Randomness versus quasi-periodicity

TL;DR

This paper explores the potential connections between randomness and quasi-periodicity in dynamical systems, aiming to understand their relations through a long-term research project on mixed cocycles.

Contribution

It provides an extended roadmap for investigating the interplay between random and quasi-periodic cocycles, building on prior work and open questions in the field.

Findings

Identifies potential relations between random and quasi-periodic systems

Proposes a long-term research framework for mixed cocycles

Addresses stability of Lyapunov exponents under noise

Abstract

This paper serves as an extended road map for our long-term project "Mixed Random-quasiperiodic Cocycles" [arXiv:2201.04745, arXiv:2109.09544, arXiv:2210.16908, 6, 7] with Pedro Duarte and Silvius Klein. Despite exhibiting totally different natures, the random world and the quasi-periodic one may still have potential relations that we are keen to reveal. This was inspired by Jiangong You's intriguing question on the stability of the Lyapunov exponent of quasi-periodic Schr\"odinger operators under random noise in 2018.