Abstract

We study the dynamics of skew product endomorphisms acting on the cylinder R/Z×R, of the form(θ,t)→(ℓθ,λt+τ(θ)) , where ℓ≥2 is an integer, λ∈(0,1) and τ:R/Z→R is a continuous function. We are interested in topological properties of the global attractor Ωλ,τ of this map. Given ℓ and a Lipschitz function τ, we show that the attractor set Ωλ,τ is homeomorphic to a closed topological annulus for all λ sufficiently close to 1. Moreover, we prove that Ωλ,τ is a Jordan curve for at most finitely many λ∈(0,1). These results rely on a detailed study of iterated "cohomological" equations of the form τ= Lλ1μ1, μ1=Lλ2μ2,..., where Lλμ=μ○mℓ- λμ and mℓ:R/Z→R/Z denotes the multiplication by ℓ map. We show the following finiteness result: each Lipschitz function τ can be written in a canonical way as,τ=Lλ1○⋯○Lλmμ, where m≥0, λ1,...,λm∈(0,1] and the Lipschitz function μ satisfies μ≠Lλρ for every continuous function ρ and every λ∈(0,1].