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.
Based on lectures by Milnor at the NATO Advanced Study Institute on Real and Complex Dynamical Systems, Hillerød, June 1993.
Partially supported by the Miller Institute of the University of California at Berkeley during the preparation of this paper.
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References
R. Bowen,On Axiom A Diffeomorphisms, Proc. Reg. Conf. Math. 35, 1978.
L. Block and J. Keesling, “Computing topological entropy of maps of the interval with three monotone pieces”,J. Statist. Phys.66 (1992) 755–774.
K. Brucks, M. Misiurewicz, and C. Tresser, “Monotonicity properties of the family of trapezoidal maps,Commun. Math. Phys.137 (1991) 1–12.
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 appear
N.J.Balmforth, E.A.Spiegel and C. Tresser, “The topological entropy of one-dimensional maps: approximation and bounds”, to appear.
A. Douady, “Topological entropy of unimodal maps”, Proceedings NATO Institute on Real and Complex Dynamical Systems, Hiller0d 1993, these proceedings.
S.P. Dawson and C. Grebogi, “Cubic maps as models of two-dimensional antimono- tonicity,Chaos,Solitons & Fractals1 (1991), 137–144.
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).
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.
S.P. Dawson, R. Galeeva, J. Milnor and C. Tresser, “Monotonicity and antimonotonicity for bimodal maps”, manuscript in preparation.
A.Douady and J.Hubbard, “A proof of Thurston’s Topological Characterization of Rational Maps”; Preprint, Institute Mittag-Leffler 1984.
A. Douady and J. H. Hubbard, “Etude dynamique des polynômes quadratiques complexes,” I (1984) & II (1985), Pub/. Mat. d’Orsay.
W. De Melo and S. Van Strien, One Dimensional Dynamics, Springer V., 1993.
P. Fatou, “Sur les équations fonctionnelles, II”,Bull. Soc. Math.France 48 (1920) 33–94.
R. Galeeva, “Kneading sequences for piecewise linear bimodal maps”, to appear.
J. Guckenheimer, “Dynamical Systems”, C.I.M.E. Lectures (J. Guckenheimer, J. Moser and S. Newhouse), Birkhäuser, Progress in Mathematics8, 1980.
A. Katok, “Lyapunov exponents, entropy and periodic orbits of diffeomorphisms”,Pub. Math.IHES 51 (1980) 137–173.
M. Lyubich, “Geometry of quadratic polynomials: moduli, rigidity, and local connectivity”, Stony Brook I.M.S. Preprint 1993/9.
J. Milnor, “Remarks on iterated cubic maps”,Experimental Math.1 (1992) 5–24.
J. Milnor, “Hyperbolic components in Spaces of Polynomial Maps (with an appendix by A. Poirier)”, Stony Brook I.M.S. Preprint 1992#3.
J. Milnor, “On cubic polynomials with periodic critical point”, in preparation.
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.
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.
C. McMullen, “Complex dynamics and renormalization”, preprint, U.C. Berkeley 1993.
M. Misiurewicz, “On non-continuity of topological entropy”,Bull. Ac Pol. Sci.,Ser. Sci. Math. Astr.Phys.19 (1971) 319–320.
M. Misiurewicz and W. Szlenk, “Entropy of piecewise monotone mappings”,Studia Math.67 (1980) 45–63
M. Misiurewicz and W. Szlenk, “Entropy of piecewise monotone mappings”,Astérisque 50 (1977) 299–310
J. Milnor and W. Thurston, “On iterated maps of the interval,Springer Lecture Notes 1342 (1988), 465–563.
S. Newhouse, “Continuity properties of entropy”,Ann. Math.129 (1989) 215–235
S. Newhouse, “Continuity properties of entropy”,Ann. Math.131 (1990) 409–410.
A. Poirier, “On post critically finite polynomials, Part II, Hubbard Trees”, Stony Brook I.M.S. Preprint 1993/7.
C. Preston, “What you need to know to knead”,Advances Math.78 (1989) 192–252.
J. Rothschild, “On the computation of topological entropy”, Thesis, CUNY 1971.
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.
J. Ringland and C. Tresser, “A genealogy for finite kneading sequences of bimodal maps of the interval”, preprint, IBM 1993.
J. Stimson, “Degree two rational maps with a periodic critical point”, Thesis, Univ. Liverpool 1993.
G. Swiatek, “Hyperbolicity is dense in the real quadratic family”, Stony Brook I.M.S. Preprint 1992/10.
Y. Yomdin, “Volume growth and entropy”, Isr. J. Math. 57 (1987) 285–300. (See also “ CI`-resolution of semialgebraic mappings”, ibid. pp. 301–317.).
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Dawson, S.P., Galeeva, R., Milnor, J., Tresser, C. (1995). A Monotonicity Conjecture for Real Cubic Maps. In: Branner, B., Hjorth, P. (eds) Real and Complex Dynamical Systems. NATO ASI Series, vol 464. Springer, Dordrecht. https://doi.org/10.1007/978-94-015-8439-5_7
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