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Lyapunov minimizing measures for expanding maps of the circle. (English) Zbl 0997.37016

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.

MSC:

37E10 Dynamical systems involving maps of the circle
37D35 Thermodynamic formalism, variational principles, equilibrium states for dynamical systems
37D20 Uniformly hyperbolic systems (expanding, Anosov, Axiom A, etc.)
37C40 Smooth ergodic theory, invariant measures for smooth dynamical systems
37A20 Algebraic ergodic theory, cocycles, orbit equivalence, ergodic equivalence relations
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