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April
2022, 27(4): 2051-2064.
doi: 10.3934/dcdsb.2021121
On the family of cubic parabolic polynomials
Departamento de Matemática, Universidade Federal de Viçosa, Viçosa, MG 36570-900, Brazil |
For a sequence $ (a_n) $ of complex numbers we consider the cubic parabolic polynomials $ f_n(z) = z^3+a_n z^2+z $ and the sequence $ (F_n) $ of iterates $ F_n = f_n\circ\dots\circ f_1 $. The Fatou set $ \mathcal{F}_0 $ is the set of all $ z\in\hat{\mathbb{C}} $ such that the sequence $ (F_n) $ is normal. The complement of the Fatou set is called the Julia set and denoted by $ \mathcal{J}_0 $. The aim of this paper is to study some properties of $ \mathcal{J}_0 $. As a particular case, when the sequence $ (a_n) $ is constant, $ a_n = a $, then the iteration $ F_n $ becomes the classical iteration $ f^n $ where $ f(z) = z^3+a z^2+z $. The connectedness locus, $ M $, is the set of all $ a\in\mathbb{C} $ such that the Julia set is connected. In this paper we investigate some symmetric properties of $ M $ as well.
Mathematics Subject Classification:
Primary: 30D05; Secondary: 37F10, 37F12.
Citation:
Alexandre Alves, Mostafa Salarinoghabi. On the family of cubic parabolic polynomials. Discrete and Continuous Dynamical Systems - B,
2022, 27
(4)
: 2051-2064.
doi: 10.3934/dcdsb.2021121
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References:
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A. Beardon, Iteration of Rational Functions, Springer-Verlag, New York, 1991. |
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A. Bonifant, J. Kiwi and J. Milnor,
Cubic polynomial maps with periodic critical orbit. II. Escape regions, Conform. Geom. Dyn., 14 (2010), 68-112.
doi: 10.1090/S1088-4173-10-00204-3. |
| [3] |
B. Branner and J. Hubbard,
The iteration of cubic polynomials. I. The global topology of parameter space, Acta Math., 160 (1988), 143-206.
doi: 10.1007/BF02392275. |
| [4] |
B. Branner, and J. Hubbard, The iteration of cubic polynomials. II. Patterns and parapatterns, Acta Math. 169, (1992) 229-325.
doi: 10.1007/BF02392761. |
| [5] |
R. Brück,
Geometric properties of Julia sets of the composition of polynomials of the form $z^2 + c_n$, Pacific J. Math., 198 (2001), 347-372.
doi: 10.2140/pjm.2001.198.347. |
| [6] |
R. Brück and M. Büger,
Generalized iteration, Computational Methods and Function Theory, 3 (2003), 201-252.
doi: 10.1007/BF03321035. |
| [7] |
R. Brück, M. Büger and S. Reitz,
Random iterations of polynomials of the form $z^2 + c_n$: Connectedness of Julia sets, Ergodic Theory Dynam. Systems, 19 (1999), 1221-1231.
doi: 10.1017/S0143385799141658. |
| [8] |
M. Büger,
On the composition of polynomials of the form $z^2+c_n$, Math. Ann., 310 (1998), 661-683.
doi: 10.1007/s002080050165. |
| [9] |
M. Büger,
Self-similarity of Julia sets of the composition of polynomials, Ergodic Theory Dynam. Systems, 17 (1997), 1289-1297.
doi: 10.1017/S0143385797086458. |
| [10] |
L. Carleson and T. Gamelin, Complex Dynamics, Springer-Verlag, New York, 1993.
doi: 10.1007/978-1-4612-4364-9. |
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G. Carrier, M. Krook and C. Pearson, Functions of a Complex Variable: Theory and Technique, Classics in Applied Mathematics, vol. 49, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2005.
doi: 10.1137/1.9780898719116. |
| [12] |
R. Mañe and L. Rocha,
Julia sets are uniformly perfect, Proceeding of the American Mathematical Society, 116 (1992), 251-257.
doi: 10.1090/S0002-9939-1992-1106180-2. |
| [13] |
J. Milnor, Dynamics in One Complex Variable, 3$^{rd}$ edition, AMS, New Jersey, 2006. |
| [14] |
J. Milnor, Cubic polynomials with periodic critical orbit, part I, in Complex Dynamics Families and Friends, A K Peters/CRC Press, Wellesley, MA, 2009,333-411.
doi: 10.1201/b10617-13. |
| [15] |
C. Pommerenke,
Uniformly perfect sets and the Poincaré metric, Arch. Math., 32 (1979), 192-199.
doi: 10.1007/BF01238490. |
| [16] |
T. Ransford, Potential Theory in the Complex Plane, Cambridge University Press, Cambridge, 1995.
doi: 10.1017/CBO9780511623776.
