Laminations from the main cubioid

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September 2016, 36(9): 4665-4702. doi: 10.3934/dcds.2016003

Laminations from the main cubioid

Alexander Blokh ^1, , Lex Oversteegen ^1, , Ross Ptacek ^2, and Vladlen Timorin ^2,

1.	Department of Mathematics, University of Alabama at Birmingham, Birmingham, AL 35294-1170
2.	Faculty of Mathematics, Laboratory of Algebraic Geometry and its Applications, National Research University Higher School of Economics, Vavilova St. 7, 112312 Moscow, Russian Federation, Russian Federation

Received May 2013 Revised January 2016 Published May 2016

Abstract
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Polynomials from the closure of the principal hyperbolic domain of the cubic connectedness locus have some specific properties, which were studied in a recent paper by the authors. The family of (affine conjugacy classes of) all polynomials with these properties is called the Main Cubioid. In this paper, we describe a combinatorial counterpart of the Main Cubioid --- the set of invariant laminations that can be associated to polynomials from the Main Cubioid.

Keywords: Complex dynamics, cubic polynomial., invariant lamination, Mandelbrot set, Julia set.

Mathematics Subject Classification: Primary: 37F20; Secondary: 37C25, 37F10, 37F5.

Citation: Alexander Blokh, Lex Oversteegen, Ross Ptacek, Vladlen Timorin. Laminations from the main cubioid. Discrete and Continuous Dynamical Systems, 2016, 36 (9) : 4665-4702. doi: 10.3934/dcds.2016003

References:

