The partial captivity condition for U(1) extensions of expanding maps on the circle

TL;DR

This paper proves that the partial captivity condition, crucial for understanding the dynamics of certain circle extensions, is generically satisfied in a smooth sense for fixed expanding maps.

Contribution

It establishes that the partial captivity condition is a generic smooth condition on the cocycle function , given a fixed expanding map, enhancing understanding of the dynamics.

Findings

Partial captivity is a r smooth generic condition.

The result applies to the dynamics of -extensions of expanding maps.

Provides a new criterion for genericity in dynamical systems.

Abstract

This paper concerns the compact group extension \[ f:\mathbb{T}^2\to \mathbb{T}^2,\quad f (x,s)= (E(x), s+\tau(x)\ \text{mod }1) \] of an expanding map $E:\mathbb{S}^1\to \mathbb{S}^1$. The dynamics of $f$ and its stochastic perturbations have previously been studied under the so-called partial captivity condition. Here we prove a supplementary result that shows that partial captivity is a $\mathscr{C}^r$ generic condition on $\tau$, once we fix $E$.