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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
##### Keywords:
expanding maps; total exponents; optimization measures
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##### References:
 [1] T. Bousch, La condition de Walters, Ann. Sci. École. Norrm. Sup. 34 (2001) 287-311. · Zbl 0988.37036 [2] T. Bousch and O. Jenkionson, Cohomology classes of dynamically non-negative $$C^{k}$$ functions, Invent. Math. 148 (2002) 207-217. · Zbl 1079.37505 [3] R. Bowen, Equilibrium states and the ergodic theory of Anosov diffeomorphisms, Lecture Note in Math. 470 , Springer Verlag, Berlin-Hedelberg-New York 1975. · Zbl 0308.28010 [4] J. Brémont, Entropy and maximizing measures of generic continuous functions, C. R. Math. Acad. Sci. Paris 346 (2008) 199-201. · Zbl 1131.37005 [5] G. Contreras, A. Lopes, and Ph. Thieullen, Lyapunov minimizing measures for expanding maps of the circle, Ergod.Th. and Dynam.t Sys. 21 (2001) 1379-1409. · Zbl 0997.37016 [6] M. Denker, Ch. Grillenberger, and K. Sigmund, Ergodic theory on compact spaces, Lecture Note in Math. 527 , Springer Verlag, Berlin-Hedelberg-New York 1976. · Zbl 0328.28008 [7] O. Jenkinson, Ergodic optimization, Discrete Contin. Dyn. Syst. 15 (2006) 197-224. · Zbl 1116.37017 [8] O. Jenkinson and I. D. Morris, Lyapunov optimizing measures for $$C^{1}$$ expanding maps, Ergod.Th. and Dynam. Sys. 28 (2008) 1849-1860. · Zbl 1213.37067 [9] K. R. Parthasarathy, On the category of ergodic measures, Illinois J. Math. 5 (1961) 648-656. · Zbl 0103.28101 [10] D. Ruelle, Thermodynamic formalism 2nd. ed. Cambridge Univ. Press, Cambridge 2004. · Zbl 1062.82001 [11] S. V. Savchenco, Homological inequalities for finite topological Markov chains, Funktsional. Anal. i Prilozhen 33 (1999) 91-93. [12] K. Sigmund, Generic properties of invariant measures for Axiom A diffeomorphisms, Invent. Math. 11 (1970) 99-109. · Zbl 0193.35502 [13] M. Shub, Endomorphisms of compact differentiable manifolds, Amer. J. Math. 91 (1969) 175-199. · Zbl 0201.56305 [14] Y. Tokunaga, Lyapunov optimizing measures for $$C^{1}$$ expanding maps of the $$n$$-torus, Master Dissertation of Hiroshima University 2010 (in Japanses). [15] P. Walters, Invariant measures and equilibrium states for some mappings which expand distances, Trans. Amer. Math. Soc. 236 (1978) 121-153. · Zbl 0375.28009 [16] P. Walters, Introduction to ergodic theory, Springer Verlag, Berlin-Hedelberg-New York 1982. · Zbl 0475.28009
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