# THEMATIC PROGRAMS

November 26, 2014

## June 22 - 26, 2009 Workshop on Finiteness Problems in Dynamical Systems

### Workshop Abstracts

Supported by

Andreas Fischer (Fields Institute)
Definable versions of theorems by Kirszbraum and Helly

Kirszbraun's Theorem states that every Lipschitz map form $S$ to $\mathbb{R}^n$, where $S\subseteq \mathbb{R}^m$, has an extension to a Lipschitz map defined on $ämathbb{R^m}$ with the same Lipschitz constant. Its proof relies on Helly's Theorem: every family of compact subsets of $\mathbb{R}^n$, having the property that each of its subfamilies consisting of at most $n+1$ sets share a common point, has a non-empty intersection.
We sketch the proof of versions of these theorems valid for definable maps and sets in arbitrary definably complete expansions of ordered fields.

Andrei Gabrielov (Purdue University)
Triangulation of monotone o-minimal families

Let $S_t,\,t\in(0,1]$, be a family of compact sets in a compact $K\subset R^n$, such that $S_u\subset S_t$ for $0<t<u$. If the family $S_t$ and the compact $K$ are definable in an o-minimal structure, we construct a triangulation of $K$ such that, for each open simplex $\sigma$, the family $\sigma\cap S_t$, for small values of $t$, is homeomorphic to one of the standard'' combinatorial families.

Tobias Kaiser (University of Passau)
On the resolution of singularities for power series with perturbation by logarithmic terms

In [1] it was shown that transition maps at non-resonant hyperbolic singularities of analytic vector fields in the plane are definable in an o-minimal structure. The proof uses the fact that the transition maps allow an asymptotic expansion and live in a quasianalytic class (see [2]). In the non-resonant case the asymptotic expansion is given by certain generalized power series. Based on this fact it was possible to define an appropriate generalization of this quasianalytic class to several variables and to develop a blow-up algorithm for resolution of singularities to obtain o-minimality in the spirit of previous work in o-minimal structures (see for example [3]). In the resonant case the asymptotic expansion is given by generalized power series where the monomials are additionally perturbed by logarithmic polynomials. The desired goal would be to develop a blow-up algorithm for resolution of singularities also in this case to obtain o-minimality of the transition maps in the resonant case and of related topics (see [4], [5]).But, the presence of logarithmic terms complicates the situation enormously. In this talk I discuss a possible approach towards a blow-up algorithm.

Literature:
[1]T.Kaiser, J.-P.Rolin and P.Speissegger: Transition maps at non-resonant hyperbolic singularities are o-minimal, Preprint, 2006,
arXiv:math/0612745, J. Reine Angew. Math. to appear
[2]Y. S. Ilyashenko, Finiteness theorems for limit cycles, Translations
of Mathematical Monographs 94 (1991)
[3]L.Van den Dries, and P. Speissegger: The real field with convergent
generalized power series, Trans. Amer. Math. Soc. 350 (1998), 4377--4421 [4]T. Kaiser: The Riemann mapping theorem for semianalytic domains and
o-minimality. Proc. Lond. Math. Soc. (3) 98, No. 2, 427-444 (2009)
[5]T.Kaiser: The Dirichlet problem in the plane with semianalytic raw
data, quasi analyticity, and o-minimal structure. Duke Math. J. 147, No.
2, (2009), 285-314.

Krzysztof Kurdyka (Université de Savoie)
On the structure of gradient extremal set of generic functions

For a real valued smooth function on a Riemannian manifold by "gradient extremal set" we mean all critical points of the restriction to the fiber of the square of norm of the gradient. For a generic (Morse) function this is set is a union of smooth curves which intersect "transversally" at critical points. A part of the gradient extremal set gradient which correspond to the local minima is call a "Talweg" or "ridge and valley lines". It can be used to estimate the length of gradient trajectories and possibly should organise dynamics of the gradient flow.

Jean-Marie Lion (Université de Rennes 1)
Pfaffian sets

We show that the Pfaffian closure of an o-minimal structure with analytic cell decomposition is model complete.

David Marin (Universitat Autonoma de Barcelona)
Unfolding of resonant saddles and the Dulac time

In this work we study unfoldings of planar vector fields in a neighborhood of a resonant saddle. We give a C k temporal normal form for the unfolding. That is, a normal form with respect to the conjugacy relation. Using our temporal normal form we determine an asymptotic development, uniform with respect to the parameters, of the Dulac time of a resonant saddle. Conjugacy relation instead of weaker equivalence relation is necessary when studying the time function. The Dulac time of a resonant saddle can be seen as the basic building block of the total period function of an unfolding of a hyperbolic polycycle. (joint work with Pavao Mardesic and Jordi Villadelprat)

Chris Miller (Ohio State University)
Expansions of o-minimal structures by trajectories of definable planar vector fields

An expansion of the real field is said to be o-minimal if every definable set has finitely many connected components. Such structures are a natural setting for studying "tame" objects of real analytic geometry such as nonoscillatory trajectories of real analytic planar vector fields. More generally, o-minimality is preserved under expanding an o-minimal structure by nonoscillatory trajectories of definable planar vector fields. But what happens when o-minimality is not preserved? In some cases, we see the best behavior that one could reasonably expect, while in others the worst possible, and we do not know at present of any other outcomes. I will make all this precise in a survey of the current state of the art.

