Inverse shadowing and related measures
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https://doi.org/10.1007/s11425-019-1609-8
Inverse shadowing and related measures Sergey G. Kryzhevich1,∗ & Sergei Yu. Pilyugin2 1Department
of Mathematical Physics, Saint Petersburg State University, Saint Petersburg 199034, Russia; 2Department of Mathematics and Computer Science, Saint Petersburg State University, Saint Petersburg 199034, Russia Email: [email protected], [email protected] Received July 18, 2019; accepted October 7, 2019
Abstract
We study various weaker forms of the inverse shadowing property for discrete dynamical systems
on a smooth compact manifold. First, we introduce the so-called ergodic inverse shadowing property (Birkhoff averages of continuous functions along an exact trajectory and the approximating one are close). We demonstrate that this property implies the continuity of the set of invariant measures in the Hausdorff metric. We show that the class of systems with ergodic inverse shadowing is quite broad; it includes all diffeomorphisms with hyperbolic nonwandering sets. Second, we study the so-called individual inverse shadowing (any exact trajectory can be traced by approximate ones, but this shadowing is not uniform with respect to the initial point of the trajectory). We demonstrate that this property is closely related to structural stability and Ω-stability of diffeomorphisms. Keywords MSC(2010)
inverse shadowing, invariant measure, hyperbolicity, axiom A, stability 37C50, 37D05
Citation: Kryzhevich S G, Pilyugin S Y. Inverse shadowing and related measures. Sci China Math, 2020, 63, https://doi.org/10.1007/s11425-019-1609-8
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Introduction
The theory of shadowing of approximate trajectories (pseudotrajectories) of dynamical systems is now a well-developed field of the global theory of dynamical systems. Let us refer to the monographs [11,12,16] concerning the basics of the modern shadowing theory. In parallel to the study of various (direct) shadowing properties, the so-called inverse shadowing properties were introduced [1, 4] and studied (see, for example, [13, 14]). Recently, several authors studied the sets of shadowable points of dynamical systems in the context of their metric properties (see [9, 10]). Classical shadowing/inverse shadowing properties are closely related to structural stability, and there are many interesting examples of systems without shadowing or inverse shadowing (IS). Here, we introduce weaker forms of inverse shadowing: (i) we study inverse shadowing “almost always”—the so-called ergodic inverse shadowing property; this idea was inspired by the approach of the paper [5] where the so-called ergodic shadowing was introduced; * Corresponding author c Science China Press and Springer-Verlag GmbH Germany, part of Springer Nature 2020 ⃝
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Kryzhevich S G et al.
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(ii) we introduce a “nonuniform” version of inverse shadowing, the so-called individual inverse shadowing. We study several properties of measures related to the introduced forms of shadowing. The rest of this paper is as f
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