Research interests: Holomorphic Dynamics, Parabolic 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

1. (joint with R. Grigorchuk) Characters and IRS's on branch groups and embeddings into hyperfinite factor, preprint [ arXiv:math.RT/2312.13124 ]
2. (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 ] 
3. (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 ] 
4. On Lebesgue measure and Hausdorff dimension of Julia sets of real periodic points of renormalization, Bulletin Polish Acad. Sci. Math. 68 (2020), 151-168.
5. (joint with S. Sutherland) On the Lebesgue measure of the Feigenbaum Julia set, Invent. math. 221 (2020), 167-202; [ arXiv: math.DS/1712.08638 ]
6. (joint with M. Yampolsky) On computational complexity of Cremer Julia sets, Fund. Math. 252 (2021), no. 3, 343-353 [ arxiv: math.DS/1907.11047 ]
7. (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 ]
8. (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 ]
9. 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 ]
10. (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 ]
11. (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 ]
12. (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 ]
13. (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 ].
14. (joint with M. Yampolsky) Poly-time Computability of the Feigenbaum Julia set, Ergodic Theory and Dynam. Sys., to appear. [ arXiv: math.DS/1404.1236 ]
15. (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 ] 
16. (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 ]
17. Computability of the Julia set. Nonrecurrent critical orbits, Discrete and Continuous Dynamical Systems, Volume 34, 7 (2014), 2751-2778. [   arXiv: math.DS/1109.2946 ]
18. (joint with K. Medynets) Finite Factor Representations of Higman-Thompson groups, Groups, Geometry, and Dynamics 8 (2014). [ arXiv: math.RT/1212.1230 ]
19. (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 ]
20. 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 ]
21. (joint with S. Favorov), A uniformly spread measure criterion, Anal. Math. Phys., Volume 1, 2-3 (2011), 115-119. [ arXiv: math.CA/0805.0959 ]
22. (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 ]
23. 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.
24. (joint with N. Nessonov) Invariant states on the wreath product, preprint [ arXiv: math/RT/0903.4987 ]
25. (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.
26. (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.
27. (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 ]
28. 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.

• 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.