Topological horseshoes

Abstract

When does a continuous map have chaotic dynamics in a set Q Q ? More specifically, when does it factor over a shift on M M symbols? This paper is an attempt to clarify some of the issues when there is no hyperbolicity assumed. We find that the key is to define a “crossing number” for that set Q Q . If that number is M M and M > 1 M>1 , then Q Q contains a compact invariant set which factors over a shift on M M symbols.