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

Abstract

This paper concerns the compact group extension \[ f:\mathbb{T}^2\to \mathbb{T}^2,\quad f (x,s)= (E(x), s+τ(x)\ \text{mod }1) \] 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 ${\mathcal{C}}^r$ generic condition on $\tau$, once we fix $E$.

9 pages; minor changes

References in corpus (1)

• On the spectra of quenched random perturbations of partially expanding maps on the torus

Cited by in corpus (3)

• Historic behaviour for nonautonomous contraction mappings
• Flat trace statistics of the transfer operator of a random partially expanding map
• Higher-dimensional attractors with absolutely continuous invariant probability