Axiom A maps are dense in the space of unimodal maps in the $C^k$ topology

TL;DR

This paper proves that Axiom A maps are densely distributed within the space of C^k unimodal maps, confirming a structure stability conjecture across various smoothness levels.

Contribution

It establishes the density of Axiom A maps in the space of C^k unimodal maps, extending the structure stability conjecture to all smoothness classes.

Findings

Axiom A maps are dense in the space of C^k unimodal maps.

The structure stability conjecture is confirmed for all k.

The result applies to smoothness levels k=1,2,...,∞,ω.

Abstract

In this paper we prove C^k structure stability conjecture for unimodal maps. In other words, we shall prove that Action A maps are dense in the space of C^k unimodal maps in the C^k topology. Here k can be 1,2,...,\infty,\omega.