-
Previous Article
Approximate controllability of the semilinear reaction-diffusion equation governed by a multiplicative control
- DCDS-B Home
- This Issue
-
Next Article
Random Möbius dynamics on the unit disc and perturbation theory for Lyapunov exponents
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 & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021121
![]()
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.
Google Scholar
|
| [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. |
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.
Google Scholar
|
| [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 $
| [1] |
Koh Katagata. On a certain kind of polynomials of degree 4 with disconnected Julia set. Discrete & Continuous Dynamical Systems, 2008, 20 (4) : 975-987. doi: 10.3934/dcds.2008.20.975 |
| [2] |
François Berteloot, Tien-Cuong Dinh. The Mandelbrot set is the shadow of a Julia set. Discrete & Continuous Dynamical Systems, 2020, 40 (12) : 6611-6633. doi: 10.3934/dcds.2020262 |
| [3] |
Luke G. Rogers, Alexander Teplyaev. Laplacians on the basilica Julia set. Communications on Pure & Applied Analysis, 2010, 9 (1) : 211-231. doi: 10.3934/cpaa.2010.9.211 |
| [4] |
Nathaniel D. Emerson. Dynamics of polynomials with disconnected Julia sets. Discrete & Continuous Dynamical Systems, 2003, 9 (4) : 801-834. doi: 10.3934/dcds.2003.9.801 |
| [5] |
Yu-Hao Liang, Wan-Rou Wu, Jonq Juang. Fastest synchronized network and synchrony on the Julia set of complex-valued coupled map lattices. Discrete & Continuous Dynamical Systems - B, 2016, 21 (1) : 173-184. doi: 10.3934/dcdsb.2016.21.173 |
| [6] |
Artem Dudko. Computability of the Julia set. Nonrecurrent critical orbits. Discrete & Continuous Dynamical Systems, 2014, 34 (7) : 2751-2778. doi: 10.3934/dcds.2014.34.2751 |
| [7] |
Volodymyr Nekrashevych. The Julia set of a post-critically finite endomorphism of $\mathbb{PC}^2$. Journal of Modern Dynamics, 2012, 6 (3) : 327-375. doi: 10.3934/jmd.2012.6.327 |
| [8] |
Rich Stankewitz. Density of repelling fixed points in the Julia set of a rational or entire semigroup, II. Discrete & Continuous Dynamical Systems, 2012, 32 (7) : 2583-2589. doi: 10.3934/dcds.2012.32.2583 |
| [9] |
Janina Kotus, Mariusz Urbański. The dynamics and geometry of the Fatou functions. Discrete & Continuous Dynamical Systems, 2005, 13 (2) : 291-338. doi: 10.3934/dcds.2005.13.291 |
| [10] |
Roger Metzger, Carlos Arnoldo Morales Rojas, Phillipe Thieullen. Topological stability in set-valued dynamics. Discrete & Continuous Dynamical Systems - B, 2017, 22 (5) : 1965-1975. doi: 10.3934/dcdsb.2017115 |
| [11] |
Lan Wen. On the preperiodic set. Discrete & Continuous Dynamical Systems, 2000, 6 (1) : 237-241. doi: 10.3934/dcds.2000.6.237 |
| [12] |
Héctor A. Tabares-Ospina, Mauricio Osorio. Methodology for the characterization of the electrical power demand curve, by means of fractal orbit diagrams on the complex plane of Mandelbrot set. Discrete & Continuous Dynamical Systems - B, 2020, 25 (5) : 1895-1905. doi: 10.3934/dcdsb.2020008 |
| [13] |
James W. Cannon, Mark H. Meilstrup, Andreas Zastrow. The period set of a map from the Cantor set to itself. Discrete & Continuous Dynamical Systems, 2013, 33 (7) : 2667-2679. doi: 10.3934/dcds.2013.33.2667 |
| [14] |
Nancy Guelman, Jorge Iglesias, Aldo Portela. Examples of minimal set for IFSs. Discrete & Continuous Dynamical Systems, 2017, 37 (10) : 5253-5269. doi: 10.3934/dcds.2017227 |
| [15] |
Răzvan M. Tudoran. Dynamical systems with a prescribed globally bp-attracting set and applications to conservative dynamics. Discrete & Continuous Dynamical Systems, 2020, 40 (5) : 3013-3030. doi: 10.3934/dcds.2020159 |
| [16] |
Yu Zhang, Tao Chen. Minimax problems for set-valued mappings with set optimization. Numerical Algebra, Control & Optimization, 2014, 4 (4) : 327-340. doi: 10.3934/naco.2014.4.327 |
| [17] |
Sanjit Chatterjee, Chethan Kamath, Vikas Kumar. Private set-intersection with common set-up. Advances in Mathematics of Communications, 2018, 12 (1) : 17-47. doi: 10.3934/amc.2018002 |
| [18] |
Maxim Arnold, Walter Craig. On the size of the Navier - Stokes singular set. Discrete & Continuous Dynamical Systems, 2010, 28 (3) : 1165-1178. doi: 10.3934/dcds.2010.28.1165 |
| [19] |
Michel Crouzeix. The annulus as a K-spectral set. Communications on Pure & Applied Analysis, 2012, 11 (6) : 2291-2303. doi: 10.3934/cpaa.2012.11.2291 |
| [20] |
Stefanie Hittmeyer, Bernd Krauskopf, Hinke M. Osinga, Katsutoshi Shinohara. How to identify a hyperbolic set as a blender. Discrete & Continuous Dynamical Systems, 2020, 40 (12) : 6815-6836. doi: 10.3934/dcds.2020295 |
2019 Impact Factor: 1.27
Tools
Metrics
Other articles
by authors
[Back to Top]