On various notions of the shadowing property in non-compact spaces.


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We discuss various notions of the shadowing property on non-compact spaces. In the first part, we discuss the shadowing property acting on general sequence spaces. We develop criteria upon the weights of the space in which shadowing of particular types of orbits occurs. Then we move on to operators acting on Fréchet spaces, and show that an system exhibits the shadowing property if its Waelbroeck spectrum misses the unit circle. Additionally, we discuss the non-uniform pseudo-orbit tracing property, a variant of the shadowing property which allows for different error tolerances depending on where a point lies in the space.



Shadowing. Pseudo-orbit tracing property.