[1]	L. Alseda, J. Llibre and M. Misiurewicz, Combinatorial Dynamics and Entropy in Dimension One, World Scientific (Advanced Series in Nonlinear Dynamics, vol. 5), Second Edition, 2000. doi: 10.1142/4205.
[2]	A. Blokh, C. Curry and L. Oversteegen, Locally connected models for Julia sets, Advances in Mathematics, 226 (2011), 1621-1661. doi: 10.1016/j.aim.2010.08.011.
[3]	A. Blokh, R. Fokkink, J. Mayer, L. Oversteegen and E. Tymchatyn, Fixed point theorems in plane continua with applications, Memoirs of the American Mathematical Society, 224 (2013), xiv+97 pp. doi: 10.1090/S0065-9266-2012-00671-X.
[4]	A. Blokh and G. Levin, Growing trees, laminations and the dynamics on the Julia set, Ergod. Th. and Dynam. Sys., 22 (2002), 63-97. doi: 10.1017/S0143385702000032.
[5]	A. Blokh, J. Malaugh, J. Mayer, L. Oversteegen and D. Parris, Rotational subsets of the circle under $z^n$, Topology and its Appl., 153 (2006), 1540-1570. doi: 10.1016/j.topol.2005.04.010.
[6]	A. Blokh, D. Mimbs, L. Oversteegen and K. Valkenburg, Laminations in the language of leaves, Trans. of the Amer. Math. Soc., 365 (2013), 5367-5391. doi: 10.1090/S0002-9947-2013-05809-6.
[7]	A. Blokh and L. Oversteegen, {Monotone images of Cremer Julia sets, Houston Journal of Mathematics, 36 (2010), 469-476.
[8]	A. Blokh, L. Oversteegen, R. Ptacek and V. Timorin, Dynamical cores of topological polynomials, Frontiers in complex dynamics, Princeton Math. Ser., Princeton Univ. Press, Princeton, NJ, 51 (2014), 27-48. doi: 10.1515/9781400851317-005.
[9]	A. Blokh, L. Oversteegen, R. Ptacek and V. Timorin, The main cubioid, Nonlinearity, 27 (2014), 1879-1897. doi: 10.1088/0951-7715/27/8/1879.
[10]	X. Buff and C. Henriksen, Julia Sets in Parameter Spaces, Commun. Math. Phys., 220 (2001), 333-375. doi: 10.1007/PL00005568.
[11]	C. Carathéodory, Über die Begrenzung einfach zusammenhängender Gebiete (German), Math. Ann., 73 (1913), 323-370. doi: 10.1007/BF01456699.
[12]	L. Carleson and T. W. Gamelin, Complex Dynamics, Springer, 1993. doi: 10.1007/978-1-4612-4364-9.
[13]	A. Douady and J. H. Hubbard, Étude dynamique des polynômes complexes I, Publications Mathématiques d'Orsay, 1984.
[14]	A. Douady and J. H. Hubbard, Étude dynamique des polynômes complexes II, Publications Mathématiques d'Orsay, 85-04, 1985.
[15]	A. Epstein and M. Yampolsky, Geography of the Cubic Connectedness Locus: Intertwining Surgery, Ann. Sci. Éc. Norm. Sup., 32 (1999), 151-185. doi: 10.1016/S0012-9593(99)80013-5.
[16]	T. Gauthier, Higher bifurcation currents, neutral cycles, and the Mandelbrot set, Indiana Univ. Math. J., 63 (2014), 917-937. doi: 10.1512/iumj.2014.63.5328.
[17]	L. Goldberg and J. Milnor, Fixed points of polynomial maps. II. Fixed point portraits, Ann. Sci. École Norm. Sup. (4), 26 (1993), 51-98.
[18]	J. Kiwi, Wandering orbit portraits, Trans. of the Amer. Math. Soc., 354 (2002), 1473-1485. doi: 10.1090/S0002-9947-01-02896-3.
[19]	J. Kiwi, $\mathbb R$eal laminations and the topological dynamics of complex polynomials, Advances in Mathematics, 184 (2004), 207-267. doi: 10.1016/S0001-8708(03)00144-0.
[20]	C. McMullen, The Mandelbrot set is universal, in: The Mandelbrot Set, Theme and Variations, ed. T. Lei, Cambridge U.K. Cambridge Univ. Press. Revised, 274 (2007), 1-17.
[21]	J. Milnor, Geometry and dynamics of quadratic rational maps, Experimental Math., 2 (1993), 37-83. doi: 10.1080/10586458.1993.10504267.
[22]	J. Milnor, Dynamics in One Complex Variable, Annals of Mathematical Studies, 160, Princeton, 2006.
[23]	J. Milnor, Cubic polynomial maps with periodic critical orbit I, in: Complex Dynamics, Families and Friends, ed. D. Schleicher, A.K. Peters (2009), 333-411. doi: 10.1201/b10617-13.
[24]	J. Milnor and A. Poirier, Hyperbolic components in spaces of polynomial maps, Contemp. Math., Conformal dynamics and hyperbolic geometry, Amer. Math. Soc., Providence, RI, 573 (2012), 183-232. arXiv:math/9202210 doi: 10.1090/conm/573/11428.
[25]	J. Milnor and W. Thurston, On iterated maps of the interval, in Dynamical systems, Lecture Notes in Math., 1342, Springer, Berlin, (1988), 465-563. doi: 10.1007/BFb0082847.
[26]	M. Misiurewicz, Horseshoes for mappings of the interval, Bull. Acad. Pol. Sci., Ser. sci. math., astr. et phys., 27 (1979), 167-169.
[27]	M. Misiurewicz and W. Szlenk, Entropy of piecewise monotone mappings, Studia Math., 67 (1980), 45-63.
[28]	C. L. Petersen and T. Lei, Analytic coordinates recording cubic dynamics, In: Complex Dynamics: Families and Friends, ed. Dierk Schleicher, Wellesley Massachusetts: A. K. Peters, Limited, (2009), 413-449. doi: 10.1201/b10617-14.
[29]	C. L. Petersen, P. Roesch and T. Lei, Parabolic slices on the boundary of $\mathcal H$,, work in progress., ().
[30]	P. Roesch, Hyperbolic components of polynomials with a fixed critical point of maximal order, Ann. Sci. école Norm. Sup. (4), 40 (2007), 901-949. doi: 10.1016/j.ansens.2007.10.001.
[31]	W. Thurston, On the geometry and dynamics of iterated rational maps, in: Complex dynamics: Families and Friends, ed. by D. Schleicher, A K Peters, (2009), 3-137. doi: 10.1201/b10617-3.
[32]	L.-S. Young, On the prevalence of horseshoes, Trans. Amer. Math. Soc., 263 (1981), 75-88. doi: 10.1090/S0002-9947-1981-0590412-0.
[33]	S. Zakeri, Dynamics of cubic Siegel polynomials, Comm. Math. Phys., 206 (1999), 185-233. doi: 10.1007/s002200050702.

