Inverse shadowing and related measures. (English) Zbl 1451.37030
The authors introduce weaker forms of inverse shadowing and investigate inverse shadowing
“almost always” – the so-called ergodic inverse shadowing (EIS) property. They also introduce a nonuniform version of inverse shadowing, the so-called individual inverse shadowing.
Let \(CM\) be the class of all the mappings with weakly continuous sets of Borel probability invariant measures. The authors prove that if \(f\in \mathrm{EIS}\), then \(f\in CM\). They study the sets of diffeomorphisms of a closed smooth manifold \(M\) having the individual inverse shadowing property on various subsets of the phase space. For a set \(A\) of diffeomorphisms, denote by Int\(^1(A)\) its interior with respect to the \(C^1\)-topology. It is proved that the set Int\(^1(\mathrm{IIS})\) coincides with the set of structurally stable diffeomorphisms, where \(\mathrm{IIS}\) is individual inverse shadowing property. Denote by \(\mathcal{I}_3\) the set of diffeomorphisms \(f\) of a smooth closed manifold \(M\). The authors prove that the set Int\(^1(\mathcal{I}_3)\) coincides with the set of \(\Omega\)-stable diffeomorphisms.
Let \(CM\) be the class of all the mappings with weakly continuous sets of Borel probability invariant measures. The authors prove that if \(f\in \mathrm{EIS}\), then \(f\in CM\). They study the sets of diffeomorphisms of a closed smooth manifold \(M\) having the individual inverse shadowing property on various subsets of the phase space. For a set \(A\) of diffeomorphisms, denote by Int\(^1(A)\) its interior with respect to the \(C^1\)-topology. It is proved that the set Int\(^1(\mathrm{IIS})\) coincides with the set of structurally stable diffeomorphisms, where \(\mathrm{IIS}\) is individual inverse shadowing property. Denote by \(\mathcal{I}_3\) the set of diffeomorphisms \(f\) of a smooth closed manifold \(M\). The authors prove that the set Int\(^1(\mathcal{I}_3)\) coincides with the set of \(\Omega\)-stable diffeomorphisms.
Reviewer: Hasan Akin (Gaziantep)
MSC:
| 37C50 | Approximate trajectories (pseudotrajectories, shadowing, etc.) in smooth dynamics |
| 37C20 | Generic properties, structural stability of dynamical systems |
| 37C05 | Dynamical systems involving smooth mappings and diffeomorphisms |
| 37B65 | Approximate trajectories, pseudotrajectories, shadowing and related notions for topological dynamical systems |
| 37D05 | Dynamical systems with hyperbolic orbits and sets |
| 37D20 | Uniformly hyperbolic systems (expanding, Anosov, Axiom A, etc.) |
| 37A05 | Dynamical aspects of measure-preserving transformations |
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