One-Parameter Families of Smooth Interval Maps: Density of Hyperbolicity and Robust Chaos

TL;DR

This paper explores the concept of robust chaos in one-parameter families of smooth interval maps, demonstrating the existence of such families and showing that hyperbolicity is dense among generic families, indicating a nuanced relationship between chaos and hyperbolicity.

Contribution

The paper introduces natural one-parameter families exhibiting robust chaos and proves that hyperbolicity is dense in generic families, clarifying their relationship.

Findings

Existence of natural one-parameter families with robust chaos

Density of hyperbolicity in generic one-parameter families

Robust chaos and hyperbolicity are not mutually exclusive

Abstract

In this note we will discuss the notion of robust chaos, and show that (i) there are natural one-parameter families with robust chaos and (ii) hyperbolicity is dense within generic one-parameter families (and so these families are not robustly chaotic).