A numerical method for constructing the hyperbolic structure of complex henon mappings

Foundations of Computational Mathematics

Volumn 6, Issue 4, 2006, Pages 427-455

  Hruska, Suzanne Lynch ^a  

^a INDIANA UNIVERSITY   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

COMPLEX PARAMETER; HENON MAPPINGS; HYPERBOLIC STRUCTURES; JULIA SET;

COMPUTATIONAL METHODS;

EID: 33846858196     PISSN: 16153375     EISSN: 16153383     Source Type: Journal    
DOI: 10.1007/s10208-006-0141-2     Document Type: Article

References (37)

1. Dynamics at Cornell, [http://www.math.cornell.edu/~dynamics].
2. 2, Invent. Math. 114(2) (1993), 277-288.
3. 2, IV, The measure of maximal entropy and laminar currents, Invent. Math. 112(1) (1993), 77-125.
4. E. Bedford and J. Smillie, Real polynomial diffeomorphisms with maximal entropy: Tangencies, Ann. Math. (2) 160(1) (2004), 1-26.
5. 2: Currents, equilibrium measure and hyperbolicity, Invent. Math. 103(1) (1991), 69-99.
6. 2, II, Stable manifolds and recurrence, J. Amer. Math. Soc. 4(4) (1991), 657-679.
7. 2, VI, Connectivity of J. Ann. of Math. (2) 148(2) (1998), 695-735.
8. M. Benedicks and L. Carleson, The dynamics of the Hénon map, Ann. of Math. (2) 133(1) (1991), 73-169.
9. Interval Computations, [http://www.cs.utep.edu/interval-comp/].
10. T. Cormen et al., Introduction to Algorithms, MIT Electrical Engineering and Computer Science Series, MIT Press, Cambridge, MA, and McGraw-Hill, New York, 1990.
11. M. Dellnitz and O. Junge. Set oriented numerical methods for dynamical systems, in Handbook of Dynamical Systems, Vol. 2, pp. 221-264, North-Holland, Amsterdam, 2002.
12. R. Devaney and Z. Nitecki, Shift automorphisms in the Hénon mapping, Comm. Math. Phys. 67(2) (1979), 137-146.
13. M. Eidenschink, Exploring Global Dynamics: A Numerical Algorithm Based on the Conley Index Theoty, PhD thesis, Georgia Institute of Technology, 1995.
14. 2 and Fatou-Bieberbach domains, Duke Math. J. 65(2) (1992), 345-380.
15. S. Friedland and J. Milnor, Dynamical properties of plane polynomial automorphisms, Ergodic Theory Dynamical Systems 9(1) (1989), 67-99.
16. K. Hoffman and R. Kunze, Linear Algebra, 2nd ed., Prentice Hall, Englewood Cliffs, NJ, 1971.
17. J. S. L. Hruska, On the numerical, constmction of hyperbolic structures for complex dynamical systems, PhD thesis, Cornell University, 2002. Download at [http://www.math.sunysb.edu/dynamics/theses/index.html].
18. S. L. Hruska, Rigorous numerical models for the dynamics of complex Hénon mappings on their chain recurrent sets, Discrete Contin. Dynam. Syst., 15(2) (2006), 529-558.
19. S. L. Hruska, Constructing an expanding metric for dynamical systems in one complex variable, Nonlinearity 18 (2005), 81-100.
20. J. Hubbard and K. Papadantonakis, Exploring the parameter space of complex Hénon mappings. J. Exper. Math., to appear.
21. 2 as dynamical systems, Acta Math. 184(2) (2000), 203-270.
22. J. H. Hubbard and R. W. Oberste-Vorth, Hénon mappings in the complex domain, I, The global topology of dynamical space, Inst. Hautes Études Sci. Publ. Math. 79 (1994), 5-46.
23. J. H. Hubbard and R. W. Oberste-Vorth, Hénon mappings in the complex domain, II, Projective and inductive limits of polynomials, in B. Branner and P. Hjorth, eds., Real and Complex Dynamical Systems, NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., Vol. 464, pp. 89-132. Kluwer Academic, Dordrecht, 1995.
24. Y. Ishii and J. Smillie, On the hyperbolicity of some complex Hénon maps, in preparation.
25. K. Mischaikow, Topological techniques for efficient rigorous computations in dynamics, Acta Numerica, 2002.
26. R. E. Moore, Interval Analysis, Prentice-Hall, Englewood Cliffs, NJ, 1966.
27. R. E. Moore, Methods and Applications of Interval Analysis, SIAM Studies in Applied Mathematics, Philadelphia, 1979.
28. S. Morosawa, Y. Nishimura, M. Taniguchi, and T. Ueda, Holomorphic Dynamics, Cambridge University Press, Cambridge, 2000.
29. S. Newhouse, Lectures on dynamical systems, in Dynamical Systems (C.I.M.E. Summer School, Bressanone, 1978), Progress in Mathematics, Vol. 8, pp. 1-114. Birkhäuser, Boston, MA, 1980.
30. S. Newhouse and J. Palis, Bifurcations of Morse-Smale dynamical systems, in Dynamical Systems (Proc. Sympos., Univ. Bahia, Salvador, 1971), pp. 303-366. Academic Press, New York, 1973.
31. R. W. Oberste-Vorth, Complex horseshoes and the dynamics of mappings of two complex variables, PhD thesis, Cornell University, 1987.
32. R. Oliva, On the combinatorics of external rays in the dynamics of the complex Hénon map, PhD thesis, Cornell University, 1998.
33. G. Osipenko, Construction of attractors and filtrations, in Conley Index Theory (Warsaw, 1997), Banach Center Publ., Vol. 47, pp. 173-192, Polish Academy of Sciences, Warsaw, 1999.
34. O. Osipenko and S. Campbell, Applied symbolic dynamics: Attractors and Filtrations, Discrete Contin. Dynam. Syst. 5(1) (1999), 43-60.
35. PROFIL/BIAS Interval Arithmetic Package. [http://www.ti3.tu-harburg.de/ Software/PROFILEnglisch.html].
36. C. Robinson, Dynamical Systems, Stability, symbolic dynamics, and chaos, 2nd ed., CRC Press, Boca Raton, FL, 1999.
37. W. Tucker, A rigorous ODE solver and Smale's 14th problem, Found. Comput. Math. 2(1) (2002), 53-117.