Marcin Kulczycki

This paper introduces the notion of a general approximation property, which encompasses many existing types of shadowing.
It is proven that there exists a metric space $X$ such that the sets of maps with many types of general approximation properties (including the classic shadowing, the $\mathcal{L}_p$-shadowing, limit shadowing, and the $s$-limit shadowing) are not dense in $C(X)$, $S(X)$, and $H(X)$ (the space of continuous self-maps of $X$, continuous surjections of $X$ onto itself, and self-homeomorphisms of $X$) and that there exists a manifold M such that the sets of maps with general approximation properties of nonlocal type (including the average shadowing property and the asymptotic average shadowing property) are not dense in $C(M)$, $S(M)$, and $H(M)$. Furthermore, it is proven that the sets of maps with a wide range of general approximation properties (including the classic shadowing, the $\mathcal{L}_p$-shadowing, and the $s$-limit shadowing) are dense in the space of continuous self-maps of the Cantor set.
A condition is given that guarantees transfer of general approximation property from a map on $X$ to the map induced by it on the hyperspace of $X$. It is also proven that the transfer in the opposite direction always takes place.

Vu Dong Tô has proven in [1] that for any mapping $f: X \to X$, where $X$ is a metric space that is not precompact, the third condition in the Devaney’s definition of chaos follows from the first two even if $f$ is not assumed to be continuous. This paper completes this result by analysing the precompact case. We show that if $X$ is either finite or perfect one can always find a map $f: X \to X$ that satisfies the first two conditions of Devaney’s chaos but not the third. Additionally, if $X$ is neither finite nor perfect there is no $f: X \to X$ that would satisfy the first two conditions of Devaney’s chaos at the same time.