In the study of deterministic processes described by the Morse-Smale systems noncompact heteroclinic curves play special role. These curves belong to intersections of stable and unstable manifolds of saddle periodic points. In particular, these curves are mathematical models of magnetic separators in the plasma eld. We consider the class of gradient-like dieomorphisms on three-dimensional manifolds such that their periodic points and a part of their invariant manifolds form disjoint tamely embedded surfaces. We prove that the number of the surfaces is nite and all of them have the same genus. The main result is presentation of the exact lower estimation for the number of heteroclinic curves of any dieomorphism from considered class. This estimation is dened by genus of surfaces and their number. In addition the paper describes the topological type of manifolds which admit considered dieomorphisms.

In this paper we consider the class G of A-dieomorphisms f , dened on a closed 3-manifold M3 . The nonwandering set is located on nite number of pairwise disjoint f -invariant 2-tori embedded in M3 . Each torus T is a union of  $W^u_{B_T}\cup W^u_{\Sigma_T}$, либо $W^s_{B_T}\cup W^s_{\Sigma_T}$, where $B_T$ -- dimensional basic set exteriorly situated on T and T is nite number of periodic points with the same Morse number. We found out that an ambient manifold which allows such dieomorphisms is homeomorphic to a quotient space $M_{\widehat J}=\mathbb T^2\times[0,1]/_\sim$, where $(z,1)\sim(\widehat J(z),0)$ for some algebraic torus automorphism b J , dened by matrix $J\in GL(2,\mathbb Z)$,  which is either hyperbolic or J = ±Id . We showed that each dieomorphism f ∈ G is semiconjugate to a local direct product of an Anosov dieomorphism and a rough circle transformation. We proved that structurally stable dieomorphism f ∈ G is topologically conjugate to a local direct product of a generalized DA-dieomorphism and a rough circle transformation. For these dieomorphisms we found the complete system of topological invariants; we also constructed a standard representative in each class of topological conjugation

We consider a class  $H(\mathbb{R}^n)$ of orientation preserving homeomorphisms of Euclidean space $\mathbb{R}^n$ such that for any homeomorphism  $h\in H(\mathbb{R}^n)$  and for any point  $x\in \mathbb{R}^n$  a condition  $\lim \limits_{n\to +\infty}h^n(x)\to O$ holds, were  $O$  is the origin. It is provided that for any $n\geq 1$  an arbitrary homeomorphism  $h\in H(\mathbb{R}^n)$  is topologically conjugated with the homothety $a_n: \mathbb{R}^n\to \mathbb{R}^n$, given by  $a_n(x_1,\dots,a_n)=(\frac12 x_1,\dots,\frac12 x_n)$. For a smooth case under the condition that all eighenvalues of the differetial of the map $h$ have absolute values smaller than one, this fact follows from the classical  theory of  dynamical systems. In the topological case for  $n\notin \{4,5\}$ this fact is proven in several works of 20th centure, but authors do not know any papers where it would be prooven for     $n\in \{4,5\}$. This paper fills this gap.

This paper deals with the study of the dynamics in the neighborhood of one-dimensional basic sets of Ck , k ≥ 1 , endomorphism satisfying axiom of A and given on surfaces. It is established that if one-dimensional basic set of endomorphism f has the type (1; 1) and is a onedimensional submanifold without boundary, then it is an attractor smoothly embedded in ambient surface. Moreover, there is a k ≥ 1 such that the restriction of the endomorphism fk to any connected component of the attractor is expanding endomorphism. It is also established that if the basic set of endomorphism f has the type (2; 0) and is a one-dimensional submanifold without boundary then it is a repeller and there is a k ≥ 1 such that the restriction of the endomorphism fk to any connected component of the basic set is expanding endomorphism.

We consider the class of continuous Morse-Smale flows defined on a topological closed manifold $M^n$ of dimension n which is not less than three, and such that the stable and unstable manifolds of saddle equilibrium states do not have intersection. We establish a relationship between the existence of such flows and topology of closed trajectories and topology of ambient manifold. Namely, it is proved that if $f^t$ (that is a continuous Morse-Smale flow from considered class) has mu sink and source equilibrium states and $\nu$ saddles of codimension one, and the fundamental group $\pi_{1}(M^n$) does not contain a subgroup isomorphic to the free product $g = 1/ 2 ( \nu−\mu  + 2)$ copies of the group of integers Z , then the flow $f^t$ has at least one periodic trajectory.

We define  a class of gradient-like diffeomorphisms that can be presented as local products of diffeomorphisms on the circle and on a surface,  provide their topological classification and  specify topology of the ambient manifold.

We describe a class of gradient-like systems on surfaces admitting topological classificcation in turms of A.G. Mayer  classification of rough systems on the circle.

We introduce the definition of  consistent equivalence of energy Morse-Bott functions for Morse-Smale flows on surfaces and state that consistent equivalence of  that functions is necessary and sufficient condition  for such flows.