The article contains a brief biography of Professor M.V. Dolov, a description of his main scientific achievements and a brief description of articles included into the Special Issues of the Journal “Mathematics and Theoretical Computer Science”, dedicated to the conference “To the 90th anniversary of M.V. Dolov” held at Nizhny Novgorod State University.

The 5th of November 2024 marked the 90th anniversary of the birth of an outstanding scientist and a remarkable person Mikhail Vasilievich Dolov. This short essay is dedicated to this event and contains an overview of some results of M.V. Dolov and other researchers at Nizhny Novgorod State University on the second part of the sixteenth Hilbert problem. 

We obtain a topological classification of smooth structurally stable flows on four-dimensional closed manifolds whose wandering set contains isolated trajectories connecting saddle equilibria (heteroclinic curves). For dimensional reasons, heteroclinic curves of such flows belong to the intersection of invariant manifolds of saddles of neighboring Morse indices. We assume that the non-wandering set of the flows under consideration consists of exactly one source, one sink, and an arbitrary number of saddles, the dimension of whose unstable manifolds is equal to 1 and 2. We obtain necessary and sufficient conditions for the topological equivalence of such flows and present an algorithm for realizing a representative in each class of topological equivalence. In particular, we show that in the considered class of flows on the sphere S 4 there exists exactly one class of topological equivalence of flows with a single heteroclinic curve and a countable set of topologically nonequivalent flows with three heteroclinic curves. The latter result contrasts with the three-dimensional situation, where for a similar class of flows there are only finitely many equivalence classes for each number of heteroclinic curves. 

Let G be the group of homeomorphisms of an n-dimensional topological manifold M with a nonempty boundary ∂M. The aim of this work is to study the effect of the nonempty boundary of the manifold M on the structure of global attractors of the homeomorphism group G. Our main result is the proof that any global attractor A of the homeomorphism group G on a manifold M with a nonempty boundary ∂M either belongs to the boundary and it can be either a proper subset of the boundary or coincide with the boundary, or it is equal to the union of the boundary with the global attractor of the group induced on the interior of the manifold M. It is shown that, generally speaking, non-global attractors do not possess this property. Various examples are constructed, including an example with two different global attractors. 

The article continues the research of the asymptotic behavior of the trajectories of the most simple skew products on multidimensional cells conducted by the authors. Here we describe the structure of nonwandering set of continuous skew products having a closed set of periodic points and such that the set of (least) periods of periodic points is unbounded. An example of a differentiable skew product with a closed set of periodic points is constructed, defined on an n-dimensional cell (n ≥ 3) and having a one-dimensional ω-limit set. Keywords: skew product, nonwandering set, Ω-blow up, ω-limit set

According to J. Nielsen and H. Hang, each class of topological conjugacy of periodic homeomorphisms of orientable compact surfaces is completely described by a finite set of data called the characteristic. For a two-dimensional sphere, exhaustive classification results with the construction of linear representatives in each conjugacy class were obtained by B. Kerekyarto. For a two-dimensional torus, similar results were obtained with the participation of the authors of this article. Here we find all the characteristics of orientation-changing periodic homeomorphisms of a two-dimensional torus. A homeomorphism representing a class of topological conjugacy is constructed for each of them. The classification of periodic homeomorphisms, in addition to being of independent interest, plays a key role in solving the Palis–Pugh problem of constructing stable arcs in the space of discrete dynamical systems, which is included in the list of 50 most important problems of dynamical systems. For all classes of gradient-like diffeomorphisms of surfaces where this problem is solved, the idea of a close connection of such systems with periodic transformations was used. Thus, the obtained result will make it possible to expand the class of systems for which the Palis problem has been solved. 

Let X be a finite tree and let f : X → X be a continuous map with zero topological entropy and an infinite minimal set M. We show that the restriction of f|M of f to M is topologically conjugate to adding-machine τα, where α = (j1, . . . , jn, 2, 2, . . .) be the sequence for ji ≥ 2 if 1 ≤ i ≤ n. We describe the topological structure of finite trees on which there exist continuous maps with zero topological entropy and an infinite minimal set M on which the map f|M is topologically conjugate to adding machine τα, where α = (j1, . . . , jn, 2, 2, . . .). At the same time, for any sequence α = (j1, . . . , ji , . . .), where ji ≥ 2 for all i ≥ 1, there exist a dendrite X that is not a finite tree and a continuous map f with zero topological entropy and an infinite minimal set M on which the map f is topologically conjugate to adding machine τα.

We also show that for any sequence α = (j1, . . . , jn, . . .), where ji ≥ 2 for all i ≥ 1, there exist a dendrite X that is not a finite tree and a continuous map f with zero topological entropy and an infinite minimal set M such that f|M is topologically conjugate to adding-machine τα.