Journal of Computational Dynamics (JCD)

Rigorous bounds for polynomial Julia sets
Pages: 113 - 137, Issue 2, December 2016

doi:10.3934/jcd.2016006      Abstract        References        Full text (957.6K)           Related Articles

Luiz Henrique de Figueiredo - IMPA, Rio de Janeiro, Brazil (email)
Diego Nehab - IMPA, Rio de Janeiro, Brazil (email)
Jorge Stolfi - Instituto de Computacão, UNICAMP, Campinas, Brazil (email)
João Batista S. de Oliveira - Faculdade de Informática, PUC-RS, Porto Alegre, Brazil (email)

1 Z. Arai, On hyperbolic plateaus of the Hénon map, Experimental Mathematics, 16 (2007), 181-188.       
2 P. Blanchard, Complex analytic dynamics on the Riemann sphere, Bulletin of the American Mathematical Society, 11 (1984), 85-141.       
3 B. Branner, The Mandelbrot set, in Amer. Math. Soc., 39 (1989), 75-105.       
4 M. Braverman, Hyperbolic Julia sets are poly-time computable, Electronic Notes in Theoretical Computer Science, 120 (2005), 17-30.       
5 M. Braverman and M. Yampolsky, Computability of Julia Sets, Springer-Verlag, 2009.       
6 R. Carniel, A quasi-cell mapping approach to the global dynamical analysis of Newton's root-finding algorithm, Applied Numerical Mathematics, 15 (1994), 133-152.       
7 L. H. de Figueiredo and J. Stolfi, Affine arithmetic: concepts and applications, Numerical Algorithms, 37 (2004), 147-158.       
8 M. Dellnitz and A. Hohmann, A subdivision algorithm for the computation of unstable manifolds and global attractors, Numerische Mathematik, 75 (1997), 293-317.       
9 M. Dellnitz and O. Junge, Set oriented numerical methods for dynamical systems, in Handbook of dynamical systems, North-Holland, 2 (2002), 221-264.       
10 R. L. Devaney and L. Keen (eds.), Chaos and Fractals: The Mathematics behind the Computer Graphics, Proceedings of Symposia in Applied Mathematics 39, AMS, 1989.       
11 A. Douady, Does a Julia set depend continuously on the polynomial?, in Complex Dynamical Systems: The Mathematics Behind the Mandelbrot and Julia Sets (ed. R. L. Devaney), Proceedings of Symposia in Applied Mathematics, AMS, 49 (1994), 91-138.       
12 M. B. Durkin, The accuracy of computer algorithms in dynamical systems, International Journal of Bifurcation and Chaos in Applied Sciences and Engineering, 1 (1991), 625-639.       
13 Z. Galias, Rigorous investigation of the Ikeda map by means of interval arithmetic, Nonlinearity, 15 (2002), 1759-1779.       
14 E. Grassmann and J. Rokne, The range of values of a circular complex polynomial over a circular complex interval, Computing, 23 (1979), 139-169.       
15 T. H. Gronwall, Some remarks on conformal representation, Annals of Mathematics, 16 (1914/15), 72-76.       
16 S. L. Hruska, Constructing an expanding metric for dynamical systems in one complex variable, Nonlinearity, 18 (2005), 81-100.       
17 S. L. Hruska, Rigorous numerical studies of the dynamics of polynomial skew products of $C^2$, in Complex dynamics, vol. 396 of Contemp. Math., AMS, 2006, 85-100.       
18 C. S. Hsu, Cell-to-cell Mapping: A Method of Global Analysis for Nonlinear Systems, Springer-Verlag, 1987.       
19 C. S. Hsu, Global analysis by cell mapping, International Journal of Bifurcations and Chaos, 2 (1992), 727-771.       
20 O. Junge, Rigorous discretization of subdivision techniques, in International Conference on Differential Equations, Vol. 1, 2 (Berlin, 1999), World Sci. Publ., 2000, 916-918.       
