Stability of chain recurrent points of continuous maps on interval

Abstract

Recently, J. Smítal and T.H. Steele [J. Difference Equ. Appl. 15 (2009), 77–86] considered the mappings C and NW, which take a function to its centre , and to its set of non-wandering points , respectively. They characterize a function at which maps C and NW are continuous, and put a problem whether the same characterization is true for the map CR, which takes an to its set of chain recurrent points . We show that the answer is positive.

Keywords:

• chain recurrent points
• Hausdorff metric
• non-wandering points
• omega-limit points
• stability under perturbation

Keywords:

• Primary 37B20
• 37E05
• 26A18
• 54H20

Acknowledgements

The research was supported, in part, by project MSM 4781305904 from the Czech Ministry of Education. The author would like to thank her supervisor Professor Marta Štefánková for fruitful discussions, Professor James A. Yorke for taking her attention to [4], and the referees for valuable comments.