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A Monotonicity Conjecture for Real Cubic Maps

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Part of the NATO ASI Series book series (ASIC, volume 464)

Abstract

This will be an outline of work in progress. We study the conjecture that the topological entropy of a real cubic map depends “monotonely” on its parameters, in the sense that each locus of constant entropy in parameter space is a connected set.

Keywords

Periodic Orbit Entropy Function Topological Entropy Order Type Hyperbolic Component 
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References

  1. [B]
    R. Bowen,On Axiom A Diffeomorphisms, Proc. Reg. Conf. Math. 35, 1978.Google Scholar
  2. [B K]
    L. Block and J. Keesling, “Computing topological entropy of maps of the interval with three monotone pieces”,J. Statist. Phys.66 (1992) 755–774.MathSciNetzbMATHCrossRefGoogle Scholar
  3. [BMT]
    K. Brucks, M. Misiurewicz, and C. Tresser, “Monotonicity properties of the family of trapezoidal maps,Commun. Math. Phys.137 (1991) 1–12.MathSciNetzbMATHCrossRefGoogle Scholar
  4. [B R]
    V. Baladi and D. Ruelle, “An extension of the theorem of Milnor and Thurston on the zeta functions of interval maps”, Ergodic Theory and Dynamical Systems, to appearGoogle Scholar
  5. [BST]
    N.J.Balmforth, E.A.Spiegel and C. Tresser, “The topological entropy of one-dimensional maps: approximation and bounds”, to appear.Google Scholar
  6. [D]
    A. Douady, “Topological entropy of unimodal maps”, Proceedings NATO Institute on Real and Complex Dynamical Systems, Hiller0d 1993, these proceedings.Google Scholar
  7. [D G]
    S.P. Dawson and C. Grebogi, “Cubic maps as models of two-dimensional antimono- tonicity,Chaos,Solitons & Fractals1 (1991), 137–144.Google Scholar
  8. [DGK]
    S.P. Dawson, C. Grebogi and H. Koçak, “A geometric mechanism for antimonotonicity in scalar maps with two critical points,” Phys. Rev. E (in press).Google Scholar
  9. [DGKKY]
    S.P. Dawson, C. Grebogi, I. Kan, H. Koçak and J.A. Yorke, “Antimonotonicity: inevitable reversals of period doubling cascades,Phys. Lett. A162 (1992) 249–254.CrossRefGoogle Scholar
  10. [DGMT]
    S.P. Dawson, R. Galeeva, J. Milnor and C. Tresser, “Monotonicity and antimonotonicity for bimodal maps”, manuscript in preparation.Google Scholar
  11. [DH1]
    A.Douady and J.Hubbard, “A proof of Thurston’s Topological Characterization of Rational Maps”; Preprint, Institute Mittag-Leffler 1984.Google Scholar
  12. [DH2]
    A. Douady and J. H. Hubbard, “Etude dynamique des polynômes quadratiques complexes,” I (1984) & II (1985), Pub/. Mat. d’Orsay.Google Scholar
  13. [dM vS]
    W. De Melo and S. Van Strien, One Dimensional Dynamics, Springer V., 1993.Google Scholar
  14. [F]
    P. Fatou, “Sur les équations fonctionnelles, II”,Bull. Soc. Math.France 48 (1920) 33–94.MathSciNetGoogle Scholar
  15. [Ga]
    R. Galeeva, “Kneading sequences for piecewise linear bimodal maps”, to appear.Google Scholar
  16. [Gu]
    J. Guckenheimer, “Dynamical Systems”, C.I.M.E. Lectures (J. Guckenheimer, J. Moser and S. Newhouse), Birkhäuser, Progress in Mathematics8, 1980.Google Scholar
  17. [K]
    A. Katok, “Lyapunov exponents, entropy and periodic orbits of diffeomorphisms”,Pub. Math.IHES 51 (1980) 137–173.MathSciNetzbMATHGoogle Scholar
  18. [L]
