Free fall motion in an invariant field of forces: the 2D-case.
This is the first part of the work studying the family of all rational maps of degree three with two superattracting fixed points. We determine the topological type of the moduli space of and give a detailed study of the subfamily consisting of maps with a critical point which is periodic of period 2. In particular, we describe a parabolic bifurcation in from Newton maps to maps with so-called exotic basins.
This is a continuation of the work [Ba] dealing with the family of all cubic rational maps with two supersinks. We prove the existence of the following parabolic bifurcation of Mandelbrot-like sets in the parameter space of this family. Starting from a Mandelbrot-like set in cubic Newton maps and changing parameters in a continuous way, we construct a path of Mandelbrot-like sets ending in the family of parabolic maps with a fixed point of multiplier 1. Then it bifurcates into two paths of Mandelbrot-like...
We consider the full group [φ] and topological full group [[φ]] of a Cantor minimal system (X,φ). We prove that the commutator subgroups D([φ]) and D([[φ]]) are simple and show that the groups D([φ]) and D([[φ]]) completely determine the class of orbit equivalence and flip conjugacy of φ, respectively. These results improve the classification found in [GPS]. As a corollary of the technique used, we establish the fact that φ can be written as a product of three involutions from [φ].
We use the Galerkin averaging method to construct a coordinate transformation putting a nonlinear PDE in Poincaré normal form up to finite order. We also give a rigorous estimate of the remainder showing that it is small as a differential operator of very high order. The abstract theorem is then applied to a quasilinear wave equation, to the water wave problem and to a nonlinear heat equation. The normal form is then used to construct approximate solutions whose difference from true solutions is...
We consider a dynamical system consisting of a pair of commuting power series under composition, one noninvertible and another nontorsion invertible, of height one with coefficients in the -adic integers. Assuming that each point of the dynamical system generates a Galois extension over the base field, we show that these extensions are in fact abelian, and, using results from the theory of the field of norms, we also show that the dynamical system must include a torsion series. From an earlier...
We overview a unified approach to the André-Oort and Manin-Mumford conjectures based on a combination of Galois-theoretic and ergodic techniques. This paper is based on recent work of Klingler, Ullmo and Yafaev on the André-Oort conjecture, and of Ratazzi and Ullmo on the Manin-Mumford conjecture.
We study ergodic properties of the class of Gaussian automorphisms whose ergodic self-joinings remain Gaussian. For such automorphisms we describe the structure of their factors and of their centralizer. We show that Gaussian automorphisms with simple spectrum belong to this class. We prove a new sufficient condition for non-disjointness of automorphisms giving rise to a better understanding of Furstenberg's problem relating disjointness to the lack of common factors. This...