Marcin Kulczycki
Reymonta 4, 30059 Krak´ow, Poland
Institute of Mathematics, Jagiellonian University
Publications:
Kulczycki M.
A Unified Approach to Theories of Shadowing
2014, vol. 19, no. 3, pp. 310317
Abstract
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 selfmaps of $X$, continuous surjections of $X$ onto itself, and selfhomeomorphisms 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 selfmaps 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. 
Kulczycki M.
Noncontinuous Maps and Devaney’s Chaos
2008, vol. 13, no. 2, pp. 8184
Abstract
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.
