Abstract
In the present paper we deal with the Milnor-Thurston world and we present elementary proofs of some results by combining dynamics, combinatory, linear algebra and entropy.
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References
Alseda, L., Llibre, J., Misiurewicz, M.: Combinatorial Dynamics and Entropy in Dimension One. Advenced Series in Nonlinear Dynamics, vol. 5, 2nd edn, pp. xvi + 15. World Scientific, River Edge (2000). ISBN:981-02-4053-8
Aranzubía, S., Labarca, R.: On the existence of bubbles of constant entropy in the lexicographical world. prepint (2012)
Bamón, R., Labarca, R., Pacifico, M.J., Mañé, R.: The explosion of singular cycles. Publ. Math. IHES 78, 207–232 (1993)
Cooper, R.D., Hoare, M.R.: Distributive processes and combinatorial dynamics. J. Stat. Phys. 20(6), 597–628 (1979)
de Melo, W., Van Strien, S.: One-Dimensional Dynamics. Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 25. Springer, Berlin/New York (1993)
Field, M.: Combinatorial dynamics. Dyn. Syst. 19(3), 217–243 (2004)
Guckenheimer, J., Williams, R.F.: Structured stability of Lorenz attractors. Publ. Math IHES 50, 59–72 (1979)
Kauffman, S., Smolin, L.: Combinatorial dynamics and time in quantum gravity. In: Kowalski-Glikman, J. (ed.) Towards Quantum Gravity: Proceedings of the XXXV International Winter School on Theoretical Physics (Polanica, 1999). Lecture Notes in Physics, vol. 541, pp. 101–129. Springer, Berlin (2000)
Labarca, R.: Bifurcation of contracting singular cycles. Ann. Scient. Ec. Norm. Sup. 4 Serie t. 28, 705–745 (1993)
Labarca, R.: A note on the topological classification of Lorenz maps on the interval. In: Blanchard, F., Maass, A., Nogueira, A. (eds.) Topics in Symbolic Dynamics and Applications. London Mathematical Society Lecture Note Series, vol. 279, pp. 229–245. Cambridge University Press (2000)
Labarca, R.: Unfolding singular cycles. Notas Soc. Mat. Chile (NS) 1, 38–71 (2001)
Labarca, R.: La Entropía Topológica, Propiedades Generales y algunos cálculos en el caso de Milnor-Thurston. XXIV Escuela Venezolana de Matemáticas. EMALCA-Venezuela 2011. Ediciones IVIC (2011)
Labarca, R., Moreira, C.: Bifurcation of the essential dynamics of Lorenz maps of the real line and the bifurcation scenario of the linear family. Sci. Sci. A Math. Sci. (NS) 7, 13–29 (2001)
Labarca, R., Moreira, C.: Bifurcations of the essential dynamics of Lorenz maps and applications to Lorenz like flows: contributions to the study of the expanding case. Bol. Soc. Bras. Mat. (NS) 32, 107–144 (2001)
Labarca, R., Moreira, C.: Essential dynamics for Lorenz maps on the real line and the lexicographical world. Annales de L’Institut H. Poncaré Analyse non Linéaire 23, 683–694 (2006)
Labarca, R., Moreira, C.: Bifurcations of the essential dynamics of Lorenz maps on the real line and the bifurcation scenario for Lorenz like flows: the contracting case. Proyecciones 29(3), 247–293 (2010)
Labarca, R., Plaza, S.: Bifurcation of discontinuous maps of the interval and palindromic numbers. Bol. Soc. Mat. Mex. (3) 7(1), 99–116 (2001)
Labarca, R., Vásquez L.: On the characterization of the kneading sequences associated to injective Lorenz maps of the interval and to orientation preserving homeomorphism of the circle. Bol. Soc. Mat. Mex. 3a Ser. 16(2), 101–116 (2010)
Labarca, R., Vásquez, L.: On the characterization of the kneading sequences associated to Lorenz maps of the interval. Bol. Soc. Bras. Mat. (NS) 43(2), 221–245 (2012)
Labarca, R., Pumariño, A., Rodriguez, J.A.: On the boundary of topological chaos for the Milnor-Thurston world. Commun. Contemp. Math. 11(9), 1049–1066 (2009)
Labarca, R., Moreira, C., Pumariño, A., Rodriguez, J.A.: On bifurcation set for symbolic dynamics in the Milnor-Thurston world. Commun. Contemp. Math. 14(4), 1250024 1–16 (2012)
Metropolis, N., Stein, M.L., Stein, P.R.: Stabe states of a nonlinear transformation. Numer. Math. 10, 1–19 (1967)
Metropolis, N., Stein, M.L., Stein, P.R.: On finite limit sets for transformations on the unit interval. J. Comb. Theory (A) 15, 25–44 (1973)
Mielnik, B.: Combinatorial dynamics. In: Proccedings of the 14th ICGTMP (Seoul, 1985), pp. 265–267. World Scientific, Singapore, (1986). 81B05
Milnor, J., Thurston, W.: On iterated maps on the interval. In: Alexander, J.C. (ed.) Dynamical Systems. Lecture Notes in Mathematics, vol. 1342, pp 465–563. Springer, Berlin (1988)
Moreira, C.: Maximal invariant sets for restriction of tent and unimodal maps. Anal. Theory Dyn. Syst. 2(2), 385–398 (2001)
Acknowledgements
Part of this paper is an outgrowth of research during a visit of the authors to IMPA (Brazil). The authors were partially supported by DICYT – USACH (Chile), PCI – IMPA (Brazil) and by the Dirección de Graduados of the USACH. We thanks IMPA and USACH for its support while preparing the present paper.
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Aranzubía, S., Labarca, R. (2013). Combinatorial Dynamics and an Elementary Proof of the Continuity of the Topological Entropy at θ =101, in the Milnor Thurston World. In: Ibáñez, S., Pérez del Río, J., Pumariño, A., Rodríguez, J. (eds) Progress and Challenges in Dynamical Systems. Springer Proceedings in Mathematics & Statistics, vol 54. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-38830-9_3
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DOI: https://doi.org/10.1007/978-3-642-38830-9_3
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