Papers and preprints:

  • On maximal subgroups of ample groups (with R. Grigorchuk)

    The paper is concerned with maximal subgroups of the ample (better known as topological full) groups of homeomorphisms of totally disconnected compact metrizable topological spaces. We describe all maximal subgroups that are stabilizers of finite sets. Under certain assumptions on the ample group (including minimality), we describe all maximal subgroups that are stabilizers of closed sets or stabilizers of partitions into clopen sets. In particular, our results apply to the ample groups associated with Cantor minimal systems.
  • On the topological full group containing the Grigorchuk group

    We consider the topological full group of a substitution subshift induced by a substitution a→aca, b→d, c→b, d→c. This group is interesting since the Grigorchuk group naturally embeds into it. We show that the topological full group is finitely generated and give a simple generating set for it.
  • On Mealy-Moore coding and images of Markov measures (with R. Grigorchuk and R. Kogan)

    We study the images of the Markov measures under transformations generated by the Mealy automata. We find conditions under which the image measure is absolutely continuous or singular relative to the Markov measure. Also, we determine statistical properties of the image of a generic sequence.
  • Automatic logarithm and associated measures (with R. Grigorchuk and R. Kogan)

    We introduce the notion of the automatic logarithm to study expanding properties of the Schreier graphs generated by an action of two finite-state Mealy automata. We establish a sufficient condition under which the automatic logarithm can be computed by a finite-state automaton. A number of examples is worked out.
  • Notes on the commutator group of the group of interval exchange transformations

    We study the group of interval exchange transformations and obtain several characterizations of its commutator group. In particular, it turns out that the commutator group is generated by elements of order 2.
  • On growth of random groups of intermediate growth (with M. G. Benli and R. Grigorchuk)

    We study the growth of typical groups from the family of p-groups of intermediate growth constructed by Grigorchuk. We find that, in the sense of category, a generic group exhibits oscillating growth with no universal upper bound. At the same time, from a measure-theoretic point of view (i.e., almost surely relative to an appropriately chosen probability measure), the growth function is bounded by a specific function of subexponential growth depending only on the measure.
  • Notes on the Schreier graphs of the Grigorchuk group

    The paper is concerned with the space of the marked Schreier graphs of the Grigorchuk group and the action of the group on this space. In particular, we describe the invariant set of the Schreier graphs corresponding to the action on the boundary of the binary rooted tree and dynamics of the group action restricted to this invariant set.
  • Automata generating free products of groups of order 2 (with D. Savchuk)

    We construct an infinite family of finite automata acting on a rooted binary tree such that the transformation group generated by the action is a free product of groups of order 2.
  • On a substitution subshift related to the Grigorchuk group

    We study the dynamics of a substitution subshift given by the substitution a->aca, b->d, c->b, d->c, which is related to the Grigorchuk group. This dynamical system is shown to be, up to a countable set, conjugate to the binary odometer.
  • On stability of periodic billiard orbits in polyhedra

    We study stability of periodic billiard orbits in polyhedra under small perturbations of the polyhedron. Isolated periodic orbits survive such perturbations. On the other hand, any 1-parameter family of periodic orbits can be completely destroyed. For a 2-parameter family, at most one periodic orbit can survive arbitrary small perturbations of the polyhedron. As a consequence, generic polyhedra admit only isolated periodic billiard orbits.
  • Automata over a binary alphabet generating free groups of even rank (with B. Steinberg and M. Vorobets)

    We construct automata over a binary alphabet with 2n states, n≥2, whose states freely generate a free group of rank 2n. Combined with previous work, this shows that a free group of every finite rank can be generated by a finite automaton over a binary alphabet. We also construct free products of cyclic groups of order two via such automata.
  • On a series of finite automata defining free transformation groups (with M. Vorobets)

    We introduce two series of finite automata starting from the so-called Aleshin and Bellaterra automata. We prove that transformations defined by all automata from the first series generate a free non-Abelian group of infinite rank while automata from the second series give rise to the free product of infinitely many groups of order 2.
  • On a free group of transformations defined by an automaton (with M. Vorobets)

    We prove that three automorphisms of a rooted binary tree defined by a certain 3-state automaton generate a free non-Abelian group of rank 3.
  • Periodic geodesics on generic translation surfaces

    We establish a number of properties of periodic geodesics on translation surfaces that hold for generic elements in their moduli space. This includes asymptotic growth for several counting functions, asymptotic distribution of directions and asymptotic distribution of areas of periodic cylinders.
  • Periodic geodesics on translation surfaces

    Masur proved that any translation surface admits a relatively short periodic geodesic, the growth of the number of periodic geodesics has an upper and lower quadratic bounds, and their directions are dense. We establish effective versions of the first two results and generalize the third.
  • On the uniform distribution of orbits of finitely generated groups and semigroups of plane isometries [scanned]

    We consider actions of free groups, free semigroups and free products of groups of order 2 on a Euclidean plane by isometries. We show that for a generic action, all orbits are uniformly distributed in the plane. The actions for which the distribution of orbits fails to be uniform are explicitly described.
  • Isospectrality and projective geometries (with A. Stepin)

    The paper contains rather concise an exposition of certain relations between isospectrality and finite projective geometries. We show how existence of spectrally equivalent while non-conjugate groups of permutations leads to a construction of isospectral domains and manifolds. Projective geometries come forward as a source of such permutation groups.
  • Billiards in rational polygons: periodic trajectories, symmetries, and d-stability [scanned]

    We consider periodic billiard orbits in rational polygons that satisfy Veech's dichotomy (in particular, triangles that can tile a regular polygon by reflection). The topics include: symmetry of a periodic orbit, the growth rate of the number of periodic orbits, stability of a periodic orbit under small perturbations of the polygon.
  • Ergodicity of billiards in polygons [scanned]

    Kerckhoff, Masur and Smillie proved that the billiard flow in a generic (in the sense of category) polygon is ergodic. We establish an effective version of their result, which allows to provide explicit examples of polygons with the ergodic billiard flow.
  • Planar structures and billiards in rational polygons: the Veech alternative [scanned]

    This paper reviews Veech's theory of translation surfaces that admit a rich group of affine symmetries. The theory leads to remarkable results on the billiard flow in certain polygons. Also, we provide new examples of polygons to which the theory applies and establish new properties of the corresponding translation surfaces.
  • On the measure of the set of periodic points of the billiard [scanned]

    We prove that for any billiard table in any dimension, periodic points of period 3 of the billiard form a set of zero measure in the phase space.