Research interests: Holomorphic Dynamics, Renormalization, Resurgence Theory, Computability and Computational Complexity in Dynamics

Representations of infinite-dimensional groups, Groups acting on rooted trees, Self-similar groups, Spectral Theory, Ergodic Theory

Recent and future events:

Simons Semester "Continued Fractions, Fractals, and Dynamics", April 27-May 31, 2026, Będlewo and Warsaw, Poland
Simons Semester "Invariant structures in groups, spaces, and algebras", January 12-February 15, 2026, Będlewo and Warsaw, Poland
Conference "On geometric complexity of Julia sets VI", August 3-8, 2025, Będlewo, Poland

Conference "Ergodic group actions and unitary representations II", June 15-19, 2025, Będlewo, Poland
Conference "On geometric complexity of Julia sets V", July 21-26, 2024, Będlewo, Poland
Conference "Ergodic group actions and unitary representations", June 16-20, 2024, Warsaw, Poland
IM PAN Dynamical Systems Seminar

Publications:

1. (joint with T. Amrutam, Y. Jiang and A. Skalski) Invariant subalgebras rigidity for von Neumann algebras of groups arising as certain semidirect products, preprint [ arXiv:math.OA/2507.12824 ]

2. (joint with A.I. Danilenko) Irreducible Koopman representations for nonsingular actions on boundaries of rooted trees, preprint [ arXiv:math.DS/2503.03894 ]

3. (joint with S. Bezuglyi and O. Karpel) Measures and dynamics on Pascal-Bratteli diagrams, preprint [ arXiv:math.DS/2411.06280  ]

4. (joint with Y. Jiang) A character approach to the ISR property, preprint [ arXiv:math.OA/2410.14517 ]

5. (joint with D. Gaidashev) Wild attractors for Fibonacci maps, preprint [ arxiv:math.DS/2406.19019 ]

6. (joint with R. Grigorchuk) Characters and IRS's on branch groups and embeddings into hyperfinite factor, preprint [ arXiv:math.RT/2312.13124 ]

7. (joint with I. Gorbovickis and W. Tucker) Lower bounds on the Hausdorff dimension of some Julia sets, Nonlinearity, 36 (2023), 2867-2893; [ arXiv: math.DS/2204.07880 ] 

8. (joint with R. Grigorchuk) On spectral properties of the Schreier graphs of the Thompson group F, Trans. of AMS, 376 (2023), no. 4, 2787-2819.; [ arXiv: math.SP/2107.01109 ] 

9. On Lebesgue measure and Hausdorff dimension of Julia sets of real periodic points of renormalization, Bulletin Polish Acad. Sci. Math. 68 (2020), 151-168.

10. (joint with S. Sutherland) On the Lebesgue measure of the Feigenbaum Julia set, Invent. math. 221 (2020), 167-202; [ arXiv: math.DS/1712.08638 ]

11. (joint with M. Yampolsky) On computational complexity of Cremer Julia sets, Fund. Math. 252 (2021), no. 3, 343-353 [ arxiv: math.DS/1907.11047 ]

12. (joint with R. Grigorchuk) On the question "Can one hear the shape of a group?" and Hulanicki type theorem for graphs, Israel J. Math. 237 (2020), no. 1, 53-74; [ arXiv:math.GR/1809.04008 ]

13. (joint with K. Medynets) On invariant random subgroups of block-diagonal limits of symmetric groups, Proc. of AMS, 147 (2019), no. 6, 2481-2494; [ arXiv: math.GR / 1711.01653 ]

14. On irreducibility of Koopman representations corresponding to measure contracting actions, Groups, Geometry and Dynamics, 12 (2018), no. 4, 1417-1427; [ arxiv.math.RT/1512.02687 ]

15. (joint with M. Yampolsky) Almost every real quadratic polynomial has a poly-time computable Julia set, Found. Comp. Math., 18 (2018), no. 5, 1233-1243; [ arXiv: math.DS / 1702.05768 ]

16. (joint with R. Grigorchuk) On diagonal actions of branch groups and the corresponding characters,  J. Funct. Anal. 274 (2018), no. 11, 3033-3055; [ arxiv:math.RT/1412.5476 ]

17. (joint with R. Grigorchuk) On spectra of Koopman, groupoid and quasi-regular representations, Journal of Modern Dynamics, Volume 11 (2017), 99-123; [ arxiv:math.RT/1510.00897 ]

18. (joint with R. Grigorchuk) On irreducibility and disjointness of Koopman and quasi-regular representations of weakly branch groups, Modern Theory of Dynamical Systems: A Tribute to Dmitry Victorovich Anosov,  AMS Cont. Math. Ser., Volume 692 (2017), 51-66; [ arxiv:math.RT/1508.05452 ].

