Abstract
The first problem which arises when studying a topological classification of A-homeomorphisms with nontrivial basic sets is the problem to find a suitable topological invariant which can adequately reflect the restriction of the diffeomorphism to the basic set as well as the embedding of the basic sets to the ambient manifold. Let \(f,~f'\) be orientation preserving A-diffeomorphisms of orientable compact manifolds \(M^n,~{M}^{\prime n}\) (possibly with boundary) respectively and let \(f,~f'\) have nontrivial basic sets \(\varLambda ,~\varLambda '\) respectively which are in the interior of the manifold. The problem is to find necessary and sufficient conditions of existence of a homeomorphism \(h:M^n\rightarrow M^{\prime n}\) such that \(hf|_{\varLambda }=f'h|_{\varLambda }\). We show that the problem of topological conjugacy for arbitrary 2-dimensional and 1-dimensional basic sets as well as 0-dimensional basic sets without pairs of conjugated points can be reduced to the analogues problem for widely situated basic sets on an orientable surface with boundary (possibly empty) which is called a canonical support of the basic set. The study of the latter is essentially based on the investigation of the asymptotic properties of the stable and unstable manifolds of the points of the basic sets on the universal cover. The suggested method solves the problem of realization of arbitrary 1-dimensional basic sets as well as 0-dimensional basic sets without pair of conjugated points by constructing a so called hyperbolic diffeomorphism on the support of the basic set such that it has an invariant locally maximal set on the intersection of two geodesic laminations. We show that the restriction of the diffeomorphism f to the basic set is a factor of the restriction of the hyperbolic diffeomorphism to the intersection of the respective laminations. If the basic set is widely situated on the torus then for it a hyperbolic automorphism of the torus (Anosov diffeomorphism) is uniquely defined and it is a factor of the initial diffeomorphism. If the diffeomorphism f is structurally stable then the support of the basic set can be constructed in such a way that the restriction of the initial diffeomorphism to it consists of exactly one nontrivial basic set and of finitely many hyperbolic periodic points belonging to the boundary of the support. This and the results on the classification of the Morse-Smale diffeomorphisms enable us to construct a complete topological invariant for important classes of structurally stable diffeomorphisms on surfaces whose non-wandering sets contain a nontrivial 1-dimensional basic set (attractor or repeller). Such a class was studied for instance in [8]. The presentation of the results in this chapter follows the papers [2, 6–12, 16, 17] and it is in many concepts related to the papers [1, 4, 5, 14, 15].
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- 1.
If the restrictions of the A-diffeomorphisms f and \(f'\) to the basic sets \(\varLambda \) and \(\varLambda '\), consisting of the same number of the periodic components \(\varLambda _1,\dots ,\varLambda _m\) and \(\varLambda '_1,\dots ,\varLambda '_m\), are topologically conjugate by a homeomorphism h then there is \(i\in \{1,\dots ,m\}\) such that \(h(\varLambda _1)=\varLambda '_i\). Then the homeomorphism \(\tilde{h}:M^n\rightarrow M^{\prime n}\) defined by \(\tilde{h}=\left\{ \begin{array}{ll} h,&{} i=1,\\ (f')^{m-i+1}h,&{}i\ne 1.\end{array}\right. \) is the topological conjugacy of the diffeomorphisms \(f^m|_{\varLambda _1}\) and \((f')^m|_{\varLambda '_1}\). Conversely, if \(\tilde{h}:M^n\rightarrow M^{\prime n}\) is a topological conjugacy between \(f^m|_{\varLambda _1}\) and \((f')^m|_{\varLambda '_1}\) then the homeomorphism \(h:M^n\rightarrow M^{\prime n}\) conjugating \(f|_{\varLambda }\) and \({f'}|_{\varLambda '}\) is defined by \(h(x)=(f')^{i-1}(\tilde{h}(f^{1-i}(x))),~x\in \varLambda _i,~i\in \{1,\dots ,m\}\).
- 2.
Dedekind cut is a partition of the real numbers (or only the rational numbers) R into two nonempty sets A and B, \(A\cup B=R\) such that for every \(a\in A\) and \(b\in B\) the inequality \(a<b\) holds. The Dedekind cut is denoted by (A, B). It is known that for each Dedekind cut (A, B) in the set of rational numbers there are three possibilities:
1) the class A has the maximal element r (then the class B has no minimal element); 2) the class B has the minimal element r (then the class A has no maximal element); 3) neither A has the maximal element nor B has the minimal element.
In the cases 1) and 2) the section (A, B) is said to define the rational number r; in the case 3) the section (A, B) is said to define the irrational number.
- 3.
By [13] any foliation without singularities on the Klein bottle must have at least one closed leaf. Therefore the Klein bottle does not admit Anosov diffeomorphisms.
- 4.
A saddle (source, sink) point on the boundary of a manifold is meant to be a point \(x\in \partial N_{\varLambda }\) for which there is a chart \(\psi :U\rightarrow \mathbb R^2_+\) such that \(\psi \) conjugates the diffeomorphism \(f^{per(x)}_{\varLambda }|_{U}\) to the restriction of the diffeomorphism \(g(x,y)=(\frac{1}{2} x,2y)\) (\(a(x,y)=(2x,2y)\), \(a^{-1}(x,y)=(2x,2y)\)) on \(\mathbb R^2_+\).
- 5.
If \(w^u_{\bar{x}}\) contains the point \(\bar{p}\in \bar{P}_\varLambda \) it is possible that there is no such element \(\gamma \). Then instead of the curve \(w^u_{\bar{x}}\) the same reasoning applies to the curve \(w^u_{\bar{q}},~\bar{q}\in \bar{P}_\varLambda \), for which \(\mu \) is one of its boundary points according to item (4).
- 6.
We say a periodic point \(x\in X\) of a homeomorphism \(\varphi :X\rightarrow X\) to be attracting (repelling), if there is a neighborhood \(U(x)\subset X\) of x such that for any point \(y\in U(x)\) the sequence \(\varphi ^{m\cdot per(x)}(y)~(\varphi ^{-m\cdot per(x)}(y))\) converges to x as \(m\rightarrow +\infty \).
- 7.
According to [3] every transformation \(\psi :\mathbb {T}^n\rightarrow \mathbb {T}^n\) which conjugates hyperbolic automorphisms of the torus \(\mathbb T^n\) is linear, that is it can be represented as a composition of an algebraic automorphism and a group shift of the torus.
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Grines, V.Z., Medvedev, T.V., Pochinka, O.V. (2016). The Classification of Nontrivial Basic Sets of A-Diffeomorphisms of Surfaces. In: Dynamical Systems on 2- and 3-Manifolds. Developments in Mathematics, vol 46. Springer, Cham. https://doi.org/10.1007/978-3-319-44847-3_9
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