Complex dynamics

Complex dynamics is the study of dynamical systems defined by iteration of functions on complex number spaces. Complex analytic dynamics is the study of the dynamics of specifically analytic functions.

Techniques

• General[1]
  • Montel's theorem
  • Poincaré metric
  • Schwarz lemma
  • Riemann mapping theorem
  • Carathéodory's theorem (conformal mapping)
  • Böttcher's equation
• Combinatorial[2]
  • Hubbard trees
  • Spider algorithm[3]
  • Tuning
  • Laminations
  • Devil's Staircase algorithm (Cantor function)
  • Orbit portraits
  • Yoccoz puzzles

Parts

• Holomorphic dynamics ( dynamics of holomorphic functions )[4]
  • in one complex variable
  • in several complex variables
• Conformal dynamics unites holomorphic dynamics in one complex variable with differentiable dynamics in one real variable.

See also

• Arithmetic dynamics
• Chaos theory
• Complex analysis
• Complex quadratic polynomial
• Fatou set
• Infinite compositions of analytic functions
• Julia set
• Mandelbrot set
• Symbolic dynamics

Notes

1. The Mandelbrot Set, Theme and Variations (London Mathematical Society Lecture Note Series) (No 274) by Tan Lei (Editor), Cambridge University Press, 2000
2. Flek, R; Keen, L (July 13, 2009), "Boundaries of bounded Fatou components of quadratic maps" (PDF), Journal of Difference Equations and Applications, retrieved 2014-12-12
3. John H. Hubbard and Dierk Schleicher (1991). "The Spider Algorithm" (PDF). Cite journal requires |journal= (help)
4. Surveys in Dynamical systems available on-line at Dynamical Systems Homepage of Institute for Mathematical Sciences SUNY at Stony Brook

References

• Alan F. Beardon, Iteration of Rational Functions: complex analytic dynamical systems, Springer, 2000, ISBN 978-0-387-95151-5
• Araceli Bonifant, Misha Lyubich, Scott Sutherland (editors), Frontiers in Complex Dynamics, Princeton University Press, 2014.
• Lennart Carleson, Theodore W. Gamelin, Complex Dynamics, Springer, 1993, ISBN 978-0-387-97942-7
• John Milnor, Dynamics in One Complex Variable (Third edition), Princeton University Press, 2006
• Shunsuke Morosawa, Y. Nishimura, M. Taniguchi, T. Ueda, Holomorphic Dynamics, Cambridge University Press, 2000, ISBN 978-0-521-66258-1