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Rigorous bounds for polynomial Julia sets
1. | IMPA, Rio de Janeiro, Brazil, Brazil |
2. | Instituto de Computacão, UNICAMP, Campinas, Brazil |
3. | Faculdade de Informática, PUC-RS, Porto Alegre, Brazil |
References:
[1] |
Z. Arai, On hyperbolic plateaus of the Hénon map, Experimental Mathematics, 16 (2007), 181-188.
doi: 10.1080/10586458.2007.10128992. |
[2] |
P. Blanchard, Complex analytic dynamics on the Riemann sphere, Bulletin of the American Mathematical Society, 11 (1984), 85-141.
doi: 10.1090/S0273-0979-1984-15240-6. |
[3] |
B. Branner, The Mandelbrot set, in Amer. Math. Soc., 39 (1989), 75-105.
doi: 10.1090/psapm/039/1010237. |
[4] |
M. Braverman, Hyperbolic Julia sets are poly-time computable, Electronic Notes in Theoretical Computer Science, 120 (2005), 17-30.
doi: 10.1016/j.entcs.2004.06.031. |
[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.
doi: 10.1016/0168-9274(94)00016-6. |
[7] |
L. H. de Figueiredo and J. Stolfi, Affine arithmetic: concepts and applications, Numerical Algorithms, 37 (2004), 147-158.
doi: 10.1023/B:NUMA.0000049462.70970.b6. |
[8] |
M. Dellnitz and A. Hohmann, A subdivision algorithm for the computation of unstable manifolds and global attractors, Numerische Mathematik, 75 (1997), 293-317.
doi: 10.1007/s002110050240. |
[9] |
M. Dellnitz and O. Junge, Set oriented numerical methods for dynamical systems, in Handbook of dynamical systems, North-Holland, 2 (2002), 221-264.
doi: 10.1016/S1874-575X(02)80026-1. |
[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.
doi: 10.1090/psapm/049/1315535. |
[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.
doi: 10.1142/S0218127491000452. |
[13] |
Z. Galias, Rigorous investigation of the Ikeda map by means of interval arithmetic, Nonlinearity, 15 (2002), 1759-1779.
doi: 10.1088/0951-7715/15/6/304. |
[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.
doi: 10.1007/BF02252094. |
[15] |
T. H. Gronwall, Some remarks on conformal representation,, Annals of Mathematics, 16 (): 72.
doi: 10.2307/1968044. |
[16] |
S. L. Hruska, Constructing an expanding metric for dynamical systems in one complex variable, Nonlinearity, 18 (2005), 81-100.
doi: 10.1088/0951-7715/18/1/005. |
[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.
doi: 10.1090/conm/396/07395. |
[18] |
C. S. Hsu, Cell-to-cell Mapping: A Method of Global Analysis for Nonlinear Systems, Springer-Verlag, 1987.
doi: 10.1007/978-1-4757-3892-6. |
[19] |
C. S. Hsu, Global analysis by cell mapping, International Journal of Bifurcations and Chaos, 2 (1992), 727-771.
doi: 10.1142/S0218127492000422. |
[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.
doi: 10.1007/s10208-004-0163-9. |
[22] |
L. Keen, Julia sets, in Proc. Sympos. Appl. Math., 39 (1989), 57-74.
doi: 10.1090/psapm/039/1010236. |
[23] |
V. Kreinovich, Interval software,, , ().
|
[24] |
D. Michelucci and S. Foufou, Interval-based tracing of strange attractors, International Journal of Computational Geometry & Applications, 16 (2006), 27-39.
doi: 10.1142/S0218195906001914. |
[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] | |
[28] |
R. E. Moore, R. B. Kearfott and M. J. Cloud, Introduction to Interval Analysis, SIAM, 2009.
doi: 10.1137/1.9780898717716. |
[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.
doi: 10.1561/0600000053. |
[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.
doi: 10.1016/j.cag.2006.08.016. |
[33] |
H.-O. Peitgen and P. H. Richter, The Beauty of Fractals: Images of Complex Dynamical Systems, Springer-Verlag, 1986.
doi: 10.1007/978-3-642-61717-1. |
[34] |
H.-O. Peitgen and D. Saupe (eds.), The Science of Fractal Images, Springer-Verlag, 1988.
doi: 10.1007/978-1-4612-3784-6. |
[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.
doi: 10.1016/j.entcs.2004.06.041. |
[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.
doi: 10.1145/780542.780570. |
[38] |
J. Rokne, The range of values of a complex polynomial over a complex interval, Computing, 22 (1979), 153-169.
doi: 10.1007/BF02253127. |
[39] |
D. Ruelle, Repellers for real analytic maps, Ergodic Theory and Dynamical Systems, 2 (1982), 99-107.
doi: 10.1017/S0143385700009603. |
[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.
doi: 10.1145/356924.356930. |
[42] |
D. Saupe, Efficient computation of Julia sets and their fractal dimension, Physica D, 28 (1987), 358-370.
doi: 10.1016/0167-2789(87)90024-8. |
[43] |
N. Steinmetz, Rational Iteration: Complex Analytic Dynamical Systems, Walter de Gruyter & Co., 1993.
doi: 10.1515/9783110889314. |
[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.
doi: 10.1137/0201010. |
[46] |
W. Tucker, The Lorenz attractor exists, C. R. Acad. Sci. Paris Sér. I Math., 328 (1999), 1197-1202.
doi: 10.1016/S0764-4442(99)80439-X. |
[47] |
J. Tupper, Reliable two-dimensional graphing methods for mathematical formulae with two free variables, in Proceedings of SIGGRAPH '01, ACM, 2001, 77-86.
