Research

My research lies at the intersection of Geometry, Topology and Dynamical Systems, with applications to Number Theory and Percolation.

For the past several years, I have been focusing on the study of the actions of countable and profinite groups on Cantor sets.

I have also been exploring the connections of my research to other areas, recently obtaining results on the applications of my work in Arithmetic Dynamics and Number Theory.

A recent direction in my research is the study of infinite interval exchange transformations and their applications to the study of flows on translation surfaces of infinite type.

Another direction is the study of group-theoretical and geometric aspects of percolation on Cayley graphs of non-abelian groups.

Read more about my research here

Publications and preprints

H. Bruin and O. Lukina, Staircase flows over an infinite interval exchange transformation, preprint.
O. Lukina, Non-Hausdorff germinal groupoids for actions of countable groups, arXiv:2309.06340.
S. Hurder and O. Lukina, Type invariants for solenoidal manifolds, arXiv:2305.00863.
S. Hurder and O. Lukina, Prime spectrum and dynamics for nilpotent actions, arXiv:2305.00896, Pacific J. Math., 327(1) 2023, 107-128.
S. Hurder and O. Lukina, Essential holonomy of Cantor actions, arXiv:2205.06285, accepted to J. Math. Soc. Japan.
M. I. Cortez and O. Lukina, Settled elements in profinite groups, arXiv:2106.00631, Advances in Mathematics, 404 2022, 108424.
H. Bruin and O. Lukina, Rotated odometers and actions on rooted trees, arXiv: 2104.05420, Fundamenta Mathematicae, 260 2023, 233--249.
H. Bruin and O. Lukina, Rotated odometers, arXiv:2101.00868, J. Lond. Math. Soc., 107(6) 2023, 1983--2024.
O. Lukina, Hausdorff dimension in graph matchbox manifolds, arXiv: 1407.0693, Topology Appl., 308 2022, 108003.
J. Alvarez Lopez, R. Barral Lijo, O. Lukina and H. Nozawa, Wild Cantor actions, arXiv:2010.00498, J. Math. Soc. Japan, 74(2) 2022, 447--472.
S. Hurder, O. Lukina and W. van Limbeek, Cantor dynamics of renormalizable groups, arXiv: 2002.01565, Groups, Geometry and Dynamics, 15(4) 2021, 1449--1487.
M. Groeger and O. Lukina, Measures and regularity of group Cantor actions, arXiv:1911.00680, Discrete Contin. Dyn. Syst. Ser. A., 41(5) 2021, 2001--2029.
S. Hurder and O. Lukina, Nilpotent group actions, arXiv:1905.07740, Proc. Amer. Math. Soc., 150(1) 2022, 289--304.
S. Hurder and O. Lukina, Limit group invariants for non-free Cantor actions, arXiv:1904.11072, Ergodic Theory Dynam. Systems, 41(6), 2021, 1751--1794.
A. Clark, S. Hurder and O. Lukina, Pro-groups and generalizations of a theorem of Bing, Topology Appl., 271 (2020), 106986, arXiv:1811.00288.
O. Lukina, Galois Groups and Cantor actions, arXiv:1809.08475, Trans. Amer. Math. Soc., 374(3) 2021, 1579--1621.
S. Hurder and O. Lukina, Orbit equivalence and classification of weak solenoids, Indiana Univ. Math. J., 69(7) 2020, 2339--2363, arXiv:1803.02098.
O. Lukina, Arboreal Cantor actions, J. Lond. Math. Soc., 99(3) 2019, 678--706, arXiv:1801.01440.
A. Clark, S. Hurder and O. Lukina, Manifold-like matchbox manifolds, Proc. Amer. Math. Soc., 147(8) 2019, 3579--3594, arXiv:1704.04402.
S. Hurder and O. Lukina, Wild solenoids, Trans. Amer. Math. Soc., 371(7) 2019, 4493--4533, arXiv:1702.03032.
J. Dyer, S. Hurder and O. Lukina, Molino theory for matchbox manifolds, Pacific J. Math., 289(1) 2017, 91--151, arXiv:1610.03896.
J. Dyer, S. Hurder and O. Lukina, Growth and homogeneity of matchbox manifolds, Indagationes Mathematicae, 28(1) 2017, 145--169, arXiv:1602.00784.
J. Dyer, S. Hurder and O. Lukina, The discriminant invariant of Cantor group actions, Topology Appl. 208 2016, 64--92, arXiv:1509.06227.
A. Clark, S. Hurder and O. Lukina, Classifying matchbox manifolds, Geometry and Topology, 23(1) 2019, 1--27, arXiv: 1311.0226.
A. Clark, S. Hurder and O. Lukina, Shape of matchbox manifolds, Indagationes Mathematicae, 25(4) 2014, 669--712, arXiv: 1308.3535.
A. Lozano-Rojo and O. Lukina, Suspensions of Bernoulli shifts, Dynamical Systems. An International Journal, 28(4) 2013, 551-566. arXiv:1204.5376.
O. Lukina, Hierarchy of graph matchbox manifolds, Topology Appl., 159(16), 2012, 3461--3485, arXiv:1107.5303.
A.D. Clark, S. Hurder and O.V.Lukina, Voronoi tessellations for matchbox manifolds, Top. Proc. 41, 2013, 167--259 arXiv:1107.1910.
A.D. Clark, R. Fokkink and O.V.Lukina, The Schreier continuum and ends, arXiv:1007.0746, Houston J. Math., 40(2), 2014, 569--599.
H.W.Broer, K.Efstathiou and O.V.Lukina, A geometric fractional monodromy theorem, Discr. Cont. Dyn. Sys. - Ser. S, 3(4), 2010, 517--532.
K. Efstathiou, O.V. Lukina and D.A. Sadovskii, Complete classification of qualitatively different perturbations of the hydrogen atom in weak near orthogonal electric and magnetic fields, J. Phys. A, 42(5), 055209, 2009.
K. Efstathiou, O.V. Lukina and D.A. Sadovskii, Most typical 1:2 resonant perturbation of the hydrogen atom by weak electric and magnetic fields, Phys. Rev. Lett., 101(25), 253003, 2008.
O.V. Lukina, F. Takens and H.W. Broer, Global properties of integrable Hamiltonian systems, Regular and Chaotic Dynamics, 13(6), 2008, 588--630.

Unpublished work

S. Hurder and O. Lukina, The prime spectrum of solenoidal manifolds, arXiv:2103.06825. This manuscript will remain unpublished. Some of its results were included into papers Type invariants for solenoidal manifolds, arXiv:2305.00863, and Prime spectrum and dynamics for nilpotent actions, arXiv:2305.00896.