2016, 3(2): 113-137. doi: 10.3934/jcd.2016006

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

Received  March 2016 Revised  August 2016 Published  November 2016

We present an algorithm for computing images of polynomial Julia sets that are reliable in the sense that they carry mathematical guarantees against sampling artifacts and rounding errors in floating-point arithmetic. We combine cell mapping based on interval arithmetic with label propagation in graphs to avoid function iteration and rounding errors. As a result, our algorithm avoids point sampling and can reliably classify entire rectangles in the complex plane as being on either side of the Julia set. The union of the rectangles that cannot be so classified is guaranteed to contain the Julia set. Our algorithm computes a refinable quadtree decomposition of the complex plane adapted to the Julia set which can be used for rendering and for approximating geometric properties such as the area of the filled Julia set and the fractal dimension of the Julia set.
Citation: Luiz Henrique de Figueiredo, Diego Nehab, Jorge Stolfi, João Batista S. de Oliveira. Rigorous bounds for polynomial Julia sets. Journal of Computational Dynamics, 2016, 3 (2) : 113-137. doi: 10.3934/jcd.2016006
References:
[1]

Z. Arai, On hyperbolic plateaus of the Hénon map,, Experimental Mathematics, 16 (2007), 181. 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. doi: 10.1090/S0273-0979-1984-15240-6.

[3]

B. Branner, The Mandelbrot set,, in Amer. Math. Soc., 39 (1989), 75. 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. 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. 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. 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. doi: 10.1007/s002110050240.

[9]

M. Dellnitz and O. Junge, Set oriented numerical methods for dynamical systems,, in Handbook of dynamical systems, 2 (2002), 221. 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, (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), 49 (1994), 91. 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. doi: 10.1142/S0218127491000452.

[13]

Z. Galias, Rigorous investigation of the Ikeda map by means of interval arithmetic,, Nonlinearity, 15 (2002), 1759. 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. 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. 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, (2006), 85. 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. doi: 10.1142/S0218127492000422.

[20]

O. Junge, Rigorous discretization of subdivision techniques,, in International Conference on Differential Equations, (1999), 916.

[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. doi: 10.1007/s10208-004-0163-9.

[22]

L. Keen, Julia sets,, in Proc. Sympos. Appl. Math., 39 (1989), 57. 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. doi: 10.1142/S0218195906001914.

[25]

J. Milnor, Remarks on iterated cubic maps,, Experimental Mathematics, 1 (1992), 5.

[26]

J. Milnor, Dynamics in One Complex Variable, vol. 160 of Annals of Mathematics Studies,, 3rd edition, (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). doi: 10.1137/1.9780898717716.

[29]

R. P. Munafo, Roundoff error,, , (1996), 2015.

[30]

D. Nehab and H. Hoppe, A fresh look at generalized sampling,, Foundations and Trends in Computer Graphics and Vision, 8 (2014), 1. 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. 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. 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), (2003), 177. doi: 10.1145/780542.780570.

[38]

J. Rokne, The range of values of a complex polynomial over a complex interval,, Computing, 22 (1979), 153. doi: 10.1007/BF02253127.

[39]

D. Ruelle, Repellers for real analytic maps,, Ergodic Theory and Dynamical Systems, 2 (1982), 99. doi: 10.1017/S0143385700009603.

[40]

S. M. Rump and M. Kashiwagi, Implementation and improvements of affine arithmetic,, Nonlinear Theory and Its Applications, 6 (2015), 341.

[41]

H. Samet, The quadtree and related hierarchical data structures,, Computing Surveys, 16 (1984), 187. doi: 10.1145/356924.356930.

[42]

D. Saupe, Efficient computation of Julia sets and their fractal dimension,, Physica D, 28 (1987), 358. 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, (1997).

[45]

R. Tarjan, Depth-first search and linear graph algorithms,, SIAM Journal on Computing, 1 (1972), 146. doi: 10.1137/0201010.

[46]

W. Tucker, The Lorenz attractor exists,, C. R. Acad. Sci. Paris Sér. I Math., 328 (1999), 1197. 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, (2001), 77. doi: 10.1145/383259.383267.

show all references

References:
[1]

Z. Arai, On hyperbolic plateaus of the Hénon map,, Experimental Mathematics, 16 (2007), 181. 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. doi: 10.1090/S0273-0979-1984-15240-6.

[3]

B. Branner, The Mandelbrot set,, in Amer. Math. Soc., 39 (1989), 75. 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. 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. 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. 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. doi: 10.1007/s002110050240.

[9]

M. Dellnitz and O. Junge, Set oriented numerical methods for dynamical systems,, in Handbook of dynamical systems, 2 (2002), 221. 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, (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), 49 (1994), 91. 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. doi: 10.1142/S0218127491000452.

[13]

Z. Galias, Rigorous investigation of the Ikeda map by means of interval arithmetic,, Nonlinearity, 15 (2002), 1759. 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. 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. 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, (2006), 85. 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. doi: 10.1142/S0218127492000422.

[20]

O. Junge, Rigorous discretization of subdivision techniques,, in International Conference on Differential Equations, (1999), 916.

[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. doi: 10.1007/s10208-004-0163-9.

[22]

L. Keen, Julia sets,, in Proc. Sympos. Appl. Math., 39 (1989), 57. 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. doi: 10.1142/S0218195906001914.

[25]

J. Milnor, Remarks on iterated cubic maps,, Experimental Mathematics, 1 (1992), 5.

[26]

J. Milnor, Dynamics in One Complex Variable, vol. 160 of Annals of Mathematics Studies,, 3rd edition, (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). doi: 10.1137/1.9780898717716.

[29]

R. P. Munafo, Roundoff error,, , (1996), 2015.

[30]

D. Nehab and H. Hoppe, A fresh look at generalized sampling,, Foundations and Trends in Computer Graphics and Vision, 8 (2014), 1. 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. 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. 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), (2003), 177. doi: 10.1145/780542.780570.

[38]

J. Rokne, The range of values of a complex polynomial over a complex interval,, Computing, 22 (1979), 153. doi: 10.1007/BF02253127.

[39]

D. Ruelle, Repellers for real analytic maps,, Ergodic Theory and Dynamical Systems, 2 (1982), 99. doi: 10.1017/S0143385700009603.

[40]

S. M. Rump and M. Kashiwagi, Implementation and improvements of affine arithmetic,, Nonlinear Theory and Its Applications, 6 (2015), 341.

[41]

H. Samet, The quadtree and related hierarchical data structures,, Computing Surveys, 16 (1984), 187. doi: 10.1145/356924.356930.

[42]

D. Saupe, Efficient computation of Julia sets and their fractal dimension,, Physica D, 28 (1987), 358. 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, (1997).

[45]

R. Tarjan, Depth-first search and linear graph algorithms,, SIAM Journal on Computing, 1 (1972), 146. doi: 10.1137/0201010.

[46]

W. Tucker, The Lorenz attractor exists,, C. R. Acad. Sci. Paris Sér. I Math., 328 (1999), 1197. 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, (2001), 77. doi: 10.1145/383259.383267.

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