Construction and applications of proximal maps for typical cocycles

TL;DR

This paper develops a method to construct periodic orbits with proximal cocycle products in subshifts of finite type, enabling broader implications for the system's dynamics.

Contribution

It introduces a construction technique for proximal maps along periodic orbits in typical cocycles over subshifts of finite type.

Findings

Periodic orbits with proximal cocycle products can be constructed for any orbit segment.

Assumptions on periodic orbits influence the dynamics of the entire subshift.

The method has applications in understanding the stability and structure of cocycles.

Abstract

For typical cocycles over subshifts of finite type, we show that for any given orbit segment, we can construct a periodic orbit such that it shadows the given orbit segment and that the product of the cocycle along its orbit is a proximal linear map. Using this result, we show that suitable assumptions on the periodic orbits have consequences over the entire subshift.