A computer verification for the value of the Topological Entropy for some special subshifts in the Lexicographical Scenario.

Authors

  • Solange Aranzubia Universidad Central.
  • Rubén Carvajal Universidad de Santiago.
  • Rafael Labarca Universidad de Santiago.

Keywords:

Lexicographical world, Topological entropy, Maximal and minimal sequences, Shift map, Lorenz Maps

Abstract

The Lorenz Attractor has been a source for many mathematical studies. Most of them deal with lower dimensional representations of its first return map. An one dimensional scenario can be modeled by the standard two parameter family of contracting Lorenz maps. The dynamics, in this case, can be modeled by a subshift in the Lexicographical model. The Lexicographical model is the set of two symbols with the topology induced by the lexicographical metric and with the lexicographical order. These subshifts are the maximal invariant set for the shift map in some interval. For some of them, the extremes of the interval are a minimal periodic sequence and a maximal periodic sequence which is an iteration of the lower extreme (by the shift map). For some of these subshifts the topological entropy is zero. In this case the dynamics (of the respective Lorenz map) is simple.Associated to any of these subshifts (let call it Λ) we consider an extension (let call it Γ) that contains Λ which also can be constructed by using an interval whose extremes can be defined by the extremes of Λ. For these extensions we present here a computer verification of the result that compute its topological entropy. As a consequence, of our results, we can say: the longer the period of the periodic sequence is then the lower complexity in the dynamics of the extension the associated map has.

Author Biographies

Solange Aranzubia, Universidad Central.

Escuela de Matemática y Estadística.

Rubén Carvajal, Universidad de Santiago.

Departamento de Matemática y Ciencia de la Computación.

Rafael Labarca, Universidad de Santiago.

Departamento de Matemática y Ciencia de la Computación.

References

Adler, R. L.; Konheim, A.G.; Mc Andrew, M.H. Topological Entropy Transactiones of the A.M.S. vol 144, pp. 309-319, (1965).

Alekseev, V. M. Quasirandom Dynamical Systems Math. USSR Sbornik - New Series Vol. 5 No 1, (1968).

Alseda, Lluís; Llibre, Jaume; Misiurewicz, Michal Combinatorial dynamics and entropy in dimension one. Second edition. Advenced Series in Nonlinear Dynamics, 5. World Scientific Publishing Co., Inc., River Edge, NJ, xvi+15, pp. ISBN: 981-02-4053-8, (2000).

Aranzubia, S.Hacia una demostración de la arco conexidad de la isentropa de entropíaa cero de la familia cuadrática de Lorenz lexicográfica: burbujas de entropía constante y cotas superiores para burbujas las de entropía cero Tesis de doctorado Universidad de Santiago de Chile, (2015)

Aranzubia, S.; Labarca R. A Formula for the Boundary of Chaos in the Lexicographical Scenario and Applications to the Bifurcation Diagram of the Standard Two Parameter Family of Quadratic Increasing-Increasing Lorenz maps. Discrete and Continuous Dynamical Systems-Series A., Vol. 38, No 4, pp. 1745-1776, (2018).

Block,L; Guckenheimer, J.; Misiurewicz, M; Young, L.S. Periodic points and topological entropy. Lectures Notes in Mathematics. No 819, Springer Verlag, (1978).

Bamón,R.; Labarca, R.; Mañé R.; Pacifico, M.J.The explosion of singular cycles Publ. Math. IHES Vol 78, pp 207-232, (1993).

Bruin, H.; Van Strien, S. Monotonicity of Entropy for Real Multimodal Maps Journal of the American Mathematical Society, Vol 28, No 1, pp. 1-61, (2015).

de Melo W., Van Strien S. Lectures on One Dimensional Dynamics Springer Verlag, (1993).

Francis, J. G. F.; The QR Transformation, I. The Computer Journal, 4(3), pp. 265-271, (1961).

The GNU Scientific Library (GSL); http://www.gnu.org/software/gsl/

Guckenheimer J., Williams R. F. Structural Stability of Lorenz Attractors Publ. Math. IHES 50, pp 59-72, (1979).

Labarca, R. Bifurcation of Contracting Singular Cycles Ann. Scient. Ec. Norm. Sup. 4eserie, t. 28, pp 705-745, (1995).

Labarca, R. A note on the topological classification of Lorenz maps on the interval. Topics in symbolic dynamics an applications, London Math. Soc. Lect. Not. Ser. 279, Cambridge Univ. Press. pp. 229-245, (2000).

