On the minimality of semigroup actions on the interval which are $C^1$-close to the identity

TL;DR

This paper constructs a counterexample showing that semigroup actions on an interval generated by two attracting maps, which are close to the identity in the $C^1$ topology, do not necessarily have the entire interval as their minimal set.

Contribution

The paper demonstrates that the known result for $C^2$-close generators does not extend to the $C^1$-topology by providing a specific counterexample.

Findings

Counterexample to minimality under $C^1$-closeness

Minimal set can be a proper subset of the interval

Contrasts with the $C^2$-case

Abstract

We consider semigroup actions on the interval generated by two attracting maps. It is known that if the generators are sufficiently $C^2$-close to the identity, then the minimal set coincides with the whole interval. In this article, we give a counterexample to this result under the $C^1$-topology.