Regularity and Stochasticity of Nonlinear Dynamical Systems by Dimitri Volchenkov & Xavier Leoncini

Author:Dimitri Volchenkov & Xavier Leoncini
Language: eng
Format: epub
Publisher: Springer International Publishing, Cham

Fig. 6.3Types of linear half-orientable horseshoes

On the other hand, if one considers endomorphisms instead of diffeomorphisms, or if one considers diffeomorphisms on non-orientable manifolds, then, as has been shown in [1], there exist infinitely many types of horseshoes; more precisely, almost all of them (with the exception of two types: orientable and non-orientable ones) belong to the class of the so called half-orientable horseshoes.

Note that while studying the (non-trivial) hyperbolic basic sets of two-dimensional maps, one can get very important information from the structure of the set of boundary points, see Definition 6.2. It is known (see [3]) that in two-dimensional case, the boundary points are periodic, and their stable and/or unstable manifold form the natural invariant border of the hyperbolic set , so that from one side of this manifold, there are no points of this set, while from the other side of the manifold there are such points.

In the present chapter, we study the problem on topological classification of Smale horseshoes in terms of the boundary points. As mentioned above, linear horseshoes will be represented by 10 different types (their boundary points are described in Proposition 6.1), and, in contrast, for half-orientable hyperbolic horseshoes, the classification provides countably many types.

Unlike the well-known orientable and non-orientable Smale horseshoes (see Fig. 6.2), half-orientable horseshoes are not so popular (some examples of linear half-orientable horseshoes are shown in Fig. 6.3, and of nonlinear ones in Figs. 6.6 and 6.11). It seems that their study begins from [1], in which paper such horseshoes were discovered for GHM.

The structure of the chapter is as follows. In Sect. 6.1, we consider mainly linear Smale horseshoes and classify them in terms of the type of their boundary periodic points (see Statement 1). Main attention in Sect. 6.2 is paid to the problems of hyperbolic of GHM (6.1). We show that dynamical behavior of GHM is related to existence of both the usual orientable/non-orientable horseshoes and also of half-orientable ones. The regions where such horseshoes exist are described in Theorem 6.1 (we cite the result from [1]), and in Lemma 6.1. Then in Sect. 6.2, we prove Theorem 6.2 on existence of countably many different types (with respect to the local topological conjugacy) half-orientable horseshoes. This fact is known from [1], but we give here totally different proof, which is geometric and constructive. By using similar geometric construction, we prove Theorem 6.3 on existence of boundary points of arbitrary (given) period. Since the proof of Theorem 6.3 is constructive, it is actually a realization theorem.

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