This article is cited in
2 papers
A Unified Approach to Theories of Shadowing
Marcin Kulczycki Institute of Mathematics, Faculty of Mathematics and Computer Science, Jagiellonian University,
ul. Lojasiewicza 6, 30-348 Kraków, Poland
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 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.
Keywords:
shadowing, average shadowing, limit shadowing, pseudo-orbit, chain-transitivity.
MSC: 37B05,
34D05,
37D45 Received: 16.09.2013
Accepted: 13.12.2013
Language: English
DOI:
10.1134/S1560354714030046
References