Lyapunov minimizing measures for expanding maps of the circle (Q2757010)

Let \(f:S^1 \longrightarrow S^1\) be a covering map of degree \(D\), orientation-preserving and expanding, i.e. \(\min_{x\in S^1} f'(x) > 1\). For \(1 < \alpha < 2\) denote by \({\mathcal F}_{\alpha +}\) the set of such maps of class \(C^\beta\), with some \(\alpha < \beta <2 \). A Lyapunov minimizing measure is a measure minimizing \(\int \ln f' d\mu\) over all \(f\)-invariant probability measures \(\mu\). It is shown that: (1) There exists an open and dense in \(C^\alpha\) topology subset of \({\mathcal F}_{\alpha +}\) such that any of its elements admits a unique Lyapunov minimizing measure supported on a periodic orbit. (2) If \(f \in {\mathcal F}_{\alpha +}\) has a Lyapunov measure not supported on a finite set of periodic points, then \(f\) is a \(C^\alpha\) limit of maps \(f_n\) from \({\mathcal F}_{\alpha +}\) admitting a unique Lyapunov minimizing measure \(\mu_n\) such that \(f_n\) restricted to supp\((\mu_n)\) is strictly ergodic and has positive topological entropy.