|
| [17] |
S. Reitz, Asymptotische iteration, Mitt. Math. Sem. Giessen 225, (1996) 1-79. |
| [18] |
P. Roesch, Cubic polynomials with a parabolic point, Ergodic Theory Dynam. Systems 30 (2010) 1843-1867.
doi: 10.1017/S0143385709000820. |
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N. Steinmetz, Rational Iteration: Complex Analytic Dynamical Systems, Walter de Gruyter & Co., Berlin,, (1993).
doi: 10.1515/9783110889314. |
show all references
References:
| [1] |
A. Beardon, Iteration of Rational Functions, Springer-Verlag, New York, 1991. |
| [2] |
A. Bonifant, J. Kiwi and J. Milnor,
Cubic polynomial maps with periodic critical orbit. II. Escape regions, Conform. Geom. Dyn., 14 (2010), 68-112.
doi: 10.1090/S1088-4173-10-00204-3. |
| [3] |
B. Branner and J. Hubbard,
The iteration of cubic polynomials. I. The global topology of parameter space, Acta Math., 160 (1988), 143-206.
doi: 10.1007/BF02392275. |
| [4] |
B. Branner, and J. Hubbard, The iteration of cubic polynomials. II. Patterns and parapatterns, Acta Math. 169, (1992) 229-325.
doi: 10.1007/BF02392761. |
| [5] |
R. Brück,
Geometric properties of Julia sets of the composition of polynomials of the form $z^2 + c_n$, Pacific J. Math., 198 (2001), 347-372.
doi: 10.2140/pjm.2001.198.347. |
| [6] |
R. Brück and M. Büger,
Generalized iteration, Computational Methods and Function Theory, 3 (2003), 201-252.
doi: 10.1007/BF03321035. |
| [7] |
R. Brück, M. Büger and S. Reitz,
Random iterations of polynomials of the form $z^2 + c_n$: Connectedness of Julia sets, Ergodic Theory Dynam. Systems, 19 (1999), 1221-1231.
doi: 10.1017/S0143385799141658. |
| [8] |
M. Büger,
On the composition of polynomials of the form $z^2+c_n$, Math. Ann., 310 (1998), 661-683.
doi: 10.1007/s002080050165. |
| [9] |
M. Büger,
Self-similarity of Julia sets of the composition of polynomials, Ergodic Theory Dynam. Systems, 17 (1997), 1289-1297.
doi: 10.1017/S0143385797086458. |
| [10] |
L. Carleson and T. Gamelin, Complex Dynamics, Springer-Verlag, New York, 1993.
doi: 10.1007/978-1-4612-4364-9. |
| [11] |
G. Carrier, M. Krook and C. Pearson, Functions of a Complex Variable: Theory and Technique, Classics in Applied Mathematics, vol. 49, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2005.
doi: 10.1137/1.9780898719116. |
| [12] |
R. Mañe and L. Rocha,
Julia sets are uniformly perfect, Proceeding of the American Mathematical Society, 116 (1992), 251-257.
doi: 10.1090/S0002-9939-1992-1106180-2. |
| [13] |
J. Milnor, Dynamics in One Complex Variable, 3$^{rd}$ edition, AMS, New Jersey, 2006. |
| [14] |
J. Milnor, Cubic polynomials with periodic critical orbit, part I, in Complex Dynamics Families and Friends, A K Peters/CRC Press, Wellesley, MA, 2009,333-411.
doi: 10.1201/b10617-13. |
| [15] |
C. Pommerenke,
Uniformly perfect sets and the Poincaré metric, Arch. Math., 32 (1979), 192-199.
doi: 10.1007/BF01238490. |
| [16] |
T. Ransford, Potential Theory in the Complex Plane, Cambridge University Press, Cambridge, 1995.
doi: 10.1017/CBO9780511623776.
|
| [17] |
S. Reitz, Asymptotische iteration, Mitt. Math. Sem. Giessen 225, (1996) 1-79. |
| [18] |
P. Roesch, Cubic polynomials with a parabolic point, Ergodic Theory Dynam. Systems 30 (2010) 1843-1867.
doi: 10.1017/S0143385709000820. |
| [19] |
N. Steinmetz, Rational Iteration: Complex Analytic Dynamical Systems, Walter de Gruyter & Co., Berlin,, (1993).
doi: 10.1515/9783110889314. |
Figure 2.
Julia set for $ z^3+ i z^2+z $
Figure 3.
The Connectedness locus of the cubic parabolic family $ f_a(z) = z^3+az^2+z $ . The black dashed line is the lemniscate $ |a^2+1|<1 $
Figure 4.
The estimate of the graph of $ f_a(z) $ when $ a>2 $ (left) and when $ \sqrt{3}<a\leq 2 $ (right)
Figure 5.
The graph of the real function $ g(x) = -x^3-0.5 x^2+x $ . In this case $ g^n(x_1)\to 0 $ and $ g^n(x_2)\to -y $ , where $ y = 0.5 $
Figure 6.
The graph of the real function $ g(x) = -x^3-2x^2+x $ . In this case $ g^n(x_1)\to 0 $ and $ g^n(x_2)\to \infty $
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