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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 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.
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.
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.
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 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.
It is well known that the Hausdorff dimension of the invariant set $\Lambda_t$ of an iterated function system ${\mathcal F}_t$ on $\mathbb{R}^n$ depending smoothly on a parameter $t$ does not vary continuously. In fact, it has been shown recently that in general it varies lower-semi-continuously. For a specific family of systems we investigate numerically the conjecture that discontinuities in the dimension only arise when in some iterate of the iterated function system two (or more) of its branches coincide. This happens in a set of co-dimension one, but which is dense. All the other points are conjectured to be points of continuity.
We discuss the dynamics of exponential maps $z\mapsto \lambda e^z$ from the point of view of dynamic rays, which have been an important tool for the study of polynomial maps. We prove existence of dynamic rays with bounded combinatorics and show that they contain all points which ``escape to infinity'' in a certain way. We then discuss landing properties of dynamic rays and show that in many important cases, repelling and parabolic periodic points are landing points of periodic dynamic rays. For the case of postsingularly finite exponential maps, this needs the use of spider theory.
We investigate closures of orbits for the action of the group of diagonal matrices acting on $SL(n,R)/SL(n,Z)$, where $n \geq 3$. It has been conjectured by Margulis that possible orbit-closures for this action are very restricted. Lending support to this conjecture, we show that any orbit-closure containing a compact orbit is homogeneous. Moreover if $n$ is prime then any orbit whose closure contains a compact orbit is either compact itself or dense. This implies a number-theoretic result generalizing an isolation theorem of Cassels and Swinnerton-Dyer for products of linear forms. We also obtain similar results for other lattices instead of $SL(n,Z)$, under a suitable irreducibility hypothesis.
Extending results of a number of authors, we prove that if $U$ is the unipotent radical of a solvable epimorphic subgroup of an algebraic group $G$, then the action of $U$ on $G/\Gamma$ is uniquely ergodic for every cocompact lattice $\Gamma$ in $G$. This gives examples of uniquely ergodic and minimal two-dimensional flows on homogeneous spaces of arbitrarily high dimension. Our main tools are Ratner classification of ergodic invariant measures for the action of a unipotent subgroup on a homogeneous space, and a simple lemma (the 'Cone Lemma') about representations of epimorphic subgroups. (revised version of July 1999)