Members

Ayreena Bakhtawar's research lies in the intersection of  Diophantine approximation, Fractal geometry and Ergodic theory and dynamical systems. She is particularly interested in exploring the Hausdorff dimension theory of the fractals and the ergodic theory of dynamical systems that comes from various versions of continued fractions. Her research also investigates  the connection between the continued fractions and Diophantine approximation.

[1] Ayreena Bakhtawar, Mumtaz Hussain, Dmitry Kleinbock and Bao-Wei Wang. Metrical properties for the weighted products of multiple partial quotients in continued fractions. Houston Journal of Mathematics, 49(1) (2023), pp.159--194. 

[2] Ayreena Bakhtawar and David Simmons. Hausdorff measure of sets of Dirichlet non-improvable matrices in higher dimensions. Research in Number Theory 9 (2023), no.3, Paper No. 54, 18 pp.

[3] Ayreena Bakhtawar. Hausdorff dimension for the set of points connected with the generalized Jarn'ik--Besicovitch set. Journal of the Australian Mathematical Society, 112(1) (2022), 1--29

[4] Ayreena Bakhtawar, Philip Bos and Mumtaz Hussain. Hausdorff dimension of an exceptional set in the theory of continued fractions. Nonlinearity, 33(6) (2020), 2615--2639

[5] Ayreena Bakhtawar, Philip Bos and Mumtaz Hussain. The sets of Dirichlet non-improvable numbers versus well-approximable numbers. Ergodic Theory and Dynamical Systems, 40(12) (2020), 3217--3235

Artem Dudko's research interests are in representation theory, dynamical systems, and related areas. He obtained many results on characters and representations of infinite discrete groups, including approximately finite groups ([1]) and Higman-Thompson groups ([2]). In several papers, including ([3)] and ([5]), Artem studied spectra of operators of representations of groups acting on various spaces. His another area of research is holomorphic dynamics, in which Artem focuses on measure-theoretic properties and complexity of Julia sets ([4]).

[1] A. Dudko and K. Medynets, On Characters of Inductive Limits of Symmetric Groups, Journal of Functional Analysis, 264 (2013), no.7, 1565-1598.

[2] A. Dudko and K. Medynets, Finite Factor Representations of Higman- Thompson groups, Groups, Geometry, and Dynamics, 8 (2014), no. 2, 375-389.

[3] A. Dudko and R. Grigorchuk, On spectra of Koopman, groupoid and quasi- regular representations, J. Mod. Dyn. 11 (2017), 99-123.

[4] A. Dudko and S. Sutherland, On the Lebesgue measure of the Feigenbaum Julia set, Invent. math. 221 (2020), 167-202.

[5] A. Dudko and R. Grigorchuk, On spectral properties of the Schreier graphs of the Thompson group F, Trans. of AMS, 376 (2023), no. 4, 2787-2819. 

Yonatan Gutman is interested in topological dynamics and ergodic theory. He has written numerous articles about mean dimension, in particular, the embedding problem in topological dynamics ([1]). He has worked within the theory of nilspaces and universal minimal spaces ([2],[3]). In recent years he is interested in the representation of dynamical systems through time-delayed measurements ([4]).

[1] Yonatan Gutman and Masaki Tsukamoto, Embedding minimal dynamical systems into Hilbert cubes. Inventiones Mathematicae 221 (2020) 113–166.

[2] Yonatan Gutman, Freddie Manners and Péter Varjú, The structure theory of Nilspaces III: Inverse limit representations and topological dynamics. Advances in Mathematics 365 (2020), 107059.

[3] Yonatan Gutman, Todor Tsankov and Andy Zucker, Universal minimal flows of homeomorphism groups of high-dimensional manifolds are not metrizable. Mathematische Annalen 379 (2021), 1605–1622.

[4] Krzysztof Barański, Yonatan Gutman and Adam Śpiewak, On the Shroer-Sauer-Ott-Yorke Predictability Conjecture for Time-Delay Embeddings. Communications in Mathematical Physics 391 (2022), no. 2, 609–641. 

Michael Levin is interested in topology and topological dynamics. 

[1] Jerzy Dydak, Michael Levin and Jeremy Sieger, Universal spaces for asymptotic dimension via factorization. Canad. Math. Bull.67 (2024), no.2, 391–402.

[2] Michael Levin, Resolving compacta by free p-adic actions. Fund. Math.255 (2021), no.2, 181–207.

[3] Michael Levin, On the unstable intersection conjecture. Geom. Topol. 22 (2018), no. 5, 2511–2532.

Chunlin Liu's primary research focus is on topological dynamical systems and ergodic theory. Additionally, he is studying probability theory, with a particular emphasis on distribution-dependent equations and nonlinear expectations. 

