Hostname: page-component-cd9895bd7-p9bg8 Total loading time: 0 Render date: 2024-12-26T05:22:54.999Z Has data issue: false hasContentIssue false
  • English
  • Français

Chain transitivity and variations of the shadowing property

Published online by Cambridge University Press:  03 July 2014

WILLIAM R. BRIAN ,
JONATHAN MEDDAUGH  and
BRIAN E. RAINES
WILLIAM R. BRIAN
Affiliation:
Department of Mathematics, Tulane University, 6823 St. Charles Ave, New Orleans, LA 70118, USA email wbrian.math@gmail.com
JONATHAN MEDDAUGH
Affiliation:
Department of Mathematics, Baylor University, Waco, TX 76798-7328, USA email jonathan˙meddaugh@baylor.edu, brian˙raines@baylor.edu
BRIAN E. RAINES
Affiliation:
Department of Mathematics, Baylor University, Waco, TX 76798-7328, USA email jonathan˙meddaugh@baylor.edu, brian˙raines@baylor.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

We show that, under the assumption of chain transitivity, the shadowing property is equivalent to the thick shadowing property. We also show that, if ${\mathcal{F}}$ is a family with the Ramsey property, then an arbitrary sequence of points in a chain transitive space can be ${\it\varepsilon}$-shadowed (for any ${\it\varepsilon}$) on a set in ${\mathcal{F}}$.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 35 , Issue 7 , October 2015 , pp. 2044 - 2052
Copyright
© Cambridge University Press, 2014 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Dastjerdi, D. A. and Hosseini, M.. Sub-shadowings. Nonlinear Anal. 72 (2010), 3759–3766.CrossRefGoogle Scholar
Akin, E.. The General Topology of Dynamical Systems (Graduate Studies in Mathematics, 1). American Mathematical Society, Providence, RI, 1993.Google Scholar
Aoki, N. and Hiraide, K.. Topological Theory of Dynamical Systems (North-Holland Math. Library, 52). North-Holland, Amsterdam, 1994.Google Scholar
Bergelson, V. and Hindman, N.. Partition regular structures contained in large sets are abundant. J. Comb. Theory (Series A) 93 (2001), 18–36.CrossRefGoogle Scholar
Blank, M. L.. Metric properties of 𝜀-trajectory of dynamical systems with stochastic behavior. Ergod. Th. & Dynam. Sys. 8 (1988), 365–378.CrossRefGoogle Scholar
Bowen, R.. Equilibrium States and Ergodic Theory of Anosov Diffeomorphisms (Lecture Notes in Mathematics, 470). Springer, Berlin, 1975.CrossRefGoogle Scholar
Fakhari, A. and Gane, F. H.. On shadowing: ordinary and ergodic. J. Math. Anal. Appl. 364 (2010), 151–155.CrossRefGoogle Scholar
Hindman, N.. Finite sums from sequences within cells of a partition of N. J. Combin. Theory (Series A) 17 (1974), 1–11.CrossRefGoogle Scholar
Oprocha, P., Dastjerdi, D. A. and Hosseini, M.. On partial shadowing of complete pseudo-orbits. J. Math. Anal. Appl. 411 (2014), 454–463.CrossRefGoogle Scholar
Alfonsín, J. L. R.. The Diophantine Frobenius Problem (Oxford Lecture Series in Mathematics and its Applications, 30). Oxford University Press, Oxford, 2005.CrossRefGoogle Scholar
Richeson, D. and Wiseman, J.. Chain recurrence rates and topological entropy. Topology Appl. 156(2) (2008), 251–261.CrossRefGoogle Scholar
Sakai, K.. Various shadowing properties for positively expansive maps. Topology Appl. 131(1) (2003), 15–31.CrossRefGoogle Scholar