Godel Diffeomorphisms

TL;DR

This paper explores the complexity of determining whether a smooth diffeomorphism is isomorphic to its inverse, linking this problem to major number theory conjectures and set-theoretic independence results.

Contribution

It establishes that the problem is as complex as famous conjectures like the Riemann Hypothesis and Goldbach's conjecture, and shows the existence of diffeomorphisms with independence from ZFC.

Findings

Determining if a diffeomorphism is isomorphic to its inverse relates to the Riemann Hypothesis.

The problem's complexity can encode Goldbach's conjecture.

Some diffeomorphisms have properties independent of ZFC.

Abstract

A basic problem in smooth dynamics is determining if a system can be distinguished from its inverse, i.e., whether a smooth diffeomorphism $T$ is isomorphic to $T^{-1}$. We show that this problem is sufficiently general that asking it for particular choices of $T$ is equivalent to the validity of well-known number theoretic conjectures including the Riemann Hypothesis and Goldbach's conjecture. Further one can produce computable diffeomorphisms $T$ such that the question of whether $T$ is isomorphic to $T^{-1}$ is independent of ZFC.