Dynamical Systems on 2- and 3-Manifolds pp 167-216 | Cite as

# The Classification of Nontrivial Basic Sets of *A*-Diffeomorphisms of Surfaces

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## 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, 7, 8, 9, 10, 11, 12, 16, 17] and it is in many concepts related to the papers [1, 4, 5, 14, 15].

## Keywords

Nontrivial Basic Set Topological Conjugacy Canonical Support Anosov Diffeomorphisms Conjugate Points## References

- 1.Anosov, D.: About one class of invariant sets of smooth dynamical systems. Proc. Int. Conf. Non-linear Oscil.
**2**, 39–45 (1970)Google Scholar - 2.Aranson, S., Grines, V.: Dynamical systems with minimal entropy on two-dimensional manifolds. Selecta Math. Sovi
**2**(2), 123–158 (1992)MathSciNetzbMATHGoogle Scholar - 3.Arov, D.: On the topological similarity of automorphisms and translations of compact commutative groups. Uspekhi Mat. Nauk
**18**(5(113)), 133–138 (1963)Google Scholar - 4.Bowen, R.: Periodic points and measures for Axiom A diffeomorphisms. Trans. Am. Math. Soc. pp. 377–397 (1971)Google Scholar
- 5.Franks, J.: Anosov diffeomorphisms. Proc. Sympos. Pure Math.
**14**, 61–94 (1970)MathSciNetCrossRefzbMATHGoogle Scholar - 6.Grines, V.: The topological conjugacy of diffeomorphisms of a two-dimensional manifold on one-dimensional orientable basic sets. I. Tr. Mosk. Mat. O.-va
**32**, 35–60 (1975)Google Scholar - 7.Grines, V.: On the topological conjugacy of diffeomorphisms of a two-dimensional manifold on one-dimensional orientable basic sets. II. Tr. Mosk. Mat. O.-va
**34**, 243–252 (1977)Google Scholar - 8.Grines, V.: On the topological classification of structurally stable diffeomorphisms of surfaces with one-dimensional attractors and repellers. Sb. Math.
**188**(4), 537–569 (1997). doi: 10.1070/SM1997v188n04ABEH000216 MathSciNetCrossRefzbMATHGoogle Scholar - 9.Grines, V.: A representation of one-dimensional attractors of \(A\)-diffeomorphisms by hyperbolic homeomorphisms. Math. Notes
**62**(1), 64–73 (1997). doi: 10.1007/BF02356065 MathSciNetCrossRefzbMATHGoogle Scholar - 10.Grines, V.: Topological classification of one-dimensional attractors and repellers of A-diffeomorphisms of surfaces by means of automorphisms of fundamental groups of supports. J. Math. Sci. (N.Y.)
**95**(5), 2523–2545 (1999)Google Scholar - 11.Grines, V.: On topological classification of A-diffeomorphisms of surfaces. J. Dyn. Control Syst.
**6**(1), 97–126 (2000)MathSciNetCrossRefzbMATHGoogle Scholar - 12.Grines, V., Kalai, K.K.: On topological equivalence of diffeomorphisms with nontrivial basic sets on two-dimensional manifolds. In: Methods of the Qualitative Theory of Differential Equations, Gorky pp. 40–49 (1988). (Russian)Google Scholar
- 13.Kneser, H.: Reguläre Kurvenscharen auf den Ringflächen. Math. Ann.
**91**(1), 135–154 (1924)MathSciNetCrossRefzbMATHGoogle Scholar - 14.Manning, A.: There are no new Anosov diffeomorphisms on tori. Am. J. Math.
**96**, 422–429 (1974)MathSciNetCrossRefzbMATHGoogle Scholar - 15.Newhouse, S.E.: On codimension one Anosov diffeomorphisms. Am. J. Math.
**92**(3), 761–770 (1970)MathSciNetCrossRefzbMATHGoogle Scholar - 16.Plykin, R.: Sources and sinks of A-diffeomorphisms of surfaces. Math. USSR, Sbornik
**23**, 233–253 (1975). doi: 10.1070/SM1974v023n02ABEH001719 CrossRefzbMATHGoogle Scholar - 17.Plykin, R.: On the geometry of hyperbolic attractors of smooth cascades. Russian Math. Surv.
**39**(6), 85–131 (1984). doi: 10.1070/RM1984v039n06ABEH003182 Google Scholar