On the dynamics of subcontinua of a tree

Abstract

Given a tree map , we study the dynamics of subcontinua of T under action of f. In particular, we prove that a subcontinuum of T is asymptotically periodic or asymptotically degenerate. As an application of this result, we show that zero topological entropy of the system (T, f) implies zero topological entropy of its functional envelope (endowed with the Hausdorff metric).

Keywords:

• dynamics of sets
• tree map
• functional envelope
• connected envelope
• topological entropy

Keywords:

• 37B45
• 37E25
• 37B40

Acknowledgements

The kind hospitality of the Institute of Mathematics of PAN and the Banach Centre is highly appreciated. The author thanks the referees for many useful corrections, remarks and suggestions. The paper was essentially written during the participation of the author in CODY Autumn in Warsaw '10.