Dynamics of three-dimensional A-diffeomorphisms with two-dimensional attractors and repellers

Abstract

In this paper, we consider a class of A-diffeomorphisms given on a 3-manifold, assuming that all the basic sets of the diffeomorphisms are two dimensional. It is known that such basic sets are either attractors or repellers and they are two types only, surface or expanding (contracting). One of the results of the paper is the proof that different types of two-dimensional basic sets do not coexist in the non-wandering set of the same 3-diffeomorphism. Also, the existence of an energy function is constructively proved for systems of the class under consideration. It is illustrated by examples that the two-dimensionality of the basic sets is essential in this matter and a decrease in the dimension can lead to the absence of the energy function for a diffeomorphism.

Keywords:

• Hyperbolicity
• energy function
• axiom A
• A-diffeomorphism
• expanding attractor

2020 Mathematics Subject Classification:

• 37D20

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1 A fact of the existence of a Lyapunov function for a wider class of dynamical systems is called the Fundamental Theorem of Dynamical Systems.

2 Let $Diff(M^n)$ is a space of diffeomorphisms given on a manifold $M^n$. A diffeomorphism $f\in Diff(M^n)$ is called Ω-stable if there exists $\epsilon >0$ such that every diffeomorphism $g\in Diff(M^n)$ such that $||g-f|{|}_{C^1}<\epsilon$ possesses the property $g{|}_{NW(g)}$ and $f{|}_{NW(g)}$ are topologically conjugated.

Additional information

Funding

The paper is supported by Russian Science Foundation (Grant No. 21-11-00010), except Sections 3 and 4 devoted to the existence of an energy function for diffeomorphisms under consideration, which was supported by the Laboratory of Dynamical Systems and Applications NRU HSE, grant of the Ministry of Science and Higher Education of the Russian Federation (ag. 075-15-2022-1101).