Dmitry Novikov (Weizmann Institute of Science)
Infinitesimal Hilbert 16th problem: recent progress

I'll describe a recent progress toward providing general constructive solution of Infinitesimal Hilbert 16th problem.

Olivier Le Gal (University of Toronto)
O-minimalty for solutions of analytic 2-dimensional linear ODEs

Daniel Panazzolo (Universidade de São Paulo)
The group of power and translations and its relation to generalized Witt algebras

The group of maps generated by the powers $x \righrarrow x^r$ and translations $x \rightarrow x + a$ appears in numerous problems in the analytic theory of differential equations and dynamical systems. In this talk, we will discuss some recent results related to two old questions: how fast the number of fixed points of a "word" can grow in terms of its length? Is this group a free product?

Limits of gradient directions at a singular point

Let $f$ be a definable $C^1$ function defined in a neighborhood of the origin in $R^n$. Using Lagrange specialization and the deformation to the normal cone we describe geometrically the space of limits at the origin of secant lines and gradient directions of $f$, understood as a subset of $P^{n-1} \times P^{n-1}$.

We apply this description to study the gradient flow of $f$.

Christiane Rousseau (Université de Montréal)
Orbital analytic classification of germs of families unfolding a codimension 1 resonant saddle or saddle-node

We consider the equivalence problem for germs of 1-parameter analytic families of planar vectors fields unfolding a resonant saddle or a saddle-node. We will describe a complete modulus under orbital equivalence. Moreover, we will give sufficient conditions for a “candidate modulus” to be indeed realizable as the modulus of a germ of family. We will characterize the moduli corresponding to families of real vector fields.

Tere Seara (Universitat Politècnica de Catalunya)
Resurgence of inner solutions for analytical perturbations of the McMillan map

In the study of the exponentially small splitting that occurs in certain perturbations of the McMillan map a sequence of "inner equations" has to be considered. An essential step in the measure of the splitting is to know some special solutions of these equations and to be able to give an asymptotic value of their difference.

The present work relies on ideas from resurgence theory: we obtain the desired solutions as Borel-Laplace sums of the formal solutions, studying the analyticity of their Borel transforms. Moreover, using 'Ecalle's alien derivations we are able to measure the discrepancy between different Borel-Laplace sums.

(Joint work with P. Martin and D. Sauzin)

Masahiro Shiota (Nagoya University)
O-minimal Hauptvermutung

Arguments on PL (=piecewise linear) topology work over any ordered field in the same way as over the real number field, and those on differential topology do over a real closed field in an o- minimal structure that expands (R,<,0,1,+,\cdot). It is known that a compact definable set is definably homeomorphic to a polyhedron.We show uniqueness of the polyhedron up to PL homeomorphism (o-minimal Hauptvermutung).We see also that many problems on PL and differential topology can be translated to those over the real number field.

Lou van den Dries (University of Illinois at Urbana-Champaign)
Asymptotic real differential algebra

Joris van der Hoeven (Université Paris-Sud)
Machine computations with transseries

In our talk, we will discuss several ways to implement computations with transseries in a computer algebra system. In particular, we will explain the new technique of meta-expansion, which is highly effective in practice, although merely heuristic from a theoretical point of view.

Alex Wilkie (University of Manchester)
Some Model Theory for Complex Analytic Functions

We consider complex analytic functions that are locally definable in a polynomially bounded o-minimal structure (for example any polynomial in z, exp(z)). This restriction seems to place quite severe constraints on the analytic sets defined by such functions and provides a way in to Zilber's conjecture, ie if a subset of the set of complex numbers is first order definable in the complex exponential field then it is either countable or co-countable.

Yosef Yomdin (The Weizmann Institute of Science)
Poincare Inversion, Moment inversion, Compositions, and Signal Processing

We present some new results and question related to the Center-Focus problem for Abel differential equation. There was recently a serious progress in understanding some aspects of this problem. In particular, F. Pakovich and M. Muzychuk completely solved the vanishing problem for polynomial moments: this is an infinitesimal version of the Center-Focus problem. On the other hand, some initial results on the "Poincare inversion problem" have been obtained. This problem asks for a characterization of all possible sequences of Taylor coefficients of the Poincare first return mapping, and for reconstruction of the original Abel equation from the given sequence of its Poincare coefficients. This generalizes the classical C-F problem (where all the coefficients besides the first one are zero). In the Poincare inversion problem, besides the usual "Composition condition" which is a conjectural Center condition, some other types of compositions arise. Closed connections have been found also with the classical Moment inversion problem (which naturally appears as the infinitesimal version of the Poincare inversion). Very encouraging connections have also been found with the "algebraic sampling" problem (i.e. the problem of reconstructing a non-linear model from a set of measurements), as it appears in Signal Processing.