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doi: 10.3934/dcdsb.2021121
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On the family of cubic parabolic polynomials

Alexandre Alves , and  Mostafa Salarinoghabi

Departamento de Matemática, Universidade Federal de Viçosa, Viçosa, MG 36570-900, Brazil

* Corresponding author: Alexandre Alves

Received  September 2020 Revised  January 2021 Early access April 2021

Fund Project: The second author is supported by PNPD-Capes of Brazil and Fundação de Amparo a Pesquisa do Estado de São Paulo (FAPESP) with process number 2019/07316-0

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.

Keywords: Complex dynamics, composition of polynomials, connectedness locus, Julia set, Fatou set.
Mathematics Subject Classification: Primary: 30D05; Secondary: 37F10, 37F12.
Citation: Alexandre Alves, Mostafa Salarinoghabi. On the family of cubic parabolic polynomials. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021121
References:
[1]

A. Beardon, Iteration of Rational Functions, Springer-Verlag, New York, 1991.  Google Scholar

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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.  Google Scholar

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B. Branner, and J. Hubbard, The iteration of cubic polynomials. II. Patterns and parapatterns, Acta Math. 169, (1992) 229-325. doi: 10.1007/BF02392761.  Google Scholar

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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.  Google Scholar

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R. Brück and M. Büger, Generalized iteration, Computational Methods and Function Theory, 3 (2003), 201-252.  doi: 10.1007/BF03321035.  Google Scholar

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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.  Google Scholar

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M. Büger, On the composition of polynomials of the form $z^2+c_n$, Math. Ann., 310 (1998), 661-683.  doi: 10.1007/s002080050165.  Google Scholar

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M. Büger, Self-similarity of Julia sets of the composition of polynomials, Ergodic Theory Dynam. Systems, 17 (1997), 1289-1297.  doi: 10.1017/S0143385797086458.  Google Scholar

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

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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.  Google Scholar

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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.  Google Scholar

[13]

J. Milnor, Dynamics in One Complex Variable, 3$^{rd}$ edition, AMS, New Jersey, 2006.  Google Scholar

[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.  Google Scholar

[15]

C. Pommerenke, Uniformly perfect sets and the Poincaré metric, Arch. Math., 32 (1979), 192-199.  doi: 10.1007/BF01238490.  Google Scholar

[16] T. Ransford, Potential Theory in the Complex Plane, Cambridge University Press, Cambridge, 1995.  doi: 10.1017/CBO9780511623776.  Google Scholar
[17]

S. Reitz, Asymptotische iteration, Mitt. Math. Sem. Giessen 225, (1996) 1-79.  Google Scholar

[18]

P. Roesch, Cubic polynomials with a parabolic point, Ergodic Theory Dynam. Systems 30 (2010) 1843-1867. doi: 10.1017/S0143385709000820.  Google Scholar

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N. Steinmetz, Rational Iteration: Complex Analytic Dynamical Systems, Walter de Gruyter & Co., Berlin,, (1993). doi: 10.1515/9783110889314.  Google Scholar

show all references

References:
[1]

A. Beardon, Iteration of Rational Functions, Springer-Verlag, New York, 1991.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

R. Brück and M. Büger, Generalized iteration, Computational Methods and Function Theory, 3 (2003), 201-252.  doi: 10.1007/BF03321035.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

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

[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.  Google Scholar

[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.  Google Scholar

[13]

J. Milnor, Dynamics in One Complex Variable, 3$^{rd}$ edition, AMS, New Jersey, 2006.  Google Scholar

[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.  Google Scholar

[15]

C. Pommerenke, Uniformly perfect sets and the Poincaré metric, Arch. Math., 32 (1979), 192-199.  doi: 10.1007/BF01238490.  Google Scholar

[16] T. Ransford, Potential Theory in the Complex Plane, Cambridge University Press, Cambridge, 1995.  doi: 10.1017/CBO9780511623776.  Google Scholar
[17]

S. Reitz, Asymptotische iteration, Mitt. Math. Sem. Giessen 225, (1996) 1-79.  Google Scholar

[18]

P. Roesch, Cubic polynomials with a parabolic point, Ergodic Theory Dynam. Systems 30 (2010) 1843-1867. doi: 10.1017/S0143385709000820.  Google Scholar

[19]

N. Steinmetz, Rational Iteration: Complex Analytic Dynamical Systems, Walter de Gruyter & Co., Berlin,, (1993). doi: 10.1515/9783110889314.  Google Scholar

Figure 1.  The annulus $ A $ separates the compact set $ E $
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Figure 2.  Julia set for $ z^3+ i z^2+z $
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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 $
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Figure 4.  The estimate of the graph of $ f_a(z) $ when $ a>2 $ (left) and when $ \sqrt{3}<a\leq 2 $ (right)
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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 $
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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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