CV

1. Description of professional career

Positions at V.Karazin Kharkiv National University, Department of Mechanics and Mathematics:

1989—1990 assistant professor,
1990—1991 junior research fellow,
1991—1993 assistant professor,
1994—2002 associate professor.

Positions at B.Verkin Institute for Low Temperature Physics and Engineering of National Academy of Sciences of Ukraine, Kharkiv.

2002—2011 senior research fellow,
2012—present time leading research fellow.

2. Description of scientific career, including the research work conducted and the scientific activity carried out

2.1. Received diplomas and degrees:

2003: Dr. hab., thesis ``Orbital approach in ergodic theory'', N. Copernicus University, Toruń, Poland (with excellence). Reviewers: K.Schmidt, T.Downarowicz, M.Lemańczyk, M.Misiurewicz.

1991: C.Sc. in physics and mathematics, PhD thesis ``Ergodic dynamical systems, their cocycles, and automorphisms compatible with cocycles", Kharkiv State University, Ukraine, under the guidance of V.Ya. Golodets.

1986: Diploma in mathematics (with honors), Kharkiv State University.

2.2. Research interests. Main research topics:

I. Ergodic theory: rank one, measurable orbit theory, spectral theory, joinings, entropy, mixing, systems of probabilistic origin, infinite measure preserving and nonsingular dynamical systems

II. Topological dynamics: dynamics in locally compact spaces, topological orbit equivalence

2.3. Description of research conducted and the scientific activity carried out.

(For all the references below (e.g. [64]) we refer to the file Publications_Alexandre_Danilenko.pdf)

Measurable orbit theory. My scientific career started in 1984 when I was a student of Kharkiv state university. My scientific advisor, Dr. V.Ya.Golodets, involved me into research in the field of ergodic theory and operator algebras. At that time, our research team, leaded by V.Ya.Golodets, was under strong influence of the progress in the theory of von Neumann algebras achieved to A.Connes (Fields medal, 1982), M.Takesaki, V.F.R.Jones (Fields medal, 1990) and others. A remarkable interaction between von Neumann algebras and measurable orbit theory was discovered and developed in works by von Neumann, H.Dye, W.Krieger, A.Connes, B.Weiss, J.Feldman, C.C.Moore and others. Contribution of the Kharkiv team (V.Golodets, S. Bezuglyi, S.Sinelshchikov, S.Gefter) into this field included the theorems on existence and uniqueness for cocycles of amenable dynamical systems with prescribed associated actions, solution of the outer conjugacy problems for subgroups in the normalizer of full groups, etc. I was given a task to refine some of these results: given a cocycle of an ergodic amenable equivalence relation, to investigate algebraic and topological properties of an associated ''symmetry'' subgroup of the normalizer of the full group of this equivalence relation and solve the corresponding outer conjugacy problem within this subgroup. I completed the task in a series of papers [64], [63], [62], [60] (some are joint with V.Golodets and S.Bezuglyi). My PhD thesis was based on those results.

After obtaining the PhD degree in 1991 I continued to work in the field of orbit theory. Impressed by a paper of S.Popa and M.Takesaki on topological and homotopical structure the unitary and automorphism groups of a factor (Comm. Math. Phys., 155(1993)] I proved measure theoretical analogues of these results in [61]: I proved more general and subtle theorems on contractibility of the full groups and their normalizers furnished with the natural Polish topologies.

Some of my works are devoted to measure theoretical analogues of V.F.R.Jones famous results on index of subfactors and M.Takesaki’s description of the cross product structure of type-III factors. In [58], [51] and [50] (the latter is a joint work with T.Hamachi) I investigated ergodic subrelations of measurable equivalence relations (as finite measure preserving as nonsingular ones), described their structure and "position"  inside the enveloping equivalence relation and provided some classification in the amenable case.

