Some consequences of the shadowing property in low dimensions (Q2925265)

Let \((X, d)\) be a metric space and \(f : X \to X\) a homeomorphism. For \(\delta > 0\), a \(\delta\)-pseudo-orbit of \(f\) is a sequence \(\{ x_n \}_{n \in \mathbb{Z}}\) such that \(d(f(x_n), x_{n+1}) < \delta\) for all \(n \in \mathbb{Z}\). For \(\epsilon > 0\), we say that the orbit of \(x\) \(\epsilon\)-shadows the given pseudo-orbit if \(d(f^n(x), x_n) < \epsilon\) for all \(n \in \mathbb{Z}\). We say that \(f\) has the shadowing property if for each \(\epsilon > 0\), there is \(\delta > 0\) such that every \(\delta\)-pseudo-orbit of \(f\) is \(\epsilon\)-shadowed by an orbit of \(f\). Note that we do not assume uniqueness of the shadowing orbit. In this paper, lots of interesting dynamical consequences of the shadowing property for surfaces and one-dimensional dynamics are obtained. In dimension two, the authors show that the shadowing property for a homeomorphism implies the existence of periodic orbits in every \(\epsilon\)-transitive class, and in contrast with the result, an example of a \(C^\infty\) Kupka-Smale diffeomorphism with the shadowing property exhibiting an aperiodic transitive class is provided. Finally, the authors consider the case of transitive endomorphisms of the circle, and it is proved that the \(\alpha\)-Hölder shadowing property with \(\alpha > \frac{1}{2}\) implies that the system is conjugate to an expanding map. To be more precise, let us list below all of the results proved by this paper.NEWLINENEWLINETheorem 1.1. Let \(S\) be a compact orientable surface, and let \(f : S \to S\) be a homeomorphism with the shadowing property. Then, for any given \(\epsilon > 0\), each \(\epsilon\)-transitive component has a periodic point.NEWLINENEWLINECorollary 1.2. If a homeomorphism of a compact surface has the shadowing property, then it has a periodic point.NEWLINENEWLINECorollary 1.3. Let \(S\) be a compact orientable surface, and let \(f : S \to S\) be a homeomorphism with the shadowing property. Then \(\overline{\text{Per}(f)}\) intersects every chain transitive class.NEWLINENEWLINETheorem 1.4. Let \(S\) be a compact orientable surface, and let \(f : S \to S\) be a Kupka-Smale diffeomorphism with the shadowing property. If there are only finitely many periodic points, then \(f\) is Morse-Smale.NEWLINENEWLINETheorem 1.5. In any compact surface \(S\), there exists a Kupka-Smale \(C^\infty\)-diffeomorphism \(f : S \to S\) with the shadowing property which has an aperiodic chain transitive component. More precisely, it has a component which is an invariant circle supporting an irrational rotation.NEWLINENEWLINETheorem 1.6. Let \(f\) be a \(C^2\) endomorphism of the circle with finitely many turning points. Suppose that \(f\) is transitive and satisfies the \(\alpha\)-Hölder shadowing property with \(\alpha > \frac{1}{2}\). Then \(f\) is conjugate to a linear expanding endomorphism.NEWLINENEWLINETheorem 1.7. Let \(f\) be a \(C^2\) orientation-preserving endomorphism of the circle with finitely many turning points. Suppose that \(f\) satisfies the \(\alpha\)-Hölder shadowing property with \(\alpha > \frac{1}{2}\). If \(f\) is \(C^r\)-robustly transitive, \(r \geq 1\), then \(f\) is an expanding endomorphism.NEWLINENEWLINEThe techniques used in this paper are explained in the text. To obtain Theorem 1.1, the authors use Conley's theory combined with a Lefschetz index argument to reduce the problem to one in the annulus or the torus. To do this, they prove a result about aperiodic \(\epsilon\)-transitive components that is unrelated to the shadowing property and may be of interest in itself. In that setting, they apply Brouwer's theory for plane homeomorphisms (which is strictly two-dimensional) to obtain the required periodic points. To prove Theorem 1.5, the authors use a construction in the annulus with a special kind of hyperbolic sets, called ``crooked horseshoes'', accumulating on an irrational rotation on the circle (with Liouville rotation number). The shadowing property is obtained for points far from the rotation due to the hyperbolicity of the system outside a neighborhood of the rotation, and near the rotation the shadowing comes from the crooked horseshoes. The main technical difficulty for this construction is to obtain a hyperbolic system arbitrarily close to the identity and with a power exhibiting a crooked horseshoe. To prove Theorem 1.6 the authors use the shadowing property to obtain a small interval containing a turning point, such that some forward iterate of the interval also contains a turning point. Assuming that the shadowing is Hölder with \(\alpha > \frac{1}{2}\), it is concluded that some forward iterate of the interval has to be contained inside the initial interval, contradicting the transitivity. Once turning points are discarded, Theorem 1.7 is proved using the fact that the dynamics preserves orientation and recurrent points can be close to periodic points by composing with a translation.