Welington de Melo

Geometric theory of dynamical systems : an introduction

1 Differentiable Manifolds and Vector Fields.- 0 Calculus in ?n and Differentiable Manifolds.- 1 Vector Fields on Manifolds.- 2 The Topology of the Space of Cr Maps.- 3 Transversality.- 4 Structural Stability.- 2 Local Stability.- 1 The Tubular Flow Theorem.- 2 Linear Vector Fields.- 3 Singularities and Hyperbolic Fixed Points.- 4 Local Stability.- 5 Local Classification.- 6 Invariant Manifolds.- 7 The ?-lemma (Inclination Lemma). Geometrical Proof of Local Stability.- 3 The Kupka-Smale Theorem.- 1 The Poincare Map.- 2 Genericity of Vector Fields Whose Closed Orbits Are Hyperbolic.- 3 Transversality of the Invariant Manifolds.- 4 Genericity and Stability of Morse-Smale Vector Fields.- 1 Morse-Smale Vector Fields Structural Stability.- 2 Density of Morse-Smale Vector Fields on Orientable Surfaces.- 3 Generalizations.- 4 General Comments on Structural Stability. Other Topics.- Appendix: Rotation Number and Cherry Flows.- References.

Rigidity of critical circle mappings I

Abstract.We prove that two C3 critical circle maps with the same rotation number in a special set ? are C1+α conjugate for some α>0 provided their successive renormalizations converge together at an exponential rate in the C0 sense. The set ? has full Lebesgue measure and contains all rotation numbers of bounded type. By contrast, we also give examples of C∞ critical circle maps with the same rotation number that are not C1+β conjugate for any β>0. The class of rotation numbers for which such examples exist contains Diophantine numbers.

Rigidity of critical circle mappings II

We prove that two C3 critical circle maps with the same rotation number in a special set ? are C1+α conjugate for some α>0 provided their successive renormalizations converge together at an exponential rate in the C0 sense. The set ? has full Lebesgue measure and contains all rotation numbers of bounded type. By contrast, we also give examples of C∞ critical circle maps with the same rotation number that are not C1+β conjugate for any β>0. The class of rotation numbers for which such examples exist contains Diophantine numbers.

Universal models for Lorenz maps

The existence of smooth families of Lorenz maps exhibiting all possible dynamical behavior is established and the structure of the parameter space of these families is described.

On Cherry flows

The purpose of this research is to describe all smooth vector fields on the torus T 2 with a finite number of singularities, no periodic orbits and no saddleconnections. In this paper we are able to complete the description within the class of vector fields which are area contracting near all singularities. In particular we give a large class of analytic vector fields on the torus T 2 which have non-trivial recurrence and also sinks.

THE MULTIPLIERS OF PERIODIC POINTS IN ONE-DIMENSIONAL DYNAMICS

It will be shown that the smooth conjugacy class of an S-unimodal map which has neither a periodic attractor nor a Cantor attractor is determined by the multipliers of the periodic orbits. This generalizes a result by M Shub and D Sullivan (1985 Expanding endomorphism of the circle revisited Ergod. Theor. Dynam. Sys. 5 285-9) for smooth expanding maps of the circle.

Rigidity of smooth critical circle maps

We prove that any two

Rigidity of critical circle maps

C^3

Structural Stability on Two-Manifolds†

critical circle maps with the same irrational rotation number of bounded type and the same odd criticality are conjugate to each other by a

The Combinatorics of One-Dimensional Endomorphisms

C^{1+\alpha}