21 W. D. Kalies, K. Mischaikow and R. C. A. M. VanderVorst, An algorithmic approach to chain recurrence, Foundations of Computational Mathematics, 5 (2005), 409-449.       
22 L. Keen, Julia sets, in Proc. Sympos. Appl. Math., 39 (1989), 57-74.       
23 V. Kreinovich, Interval software, http://cs.utep.edu/interval-comp/intsoft.html.
24 D. Michelucci and S. Foufou, Interval-based tracing of strange attractors, International Journal of Computational Geometry & Applications, 16 (2006), 27-39.       
25 J. Milnor, Remarks on iterated cubic maps, Experimental Mathematics, 1 (1992), 5-24.       
26 J. Milnor, Dynamics in One Complex Variable, vol. 160 of Annals of Mathematics Studies, 3rd edition, Princeton University Press, 2006.       
27 R. E. Moore, Interval Analysis, Prentice-Hall, 1966.       
28 R. E. Moore, R. B. Kearfott and M. J. Cloud, Introduction to Interval Analysis, SIAM, 2009.       
29 R. P. Munafo, Roundoff error, http://mrob.com/pub/muency/roundofferror.html, 1996, Accessed: 2015-12-09.
30 D. Nehab and H. Hoppe, A fresh look at generalized sampling, Foundations and Trends in Computer Graphics and Vision, 8 (2014), 1-84.
31 G. Osipenko, Dynamical Systems, Graphs, and Algorithms, vol. 1889 of Lecture Notes in Mathematics, Springer-Verlag, 2007.       
32 A. Paiva, L. H. de Figueiredo and J. Stolfi, Robust visualization of strange attractors using affine arithmetic, Computers & Graphics, 30 (2006), 1020-1026.
33 H.-O. Peitgen and P. H. Richter, The Beauty of Fractals: Images of Complex Dynamical Systems, Springer-Verlag, 1986.       
34 H.-O. Peitgen and D. Saupe (eds.), The Science of Fractal Images, Springer-Verlag, 1988.       
35 M. S. Petković and L. D. Petković, Complex Interval Arithmetic and Its Applications, Wiley-VCH Verlag, 1998.       
36 R. Rettinger, A fast algorithm for {Julia} sets of hyperbolic rational functions, Electronic Notes in Theoretical Computer Science, 120 (2005), 145-157.       
37 R. Rettinger and K. Weihrauch, The computational complexity of some Julia sets, in Proceedings of the 35th Annual ACM Symposium on Theory of Computing (STOC 2003), ACM, 2003, 177-185.       
38 J. Rokne, The range of values of a complex polynomial over a complex interval, Computing, 22 (1979), 153-169.       
39 D. Ruelle, Repellers for real analytic maps, Ergodic Theory and Dynamical Systems, 2 (1982), 99-107.       
40 S. M. Rump and M. Kashiwagi, Implementation and improvements of affine arithmetic, Nonlinear Theory and Its Applications, IEICE, 6 (2015), 341-359.
41 H. Samet, The quadtree and related hierarchical data structures, Computing Surveys, 16 (1984), 187-260.       
42 D. Saupe, Efficient computation of Julia sets and their fractal dimension, Physica D, 28 (1987), 358-370.       
43 N. Steinmetz, Rational Iteration: Complex Analytic Dynamical Systems, Walter de Gruyter & Co., 1993.       
44 C. M. Stroh, Julia Sets of Complex Polynomials and Their Implementation on the Computer, Master's thesis, University of Linz, 1997.
45 R. Tarjan, Depth-first search and linear graph algorithms, SIAM Journal on Computing, 1 (1972), 146-160.       
46 W. Tucker, The Lorenz attractor exists, C. R. Acad. Sci. Paris Sér. I Math., 328 (1999), 1197-1202.       
47 J. Tupper, Reliable two-dimensional graphing methods for mathematical formulae with two free variables, in Proceedings of SIGGRAPH '01, ACM, 2001, 77-86.

Go to top