• Yu. S. Ilyashenko (HSE University)
  Finiteness theorems for limit cycles
• Mario Ponce (Pontificia Universidad Católica de Chile, UC)
  Dynamical aspects of skew products over minimal dynamics
• Marcelo Viana (IMPA)
  Continuity of Lyapunov exponents

CONTENTS

Abstract. A polynomial vector field in the plain has but a finite number of limit cycles. Limit cycles of an analytic vector field cannot accumulate to a polycycle of this field. The first theorem is an immediate corollary of the second one. My current proof consists of the three parts:
   
1. Reduction from the ODE to complex analysis
2. Proofs of the theorems above for vector fields with the so called ``alternant'' polycycles (those in which the maps TO and FROM central manifolds of the saddlenodes alternate)
3. Proof for the general case.

Abstract. We will introduce the skew-product dynamics, especially those whose transformation in the fibre corresponds to an isometry. We will show how these simple dynamics give rise to several phenomena that are governed both by the arithmetic properties of the rotation numbers, as well as by geometric properties that force some rigid dynamic behaviours. In particular we will study the dynamics associated to invariant sections, to bounded orbits, among other phenomena. These skew-products constitute a geometric model for the resolution of linearised equations of more complex dynamics. We will investigate about the geometric and dynamic methods that allow us to solve such equations.

Abstract. I will introduce the notions of linear cocycle and Lyapunov exponent, and will sketch of the proof of the fact that the Lyapunov exponents of random products of matrices vary continuously with the data. This is based on joint work with C. Bocker, A. Avila and A. Eskin.