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Pseudo-orbit shadowing in the $$C^1$$ topology. (English) Zbl 1132.37014
The problem of shadowing of approximate orbits plays a central role in the numerical simulations of dynamical systems. Refering to discrete dynamical systems of diffeomorphisms, the authors prove that the shadowing property does not hold for diffeomorphisms in an open and dense subset of the set of $$C^1$$-robustely nonhyperbolic transitive diffeomorphism.

MSC:
 37C50 Approximate trajectories (pseudotrajectories, shadowing, etc.) in smooth dynamics 37C20 Generic properties, structural stability of dynamical systems 37C25 Fixed points and periodic points of dynamical systems; fixed-point index theory, local dynamics 37D30 Partially hyperbolic systems and dominated splittings
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