[1] Chunlin Liu, Feng Tan and Jianhua Zhang. Mean Li-Yorke chaos along any infinite sequence for infinite-dimensional random dynamical systems.  J. Differential Equations 403 (2024), 548–575.

[2] Chunlin Liu and Fagner B. Rodrigues.  Metric Mean Dimension via Preimage Structures. J. Stat. Phys. 191 (2024), no. 2, 31.

[3] Jie Li, Chunlin Liu, Siming Tu and Tao Yu. Sequence entropy tuples and mean sensitive tuples. Ergodic Theory Dynam. Systems 44 (2024), no. 1, 184--203.

[4] Chunlin Liu and Leiye Xu. Directional Kronecker algebra for Z^q-actions. Ergodic Theory Dynam. Systems 43 (2023), no. 4, 1324--1350. 

[5] Chunlin Liu and Xiaomin Zhou. Directional entropy dimension of topological dynamical systems. J. Differential Equations 333 (2022), 332--360.

Feliks Przytycki is working in iteration of holomorphic maps in

complex dimension 1 and maps of interval, and in smooth hyperbolic dynamics, often in relation with geometric measure theory and thermodynamic formalism. Several of  his results concern estimates of topological entropy and Lyapunov exponents.

[1] Reza Mohammadpour, Feliks Przytycki, Michał Rams, Hausdorff and packing dimensions and measures for nonlinear transversally non-conformal thin solenoids, Ergodic Theory and Dynamical Systems 42 (2022), 3458-3489.

[2] Feliks Przytycki, Thermodynamic formalism methods in

one-dimensional real and complex dynamics, Proceedings of the International Congress of Mathematicians 2018,

Rio de Janeiro, Vol.2, pp. 2081-2106.

[3] Feliks Przytycki, Juan Rivera-Letelier, Geometric pressure for

multimodal maps of the interval, Memoirs of the American Mathematical Society 1246 (2019) 1-81.

[4] Feliks Przytycki, Mariusz Urbański, Conformal Fractals: Ergodic

Theory Methods, Cambridge University Press, 2010.

[5] Feliks Przytycki, Juan Rivera-Letelier, Stanislav Smirnov,

Equivalence and topological invariance of conditions for non-uniform hyperbolicity in iteration of rational maps, Inventiones mathematicae 151 (2003), 29-63.

Michał Rams is interested in the geometric aspects of dynamical

systems. Among other fields, this includes commutative and

noncommutative thermodynamic formalism, fractals and multifractal

formalism, and metric number theory. His particular interest lies in

the nonuniformly hyperbolic phenomena in different areas of dynamical systems, from interval maps to higher dimensional smooth dynamics.

[1] The Lyapunov spectrum of some parabolic systems, with K. Gelfert,Erg. Th. and Dyn. Sys. 29 (2009), 919-940.

[2] Projections of Fractal Percolations, with K. Simon, Erg. Th. and

Dyn. Sys. 35 (2015), 530-545.

[3] Multifractal analysis for expanding interval maps with infinitely

many branches, with A-H. Fan, T. Jordan, L. Liao, Trans. Amer. Math.

Soc. 367 (2015), 1847-1870.

[4] Shrinking targets on Bedford-McMullen carpets, with B. Barany,

Proc. of the London Math. Soc. 117 (2018), 951-995.

[5] Entropy spectrum of Lyapunov exponents for nonhyperbolic step-skew products and elliptic cocycles, with L. Diaz and K. Gelfert,

Communications in Math. Phys. 367 (2019), 351-416.

[6] Mass Transference Principle: From Balls to Arbitrary Shapes, with H. Koivusalo, Int. Math. Res. Not. 8 (2021), 6315-6330.

Adam Śpiewak is interested in dynamical systems and fractal geometry. He studies geometric properties of measures arising in dynamical systems and iterated function systems [1]. He is also interested in mean dimension (including its applications to compression [2]) and works on developing a probabilistic theory of time-delayed embeddings [3].

[1] Balázs Bárány, Károly Simon, Boris Solomyak and Adam Śpiewak,

Typical absolute continuity for classes of dynamically defined measures. Adv. in Math. 399 (2022), 108258.

[2] Yonatan Gutman and Adam Śpiewak, Metric mean dimension and analog compression. IEEE Transactions on Information Theory 66 (2020), no. 11, 6977–6998.

[3] Krzysztof Barański, Yonatan Gutman and Adam Śpiewak, Prediction of dynamical systems from time-delayed measurements with self-intersections. J. Math. Pures Appl. 186 (2024), 103-149.