Abstract. When does a continuous map have chaotic dynamics in a set Q? More specifically, when does it factor over a shift on 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. If that number is M and M > 1, then Q contains a compact invariant set which factors over a shift on M symbols.
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