On the topology of the chain recurrent set of a dynamical system

Seyyed Alireza Ahmadi

Iran, Islamic Republic of

University of Sistan and Baluchestan

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Accepted: 2014-06-05

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Published: 2014-06-12

DOI: https://doi.org/10.4995/agt.2014.3050
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Keywords:

chain recurrent, chain transitive, chain component, inverse limit space.

Supporting agencies:

This research was not funded

Abstract:

n this paper we associate a pseudo-metric to a dynamical system on a compact metric space. We show that this pseudo-metric is identically zero if and only if the system is chain transitive. If we associate this pseudo-metric to the identity map, then we can present a  characterization for connected and totally disconnected metric spaces.
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References:

N. Aoki and K. Hiraide, Topological Theory of Dynamical Systems, Recent Advances. North-Holland Math. Library 52. (North-Holland, Amsterdam 1994)

K. Athanassopoulos, One-dimensional chain recurrent sets of flows in the 2-sphere, Math. Z. 223 (1996), 643-649.

(http://dx.doi.org/10.1007/PL00004279)

F. Balibrea, J. S. Cánovas and A. Linero, New results on topological dynamics of antitriangular maps, App. Gen. Topol. 2 (2001), 51-61.

C. Fujita and H. Kato, Almost periodic points and minimal sets in topological spaces, App. Gen. Topol. 10 (2009), 239-244.

(http://dx.doi.org/10.4995/agt.2009.1737)

D. Richeson and J. Wiseman, Chain recurrence rates and topological entropy, Topology Appl. 156 (2008), 251-261.

(http://dx.doi.org/10.1016/j.topol.2008.07.005)

K. Sakai, $C^1$-stably shadowable chain components, Ergodic Theory Dyn. Syst. 28 (2008), 987-1029.

T. Shimomura, On a structure of discrete dynamical systems from the view point of chain components and some applications, Japan. J. Math. (NS) 15 (1989), 99-126.

X. Wen, S. Gan and L. Wen, $C^1$-stably shadowable chain components are hyperbolic, J. Differ. Equations 246 (2009), 340-357.

(http://dx.doi.org/10.1016/j.jde.2008.03.032)

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