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.