show all references

References:

[1]	L. Alseda, J. Llibre and M. Misiurewicz, Combinatorial Dynamics and Entropy in Dimension One, World Scientific (Advanced Series in Nonlinear Dynamics, vol. 5), Second Edition, 2000. doi: 10.1142/4205.
[2]	A. Blokh, C. Curry and L. Oversteegen, Locally connected models for Julia sets, Advances in Mathematics, 226 (2011), 1621-1661. doi: 10.1016/j.aim.2010.08.011.
[3]	A. Blokh, R. Fokkink, J. Mayer, L. Oversteegen and E. Tymchatyn, Fixed point theorems in plane continua with applications, Memoirs of the American Mathematical Society, 224 (2013), xiv+97 pp. doi: 10.1090/S0065-9266-2012-00671-X.
[4]	A. Blokh and G. Levin, Growing trees, laminations and the dynamics on the Julia set, Ergod. Th. and Dynam. Sys., 22 (2002), 63-97. doi: 10.1017/S0143385702000032.
[5]	A. Blokh, J. Malaugh, J. Mayer, L. Oversteegen and D. Parris, Rotational subsets of the circle under $z^n$, Topology and its Appl., 153 (2006), 1540-1570. doi: 10.1016/j.topol.2005.04.010.
[6]	A. Blokh, D. Mimbs, L. Oversteegen and K. Valkenburg, Laminations in the language of leaves, Trans. of the Amer. Math. Soc., 365 (2013), 5367-5391. doi: 10.1090/S0002-9947-2013-05809-6.
[7]	A. Blokh and L. Oversteegen, {Monotone images of Cremer Julia sets, Houston Journal of Mathematics, 36 (2010), 469-476.
[8]	A. Blokh, L. Oversteegen, R. Ptacek and V. Timorin, Dynamical cores of topological polynomials, Frontiers in complex dynamics, Princeton Math. Ser., Princeton Univ. Press, Princeton, NJ, 51 (2014), 27-48. doi: 10.1515/9781400851317-005.
[9]	A. Blokh, L. Oversteegen, R. Ptacek and V. Timorin, The main cubioid, Nonlinearity, 27 (2014), 1879-1897. doi: 10.1088/0951-7715/27/8/1879.
[10]	X. Buff and C. Henriksen, Julia Sets in Parameter Spaces, Commun. Math. Phys., 220 (2001), 333-375. doi: 10.1007/PL00005568.
[11]	C. Carathéodory, Über die Begrenzung einfach zusammenhängender Gebiete (German), Math. Ann., 73 (1913), 323-370. doi: 10.1007/BF01456699.
[12]	L. Carleson and T. W. Gamelin, Complex Dynamics, Springer, 1993. doi: 10.1007/978-1-4612-4364-9.
[13]	A. Douady and J. H. Hubbard, Étude dynamique des polynômes complexes I, Publications Mathématiques d'Orsay, 1984.
[14]	A. Douady and J. H. Hubbard, Étude dynamique des polynômes complexes II, Publications Mathématiques d'Orsay, 85-04, 1985.
[15]	A. Epstein and M. Yampolsky, Geography of the Cubic Connectedness Locus: Intertwining Surgery, Ann. Sci. Éc. Norm. Sup., 32 (1999), 151-185. doi: 10.1016/S0012-9593(99)80013-5.
[16]	T. Gauthier, Higher bifurcation currents, neutral cycles, and the Mandelbrot set, Indiana Univ. Math. J., 63 (2014), 917-937. doi: 10.1512/iumj.2014.63.5328.
