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

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