Remarks on Iterated Cubic Maps

REFERENCES

• Benedicks , M. and Carleson , L. 1985 . “On the iterations of 1 – ax2 on (–1, 1)” . Ann. of Math. , 122 : 1 – 25 . [Benedicks and Carleson 1985]
   Web of Science ®Google Scholar
• Blanchard , P. 1984 . “Complex analytic dynamics on the Riemann sphere” . Bull. Amer. Math. Soc. , 11 : 85 – 141 . [Blanchard 1984]
   Web of Science ®Google Scholar
• Blanchard , P. 1986 . “Disconnected Julia sets”. ” . In Chaotic Dynamics and Fractals Edited by: Barnsley , M. and Demko , S. Orlando : Academic Press. . [Blanchard 1986]
   Google Scholar
• Blanchard , P. , Devaney , R. and Keen , L. 1991 . “The dynamics of complex polynomials and automorphisms of the shift” . Invent. Math. , 104 : 545 – 580 . [Blanchard et al. 1991]
   Web of Science ®Google Scholar
• Block , L. and Koesling , J. “Computing the topological entropy of maps of the interval with three monotone pieces” [Block and Keesling], (to appear)
   Google Scholar
• Branner , B. 1985 . “Iterations by odd functions with two extrcma” . J. Math. Anal. and Appl. , 105 : 276 [Branner 1985]
   Web of Science ®Google Scholar
• Branner , B. 1986 . “The parameter space for cubic polynomials”. ” . In Chaotic Dynamics and Fractals Edited by: Barnsley , M. and Demko , S. 169 – 179 . Orlando : Academic Press. . [Branner 1986]
   Google Scholar
• Branner , B. 1989 . “The Mandelbrot set”. ” . In Chaos and Fractals Edited by: Devaney , R. and Keen , L. 75 – 105 . Providence , RI : American Mathematical Society. . [Branner 1989], Proc. Symp. Applied Math. 39
   Google Scholar
• Branner , B. and Douady , A. “Surgery on complex polynomials” . Holomorphic Dynamics: Proceedings Symposium on Dynamical Systems . 1986 , Mexico . Edited by: Gomez-Mont , X. pp. 11 – 72 . Berlin : Springer-Verlag. . [Branner and Douady 1988], Lecture Notes in Math. 1345
   Google Scholar
• Branner , B. and Hubbard , J. H. 1988 . “The iteration of cubic polynomials, Part 1: The global topology of parameter space” . Acta Math. , 160 : 143 – 206 . [Branner and Hubbard 1988], (Part 2 to appear.)
   Web of Science ®Google Scholar
• Bredon , G. 1972 . Introduction to Compact Transformation Groups New York : Academic Press. . [Bredon 1972]
   Google Scholar
• Brown , M. 1960 . “A proof of the generalized Schoenflies theorem” . Bull. Amer. Math. Soc. , 66 : 74 – 76 . [Brown 1960]
   Google Scholar
• Brown , M. 1961 . “The monotone union of open n-cells is an open n-cell” . Proc. Amer. Math. Soc. , 12 : 812 – 814 . [Brown 1961]
   Google Scholar
• Chavoya-Aceves , O. , Angulo-Brown , F. and Piña , E. 1985 . “Symbolic dynamics of the cubic map” . Physica , D14 : 374 – 386 . [Chavoya-Aceves et al. 1985]
   Google Scholar
• Devaney , R. , Goldberg , L. and Hubbard , J. H. “A dynamical approximation to the exponential map by polynomials” [Devaney et al.], (to appear)
   Google Scholar
• Dewdney , A. K. 1987 . “Computer Recreations” . Scient. Amer. , Nov. : 144 [Dewdney 1987]
   Google Scholar
• Douady , A. 1983 . “Systèmes dynamiques ho-lomorphes” . Astérisque , 105–106 : 39 – 63 . [Douady 1983], Sém. Bourbaki 599
   Google Scholar
• Douady , A. and Hubbard , J. H. 1982 . “Itérations des polynômes quadratiques complexes” . Comptes R. Acad. Sci. Paris , 294 : 123 – 126 . [Douady and Hubbard 1982]
   Google Scholar
• Douady , A. and Hubbard , J. H. 1984 . “Etude dynamique des polynômes complexes I” . Publ. Math. Orsay , II [Douady and Hubbard 1984], (1985)
   Google Scholar
• Douady , A. and Hubbard , J. H. 1985 . “On the dynamics of polynomial-like mappings” . Ann. Sci. Ec. Norm. Sup. (Paris) , 18 : 287 [Douady and Hubbard 1985]
   Web of Science ®Google Scholar
• El Hamouly , H. and Mira , C. 1982 . “Singularités dues au feuilletage du plan des bifurcations d'un difféoinorphisme bi-dimensionel” . C. R. Acad. Sci. Paris , 294 : 387 [El Hamouly and Mira 1982]
   Google Scholar
• Fisher , Y. 1988 . “Exploring the Mandelbrot set”. ” . In The Science of Fractal Images Edited by: Peitgen , H.-O. and Saupe , D. 287 – 296 . New York : Springcr-Verlag. . [Fisher 1988]
   Google Scholar
• Friedland , S. and Milnor , J. 1989 . “Dynamical properties of plane polynomial automorphisms” . Ergodic Theory and Dynamical Systems , 9 : 67 – 99 . [Friedland and Milnor 1989]
   Web of Science ®Google Scholar
• Friedman , B. and Tresser , C. 1986 . “Comb structure in hairy boundaries: Some transition problems for circle maps” . Phys. Lett. , A117 : 15 [Friedman and Tresser 1986]
   Google Scholar
