Thermodynamic Formalism for General Iterated Function Systems with Measures

• Jader E. Brasil ^[1] ; Elismar R. Oliveira ^[1] ; Rafael Rigão Souza ^[1]
  1. [1] Instituto de Matemática e Estatística - UFRGS
• Localización: Qualitative theory of dynamical systems, ISSN 1575-5460, Vol. 22, Nº 1, 2023
• Idioma: inglés
• Enlaces
  • Texto completo
• Resumen

  • This paper introduces a theory of Thermodynamic Formalism for Iterated Function Systems with Measures (IFSm). We study the spectral properties of the Transfer and Markov operators associated to a IFSm. We introduce variational formulations for the topological entropy of holonomic measures and the topological pressure of IFSm given by a potential. A definition of equilibrium state is then natural and we prove its existence for any continuous potential. We show, in this setting, a uniqueness result for the equilibrium state requiring only the Gâteaux differentiability of the pressure functional.

• Referencias bibliográficas
  • 1. Aguiar, D., Cioletti, L., Ruviaro, R.: A variational principle for the specific entropy for symbolic systems with uncountable alphabets....
  • 2. Ambrosio, Luigi, Gigli, Nicola, Savaré, Giuseppe: Gradient Flows in Metric Spaces and in the Space of Probability Measures. Springer Science...
  • 3. Barnsley, Michael F., Demko, Stephen: Iterated function systems and the global construction of fractals. Proceed. Royal Soc. Lond. A. Math....
  • 4. Barnsley, Michael F., Demko, Stephen G., Elton, John H., Geronimo, Jeffrey S.: Invariant measures for markov processes arising from iterated...
  • 5. Berinde, Vasile, Pacurar, Madalina: The role of the Pompeiu-Hausdorff metric in fixed point theory. Creat. Math. Inform. 22(2), 143–150...
  • 6. Charalambos, D., Aliprantis, Border: Infinite Dimensional Analysis: A Hitchhiker’s Guide. SpringerVerlag Berlin, Heidelberg GmbH &...
  • 7. Chi¸tescu, Ion, Ioana, Loredana, Miculescu, Radu: Type A sets and the attractors of infinite iterated function systems. Result. Math. 66(3–4),...
  • 8. Cioletti, Leandro, Oliveira, Elismar R.: Thermodynamic formalism for iterated function systems with weights, Preprint, arXiv:1707.01892,...
  • 9. Cioletti, Leandro, Oliveira, Elismar R.: Applications of variable discounting dynamic programming to iterated function systems and related...
  • 10. Dellacherie, Claude, Meyer, Paul-André: em Probabilities and potential, of North-Holland mathematics studies, Vol. 29 (1978)
  • 11. Doob, J.L.: Asymptotic properties of markoff transition prababilities Trans. Amer. Math. Soc. 63, 393–421 (1948)
  • 12. Dumitru, Dan: Attractors of infinite iterated function systems containing contraction type functions. An. ¸Stiin¸t. Univ. Al. I. Cuza...
  • 13. Elton, John H.: An ergodic theorem for iterated maps. Ergod. Theor. Dyn. Syst. 7(4), 481–488 (1987)
  • 14. Fan, Ai Hua, Lau, Ka-Sing.: Iterated function system and ruelle operator. J. Math. Anal. Appl. 231(2), 319–344 (1999)
  • 15. Fernau, Henning: Infinite iterated function systems. Math. Nachr. 170, 79–91 (1994)
  • 16. Gupta, Aman, Shaju, Cyril, Pratibha & Kamal. A study of the international stock market behavior during COVID-19 pandemic using a driven...
  • 17. Gwó´zd´z-Lukawska, Gertruda, Jachymski, Jacek: The Hutchinson-Barnsley theory for infinite iterated function systems. Bull. Aust. Math....
  • 18. Hanus, Pawel, Mauldin, R., Urba ´nski, Mariusz: Thermodynamic formalism and multifractal analysis of conformal infinite iterated function...
  • 19. Hille, Martial R.: Remarks on limit sets of infinite iterated function systems. Monatsh. Math. 168(2), 215–237 (2012)
  • 20. Hutchinson, John E.: Fractals and self similarity. Indiana Univ. Math. J. 30(5), 713–747 (1981)
  • 21. Käenmäki, Antti: On natural invariant measures on generalised iterated function systems. Ann. Acad. Sci. Fenn. Math 29(2), 419–458 (2004)
  • 22. Lasota, Andrzej, Myjak, Józef., Szarek, Tomasz: Markov operators with a unique invariant measure. J. Math. Anal. Appl. 276(1), 343–356...
  • 23. Lopes, Artur O., Mengue, Jairo K., Mohr, Joana, Souza, Rafael R.: Entropy and variational principle for one-dimensional lattice systems...
  • 24. Lopes, Artur O., Oliveira, Elismar R.: Entropy and variational principles for holonomic probabilities of IFS. Discret. & Contin. Dyn....
  • 25. Richard Daniel Mauldin and Mariusz Urba ´nski: Parabolic iterated function systems. Ergod. Theory Dynamic. Syst. 20(5), pp 1423–1447 (2000)
  • 26. Mauldin, Richard Daniel, Urba ´nski, Mariusz: Dimensions and measures in infinite iterated function systems. Proc. Lond. Math. Soc. (3)...
  • 27. Mihail, Alexandru, Miculescu, Radu: The shift space for an infinite iterated function system. Math. Rep. (Bucur.) 11((61)(1)), 21–32 (2009)
  • 28. Morán, Manuel: Hausdorff measure of infinitely generated self-similar sets. Monatsh. Math. 122(4), 387–399 (1996)
  • 29. Ruelle, David: A variational formulation of equilibrium statistical mechanics and the Gibbs phase rule. Commun. Math. Phys. 5(5), 324–329...
  • 30. Ruelle, David: Statistical mechanics of a one-dimensional lattice gas. Commun. Math. Phys. 9(4), 267–278 (1968)
  • 31. Simon, Károly., Solomyak, Boris, Urba ´nski, Mariusz: Invariant measures for parabolic ifs with overlaps and random continued fractions....
  • 32. Stenflo, Örjan: Uniqueness of invariant measures for place-dependent random iterations of functions. In: Fractals in Multimedia, Springer,...
  • 33. Walters, Peter: Ruelle’s operator theorem and g-measures. Transact. Am. Math. Soc. 214, 375–387 (1975)