Labarca, R. Unfolding singular cycles. Notas Soc. Mat. Chile (N5) No 1, pp. 38-71, (2001).

Labarca, R,La Entropíaa topológica, propiedades generales y algunos cálculos en el caso del shift de Milnor-Thurston.Con la colaboración de Solange Aranzubia V. y Erick Inda R. Libro de 150 páginas publicado en la serie de las Escuelas Venezolanas de Matemáticas, con comité editorial, de Ediciones IVIC. Instituto Venezolano de Investigaciones Científicas, (2011)

Labarca,R.; Moreira C. Bifurcation of the Essential Dynamics of Lorenz Maps of the real Line and the Bifurcation Scenario for the Linear family Scientia Ser A Math. Sci (N-S), Vol. 7, pp. 13-29, (2001).

Labarca, R.; Moreira C. Bifurcation of the Essential Dynamics of Lorenz Maps and applications to Lorenz Like Flows: Contributions to the study of the Expanding Case. Bol. Soc. Bros. Mat. (N-S), Vol. 32, pp. 107-144, (2001).

Labarca, R.; Moreira C. Essential Dynamics for Lorenz Maps on the real line and Lexicographical World. Ann. de L’Institut H. Poincaré Anal. non Linneaire, Vol. 23, pp. 683-694, (2006).

Labarca, R.; Moreira, C.Bifurcation of the essential dynamics of Lorenz Maps on the real line and the bifurcation scenario for Lorenz like flows: The Contracting Case Proyecciones. Journal of Mathematics, Vol 29, No. 3, pp. 241-289 (2010).

Labarca R., Plaza S. Bifurcations of discontinuous maps of the interval and palindromic numbers. Boletín de la Sociedad Matemática Mexicana(3), Vol 5., pp. 87-104, (2001)

Labarca, R.; San Martín, B. Prevalence of hyperbolicity for complex singular cycles. Bol. Soc. Brasil. Mat. (N5) 28, No 2 pp. 343-362, (1997).

Labarca, R.; Vásquez, L. On the Characterization of the kneading sequences associated to Lorenz maps of the interval and to orientation preserving homeomorphism of the circle Boletín de la Sociedad Matemática Méxicana (3), Vol. 16 No. 2, pp. 103-118, (2010).

Labarca R., Vásquez L. On the Characterization of the Kneading Sequences Associated to Lorenz Maps of the Interval. Bol. Soc. Bras. Mat.(NS), Vol. 43 N o 2, pp. 221-245, (2012).

Moreira, C. Maximal invariant sets for restriction of tent and unimodal maps. Qual. Thory Dyn. Syst, 2, N o2, pp. 385-398, (2001).

Metropolis, N.; Stein, M. L. ; Stein, P.R. “Stabe States of a nonlinear transformation.”. Numerisch Mathematik 10, pp. 1-19, (1967).

Metropolis, N.; Stein, M. L.; Stein, P. R. “On finite limit sets for transformations on the unit interval.”. Journal of combinatorial Theory (A) 15, pp. 25-44, (1973).

Milnor, J. Remarks on iterated cubic maps Experiment. Math. 1, No 1, pp. 5-24 MR1181083, (1992).

Milnor, J. ; Thurston, W. On iterated maps on the interval. Lect. Notes in Math 1342 pp 465-563 Springer Verlag, (1988).

Misiurewicz, M. On non continuity of topological entropy Bull. Acad. Polon. Sci. Ser. Sci. Math. Astronom. Phys. 19, pp. 319-320, (1971).

Sait Pierre, Matthias Topological and measurable dynamics of Lorenz Maps. Dissertationes Mathematicae (ROZPRAWY MATEMATYCZNE) Polska Akademie Nauk, Institut Matematiczny. CCCLXXXII, (1999).

Silnikov, L.P.A Contribution to the problem of the structure of an extended neighborhood of a rough equilibrium state of saddle-focus type, Math. USSR Sbornik, Vol. 10 No 1, (1970).

Smale, S. Differentiable dynamical systems. Bull. Amer. Math. Soc, 73, pp. 747-819, (1967).

Zacks, Mikhail A. Scaling Properties and renormalization invariants for the “homoclinic quasiperiodicity”. Physyca D, Vol 92, pp. 300-316, (1993).

Published

2018-09-24

How to Cite

[1]
S. Aranzubia, R. Carvajal, and R. Labarca, “A computer verification for the value of the Topological Entropy for some special subshifts in the Lexicographical Scenario.”, Proyecciones (Antofagasta, On line), vol. 37, no. 3, pp. 439-477, Sep. 2018.

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Artículos