• Guckenheimer , J. 1979 . “Sensitive dependence to initial conditions for one dimensional maps” . Commun. Math. Phys. , 70 : 133 – 160 . [Guckenheimer 1979]
   Web of Science ®Google Scholar
• Jakobson , M. 1981 . “Absolutely continuous invariant measures for one-parameter families of one-dimensional maps” . Commun. Math. Phys. , 81 : 39 – 88 . [Jakobson 1981]
   Web of Science ®Google Scholar
• Lavaurs , P. 1989 . “Systémes dynamiques ho-lomorphes: Explosion de points périodiques para-boliques” , Thesis Orsay : Univ. Paris-Sud. . [Lavaurs 1989]
   Google Scholar
• MacKay , R. and Tresser , C. “Boundary of chaos for bimodal maps of the interval” . J. London Math. Soc. , [MacKay and Tresser], (to appear)
   Web of Science ®Google Scholar
• MacKay , R. and Tresser , C. 1987 . “Some flesh on the skeleton: The bifurcation structure of bimodal maps” . Physica , D27 : 412 – 422 . [MacKay and Tresser 1987]
   Google Scholar
• McMullen , C. 1988 . “Automorphisms of rational maps”. ” . In Holomorphic Functions and Moduli I Edited by: Drasin , D. 31 – 60 . New York : Springer-Verlag. . [McMullen 1988]
   Google Scholar
• Milnor , J. 1988 . “Nonexpansive Hénon maps” . Advances in Math. , 69 : 109 – 114 . [Milnor 1988]
   Web of Science ®Google Scholar
• Milnor , J. 1989 . “Self-similarity and hairiness in the Mandelbrot set”. ” . In Computers in Geometry and Topology Edited by: Tangora , M. 211 – 257 . New York : Dekker. . [Milnor 1989], Lecture Notes Pure Appl. Math. 114
   Google Scholar
• Milnor , J. 1990/5 . Dynamics in One Complex Variable Introductory Lectures. . [Milnor 1990], Stony Brook IMS preprints
   Google Scholar
• Milnor , J. 1992/3 . “Hyperbolic components in spaces of polynomial maps” [Milnor 1992], Stoney Brook IMS prepints
   Google Scholar
• Milnor , J. “On cubic polynomials with periodic critical point” [Milnor], (in preparation)
   Google Scholar
• Milnor , J. and Thurston , W. 1988 . “Iterated maps of the interval”. ” . In Dynamical Systems (Maryland 1986–87) Edited by: Alexander , J. 1342 Berlin : Springer-Verlag. . [Milnor and Thurston 1988], Lecture Notes in Math
   Google Scholar
• Misiurewicz , M. and Szlenk , W. 1977 . “Entropy of piecewise monotone mappings” . Astérisque , 50 : 299 – 310 . [Misiurewicz and Szlenk 1977], Revised version in Studio Math. 67(1980), 45–63
   Google Scholar
• Piña , E. 1984 . “Comment on ‘Study of a one-dimensional map with multiple basins'” . Phys. Rev. , A30 : 2132 – 2133 . [Piña 1984]
   Google Scholar
• Rees , M. 1990 . “Components of degree two hyperbolic rational maps” . Invent. Math. , 100 : 357 – 382 . [Rees 1990]
   Web of Science ®Google Scholar
• Rees , M. 1992 . “A partial description of parameter space of rational maps of degree two” . Acta Math. , 168 : 11 – 87 . [Rees 1992], Part 1, Part 2: Stony Brook IMS preprints 1991/4
   Web of Science ®Google Scholar
• Ringland , J. and Schell , M. “Genealogy and bifurcation skeleton for cycles of the iterated two-extremum map of the interval” [Ringland and Schell], (to appear)
   Google Scholar
• Ringland , J. and Schell , M. 1989 . “The Farey tree embodied–in bimodal maps of the interval” . Phys. Lett. , A136 : 379 – 386 . [Ringland and Schell 1989]
   Google Scholar
• Ringland , J. and Schell , M. 1990 . “Universal geometry in the parameter plane of maps of the interval” . Europhys. Lett. , 12 : 595 – 601 . [Ringland and Schell 1990]
   Google Scholar
• Ringland , J. , Issa , N. and Schell , M. 1990 . “From U sequence to Farey sequence: A unification of one-parameter scenarios” . Phys. Rev. , A41 : 4223 – 4235 . [Ringland et al. 1990]
   Google Scholar
• Rogers , T. and Whitley , D. 1983 . “haos in the cubic mapping” . Math. Modelling , 4 : 9 – 25 . [Rogers and Whitley 1983]
   Web of Science ®Google Scholar
• Rothschild , J. 1971 . “On the computation of topological entropy” , Thesis New York : CUNY. . [Rothschild 1971]
   Google Scholar
• Rychlik , M. 1988 . “Another proof of Jakobson's theorem and related results” . Erg. Theory and Dyn. Sys. , 8 : 93 – 109 . [Rychlik 1988]
   Google Scholar
• Skjolding , H. , Branner , B. , Christiansen , P. and Jensen , H. 1983 . “Bifurcations in discrete dynamical systems with cubic maps” . Siam J. Appl. Math. , 43 : 520 – 534 . [Skjolding et al. 1983]
   Web of Science ®Google Scholar
• Winters , R. “Of tricorns and biquadratics” Boston University. . [Winters], in preparation
   Google Scholar
• Wittner , B. 1988 . “On the bifurcation loci of rational maps of degree two” , Thesis Ithaca : Cornell University. . [Wittner 1988]
   Google Scholar