It should be noted that the aforementioned my results are of strong orbital nature. Though they belong formally to measure theoretic ergodic theory, they are closer indeed to the operator algebras. From the purely ergodic theory point of view, those results seemed to be somewhat abstract and useless for applications to the classical problems of dynamical systems. However, this was a wrong point of view. In the 90s my attention was attracted to several papers devoted to lifting problems from the centralizer of a dynamical system to the centralizer of a locally compact extension of the system:

This approach turned out to be very effective. The lifting problems from the aforementioned papers were solved in much greater generality in my works [54], [56], [57] and my joint work with M.Lemańczyk [55].

Later in my career I returned couple of times to measurable orbit theory:

Entropy theory. The classical entropy theory deals with probability preserving actions of the group of integers. An important problem is to extend the classical theory to more general case of amenable group actions. While extending the theorem on equivalence of the CPE-property and uniform mixing, D.Rudolph and B.Weiss (in Annals of Mathematics, Vol. 151 (2000)) discovered that the relative entropy is an invariant of the orbit equivalence. This paper had a strong influence on me. I realized that the role of the orbit theory is underestimated in the entropy theory. The key observation that the measurable orbit theory can be applied there, however only in the relative setting. I developed an alternative approach to the entropy theory. I showed that the entropy theory can be constructed in a different way, entirely in the language of equivalence relations and their cocycles. Many conditional concepts such as relative Bernoulli, relative CPE, relative generator, etc. (with respect to an invariant sigma-sub-algebra of measurable subsets) are invariants for the factor orbit equivalence. This yields, in turn, that many important theorems in the entropy theory of amenable group actions can be deduced easily from their classical Z-analogues in a way bypassing the standard usage of the Ornstein-Weiss technique involving the Rokhlin lemma, Shannon-McMillan theorem and individual ergodic theorem for amenable group actions. This is done in [49] and [46]. Following this orbital approach, jointly with D.Rudolph (my visit to the University of Maryland was supported by a CRDF grant) we developed a conditional entropy theory for infinite measure preserving actions relative to a sigma-finite factor [35]. Using this theory, we answered a question of U.Krengel (since 1967) about the values of the Krengel entropy of the Cartesian product of two transformations.

Rank-one and (C,F)-actions. In 2000, when studying works by C.Silva and his students on rank-one infinite measure preserving transformations, I got an idea that the standard geometric cutting-and-stacking construction of rank-one transformations can be generalized to rank-one actions of more general group in purely algebraic way. I call it the (C,F)-construction [48]. As appeared, a similar, but non-equivalent, construction was suggested earlier by A.del Junco. I used this construction in a number of subsequent papers to produce rank-one dynamical systems with many additional properties. For instance, it is well known that the weak mixing for probability preserving actions can be stated in variety of ways. Also, the ergodic probability preserving systems possess strong recurrence properties: they are multiply recurrent, polynomially recurrent, etc. Neither of these two facts holds for infinite measure preserving systems. Hence, a lot of question arise: are there exist examples (or counterexamples) of infinite measure preserving systems with various combinations of weak mixing and multiple recurrence properties? Is it possible to generalize these results to actions of more general (non-commutative) groups? Is it possible to produce similar examples in the category of purely nonsingular actions, i.e. actions without equivalent invariant sigma-finite measure? Some of these questions were answered partially in works by J.Aaronson, S.Eigen, A.Hajian, K.Halverson, T.Adams, N.Friedman, C.E.Silva (in collaboration with many his students). However, it turned out that the (C,F)-construction is the most suited to solve all of these problems in the utmost generality. This is shown in my works [48], [45], [44] (joint with C.E. Silva), [36] (joint with A.Solomko), [29] (joint with K.Park), [18].

It is a classical result of del Junco that each irrational rotation on the circle is of rank one. He proved this fact and stated a problem of finding an explicit cutting-and-stacking algorithm for irrational rotations. In collaboration with my student M. Vieprik, we use the (C,F)-construction to answer this question for all well approximable rotations [3], i.e. for almost all irrationals with respect to Lebesgue measure. In a subsequent joint work with him [1] we describe explicitly all odometer factors of arbitrary nonsingular rank-one actions. This generalize significantly results obtained in recent works by M.Foreman, S.Gao, A.Hill, C.E.Silva and B.Weiss (Israel J.Math.,2023) and A.Johnson and D.McClendon (Indag.Math, 2024).

