Dense chaos for continuous interval maps

Abstract

A continuous map $f$ from a compact interval $I$ into itself is densely (resp. generically) chaotic if the set of points $(x,y)$ such that ${\mathrm{limsup}}_{n\to +\infty }|f^n(x)-f^n(y)|>0$ and ${\mathrm{liminf}}_{n\to +\infty }|f^n(x)-f^n(y)|=0$ is dense (resp. residual) in $I\times I$. We prove that if the interval map $f$ is densely but not generically chaotic then there is a descending sequence of invariant intervals, each of which containing a horseshoe for $f^2$. It implies that every densely chaotic interval map is of type at most $6$ for Sharkovsky's order (that is, there exists a periodic point of period $6$), and its topological entropy is at least $\log 2/2$. We show that equalities can be realised.

Published in 2005

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