doi: 10.1145/383259.383267. |
show all references
References:
[1] |
Z. Arai, On hyperbolic plateaus of the Hénon map, Experimental Mathematics, 16 (2007), 181-188.
doi: 10.1080/10586458.2007.10128992. |
[2] |
P. Blanchard, Complex analytic dynamics on the Riemann sphere, Bulletin of the American Mathematical Society, 11 (1984), 85-141.
doi: 10.1090/S0273-0979-1984-15240-6. |
[3] |
B. Branner, The Mandelbrot set, in Amer. Math. Soc., 39 (1989), 75-105.
doi: 10.1090/psapm/039/1010237. |
[4] |
M. Braverman, Hyperbolic Julia sets are poly-time computable, Electronic Notes in Theoretical Computer Science, 120 (2005), 17-30.
doi: 10.1016/j.entcs.2004.06.031. |
[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.
doi: 10.1016/0168-9274(94)00016-6. |
[7] |
L. H. de Figueiredo and J. Stolfi, Affine arithmetic: concepts and applications, Numerical Algorithms, 37 (2004), 147-158.
doi: 10.1023/B:NUMA.0000049462.70970.b6. |
[8] |
M. Dellnitz and A. Hohmann, A subdivision algorithm for the computation of unstable manifolds and global attractors, Numerische Mathematik, 75 (1997), 293-317.
doi: 10.1007/s002110050240. |
[9] |
M. Dellnitz and O. Junge, Set oriented numerical methods for dynamical systems, in Handbook of dynamical systems, North-Holland, 2 (2002), 221-264.
doi: 10.1016/S1874-575X(02)80026-1. |
[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.
doi: 10.1090/psapm/049/1315535. |
[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.
doi: 10.1142/S0218127491000452. |
[13] |
Z. Galias, Rigorous investigation of the Ikeda map by means of interval arithmetic, Nonlinearity, 15 (2002), 1759-1779.
doi: 10.1088/0951-7715/15/6/304. |
[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.
doi: 10.1007/BF02252094. |
[15] |
T. H. Gronwall, Some remarks on conformal representation,, Annals of Mathematics, 16 (): 72.
doi: 10.2307/1968044. |
[16] |
S. L. Hruska, Constructing an expanding metric for dynamical systems in one complex variable, Nonlinearity, 18 (2005), 81-100.
doi: 10.1088/0951-7715/18/1/005. |
[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.
doi: 10.1090/conm/396/07395. |
[18] |
C. S. Hsu, Cell-to-cell Mapping: A Method of Global Analysis for Nonlinear Systems, Springer-Verlag, 1987.
doi: 10.1007/978-1-4757-3892-6. |
[19] |
C. S. Hsu, Global analysis by cell mapping, International Journal of Bifurcations and Chaos, 2 (1992), 727-771.
doi: 10.1142/S0218127492000422. |
[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.
doi: 10.1007/s10208-004-0163-9. |
[22] |
L. Keen, Julia sets, in Proc. Sympos. Appl. Math., 39 (1989), 57-74.
doi: 10.1090/psapm/039/1010236. |
[23] |
V. Kreinovich, Interval software,, , ().
|
[24] |
D. Michelucci and S. Foufou, Interval-based tracing of strange attractors, International Journal of Computational Geometry & Applications, 16 (2006), 27-39.
doi: 10.1142/S0218195906001914. |
[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] | |
[28] |
R. E. Moore, R. B. Kearfott and M. J. Cloud, Introduction to Interval Analysis, SIAM, 2009.
doi: 10.1137/1.9780898717716. |
[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.
doi: 10.1561/0600000053. |
[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.
doi: 10.1016/j.cag.2006.08.016. |
[33] |
H.-O. Peitgen and P. H. Richter, The Beauty of Fractals: Images of Complex Dynamical Systems, Springer-Verlag, 1986.
doi: 10.1007/978-3-642-61717-1. |
[34] |
H.-O. Peitgen and D. Saupe (eds.), The Science of Fractal Images, Springer-Verlag, 1988.
doi: 10.1007/978-1-4612-3784-6. |
[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.
doi: 10.1016/j.entcs.2004.06.041. |
[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.
doi: 10.1145/780542.780570. |
[38] |
J. Rokne, The range of values of a complex polynomial over a complex interval, Computing, 22 (1979), 153-169.
doi: 10.1007/BF02253127. |
[39] |
D. Ruelle, Repellers for real analytic maps, Ergodic Theory and Dynamical Systems, 2 (1982), 99-107.
doi: 10.1017/S0143385700009603. |
[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.
doi: 10.1145/356924.356930. |
[42] |
D. Saupe, Efficient computation of Julia sets and their fractal dimension, Physica D, 28 (1987), 358-370.
doi: 10.1016/0167-2789(87)90024-8. |
[43] |
N. Steinmetz, Rational Iteration: Complex Analytic Dynamical Systems, Walter de Gruyter & Co., 1993.
doi: 10.1515/9783110889314. |
[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.
doi: 10.1137/0201010. |
[46] |
W. Tucker, The Lorenz attractor exists, C. R. Acad. Sci. Paris Sér. I Math., 328 (1999), 1197-1202.
doi: 10.1016/S0764-4442(99)80439-X. |
[47] |
J. Tupper, Reliable two-dimensional graphing methods for mathematical formulae with two free variables, in Proceedings of SIGGRAPH '01, ACM, 2001, 77-86.
doi: 10.1145/383259.383267. |
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