[17]	L. Goldberg and J. Milnor, Fixed points of polynomial maps. II. Fixed point portraits, Ann. Sci. École Norm. Sup. (4), 26 (1993), 51-98.
[18]	J. Kiwi, Wandering orbit portraits, Trans. of the Amer. Math. Soc., 354 (2002), 1473-1485. doi: 10.1090/S0002-9947-01-02896-3.
[19]	J. Kiwi, $\mathbb R$eal laminations and the topological dynamics of complex polynomials, Advances in Mathematics, 184 (2004), 207-267. doi: 10.1016/S0001-8708(03)00144-0.
[20]	C. McMullen, The Mandelbrot set is universal, in: The Mandelbrot Set, Theme and Variations, ed. T. Lei, Cambridge U.K. Cambridge Univ. Press. Revised, 274 (2007), 1-17.
[21]	J. Milnor, Geometry and dynamics of quadratic rational maps, Experimental Math., 2 (1993), 37-83. doi: 10.1080/10586458.1993.10504267.
[22]	J. Milnor, Dynamics in One Complex Variable, Annals of Mathematical Studies, 160, Princeton, 2006.
[23]	J. Milnor, Cubic polynomial maps with periodic critical orbit I, in: Complex Dynamics, Families and Friends, ed. D. Schleicher, A.K. Peters (2009), 333-411. doi: 10.1201/b10617-13.
[24]	J. Milnor and A. Poirier, Hyperbolic components in spaces of polynomial maps, Contemp. Math., Conformal dynamics and hyperbolic geometry, Amer. Math. Soc., Providence, RI, 573 (2012), 183-232. arXiv:math/9202210 doi: 10.1090/conm/573/11428.
[25]	J. Milnor and W. Thurston, On iterated maps of the interval, in Dynamical systems, Lecture Notes in Math., 1342, Springer, Berlin, (1988), 465-563. doi: 10.1007/BFb0082847.
[26]	M. Misiurewicz, Horseshoes for mappings of the interval, Bull. Acad. Pol. Sci., Ser. sci. math., astr. et phys., 27 (1979), 167-169.
[27]	M. Misiurewicz and W. Szlenk, Entropy of piecewise monotone mappings, Studia Math., 67 (1980), 45-63.
[28]	C. L. Petersen and T. Lei, Analytic coordinates recording cubic dynamics, In: Complex Dynamics: Families and Friends, ed. Dierk Schleicher, Wellesley Massachusetts: A. K. Peters, Limited, (2009), 413-449. doi: 10.1201/b10617-14.
[29]	C. L. Petersen, P. Roesch and T. Lei, Parabolic slices on the boundary of $\mathcal H$,, work in progress., ().
[30]	P. Roesch, Hyperbolic components of polynomials with a fixed critical point of maximal order, Ann. Sci. école Norm. Sup. (4), 40 (2007), 901-949. doi: 10.1016/j.ansens.2007.10.001.
[31]	W. Thurston, On the geometry and dynamics of iterated rational maps, in: Complex dynamics: Families and Friends, ed. by D. Schleicher, A K Peters, (2009), 3-137. doi: 10.1201/b10617-3.
[32]	L.-S. Young, On the prevalence of horseshoes, Trans. Amer. Math. Soc., 263 (1981), 75-88. doi: 10.1090/S0002-9947-1981-0590412-0.
[33]	S. Zakeri, Dynamics of cubic Siegel polynomials, Comm. Math. Phys., 206 (1999), 185-233. doi: 10.1007/s002200050702.

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