In 1986 J. Milnor introduced a concept of directional entropy. Since then several other concepts of ergodic theory got a ``directional’’ counterparts: ergodicity, mixing or recurrence (for infinite measure preserving systems) for Z2-actions. I used the (C,F)-construction to investigate directional recurrence and directional rigidity of infinite measure preserving actions of lattices in nilpotent groups [14].

In [17] I generalized the (C,F)-construction. If the original one gives actions of rank one, the generalized one produces actions of finite rank. I use the generalized (C,F)-actions to estimate an asymptotic growth of some specific ergodic averages of infinite measure preserving actions. Also some applications are related to finite measure preserving systems such as Bratelli-Vershik maps, ergodic interval exchange transformations, etc.

Spectral theory: spectral multiplicity problem; spectral characterization of odometer actions of Heisenberg group. A unitary operator in a separable Hilbert space is completely determined (up to unitary equivalence) by the maximal spectral type and the spectral multiplicity function. However, it is a difficult open problem to describe which measures and mappings can appear as the maximal spectral type and the spectral multiplicity function respectively for the Koopman operators associated with ergodic transformations. Consider a weak version of this problem, called the spectral multiplicity problem: which subsets of positive integers can appear as the essential values of the spectral multiplicity function of the Koopman operator associated with an ergodic transformation? This problem is also open. However, there are some partial solutions obtained by V.Oseledets, A.Robinson, M.Lemańczyk, J.Kwiatkowski, P.Liardet, G.Goodson, A.Katok, O.Ageev, V.Ryzhikov. I became interested in this problem when a paper by O.Ageev on homogeneous spectrum was published in Invent.Math (2005). He developed an interesting technique of auxiliary actions to answer affirmatively Rokhlin’s question on existence of ergodic transformations with homogeneous spectrum. I refined his techniques and combined them with techniques developed by other authors to obtain realizations for several new families of subsets. I also studied the spectral multiplicity problem "with restrictions": i.e. inside the class of mixing transformations or inside the Gaussian transformations. These results were published in [41], [33], [25], [23]. Some of these results were extended to actions of other Abelian locally compact groups in my joint works with M.Lemanczyk [22] and A.Solomko [31]. Somewhat surprisingly, we solve the spectral multiplicity problem completely in the class of ergodic (and also zero-type ergodic) infinite measure preserving transformations [32], [27] (jointly with V.Ryzhikov).

As the 3-dimentional real Heisenberg group is of type one, i.e. the set of irreducible unitary representations of it is well defined (and described explicitly), one can ask about spectral properties of the measure preserving actions of this group. We studied spectral properties of odometer actions of the 3-dimentional real Heisenberg groups, specified some spectral numerical invariants and showed that the odometers are not spectrally determined in my joint work with M.Leman’czyk [19].

Mixing. There is an open problem in ergodic theory: to construct 0-entropy mixing actions for arbitrary amenable groups. It was stated by D.Rudolph. Since each rank-one action is of 0-entropy, we can consider a finer version of his problem: which amenable groups admit mixing rank-one actions? Are there explicit constructions for such actions? I contributeed partly to solution of these problems. Combining the (C,F)-construction and a modification of Ornstein’s random spacer technique I constructed mixing rank-one actions for

No spacers are used in the (C,F)-construction of mixing actions of these groups. Instead, I developed a new technique which may be called a random rotation method. A weak point of this (as well as Ornstein’s) random method is that the parameters of the construction are not defined explicitly. Different, technique was used in my joint work with C.Silva [39] to construct mixing rank-one actions for all closed subgroups of finitely dimentional vector spaces. It is a development of T.Adams and C.Silva technique from their paper [Ergodic Theory Dynam. Systems, 19, 1999] on staircase actions. All parameters of this construction are defined explicitly.

