Ramsey Shadowing and minimal points

Brian, W.

doi:10.1090/proc/12884

Ramsey Shadowing and minimal points
HTML articles powered by AMS MathViewer

by W. R. Brian

Proc. Amer. Math. Soc. 144 (2016), 2697-2703

DOI: https://doi.org/10.1090/proc/12884

Published electronically: October 5, 2015

PDF | Request permission

Abstract:

We say that a dynamical system $X$ has the Ramsey shadowing property if an arbitrary sequence of points in $X$ can be shadowed on a set that is “large” in the sense of Ramsey theory. Our main theorem states that this property is equivalent to the existence of a dense set of minimal points.

References

Dawoud Ahmadi Dastjerdi and Maryam Hosseini, Sub-shadowings, Nonlinear Anal. 72 (2010), no. 9-10, 3759–3766. MR 2606819, DOI 10.1016/j.na.2010.01.014
Ethan Akin, Recurrence in topological dynamics, The University Series in Mathematics, Plenum Press, New York, 1997. Furstenberg families and Ellis actions. MR 1467479, DOI 10.1007/978-1-4757-2668-8
Joseph Auslander, On the proximal relation in topological dynamics, Proc. Amer. Math. Soc. 11 (1960), 890–895. MR 164335, DOI 10.1090/S0002-9939-1960-0164335-7
Vitaly Bergelson, Minimal idempotents and ergodic Ramsey theory, Topics in dynamics and ergodic theory, London Math. Soc. Lecture Note Ser., vol. 310, Cambridge Univ. Press, Cambridge, 2003, pp. 8–39. MR 2052273, DOI 10.1017/CBO9780511546716.004
Andreas Blass, Ultrafilters: where topological dynamics $=$ algebra $=$ combinatorics, Topology Proc. 18 (1993), 33–56. MR 1305122
W. R. Brian, J. Meddaugh, and B. E. Raines, “Chain transitivity and variations of the shadowing property,” to appear in Ergodic Theory and Dynamical Systems (2014).
S. Glasner, Divisible properties and the Stone-Čech compactification, Canadian J. Math. 32 (1980), no. 4, 993–1007. MR 590662, DOI 10.4153/CJM-1980-077-x
Neil Hindman and Dona Strauss, Algebra in the Stone-Čech compactification, De Gruyter Expositions in Mathematics, vol. 27, Walter de Gruyter & Co., Berlin, 1998. Theory and applications. MR 1642231, DOI 10.1515/9783110809220
Jan van Mill, An introduction to $\beta \omega$, Handbook of set-theoretic topology, North-Holland, Amsterdam, 1984, pp. 503–567. MR 776630
Piotr Kościelniak, Marcin Mazur, Piotr Oprocha, and PawełPilarczyk, Shadowing is generic—a continuous map case, Discrete Contin. Dyn. Syst. 34 (2014), no. 9, 3591–3609. MR 3190995, DOI 10.3934/dcds.2014.34.3591
Marcin Kulczycki, Dominik Kwietniak, and Piotr Oprocha, On almost specification and average shadowing properties, Fund. Math. 224 (2014), no. 3, 241–278. MR 3194417, DOI 10.4064/fm224-3-4
Marcin Mazur and Piotr Oprocha, S-limit shadowing is $\scr {C}^0$-dense, J. Math. Anal. Appl. 408 (2013), no. 2, 465–475. MR 3085044, DOI 10.1016/j.jmaa.2013.06.004
Brett Stanley, Bounded density shifts, Ergodic Theory Dynam. Systems 33 (2013), no. 6, 1891–1928. MR 3122156, DOI 10.1017/etds.2013.38