19. (joint with M. Yampolsky) Poly-time Computability of the Feigenbaum Julia set, Ergodic Theory and Dynam. Sys., to appear. [ arXiv: math.DS/1404.1236 ]

20. (joint with D. Sauzin) On the resurgent approach to Ecalle-Voronin's invariants, Comptes Rendus Mathematique, Volume 353, 3 (2015), 265–271. [ arXiv: math.DS/1307.8095 ] 

21. (joint with D. Sauzin) The resurgent character of the Fatou coordinates of a simple parabolic germ, Comptes Rendus Mathematique, Volume 352, 3 (2014), 255-261. [ arXiv: math.DS/1307.8093 ]

22. Computability of the Julia set. Nonrecurrent critical orbits, Discrete and Continuous Dynamical Systems, Volume 34, 7 (2014), 2751-2778. [   arXiv: math.DS/1109.2946 ]

23. (joint with K. Medynets) Finite Factor Representations of Higman-Thompson groups, Groups, Geometry, and Dynamics 8 (2014). [ arXiv: math.RT/1212.1230 ]

24. (joint with K. Medynets) On Characters of Inductive Limits of Symmetric Groups, Journal of Functional Analysis 264 (2013), no.7, 1565-1598. [ arXiv: math.RT/1105.6325 ]

25. Characters on the Full Group of an Ergodic Hyperfinite Equivalence Relation, Journal of Functional Analysis 261 (2011), no. 6, 1401-1414. [ arXiv: math.RT/1005.4289 ]

26. (joint with S. Favorov), A uniformly spread measure criterion, Anal. Math. Phys., Volume 1, 2-3 (2011), 115-119. [ arXiv: math.CA/0805.0959 ]

27. (joint with T. Banakh and D. Repovs) Symmetric monochromatic subsets in colorings of the Lobachevsky plane, Discrete Math. Theor. Comput. Sci. 12 (2010), no. 1 12-20. [ arXiv: math.CO/0804.1335 ]

28. A description of characters on generalized groups of motions associated with GL(\infty,F_q), (Russian) Dopov. Nats. Akad. Nauk Ukr. Mat. Prirodozn. Tekh. Nauki 2008, no. 10, 14-19.

29. (joint with N. Nessonov) Invariant states on the wreath product, preprint [ arXiv: math/RT/0903.4987 ]

30. (joint with N. Nessonov), Characters of projective representations of the infinite generalized symmetric group, (Russian) Mat. Sb. 199 (2008), no. 10, 3-32; translation in Sb. Math. 199 (2008), no. 9-10 , 1421-1450.

31. (joint with N. Nessonov), Projective characters of the infinite generalized symmetric group, (Russian) Funktsional. Anal. i Prilozhen. 42 (2008), no. 1, 82-85; translation in Funct. Anal. Appl. 42 (2008), no. 1, 69-71.

32. (joint with N. Nessonov), A description of characters on the infinite wreath product, Methods Funct. Anal. Topology 13 (2007), no. 4, 301-317. [ arXiv: math.RT/0510597 ]

33. Tame representations of the group GL(\infty,F_q) (Russian) Algebra i Analiz 18 (2006), no. 2, 57-79; translation in St. Petersburg Math. J. 18 (2007), no. 2, 223-239.

