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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

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 & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

A. Blokh and L. Oversteegen, {Monotone images of Cremer Julia sets, Houston Journal of Mathematics, 36 (2010), 469-476.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

X. Buff and C. Henriksen, Julia Sets in Parameter Spaces, Commun. Math. Phys., 220 (2001), 333-375. doi: 10.1007/PL00005568.  Google Scholar

[11]

C. Carathéodory, Über die Begrenzung einfach zusammenhängender Gebiete (German), Math. Ann., 73 (1913), 323-370. doi: 10.1007/BF01456699.  Google Scholar

[12]

L. Carleson and T. W. Gamelin, Complex Dynamics, Springer, 1993. doi: 10.1007/978-1-4612-4364-9.  Google Scholar

[13]

A. Douady and J. H. Hubbard, Étude dynamique des polynômes complexes I, Publications Mathématiques d'Orsay, 1984.  Google Scholar

[14]

A. Douady and J. H. Hubbard, Étude dynamique des polynômes complexes II, Publications Mathématiques d'Orsay, 85-04, 1985.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

J. Kiwi, Wandering orbit portraits, Trans. of the Amer. Math. Soc., 354 (2002), 1473-1485. doi: 10.1090/S0002-9947-01-02896-3.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[21]

J. Milnor, Geometry and dynamics of quadratic rational maps, Experimental Math., 2 (1993), 37-83. doi: 10.1080/10586458.1993.10504267.  Google Scholar

[22]

J. Milnor, Dynamics in One Complex Variable, Annals of Mathematical Studies, 160, Princeton, 2006.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[26]

M. Misiurewicz, Horseshoes for mappings of the interval, Bull. Acad. Pol. Sci., Ser. sci. math., astr. et phys., 27 (1979), 167-169.  Google Scholar

[27]

M. Misiurewicz and W. Szlenk, Entropy of piecewise monotone mappings, Studia Math., 67 (1980), 45-63.  Google Scholar

[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.  Google Scholar

[29]

C. L. Petersen, P. Roesch and T. Lei, Parabolic slices on the boundary of $\mathcal H$,, work in progress., ().   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[33]

S. Zakeri, Dynamics of cubic Siegel polynomials, Comm. Math. Phys., 206 (1999), 185-233. doi: 10.1007/s002200050702.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

A. Blokh and L. Oversteegen, {Monotone images of Cremer Julia sets, Houston Journal of Mathematics, 36 (2010), 469-476.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

X. Buff and C. Henriksen, Julia Sets in Parameter Spaces, Commun. Math. Phys., 220 (2001), 333-375. doi: 10.1007/PL00005568.  Google Scholar

[11]

C. Carathéodory, Über die Begrenzung einfach zusammenhängender Gebiete (German), Math. Ann., 73 (1913), 323-370. doi: 10.1007/BF01456699.  Google Scholar

[12]

L. Carleson and T. W. Gamelin, Complex Dynamics, Springer, 1993. doi: 10.1007/978-1-4612-4364-9.  Google Scholar

[13]

A. Douady and J. H. Hubbard, Étude dynamique des polynômes complexes I, Publications Mathématiques d'Orsay, 1984.  Google Scholar

[14]

A. Douady and J. H. Hubbard, Étude dynamique des polynômes complexes II, Publications Mathématiques d'Orsay, 85-04, 1985.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

J. Kiwi, Wandering orbit portraits, Trans. of the Amer. Math. Soc., 354 (2002), 1473-1485. doi: 10.1090/S0002-9947-01-02896-3.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[21]

J. Milnor, Geometry and dynamics of quadratic rational maps, Experimental Math., 2 (1993), 37-83. doi: 10.1080/10586458.1993.10504267.  Google Scholar

[22]

J. Milnor, Dynamics in One Complex Variable, Annals of Mathematical Studies, 160, Princeton, 2006.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[26]

M. Misiurewicz, Horseshoes for mappings of the interval, Bull. Acad. Pol. Sci., Ser. sci. math., astr. et phys., 27 (1979), 167-169.  Google Scholar

[27]

M. Misiurewicz and W. Szlenk, Entropy of piecewise monotone mappings, Studia Math., 67 (1980), 45-63.  Google Scholar

[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.  Google Scholar

[29]

C. L. Petersen, P. Roesch and T. Lei, Parabolic slices on the boundary of $\mathcal H$,, work in progress., ().   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[33]

S. Zakeri, Dynamics of cubic Siegel polynomials, Comm. Math. Phys., 206 (1999), 185-233. doi: 10.1007/s002200050702.  Google Scholar

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