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Measures with maximum total exponent and generic properties of \(C^1\) expanding maps. (English) Zbl 1347.37056
Let \(M\) be a \(C^\infty\) closed manifold, and let \(C^r(M)\) be the space of \(C^r\)-maps of \(M\) endowed with the \(C^r\)-topology \((r \geq 1)\). In this paper, the authors prove that \(C^1\)-generically, every expanding map in \(C^1(M)\) has a unique invariant Borel probability measure of maximum total exponent which is fully supported and of zero entropy. Moreover, the authors show that for \(r \geq 2\), \(C^r\)-generically, every expanding map in \(C^r(M)\) does not have a fully supported measure of maximum total exponent. This generalizes the results obtained in [O. Jenkinson and I. D. Morris, Ergodic Theory Dyn. Syst. 28, No. 6, 1849–1860 (2008; Zbl 1213.37067)] on the unit circle \(S^1\).

MSC:
37D20 Uniformly hyperbolic systems (expanding, Anosov, Axiom A, etc.)
37D35 Thermodynamic formalism, variational principles, equilibrium states for dynamical systems
37A05 Dynamical aspects of measure-preserving transformations
37C40 Smooth ergodic theory, invariant measures for smooth dynamical systems
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