Invited talks:

October 16, 2024, Characters and factor representations of infinite discrete groups: methods and phenomena, Group actions and representations together with Random Matrix Theory, Warsaw, Poland

October 9, 2024, On spectra of self-similar groups and graphs, IM PAN Colloquium, Warsaw, Poland

May 13, 2024, On spectral measures of self-similar graphs, Groups and Graphs Seminar, Budapest, Hungary

March 21, 2023, On spectral properties of the Schreier graphs of the Thompson group F, Groups of Dynamical Origins, Automata and Spectra, Les Diablerets, Switzerland

November 28, 2022, From invariant ergodic measures to indecomposable characters on full groups, Group Actions: Dynamics, Measure, Topology University of Munster, Germany

October 25, 2022, On Hausdorff dimension of Julia sets via a computer-assisted proof, Dynamics Seminar, IST Austria

May 31, 2022, New estimates from below on the Hausdorff dimension of some Julia sets, Workshop in Dynamics, Uppsala University, Sweden

June 16, 2021, On Hausdorff dimension of Julia sets of real Feigenbaum maps, Conference "The 9th Visegrad Conference, Dynamical Systems", Prague, Czech Republic (on-line)

May 13, 2019, On computational complexity of Julia sets, Mathematics Colloquium at Jacobs University, Bremen, Germany.

January 28, 2019, Can one hear the shape of a group?, Alfred Renyi Institute of Mathematics, Budapest, Hungary.

January 17, 2019, On computational complexity of Cremer Julia sets, Ergodic Theory and Dynamical Systems Seminar, University of Warwick, Coventy, UK.

January 15, 2019, On Hausdorff dimension of the Feigenbaum Julia set, Dynamical Systems Seminar, Imperial College London, UK.

July 23, 2018, On Hausdorff dimension of the Feigenbaum Julia set, Dynamical Systems Seminar, University of Barcelona, Spain.

July 5, 2018, On computability and computational complexity of Julia sets, Workshop "New Developments in Complex Analysis and Function Theory", University of Crete, Heraklion, Greece.

March 27, 2018, On the Lebesgue measure of the Feigenbaum Julia set, Workshop "Algorithmic Questions in Dynamical Systems", Institut de Mathématiques de Toulouse, France.

August 14, 2017, On the Julia set of the Feigenbaum quadratic polynomial, Conference "Just a little calculation in dynamics", Bedlewo, Poland.

April 5, 2016, On computational aspects of the Feigenbaum fixed point of period-doubling renormalization,  Workshop "Computation in Dynamics", ICERM, Brown University, USA.

March 20, 2016, On measure of the Feigenbaum fixed point Julia set,  AMS special session "Holomorphic Dynamics", Spring Eastern Sectional Meeting, Stony Brook University, USA.

February 26, 2016, On spectra of Koopman, groupoid and quasi-regular representations, New York Applied Algebra Collogquium, CUNY Graduate Center, USA.

November 12, 2015, On Koopman, groupoid and quasi-regular representations and weakly branch groups, Conference "Geometric and Probabilistic Methods in Group Theory and Dynamical Systems", Texas A&M University, USA.

September 12, 2014, On diagonal actions of branch groups and the corresponding characters, Linear Analysis Seminar, Texas A&M University, USA.

September 10, 2014, Representations associated with group actions and spectra of Hecke type operators, Groups and Dynamics Seminar, Texas A&M University, USA.  

February  25, 2014, On characters of approximately finite groups and Higman-Thompson groups, Conference "Groups acting on rooted trees and around", Paris, France.

December 8, 2013, The Julia set of the Feigenbaum map is poly-time computable, Section "Holomorphic dynamics and related topics", CMS Winter meeting, Ottawa, Canada.

October 17, 2013, On the resurgent approach to local dynamics of simple parabolic germs, Workshop "Topological and Combinatorial Problems in One-dimensional Complex Dynamics", Pisa, Italy.

January 21, 2013, Dynamics of parabolic germs via Resurgence Theory, Dynamical Systems Seminar, University of Toronto, Canada.

February 16, 2012, Computability of Julia sets: a survey, Ergodic Theory and Probability Seminar, The Ohio State University, USA.

September 9, 2011, On characters of inductive limits of symmetric groups, Workshop on Symmetric Functions and its Applications, City University of New York, USA.

May 4, 2011, Ecalle-Voronin invariants and resurgent functions, Dynamical Systems Seminar, Centro di Ricerca Matematica Ennio de Giorgi, Pisa, Italy.