# Dense chaos for continuous interval maps

## TL;DR

This paper characterizes densely chaotic continuous interval maps, showing they contain invariant intervals with horseshoes, have periodic points of period 6, and possess a minimum topological entropy, thus advancing the understanding of chaos in interval dynamics.

## Contribution

It proves that densely but not generically chaotic maps have invariant intervals with horseshoes and establishes bounds on their periodic points and entropy.

## Key findings

- Existence of invariant intervals with horseshoes in densely chaotic maps
- Densely chaotic maps have a periodic point of period 6
- Topological entropy is at least log 2/2

## 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 $\limsup_{n\to+\infty}|f^n(x)-f^n(y)|>0$ and $\liminf_{n\to+\infty} |f^n(x)-f^n(y)|=0$ is dense (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.