研究了向量场的不动点、向量场的奇点及其轨道的ε-邻域，
Fixed points of diffeomorphisms, singularities of vector fields and epsilon-neighborhoods of their orbits, the thesis
摘要：
本文从离散轨道的分形性质出发，研究了由这种自同态迭代生成的自同态。集合的分形性质的概念是指盒维数、闵可夫斯基容度及其适当的推广，或者在更广泛的意义上，指集合的ε邻域，对于参数ε的小的正值。本文的第一部分研究了实线同态的不动点的重数与其轨道的ε-邻域长度的渐近性态之间的关系。我们建立双射对应。在不动点上，这些超同态可以是可微的，也可以是不可微的。将所得结果应用于多项式域的某些平面极限周期集的循环性问题，这些周期集的第一回映射是实线同态。在论文的第二部分中，我们考虑复同态f：C->C的形式类和解析类与其轨道的ε-邻域之间的关系。结果表明，形式类可以从任何轨道的广义分形性质中读出。结果直接应用于C^2中复鞍的形式分类。另一方面，关于解析分类，我们指出了从轨道的ε-邻域中读取解析类的问题。
英文摘要：
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.
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