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## On the family of cubic parabolic polynomials

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

* Corresponding author: Alexandre Alves

Received  September 2020 Revised  January 2021 Published  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.

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 [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

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##### 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
The annulus $A$ separates the compact set $E$
Julia set for $z^3+ i z^2+z$
The Connectedness locus of the cubic parabolic family $f_a(z) = z^3+az^2+z$. The black dashed line is the lemniscate $|a^2+1|<1$
The estimate of the graph of $f_a(z)$ when $a>2$ (left) and when $\sqrt{3}<a\leq 2$ (right)
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 $y = 0.5$
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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