Topological chaos: what may this mean?

Abstract

We confront existing definitions of chaos with the state of the art in topological dynamics. The article does not propose any new definition of chaos but, starting from several topological properties that can be reasonably called chaotic, tries to sketch a theoretical view of chaos. Among the main ideas in this article are the distinction between overall chaos and partial chaos, and the fact that some dynamical properties may be considered more chaotic than others.

Keywords:

• chaos
• order
• partial chaos
• scrambled set
• sensitivity
• weak mixing

2000 Mathematics Subject Classification:

• Primary 37B05
• Secondary 54H20

Acknowledgements

Wen Huang allowed me to include in this article Example 1, which we found during a discussion: I want to express my gratitude to him. I want to thank the organizers of the Visegrad conference on dynamical systems (Štrbské Pleso, June 2007) at which this example was found and several changes in this article were suggested to me when listening to talks and discussing with participants. I am also grateful to the referee of JDEA, who in his/her turn proposed Example 1. Finally, I would like to thank S. Galatolo for sending me his article [26].