Joinings. My interest to joinings arose thanks to collaboration with M.Lemańczyk, who is a famous expert in this field. All the problems about joinings of probability preserving systems that I studied in my papers were communicated to me by him. Of course, joinings and disjointness are important concepts in ergodic theory. They were introduced by D.Rudolph and H. Furstenberg and studied in depth by W.Veech, D.Rudolph, A.del Junco, J.-P.Thouvenot, E.Glasner, B.Weiss, M.Lemanczyk. Knowing the self-joinings of a dynamical system we can control some dynamical properties of these systems such as factors, centralizer, etc. Several classes of transformations were introduced via joining: MSJ, simple, quasi-simple, distally simple, etc.. In [40], I introduced a new class of systems which I called near simple systems. I investigated properties of near simple actions and proved an analog of the Veech theorem on factors for them. The main motivation for this work was to to construct a transformation which is quasi-simple but disjoint from any simple action. This answered a question by J-P.Thouvenot.

I wrote several more articles on joinings:

Nonsingular systems of probabilistic origin: Bernoulli, Markov, Poisson, Gaussian. In my opinion, the most significant advances in nonsingular ergodic theory over the last 15 years have been related to the study of nonsingular versions of systems of probabilistic origin: Bernoulli and Markov shifts, Gaussian transformations and Poisson suspensions.

U.Krengel in 1969 isolated a class of nonsingular Bernoulli shifts that do not admit an equivalent finite invariant measure. However a real progress in studying such systems began since 2010 thanks to works by Z.Kosloff and afterwards S.Vaes, T.Berendschot, J.Wahl. I was deeply impressed by those works and joined to this research direction. I solved a problem by Z.Kosloff about extension his results on Krieger’s type of ergodic nonsingular Bernoulli Z-shifts to Bernoulli shiftwise actions of arbitrary amenable groups [12]. To prove this result I had to generalize the pointwise nonsingular ergodic theorem to actions of amenable groups. Of course, it is known that this theorem can not hold for all integrable functions (according to a general result by M.Hochman). However, I showed that it holds for a countable family of integrable functions. Only a countable family of functions are needed to extend Kosloff’s theorem.

Jointly with M.Lemańczyk we extended Krengel’s class of nonsingular Bernoulli shifts by considering non-balanced Bernoulli measures and their natural extensions. We proved that in that case, the Maharam extension of the Bernoulli shift has property K (according to the definition of C.Silva and P.Thieullen). We also proved similar theorems for a class of nonsingular Markov shifts [11].

Poissonian suspensions are an important class of dynamical systems of probabilistic origin. I had an idea to construct and study nonsingular analogues of these systems. However my probabilistic background at that time was not enough to fulfill this task. Therefore, I decided to collaborate with Z.Kosloff and E.Roy who are strong experts in probability theory. Jointly, we developed a theory of nonsingular Poisson suspensions in works [4], [8], [6]. We found the largest subgroup of nonsingular transformations for which nonsingular suspensions are well defined as nonsingular transformations. We introduced a Polish topology on the group and studied algebro-topological properties of this group. Some criteria for conservativeness of the action were found. We showed that the dynamical properties of the suspensions depends not only on dynamical properties of the underlying system but also on the choice of measure inside the equivalent class of the underlying density and discovered a phenomenon of faze transition there. We found typical (in Baire category sense) ergodic properties of nonsingular Poisson suspensions (including their Krieger type). Applications of nonsingular Poisson suspensions are found to Kazhdan property (T), Haagerup property for locally compact Polish groups, Furstenberg entropy, stationary actions. We also constructed ergodic nonsingular Poisson suspensions and ergodic Bernoulli actions of arbitrary Krieger type.

Following the work of Arano, Isono, Marracchi on nonsingular Gaussian systems, jointly with M. Lemańczyk, we investigated dynamical properties of that systems and obtained some applications to Gaussian cocycles over mildly mixing transformations [7]. In particular, we partly answered a question about ergodicity of Gaussian cocycles from [M.Leman’czyk, E.Lesigne, D.Skrenty, Aequationes Math. 61(2001)]. We also compared nonsingular Gaussian and Poisson systems, found some similarities as well as drastic differences between them. It is worthy to note that we also introduced a new class of nonsingular dynamical systems (we call it IDPFT). It is usefull when investigating ergodicity of nonsingular Poisson and Gaussian actions.

