# The boundary of chaos for interval mappings

## TL;DR

This paper proves a longstanding conjecture that the transition to chaos in interval maps occurs solely through period doubling bifurcations, and explores the boundary structure of maps with positive entropy.

## Contribution

It confirms the conjecture for smooth interval maps and analyzes the boundary's topological structure in various families.

## Key findings

- Transition to chaos occurs only via period doubling bifurcations.
- Boundary of positive entropy maps is locally connected in fixed critical point families.
- Boundary is a cellular set for analytic mappings.

## Abstract

A goal in the study of dynamics on the interval is to understand the transition to positive topological entropy. There is a conjecture from the 1980's that the only route to positive topological entropy is through a cascade of period doubling bifurcations. We prove this conjecture in natural families of smooth interval maps, and use it to study the structure of the boundary of mappings with positive entropy. In particular, we show that in families of mappings with a fixed number of critical points the boundary is locally connected, and for analytic mappings that it is a cellular set.