Artem Dudko

Finite factor representations of Higman–Thompson groups

We prove that the only finite factor-representations of the Higman-Thompson groups

The resurgent character of the Fatou coordinates of a simple parabolic germ

\{F_{n,r}\}

Poly-time computability of the Feigenbaum Julia set

,

Almost Every Real Quadratic Polynomial has a Poly-time Computable Julia Set

\{G_{n,r}\}

On characters of inductive limits of symmetric groups

are the regular representations and scalar representations arising from group abelianizations. As a corollary, we obtain that any measure-preserving ergodic action of a simple Higman-Thompson group must be essentially free. Finite factor representations of other classes of groups are also discussed.

On the resurgent approach to Écalle–Voronin's invariants

Abstract Given a holomorphic germ at the origin of C with a simple parabolic fixed point, the local dynamics is classically described by means of pairs of attracting and repelling Fatou coordinates and the corresponding pairs of horn maps, of crucial importance for Ecalle-Voronins classification result and the definition of the parabolic renormalization operator. We revisit Ecalles approach to the construction of Fatou coordinates, which relies on Borel–Laplace summation, and give an original and self-contained proof of their resurgent character.

On diagonal actions of branch groups and the corresponding characters

We present the first example of a poly-time computable Julia set with a recurrent critical point: we prove that the Julia set of the Feigenbaum map is computable in polynomial time.

On spectra of Koopman, groupoid and quasi-regular representations

We prove that Collet–Eckmann rational maps have poly-time computable Julia sets. As a consequence, almost all real quadratic Julia sets are poly-time.