This project is funded by National Science Centre, Poland, grant No. 2016/23/P/ST1/04088. It is carried out under POLONEZ programme which has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No. 665778.

**Principal investigator:** Dr. Artem Dudko, IMPAN, Warsaw.

**Co-investigator: **Prof. dr habil. Feliks Przytycki, Director of IMPAN, Warsaw.

**Research assistants: **Dr. David Marti-Pete and Dr. Dyi-Shing Ou, IMPAN, Warsaw.

**Publications and preprints:**

- On Lebesgue measure and Hausdorff dimension of real Feigenbaum periodic points of renormalization, manuscript in preparation
- (joint with M. Yampolsky) On computational complexity of Cremer Julia sets, accepted for publication at Fundamenta Mathematicae [ arxiv: math.DS/1907.11047 ]
- (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/711.01653 ]
- (joint with R. Grigorchuk) On the question "Can one hear the shape of a group?" and Hulanicki type theorem for graphs, accepted for publication in Israel Journal of Mathematics; [ arXiv: math.GR/1809.04008 ]
- (joint with S. Sutherland) On the Lebesgue measure of the Feigenbaum Julia set, accepted for publication at Inventiones Mathematicae [ arXiv: math.DS/1712.08638 ]
- (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 ]

**Talks:**

- May 13, 2019, On computational complexity of Julia sets, Mathematics Colloquium at Jacobs University, Bremen, Germany.
- February 18, 2019, Minicourse on Hausdorff dimension of the Feigenbaum Julia set, Holomorphic Dynamics Learning Seminar, IMPAN, Warsaw.
- 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.
- October 22, 2018, On Hausdorff dimension of the Feigenbaum Julia set, Dynamical Systems Seminar at IMPAN, Warsaw, Poland.
- 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.
- April 26, 2018, On the Lebesgue measure of the Feigenbaum Julia set, Ergodic Theory and Dynamics Seminar, Wroclaw, Poland.
- March 27, 2018, On the Lebesgue measure of the Feigenbaum Julia set, Workshop "Algorithmic Questions in Dynamical Systems", Institut de Mathématiques de Toulouse, France.
- November 13, 2017, On the Julia set of the Feigenbaum quadratic polynomial, Dynamical Systems Seminar at IMPAN, Warsaw, Poland.
- September 18, 2017, Wild attractors for Henon-like maps, Dynamical Systems Seminar at IMPAN, Warsaw, Poland.

**Conference and workshop participation:**

- Dynamics, measures and dimensions, Będlewo, Poland, April 7-12, 2019.
- Zagreb Dynamical Systems Workshop, University of Zagreb, Croatia, October 22-26, 2018.
- New Developments in Complex AnCalysis and Function Theory, University of Crete, Heraklion, Greece, July 2-6, 2018.
- Algorithmic Questions in Dynamical Systems, Institut de Mathématiques de Toulouse, France, March 26-29, 2018.
- On geometric complexity of Julia sets, Będlewo, Poland, March 18-23, 2018.
- Complex dynamics and quasi-conformal surgery (in memory of Tan Lei), Université d'Angers, October 23-25, 2017.

**Other activities within or closely related to the project:**

- Co-organizing the conference "On geometric complexity of Julia sets-II", Będlewo, Poland, July 12-17, 2020.
- Co-organized the conference "Dynamics, measures and dimensions", Będlewo, Poland, April 7-12, 2019.
- Co-organizing Holomorphic Dynamics Learning Seminar at IMPAN.
- Co-organizing Dynamical Systems Seminar at IMPAN.
- Co-organized the conference "On geometric complexity of Julia sets", Będlewo, Poland, March 18-23, 2018.