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1.
Weak forms of shadowing in topological dynamics
Sergey Kryzhevich, Danila Cherkashin, 2017, original scientific article

Abstract: We consider continuous maps of compact metric spaces. It is proved that every pseudotrajectory with sufficiently small errors contains a subsequence of positive density that is point-wise close to a subsequence of an exact trajectory with the same indices. Also, we study homeomorphisms such that any pseudotrajectory can be shadowed by a finite number of exact orbits. In terms of numerical methods this property (we call it multishadowing) implies possibility to calculate minimal points of the dynamical system. We prove that for the non-wandering case multishadowing is equivalent to density of minimal points. Moreover, it is equivalent to existence of a family of $\varepsilon$-networks ($\varepsilon > 0$) whose iterations are also $\varepsilon$-networks. Relations between multishadowing and some ergodic and topological properties of dynamical systems are discussed.
Found in: ključnih besedah
Keywords: Topological dynamics, minimal points, invariant measure, shadowing, chain recurrence, $\varepsilon$-networks, syndetic sets
Published: 27.07.2017; Views: 3069; Downloads: 0
.pdf Fulltext (499,38 KB)

2.
Sets of Invariant Measures and Cesaro Stability
Sergey Kryzhevich, 2017, original scientific article

Abstract: We take a space X of dynamical systems that could be: homeomorphisms or continuous maps of a compact metric space K or diffeomorphisms of a smooth manifold or actions of an amenable group. We demonstrate that a typical dynamical system of X is a continuity point for the set of probability invariant measures considered as a function of a map, let Y be the set of all such continuity points. As a corollary we prove that for typical dynamical systems average values of continuous functions calculated along trajectories do not drastically change if the system is perturbed.
Found in: ključnih besedah
Summary of found: ...ergodic theory, invariant measures, shadowing, stability, tolerance stability, topological dynamics...
Keywords: ergodic theory, invariant measures, shadowing, stability, tolerance stability, topological dynamics
Published: 02.10.2017; Views: 2460; Downloads: 0
.pdf Fulltext (529,51 KB)

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