Fixed points of diffeomorphisms, singularities of vector fields and epsilon-neighborhoods of their orbits, the thesis

TL;DR

This thesis explores how fractal properties of orbits, such as box dimension and Minkowski content, can identify fixed points and classify complex diffeomorphisms, linking geometric orbit features to dynamical and analytic properties.

Contribution

It establishes a bijective relation between fixed point multiplicity and epsilon-neighborhoods' asymptotics, and shows how fractal properties determine formal and analytic classes of diffeomorphisms.

Findings

Fixed point multiplicity corresponds to epsilon-neighborhood asymptotics.

Fractal properties of orbits encode formal class information.

Potential to determine analytic class from orbit neighborhoods.

Abstract

The thesis deals with recognizing diffeomorphisms from fractal properties of discrete orbits, generated by iterations of such diffeomorphisms. The notion of fractal properties of a set refers to the box dimension, the Minkowski content and their appropriate generalizations, or, in wider sense, to the epsilon-neighborhoods of sets, for small, positive values of parameter epsilon. In the first part of the thesis, we consider the relation between the multiplicity of the fixed point of a real-line diffeomorphism, and the asymptotic behavior of the length of the epsilon-neighborhoods of its orbits. We establish the bijective correspondence. At the fixed point, the diffeomorphisms may be differentiable or nondifferentiable. The results are applied to the question of cyclicity of some planar limit periodic sets of polynomial fields, whose first return maps are real-line diffeomorphisms. In the second part of the thesis, we consider the relation between formal and analytic classes of complex diffeomorphisms f:C->C and epsilon-neighborhoods of their orbits. It is shown that the formal class can be read from generalized fractal properties of any orbit. The results are directly applied to formal classification of complex saddles in C^2. On the other hand, concerning the analytic classification, we state problems in reading the analytic class of a diffeomorphisms from epsilon-neighborhoods of orbits.