Chaotic extensions of continuous maps on compact manifolds

Abstract

Let X be a compact connected Lipschitz manifold, with or without boundary. We show that for every continuous map $f:X\to X$ and every nowhere dense closed subset K of X, there is a topologically transitive continuous map $g:X\to X$ having a dense set of periodic points in X such that ${g|}_K{=f|}_K$. Combined with a deep theorem of Sullivan, our result extends to all compact connected topological manifolds of dimension $\ne 4$.

Keywords:

• Chaotic map
• compact Lipschitz manifold
• extension of continuous maps

AMS Subject Classifications:

• 54H20
• 57N15

Acknowledgements

I thank the referee for various valuable suggestions, and I thank Professor Fredric D. Ancel for some helpful remarks related to Proposition 1.

Notes

No potential conflict of interest was reported by the author.