Contributions to a rigidity conjecture (Q1273482)

Let \(p\) and \(q\) be two relatively prime positive integers and \(\mu\) a Borel probability measure invariant and ergodic under the semigroup generated by the actions of \(z^p\) and \(z^q\) on the unit circle \(| z| =1\). An open conjecture by Furstenberg states that \(\mu\) is either atomic or the Lebesgue measure. This paper discusses known partial results in this direction and generalizes them to nonlinear expanding differentiable maps of the circle \(S^1\). Let \(f\) and \(g\) be two commuting \(C^1\) expanding maps of \(S^1\) whose degrees \(p\) and \(q\) are relatively prime, and \(\mu\) an \(f,g\)-invariant measure, ergodic for the semigroup generated by \(f\) and \(g\). Then either \(\mu\) is the (unique) measure of maximal entropy for \(f\) and \(g\) or its entropy is zero with respect to both \(f\) and \(g\). If the (common) conjugacy between \(f\), \(g\) and \(z^p\), \(z^q\), respectively, is differentiable, then the unique measure of maximal entropy for \(f\) and \(g\) is absolutely continuous. If \(\mu\) is nonatomic and mixing for either \(f\) or \(g\), then \(\mu\) is the measure of maximal entropy. Lastly, any closed invariant subset under both \(f\) and \(g\) is either finite or \(S^1\).