Dense chaos for continuous interval maps

Nonlinearity, 18, 1691-1698, 2005.


A continuous map f from a compact interval I into itself is densely chaotic if the set of points (x,y) such that $\limsup_{n\to+\infty}|f^n(x)-f^n(y)|>0$ and $\liminf_{n\to+\infty}|f^n(x)-f^n(y)|=0$ is dense in I 2. We show that if f is a densely chaotic interval map then f 2 has a horseshoe, which implies that its topological entropy is at least log 2/2 and f is of type at most 6 for Sharkovskii's order (that is, there exists a periodic point of period 6).

Paper: [arXiv:1901.01064] [pdf (published paper)]