Existence and approximation of attractors for nonlinear coupled lattice wave equations
A cross-infection model with diffusion and incubation period
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
Figure(6)
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
A. Beardon, Iteration of Rational Functions, Springer-Verlag, New York, 1991. Google Scholar
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
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
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
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
R. Brück and M. Büger, Generalized iteration, Computational Methods and Function Theory, 3 (2003), 201-252. doi: 10.1007/BF03321035. Google Scholar
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
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
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
L. Carleson and T. Gamelin, Complex Dynamics, Springer-Verlag, New York, 1993. doi: 10.1007/978-1-4612-4364-9. Google Scholar
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
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
J. Milnor, Dynamics in One Complex Variable, 3$^{rd}$ edition, AMS, New Jersey, 2006. Google Scholar
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
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
S. Reitz, Asymptotische iteration, Mitt. Math. Sem. Giessen 225, (1996) 1-79. Google Scholar
P. Roesch, Cubic polynomials with a parabolic point, Ergodic Theory Dynam. Systems 30 (2010) 1843-1867. doi: 10.1017/S0143385709000820. Google Scholar
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 $
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 $
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
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
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
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
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
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
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
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
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
Lan Wen. On the preperiodic set. Discrete & Continuous Dynamical Systems, 2000, 6 (1) : 237-241. doi: 10.3934/dcds.2000.6.237
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
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
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
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
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
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
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
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
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
Alexandre Alves Mostafa Salarinoghabi