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In this paper, we will study Newton's method for solving two simultaneous quadratic equations in two variables. Presumably, there is no need to motivate a study of Newton's method, in one or several variables. The algorithm is of immense importance, and understanding its behavior is of obvious interest. It is perhaps harder to motivate the case of two simultaneous quadratic equations in two variables, but this is the simplest non-degenerate case.
In this paper, new techniques for studying the dynamics of families of surface homeomorphisms are introduced. Two dynamical deformation theories are presented --- one for surface homeomorphisms, called pruning, and another for graph endomorphisms, called kneading --- both giving conditions under which all of the dynamics in an open set can be destroyed, while leaving the dynamics unchanged elsewhere. These theories are then used to give a proof of Thurston's classification theorem for surface homeomorphisms up to isotopy.
In the near future, all the human genes will be identified. But understanding the functions coded in the genes is a much harder problem. For example, by using block entropy, one has that the DNA code is closer to a random code then written text, which in turn is less ordered then an ordinary computer code; see schmitt. Instead of saying that the DNA is badly written, using our programming standards, we might say that it is written in a different style --- an evolutionary style. We will suggest a way to search for such a style in a quantified manner by using an artificial life program, and by giving a definition of general codes and a definition of style for such codes.
We give a complete classification of hyperbolic components in the space of iterated maps $z\mapsto \lambda\exp(z)$, and we describe a preferred parametrization of those components. This leads to a complete classification of all exponential maps with attracting dynamics.
We construct an open set $\mathcal{U}$ of rational foliations of arbitrarily fixed degree $d \ge 2$ by curves in $\mathbb{C}\mathbb{P}^n$ such that any foliation $\mathcal{F}\in\mathcal{U}$ has a finite number of singularities and satisfies the following chaotic properties.
Minimality: any leaf (curve) is dense in $\mathbb{C}\mathbb{P}^n$.
Ergodicity: any Lebesgue measurable subset of leaves has zero or total Lebesgue measure.
Entropy: the topological entropy is strictly positive even far from singularities.
Rigidity: if $\mathcal{F}$ is conjugate to some $\mathcal{F}'\in\mathcal{U}$ by a homeomorphism close to the identity, then they are also conjugate by a projective transformation.
The main analytic tool employed in the construction of these foliations is the existence of several pseudo-flows in the closure of pseudo-groups generated by perturbations of elements in $\text{Diff}(\mathbb{C}^n,0)$ on a fixed ball.
Let $0< \theta <1$ be an irrational number with continued fraction expansion $\theta=[a_1, a_2, a_3, \ldots]$, and consider the quadratic polynomial $P_\theta : z \mapsto e^{2\pi i \theta} z +z^2$. By performing a trans-quasiconformal surgery on an associated Blaschke product model, we prove that if $$\log a_n = {\mathcal O} (\sqrt{n})\ \operatorname{as}\ n \to \infty ,$$ then the Julia set of $P_\theta$ is locally-connected and has Lebesgue measure zero. In particular, it follows that for almost every $0< \theta < 1$, the quadratic $P_\theta$ has a Siegel disk whose boundary is a Jordan curve passing through the critical point of $P_\theta$. By standard renormalization theory, these results generalize to the quadratics which have Siegel disks of higher periods.
We show that a Kleinian surface group, or hyperbolic 3-manifold with a cusp-preserving homotopy-equivalence to a surface, has bounded geometry if and only if there is an upper bound on an associated collection of coefficients that depend only on its end invariants. Bounded geometry is a positive lower bound on the lengths of closed geodesics. When the surface is a once-punctured torus, the coefficients coincide with the continued fraction coefficients associated to the ending laminations.
The earthquake flow and the Teichmüller horocycle flow are flows on bundles over the Riemann moduli space of a surface, and are similar in many respects to unipotent flows on homogeneous spaces of Lie groups. In analogy with results of Margulis, Dani and others in the homogeneous space setting, we prove strong nondivergence results for these flows. This extends previous work of Veech. As corollaries we obtain that every closed invariant set for the earthquake (resp. Teichmüller horocycle) flow contains a minimal set, and that almost every quadratic differential on a Teichmüller horocycle orbit has a uniquely ergodic vertical foliation.
We study dynamics of rational maps that satisfy a decay of geometry condition. Well known conditions of non-uniform hyperbolicity, like summability condition with exponent one, imply this condition. We prove that Julia sets have zero Lebesgue measure, when not equal to the whole sphere, and in the polynomial case every connected component of the Julia set is locally connected. We show how rigidity properties of quasi-conformal maps that are conformal in a big dynamically defined part of the sphere, apply to dynamics. For example we give a partial answer to a problem posed by Milnor about Thurston's algorithm and we give a proof that the Mandelbrot set, and its higher degree analogues, are locally connected at parameters that satisfy the decay of geometry condition. Moreover we prove a theorem about similarities between the Mandelbrot set and Julia sets. In an appendix we prove a rigidity property that extends a key situation encountered by Yoccoz in his proof of local connectivity of the Mandelbrot set at at most finitely renormalizable parameters.