    M. Lyubich, “Geometry of quadratic polynomials: moduli, rigidity, and local connectivity”, Stony Brook I.M.S. Preprint 1993/9.Google Scholar
  19. [Ml]
    J. Milnor, “Remarks on iterated cubic maps”,Experimental Math.1 (1992) 5–24.MathSciNetzbMATHGoogle Scholar
  20. [M2]
    J. Milnor, “Hyperbolic components in Spaces of Polynomial Maps (with an appendix by A. Poirier)”, Stony Brook I.M.S. Preprint 1992#3.Google Scholar
  21. [M3]
    J. Milnor, “On cubic polynomials with periodic critical point”, in preparation.Google Scholar
  22. [Ma T]
    R.S MacKay and C. Tresser,`Boundary of topological chaos for bimodal maps of the interval,J. London Math. Soc.37 (1988), 164–81; “Some flesh on the skeleton: the bifurcation structure of bimodal maps”,Physica 27D (1987) 412–422.MathSciNetGoogle Scholar
  23. [Mcl]
    C. McMullen, “Automorphisms of rational maps”, pp. 31–60 ofHolomorphic Functions and Moduli I, ed. Drasin, Earle, Gehring, Kra & Marden; MSRI Publ. 10, Springer 1988.Google Scholar
  24. [Mc2]
    C. McMullen, “Complex dynamics and renormalization”, preprint, U.C. Berkeley 1993.Google Scholar
  25. [Mis]
    M. Misiurewicz, “On non-continuity of topological entropy”,Bull. Ac Pol. Sci.,Ser. Sci. Math. Astr.Phys.19 (1971) 319–320.Google Scholar
  26. [M Sz]
    M. Misiurewicz and W. Szlenk, “Entropy of piecewise monotone mappings”,Studia Math.67 (1980) 45–63Google Scholar
  27. M. Misiurewicz and W. Szlenk, “Entropy of piecewise monotone mappings”,Astérisque 50 (1977) 299–310MathSciNetGoogle Scholar
  28. [MTh]
    J. Milnor and W. Thurston, “On iterated maps of the interval,Springer Lecture Notes 1342 (1988), 465–563.MathSciNetGoogle Scholar
  29. [N]
    S. Newhouse, “Continuity properties of entropy”,Ann. Math.129 (1989) 215–235Google Scholar
  30. S. Newhouse, “Continuity properties of entropy”,Ann. Math.131 (1990) 409–410.Google Scholar
  31. [Po]
    A. Poirier, “On post critically finite polynomials, Part II, Hubbard Trees”, Stony Brook I.M.S. Preprint 1993/7.Google Scholar
  32. [Pr]
    C. Preston, “What you need to know to knead”,Advances Math.78 (1989) 192–252.MathSciNetzbMATHCrossRefGoogle Scholar
  33. [Ro]
    J. Rothschild, “On the computation of topological entropy”, Thesis, CUNY 1971.Google Scholar
  34. [R S]
    J. Ringland and M. Schell, “Genealogy and bifurcation skeleton for cycles of the iterated two-extremum map of the interval”,SIAM J. Math. Anal.22 (1991) 1354–1371.MathSciNetzbMATHGoogle Scholar
  35. [R T]
    J. Ringland and C. Tresser, “A genealogy for finite kneading sequences of bimodal maps of the interval”, preprint, IBM 1993.Google Scholar
  36. [St]
    J. Stimson, “Degree two rational maps with a periodic critical point”, Thesis, Univ. Liverpool 1993.Google Scholar
  37. [Sw]
    G. Swiatek, “Hyperbolicity is dense in the real quadratic family”, Stony Brook I.M.S. Preprint 1992/10.Google Scholar
  38. [Y]
    Y. Yomdin, “Volume growth and entropy”, Isr. J. Math. 57 (1987) 285–300. (See also “ CI`-resolution of semialgebraic mappings”, ibid. pp. 301–317.).Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 1995

Authors and Affiliations

  1. 1.Los Alamos National LaboratoryC.N.L.S.Los AlamosUSA
  2. 2.Mathematics Dept.Northwestern Univ.EvanstonUSA
  3. 3.Institute for Mathematical SciencesSUNYStony BrookUSA
  4. 4.I.B.M.Yorktown HeightsUSA

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