Hiroshima Mathematical Journal

Measures with maximum total exponent and generic properties of $C^{1}$ expanding maps

Takehiko Morita and Yusuke Tokunaga

Full-text: Open access


We show that a generic $C^{1}$ expanding map on a compact Riemannian manifold has a unique measure of maximum total exponent which is fully supported and of zero entropy. We also show that for $r\ge 2$ a generic $C^{r}$ expanding map does not have fully supported measures of maximum total exponent.

Article information

Hiroshima Math. J., Volume 43, Number 3 (2013), 351-370.

First available in Project Euclid: 7 January 2014

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 137D35
Secondary: 37C20: Generic properties, structural stability 37C40: Smooth ergodic theory, invariant measures [See also 37Dxx]

Expanding map total exponent optimization measure


Morita, Takehiko; Tokunaga, Yusuke. Measures with maximum total exponent and generic properties of $C^{1}$ expanding maps. Hiroshima Math. J. 43 (2013), no. 3, 351--370. doi:10.32917/hmj/1389102580. https://projecteuclid.org/euclid.hmj/1389102580

Export citation


  • T. Bousch, La condition de Walters, Ann. Sci. École. Norrm. Sup. 34 (2001) 287–311.
  • T. Bousch and O. Jenkionson, Cohomology classes of dynamically non-negative $C^{k}$ functions, Invent. Math. 148 (2002) 207–217.
  • R. Bowen, Equilibrium states and the ergodic theory of Anosov diffeomorphisms, Lecture Note in Math. 470, Springer Verlag, Berlin-Hedelberg-New York 1975.
  • J. Brémont, Entropy and maximizing measures of generic continuous functions, C. R. Math. Acad. Sci. Paris 346 (2008) 199–201.
  • 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.
  • M. Denker, Ch. Grillenberger, and K. Sigmund, Ergodic theory on compact spaces, Lecture Note in Math. 527, Springer Verlag, Berlin-Hedelberg-New York 1976.
  • O. Jenkinson, Ergodic optimization, Discrete Contin. Dyn. Syst. 15 (2006) 197–224.
  • O. Jenkinson and I. D. Morris, Lyapunov optimizing measures for $C^{1}$ expanding maps, Ergod.Th. and Dynam. Sys. 28 (2008) 1849–1860.
  • K. R. Parthasarathy, On the category of ergodic measures, Illinois J. Math. 5 (1961) 648–656.
  • D. Ruelle, Thermodynamic formalism 2nd. ed. Cambridge Univ. Press, Cambridge 2004.
  • S. V. Savchenco, Homological inequalities for finite topological Markov chains, Funktsional. Anal. i Prilozhen 33 (1999) 91–93.
  • K. Sigmund, Generic properties of invariant measures for Axiom A diffeomorphisms, Invent. Math. 11 (1970) 99–109.
  • M. Shub, Endomorphisms of compact differentiable manifolds, Amer. J. Math. 91 (1969) 175–199.
  • Y. Tokunaga, Lyapunov optimizing measures for $C^{1}$ expanding maps of the $n$-torus, Master Dissertation of Hiroshima University 2010 (in Japanses).
  • P. Walters, Invariant measures and equilibrium states for some mappings which expand distances, Trans. Amer. Math. Soc. 236 (1978) 121–153.
  • P. Walters, Introduction to ergodic theory, Springer Verlag, Berlin-Hedelberg-New York 1982.