On the topology of solenoidal attractors of the cylinder

Abstract

We study the dynamics of skew product endomorphisms acting on the cylinder R/Z x R, of the form (theta, t) -> (l theta, gimel t + tau(theta)),

where l >= 2 is an integer, gimel is an element of (0, 1) and tau : R/Z -> R is a continuous function. We are interested in topological properties of the global attractor Omega(gimel,tau) of this map. Given l and a Lipschitz function tau, we show that the attractor set Omega(gimel,tau) is homeomorphic to a closed topological annulus for all gimel sufficiently close to 1. Moreover, we prove that Omega(gimel,tau) is a Jordan curve for at most finitely many gimel is an element of (0, 1).

These results rely on a detailed study of iterated "cohomological" equations of the form tau = L gimel(1)mu(1),mu(1) = L gimel(2)mu(2),..., here L gimel mu = mu circle...circle m(l) - gimel mu and m(l) :R/Z -+ R/Z denotes the multiplication by l map. We show the following finiteness result: each Lipschitz function tau can be written in a canonical way as,

tau = L gimel(1) circle...circle L gimel(m)mu,

where m >= 0, gimel(1),...gimel(m) is an element of(0, 1] and the Lipschitz function mu satisfies mu = L gimel p for every continuous function p and every gimel is an element of (0,1].