**Abstract** In this talk, based on joint works [1,2,3], we present some results connecting dynamic system with geometric function theory.

In the first part of the talk, we study the problem of characterizing membership of normalized holomorphic functions of the disk to the class of infinitesimal generators and some its subclasses as well as dynamical properties of generated semigroups. Presenting results include analytic extension in the semigroup parameter and the uniform convergence. Our approach is based on so-called *filtrations* of the class of infinitesimal generators.

In the second part we introduce and study a question that can be interpreted as *an inverse Fekete-Szegö problem*. This problem links to the first part of the talk. We introduce new filtration classes using the non-linear differential operator

$\alpha\cdot \dfrac{f(z)}{z}+\beta\cdot \dfrac{zf'(z)}{f(z)}+(1-\alpha-\beta)\cdot \left[1+\dfrac{zf''(z)}{f'(z)}\right]$,

and establish certain properties of these classes. Sharp upper bounds of the absolute value of the Fekete-Szegöo functional over some filtration classes are found. We also present open problems for further study.

[1] F. Bracci, M. D. Contreras, S. Díaz-Madrigal, M. Elin and D. Shoikhet, *Filtrations of infinitesimal generators*, Funct. Approx. Comment. Math. **59** (2018).

[2] M. Elin, D. Shoikhet, and T. Sugawa, *Filtration of semi-complete vector fields revisited*, in: Trends in Math., Birkhäuser/Springer, Cham, 2018.

[3] M. Elin, F. Jacobzon and N. Tuneski, *The Fekete-Szegö problem and filtration of generators*, Rendiconti del Circolo Matematico di Palermo Series 2.