20 research outputs found
Lojasiewicz exponent of families of ideals, Rees mixed multiplicities and Newton filtrations
We give an expression for the {\L}ojasiewicz exponent of a wide class of
n-tuples of ideals in \O_n using the information given by a
fixed Newton filtration. In order to obtain this expression we consider a
reformulation of {\L}ojasiewicz exponents in terms of Rees mixed
multiplicities. As a consequence, we obtain a wide class of semi-weighted
homogeneous functions for which the
{\L}ojasiewicz of its gradient map attains the maximum possible
value.Comment: 25 pages. Updated with minor change
Dissipation time and decay of correlations
We consider the effect of noise on the dynamics generated by
volume-preserving maps on a d-dimensional torus. The quantity we use to measure
the irreversibility of the dynamics is the dissipation time. We focus on the
asymptotic behaviour of this time in the limit of small noise. We derive
universal lower and upper bounds for the dissipation time in terms of various
properties of the map and its associated propagators: spectral properties,
local expansivity, and global mixing properties. We show that the dissipation
is slow for a general class of non-weakly-mixing maps; on the opposite, it is
fast for a large class of exponentially mixing systems which include uniformly
expanding maps and Anosov diffeomorphisms.Comment: 26 Pages, LaTex. Submitted to Nonlinearit
The combinatorics of plane curve singularities. How Newton polygons blossom into lotuses
This survey may be seen as an introduction to the use of toric and tropical
geometry in the analysis of plane curve singularities, which are germs
of complex analytic curves contained in a smooth complex analytic surface .
The embedded topological type of such a pair is usually defined to be
that of the oriented link obtained by intersecting with a sufficiently
small oriented Euclidean sphere centered at the point , defined once a
system of local coordinates was chosen on the germ . If one
works more generally over an arbitrary algebraically closed field of
characteristic zero, one speaks instead of the combinatorial type of .
One may define it by looking either at the Newton-Puiseux series associated to
relative to a generic local coordinate system , or at the set of
infinitely near points which have to be blown up in order to get the minimal
embedded resolution of the germ or, thirdly, at the preimage of this
germ by the resolution. Each point of view leads to a different encoding of the
combinatorial type by a decorated tree: an Eggers-Wall tree, an Enriques
diagram, or a weighted dual graph. The three trees contain the same
information, which in the complex setting is equivalent to the knowledge of the
embedded topological type. There are known algorithms for transforming one tree
into another. In this paper we explain how a special type of two-dimensional
simplicial complex called a lotus allows to think geometrically about the
relations between the three types of trees. Namely, all of them embed in a
natural lotus, their numerical decorations appearing as invariants of it. This
lotus is constructed from the finite set of Newton polygons created during any
process of resolution of by successive toric modifications.Comment: 104 pages, 58 figures. Compared to the previous version, section 2 is
new. The historical information, contained before in subsection 6.2, is
distributed now throughout the paper in the subsections called "Historical
comments''. More details are also added at various places of the paper. To
appear in the Handbook of Geometry and Topology of Singularities I, Springer,
202