Abstract
A Ruelle-expanding map is an open continuous transformation defined on a compact metric space which expands distances locally. For such dynamical systems, we will explain why: (a) the zeta function is rational; (b) the topological entropy is equal to the exponential growth rate of the periodic points; (c) the topological entropy is positive unless the domain of the map is finite. These properties have been remarked in the work of D. Ruelle but without entering into all the necessary details; the aim of this text is precisely to provide them.
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References
Artin, M., Mazur, B.: On periodic points. Ann. Math. 81, 82–99 (1965)
Bowen, R., Lanford, O.E., III.: Zeta functions of restrictions of the shift transformation. In: Proceedings of Symposia in Pure Mathematics, vol. XIV, pp. 43–49. American Mathematical Society, Providence (1970)
Bowen, R.: Periodic points and measures for Axiom A diffeomorphisms. Trans. Am. Math. Soc. 154, 377–397 (1971)
Bowen, R.: Equilibrium states and the ergodic theory of Anosov diffeomorphisms, Lecture Notes in Mathematics, vol. 470. Springer, New York (1975)
Epstein, D., Shub, M.: Expanding endomorphisms of flat manifolds. Topology 7, 139–141 (1968)
Jiang, B., Llibre, J.: Minimal sets of periods for torus maps. Discrete Contin. Dyn. Syst. 4, 301–320 (1998)
Katok, A., Hasselblatt, B.: Introduction to the modern theory of dynamical systems, Encyclopedia of Mathematics and its Applications, vol. 54. Cambridge University Press, Cambridge (1995)
Manning, A.: Axiom A diffeomorphisms have rational zeta functions. Bull. Lond. Math. Soc. 3, 215 (1971)
Misiurewicz, M., Szlenk, W.: Entropy of piecewise monotone mappings, Studia Mathematica, vol. LXVII, pp. 45–63 (1980)
Ruelle, D.: Statistical mechanics of a one dimensional lattice gas. Commun. Math. Phys. 9, 267–278 (1968)
Ruelle, D.: Zeta-functions for expanding maps and Anosov flows. Invent. Math. 34(3), 231–242 (1976)
Ruelle, D.: Thermodynamic formalism—the mathematical structures of classical equilibrium statistical mechanics, Encyclopedia of Mathematics and its Applications, vol. 5. Addison-Wesley, Reading (1978)
Ruelle, D.: The thermodynamic formalism for expanding maps. Commun. Math. Phys. 125(2), 239–262 (1989)
Ruelle, D.: An extension of the theory of Fredholm determinants. Publications Mathématiques de l’IHES 72, 175–193 (1990)
Shub, M.: Endomorphisms of compact differentiable manifolds. Am. J. Math. 91, 129–155 (1969)
Shub, M.: Global Stability of Dynamical Systems. Springer, New York (1987)
Tauraso, R.: Sets of periods for expanding maps on flat manifolds. Monatshefte für Mathematik 128(2), 151–157 (1999)
Walters, P.: An Introduction to Ergodic Theory. Springer, New York (1975)
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Research partially funded by the European Regional Development Fund through the program COMPETE and by the Portuguese Government through FCT-Fundação para a Ciência e a Tecnologia under the project PEst-C/MAT/UI0144/2013 and the grant SFRH/BD/33092/2007.
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de Carvalho, M.P., Magalhães, M.A. Periodic Points of Ruelle-Expanding Maps. Qual. Theory Dyn. Syst. 13, 215–251 (2014). https://doi.org/10.1007/s12346-014-0115-y
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DOI: https://doi.org/10.1007/s12346-014-0115-y