Quasi-linearization of parameter-depending germs of vector fields

In this paper, we study normalizationand quasi-linearization of a family of germs of hyperbolic vector fields at the origin. We show that, when the eigenvalues of these systems satisfy the so-called strongly 1-resonant condition, i.e. there exists a relation of the form (r, λ) = 0, then they can be simplified, in the context of orbital equivalence, to a normal form which can be integrated in an explicit and convenient way. More precisely, given a family of germs of sufficiently smooth vector fields having a generic nonlinear part, with strongly 1-resonant, then for any integer k, there is a neighbourhood at the origin of the parameter space such that for any , is C ^k equivalent to a system having only one resonant term on each component.     

Smooth moduli-free normal forms of hyperbolic germs of diffeomorphisms

In this paper, we study the smooth classifications of germs of diffeomorphisms near a hyperbolic fixed point based on the smooth moduli-free polynomial normal forms of the corresponding diffeomorphisms and give the following result. On , n ≤ 5, with two kinds of exceptions, any two hyperbolic germs of diffeomorphisms with generic nonlinear parts are at least C ¹ conjugated if and only if their linear parts are similar.     

Homeomorphisms of the Sierpinski curve with periodic properties

In this paper, we study the three following types of homeomorphisms of the Sierpinski curve of the two sphere : pointwise periodic, periodic, and almost periodic, and we prove that they are equivalent. We show that a subgroup of homeomorphisms whose orbits are all finite, is a finite subgroup.