The Ingram Conjecture

TL;DR

This paper proves the Ingram Conjecture, demonstrating that inverse limit spaces of tent maps with different slopes are topologically distinct, and characterizes their self-homeomorphisms as pseudo-isotopic to powers of the shift.

Contribution

It establishes the Ingram Conjecture for all tent maps with slopes in [1, 2] and describes the structure of their self-homeomorphisms.

Findings

Inverse limit spaces of tent maps with different slopes are non-homeomorphic.

Self-homeomorphisms are pseudo-isotopic to powers of the shift.

The proof provides a detailed structural understanding of these spaces.

Abstract

We prove the Ingram Conjecture, i.e., we show that the inverse limit spaces of every two tent maps with different slopes in the interval [1, 2] are non-homeomorphic. Based on the structure obtained from the proof, we also show that every self-homeomorphism of the inverse limit space of the tent map is pseudo-isotopic, on the core, to some power of the shift homeomorphism.