Topological dynamics in locally compact spaces. In 1995, T.Giordano, I.F.Putnam and C.Fr.Skau [J.Reine Angew. Math., 469] obtained a remarkable classification of minimal homeomorphisms on (compact) Cantor spaces up to several different variations of topological orbit equivalence. Their work became very popular among experts in dynamical systems. An author, C.Skau, delivered a talk on their results at our seminar in Kharkiv. During his talk I realized that those results can be extended to the case of locally compact non-compact Cantor spaces. I wrote a paper on the strong topological orbit equivalence of minimal homeomorphisms on locally compact Cantor spaces and published it [47].

My version of the (C,F)-construction [48] determines topological group actions on locally compact spaces. These actions are amenable in topological sense. Thus, just the definition of (C,F)-actions leaded to solution of a difficult problem in topological dynamics posed by J.Kellerhals, N.Monod, M.Rordam (Doc. Math., 2013) about existence of Radon uniquely ergodic amenable actions of arbitrary countable groups on locally compact Cantor spaces [10].

I also note that the (C,F)-construction is useful to construct locally compact topological models for measurable rank-one actions of groups and their factors [10], [1]. Classification of topological (C,F)-actions up to topological isomorphism is given in [10] in terms of the (C,F)-parameters.

Other topics: actions of non-locally compact groups, Furstenberg entropy, self-similarities.

