Ultrafilters and Ramsey-type shadowing phenomena in topological dynamics (Q1617914)

Some variants of the Ramsey shadowing property are being explored. The following are the main results in this paper. \par Theorem 1. A dynamical system $(X,f)$ has anchored Ramsey shadowing iff it is distal. \par Theorem 2. A dynamical system $(X,f)$ has asymptotic Ramsey shadowing iff $X$ is finite and $f$ is a permutation of $X$. \par Theorem 3. If ${\mathcal F}$ is a $P$-family, then anchored ${\mathcal F}$-Ramsey shadowing is equivalent with asymptotic ${\mathcal F}$-Ramsey shadowing. \par Theorem 4. A dynamical system has ${\mathcal F}_\infty$-Ramsey shadowing iff for every finite coloring of $N$, there exists a monochromatic $A\subseteq N$ with dense limits. \par Theorem 5. If the dynamical system $(X,f)$ has ${\mathcal F}_\infty$-Ramsey shadowing then it has a dense set of recurrent points. \par Theorem 6. If the dynamical system $(X,f)$ has asymptotic shadowing and is chain recurrent, then it has asymptotic ${\mathcal F}_\infty$-Ramsey shadowing. \par Further aspects deriving from these results are also discussed.