are to some degree understood. When $\alpha$ is different from one, the dynamics is no longer conformal. In particular, the dynamics is not completely determined by the orbit of the critical point. Nevertheless, for many values of the parameter c, the dynamics has strong similarities to that of the quadratic family. For other parameter values the dynamics is dominated by 2 dimensional behavior: saddles and the like. The objects of study are Julia sets, filled-in Julia sets and the connectedness locus. These are defined in analogy to the conformal case. The main drive in this study is to see to what extent the results in the conformal case generalize to that of maps which are topologically like quadratic maps (and when $\alpha$ is close to one, close to being quadratic).

*On the Quasisymmetrical Classification of Infinitely Renormalizable Maps: I. Maps with Feigenbaum's Topology*

A semigroup (dynamical system) generated by $C^{1+\alpha}$-contracting mappings is considered. We call a such semigroup regular if the maximum $K$ of the conformal dilatations of generators, the maximum $l$ of the norms of the derivatives of generators and the smoothness $\alpha$ of the generators satisfy a compatibility condition $K< 1/l^{\alpha}$. We prove the {\em geometric distortion lemma} for a regular semigroup generated by $C^{1+\alpha}$-contracting mappings.

*On the Quasisymmetrical Classification of Infinitely Renormalizable Maps: II. Remarks on Maps with a Bounded Type Topology*

We use the upper and lower potential functions and Bowen's formula estimating the Hausdorff dimension of the limit set of a regular semigroup generated by finitely many $C^{1+\alpha}$-contracting mappings. This result is an application of the geometric distortion lemma in the first paper at this series.

*On the Realization of Fixed Point Portraits (an addendum to Goldberg & Milnor: Fixed Point Portraits)*

We establish that every formal critical portrait (as defined by Goldberg and Milnor), can be realized by a postcritically finite polynomial.

*Periodic Orbits for Hamiltonian systems in Cotangent Bundles*

We prove the existence of at least $cl(M)$ periodic orbits for certain time dependant Hamiltonian systems on the cotangent bundle of an arbitrary compact manifold $M$. These Hamiltonians are not necessarily convex but they satisfy a certain boundary condition given by a Riemannian metric on $M$. We discretize the variational problem by decomposing the time 1 map into a product of "symplectic twist maps". A second theorem deals with homotopically non trivial orbits in manifolds of negative curvature.

*On Removable Sets for Sobolev Spaces in the Plane*

Let $K$ be a compact subset of $\bar{\textbf{C}} ={\textbf{R}}^2$ and let $K^c$ denote its complement. We say $K\in HR$, $K$ is holomorphically removable, if whenever $F:\bar{\textbf{C}} \to\bar{\textbf{C}}$ is a homeomorphism and $F$ is holomorphic off $K$, then $F$ is a Möbius transformation. By composing with a Möbius transform, we may assume $F(\infty )=\infty$. The contribution of this paper is to show that a large class of sets are $HR$. Our motivation for these results is that these sets occur naturally (e.g. as certain Julia sets) in dynamical systems, and the property of being $HR$ plays an important role in the Douady-Hubbard description of their structure.

*Geometric Finiteness and Uniqueness for Kleinian Groups with Circle Packing Limit Sets*

In this paper, we assume that $G$ is a finitely generated torsion free non-elementary Kleinian group with $\Omega(G)$ nonempty. We show that the maximal number of elements of $G$ that can be pinched is precisely the maximal number of rank 1 parabolic subgroups that any group isomorphic to $G$ may contain. A group with this largest number of rank 1 maximal parabolic subgroups is called *maximally parabolic*. We show such groups exist. We state our main theorems concisely here.

Theorem I. The limit set of a maximally parabolic group is a circle packing; that is, every component of its regular set is a round disc.

Theorem II. A maximally parabolic group is geometrically finite.

Theorem III. A maximally parabolic pinched function group is determined up to conjugacy in $PSL(2,{\bf C})$ by its abstract isomorphism class and its parabolic elements.

*Conformal Dynamics Problem List*

A list of unsolved problems was given at the Conformal Dynamics Conference which was held at SUNY Stony Brook in November 1989. Problems were contributed by Ben Bielefeld, Adrien Douady, Curt McMullen, Jack Milnor, Misuhiro Shishikura, Folkert Tangerman, and Peter Veerman.

*Measurable Dynamics of S-Unimodal Maps of the Interval*

In this paper we sum up our results on one-dimensional measurable dynamics reducing them to the S-unimodal case (compare Appendix 2). Let f be an S-unimodal map of the interval having no limit cycles. Then f is ergodic with repect to the Lebesque measure, and has a unique attractor A in the sense of Milnor. This attractor coincides with the conservative kernel of f. There are no strongly wandering sets of positive measure. If f has a finite a.c.i. (absolutely continuous invariant) measure u, then it has positive entropy. This result is closely related to the following: the measure of Feigenbaum-like attractors is equal to zero. Some extra topological properties of Cantor attractors are studied.