2.3. Participation in research funded by grants

2.4. Research Identifiers.

ORCID: 0000-0002-3198-9013
MathSciNet: ID 265198
Scopus ID: 7006631242

3. Other relevant information:

3.1. Invited conference lectures.

  1. Ergodic group actions and unitary representations, Warsaw, Poland, June 2024, invited speaker

  2. First Dynamical Systems Summer Meeting, Będlewo, Poland, August, 2021 (online talk)

  3. Dynamics, Equations and Applications, Krakow, Poland, 2019, invited speaker.

  4. Ergodic theory meeting, Jerusalem, Israel, 2019, invited speaker.

  5. Ergodic aspects of modern dynamics, Będlewo, Poland, June, 2018, invited speaker.

  6. Spectral Theory of Dynamical Systems and related topics, CIRM, Marseille, 2016, invited speaker.

  7. Ergodic theory of dynamical systems, Będlewo, Poland, November, 2015, invited speaker

  8. Activity ``Dynamics and Numbers'', MPIM, Bonn, Germany, July, 2014, a talk.

  9. II International Conference ``Analysis and Mathematical Physics", Kharkiv, Ukraine, June, 2014, a talk.

  10. Ergodic Theory and Dynamical Systems, Toruń, Poland, May, 2014, invited talk

  11. Workshop on Ergodic Theory, Williams College, Williamstown, MA, USA, July, 2012, invited talk

  12. Infinite Ergodic Theory, Surrey University, Guildford, UK, May, 2012, invited lecture

  13. Workshop in Ergodic Theory and Dynamical systems, Chapel Hill, North Carolina, USA, 2012, invited talk.

  14. Complex analysis and its applications, Kharkiv, Ukraine, 2011, invited talk

  15. Workshop on Spectral Theory, Warsaw, Poland, 2010, invited lecture

  16. Workshop on Dynamics, Pingree Park, Colorado, USA, 2010, invited lecture

  17. Colloque Proprietes des systemes dynamiques et marches aleatoires, Roscoff, France, 2010, invited lecture

  18. Spring School of Dynamical Systems, Będlewo, Poland, 2010, invited speaker

  19. Progress in Dynamics, Paris, France, 2009, invited speaker

  20. Dynamical Numbers, Bonn, Germany, 2009, invited lecture

  21. Dynamical Systems and Randomness, Paris, France, 2009, invited speaker

  22. Dynamics Workshop, Pingree Park, Colorado, USA, 2008, invited speaker

  23. Geometry and Dynamics of Groups and Spaces, Bonn, Germany, 2006, invited talk

  24. Workshop on Ergodic Theory and Dynamical Systems, Szklarska Poręba, Poland, 2006, invited speaker

  25. Ecole Plurithematique de Theorie Ergodique II, CIRM, Marseille, France, 2006, invited speaker

  26. Algebraic and Topological Dynamics, Bonn, Germany, 2004, invited talk

  27. Ergodic Theory and Topological Dynamics, Toruń, Poland, 2002, invited talk

  28. Ergodic Theory and Dynamical Systems, Villetaneuse, France, 2001, invited talk

  29. Mini-workshop on Ergodic Theory and Operator Algebras, Fukuoka, Japan, 2001, invited lecture

  30. Mathematics in the New Millennium, Seoul, Korea, 2000, invited talk

  31. Ergodic Theory and Dynamical Systems, Toruń, Poland, 2000, invited talk

  32. Dynamical Systems and Ergodic Theory, Katsiveli, Crimea, Ukraine, 2000, invited talk

  33. Conference on Ergodic Theory and Dynamical Systems, Szklarska Poręba, Poland, 1997, invited talk

  34. Ergodic Theory and Symbolic Dynamics, Churanov, Czech Republic, 1996, invited talk.

3.2. Organization of scientific conferences

  1. A member of the Organizing Committee of the  UK-Poland-Ukraine  Scientific Meeting, Section ”Group actions and representations - analytic and geometric aspects”, October 2024, Warsaw.

  2. A member of the Organizing Committee of the conference “Dynamical Systems and Ergodic Theory”, Katsiveli, Crimea, Ukraine, 2000.

3.3. Promoted doctors or staff training

  1. A.V.Solomko, PhD thesis title: Measurable rank-one actions and its applications, defended at B.Verkin Institute for Low Temperature Physics and Engineering of the National Academy of Sciences of Ukraine, Kharkiv, 2013.

  2. I am  currently the superwisor  of M.I. Vieprik, magister. We have 2 joint papers published (in Studia Math. and Erg.Theory & Dynam Syst.).

 3.4. Prizes and awards:

  1. State Prize of Ukraine in the field of Science and Technology (2010) for “Theory of dynamical systems: modern methods and applications”. Jointly with S.Bezuglyi, I.Chueshov, V.Fedorenko, S.Kolyada, V.Korobov, Yu.Maistrenko, E.Romanenko, A.Sharkovsky, A.Teplinsky.

  2. Lavrentyev Prize of the National Academy of Sciences of Ukraine (2019) for “Development of new methods in the theory of dynamical systems, group actions and representation theory”. Jointly with Yu. Drozd and V. Koshmanenko.

 3.5. Educational activity:

  1. V.Karazin Kharkiv National University, Ukraine: 11 years of full-time professorship in 1989—2001 (first as assistant professor then as associate professor) at the department of mechanics and mathematics.  I taught the following courses: calculus, algebra, ergodic theory, C*-algebras.
             Also, I taught short courses on ergodic theory, amenable groups, van der Waerden theorem at V.Karazin Kharkiv National university in 2006-2020.

  2. Ajou University, Suwon, South Korea:  lectures and exercises on topological dynamics (2000-2001) and a course on measurable orbit theory (2001).

  3. Rouen University, France. I taught a course in symbolic dynamics (2018).

  4. N.Copernicus University, Torun’, Poland. I taught a course on amenable groups and Haar measure on locally compact groups for doctorate students (2013).

 3.6. Organizational and popularizing activities:

  1. An editor of Mathematical Bulletin of the Shevchenko Scientific Society, Section  "Topological Dynamics and Ergodic Theory", Ukraine.

  2. A member of the Scientific  Council  of Mathematical Department of  B.Verkin Institute for Low Temperature Physics and Engineering of the National Academy of Sciences of Ukraine. Kharkiv, Ukraine.

  3. Member of the Mathematics Discipline Council, N.Copernicus University, Torun’, Poland.

  4. Reviewer for MathSciNet. Author of 227 reviews.

3.7. Research teams led by the candidate. Coordinating research work.

Being the Principal Investigator of the Ukrainian team of the U.S. Civilian Research and Development Foundation (CRDF) grant UM1-2546-KH-03 in 2004-2005, I led this research team and coordinated the research work of this team.

3.8. Scientific visits.

3.9. Presentations at seminars.