TY - JOUR

T1 - Inverse shadowing and related measures

AU - Kryzhevich, Sergey G.

AU - Pilyugin, Sergei Yu.

N1 - Publisher Copyright: © 2020, Science China Press and Springer-Verlag GmbH Germany, part of Springer Nature.

PY - 2020/9/1

Y1 - 2020/9/1

N2 - 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 omega-stability of diffeomorphisms.

AB - 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 omega-stability of diffeomorphisms.

KW - inverse shadowing

KW - invariant measures

KW - hyperbolicity

KW - Axiom A

KW - STABILITY

KW - 37C50

KW - invariant measure

KW - axiom A

KW - 37D05

KW - stability

UR - https://www.researchgate.net/publication/334506887_Inverse_shadowing_and_related_measures

UR - http://www.scopus.com/inward/record.url?scp=85088380057&partnerID=8YFLogxK

UR - https://www.mendeley.com/catalogue/290942a4-8cfa-3562-85d0-0ffd6a4927ed/

U2 - 10.1007/s11425-019-1609-8

DO - 10.1007/s11425-019-1609-8

M3 - Article

VL - 63

SP - 1825

EP - 1836

JO - Science China Mathematics

JF - Science China Mathematics

SN - 1674-7283

IS - 9

ER -