Mathematics of Complexity and Dynamical Systems

2011 Edition
| Editors: Robert A. Meyers (Editor-in-Chief)

Topological Dynamics

  • Ethan Akin
Reference work entry
DOI: https://doi.org/10.1007/978-1-4614-1806-1_111

Article Outline

Glossary

Definition of the Subject

Introduction and History

Dynamic Relations, Invariant Sets and Lyapunov Functions

Attractors and Chain Recurrence

Chaos and Equicontinuity

Minimality and Multiple Recurrence

Future Directions

Cross References

Bibliography

Keywords

Entropy Manifold Radon Hunt Undercut 
This is a preview of subscription content, log in to check access

References

  1. 1.
    Akin E (1993) The General Topology of Dynamical Systems. Am Math SocGoogle Scholar
  2. 2.
    Akin E (1996) Dynamical systems: the topological foundations. In: Aulbach B, Colonius F (eds) Six Lectures on Dynamical Systems. World Scientific, Singapore, pp 1–43CrossRefGoogle Scholar
  3. 3.
    Akin E (1996) On chain continuity. Discret Contin Dyn Syst 2:111–120MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Akin E (1997) Recurrence in Topological Dynamics: Furstenberg Families and Ellis Actions. Plenum Press, New YorkMATHCrossRefGoogle Scholar
  5. 5.
    Akin E (2004) Lectures on Cantor and Mycielski sets for dynamical systems. In: Assani I (ed) Chapel Hill Ergodic Theory Workshops. Am Math Soc, Providence, pp 21–80CrossRefGoogle Scholar
  6. 6.
    Akin E, Auslander J, Berg K (1996) When is a transitive map chaotic? In: Bergelson V, March K, Rosenblatt J (eds) Conference in Ergodic Theory and Probability. Walter de Gruyter, Berlin, pp 25–40Google Scholar
  7. 7.
    Akin E, Glasner E (2001) Residual properties and almost equicontinuity. J d'Analyse Math 84:243–286Google Scholar
  8. 8.
    Akin E, Hurley M, Kennedy JA (2003) Dynamics of Topologically Generic Homeomorphisms. Memoir, vol 783. Am Math Soc, ProvidenceGoogle Scholar
  9. 9.
    Alexopoulos J (1991) Contraction mappings in a compact metric space. Solution No.6611. Math Assoc Am Mon 98:450MathSciNetCrossRefGoogle Scholar
  10. 10.
    Alligood KT, Sauer TD, Yorke JA (1996) Chaos: An Introduction to Dynamical Systems. Spring Science Business Media, New YorkMATHGoogle Scholar
  11. 11.
    Almgren FJ (1966) Plateau's Problem. Benjamin WA, Inc., New YorkGoogle Scholar
  12. 12.
    Alpern SR, Prasad VS (2001) Typical dynamics of volume preserving homeomorphisms. Cambridge University Press, CambridgeCrossRefGoogle Scholar
  13. 13.
    Arhangel'skii AV, Pontryagin LS (eds) (1990) General topology I. Springer, BerlinGoogle Scholar
  14. 14.
    Arhangel'skii AV (ed) (1995) General Topology III. Springer, BerlinMATHGoogle Scholar
  15. 15.
    Arhangel'skii AV (ed) (1996) General Topology II. Springer, BerlinMATHGoogle Scholar
  16. 16.
    Auslander J (1964) Generalized recurrence in dynamical systems. In: Contributions to Differential Equations, vol 3. John Wiley, New York, pp 55–74Google Scholar
  17. 17.
    Auslander J (1988) Minimal Flows and Their Extensions. North-Holland, AmsterdamMATHGoogle Scholar
  18. 18.
    Auslander J, Bhatia N, Siebert P (1964) Attractors in dynamical systems. Bol Soc Mat Mex 9:55–66MATHGoogle Scholar
  19. 19.
    Auslander J, Siebert P (1963) Prolongations and generalized Liapunov functions. In: International Symposium on Nonlinear Differential Equations and Nonlinear Mechanics. Academic Press, New York, pp 454–462CrossRefGoogle Scholar
  20. 20.
    Auslander J, Siebert P (1964) Prolongations and stability in dynamical systems. Ann Inst Fourier 14(2):237–268MathSciNetMATHCrossRefGoogle Scholar
  21. 21.
    Auslander J, Yorke J (1980) Interval maps, factors of maps and chaos. Tohoku Math J 32:177–188MathSciNetMATHCrossRefGoogle Scholar
  22. 22.
    Banks J, Brooks J, Cairns G, Davis G, Stacey P (1992) On Devaney's Definition of Chaos. Am Math Mon 99:332–333MathSciNetMATHCrossRefGoogle Scholar
  23. 23.
    Barrow-Green J (1997) Poincaré and the Three Body Problem. American Mathematical Society, ProvidenceGoogle Scholar
  24. 24.
    Bergelson V, Leibman A (1996) Polynomial extensions of Van der Waerden's and Szemeredi's theorems. J Am Math Soc 9:725–753MathSciNetMATHCrossRefGoogle Scholar
  25. 25.
    Bing RH (1988) The Collected Papers of R. H. Bing. American Mathematical Society, ProvidenceMATHGoogle Scholar
  26. 26.
    Brown M (ed) (1991) Continuum Theory and Dynamical Systems. American Mathematical Society, ProvidenceMATHGoogle Scholar
  27. 27.
    Bhatia N, Szego G (1970) Stability Theory of Dynamical Systems. Springer, BerlinMATHCrossRefGoogle Scholar
  28. 28.
    Coddington E, Levinson N (1955) Theory of Ordinary Differential Equations. McGraw-Hill, New YorkMATHGoogle Scholar
  29. 29.
    Conley C (1978) Isolated Invariant Sets and the Morse Index. American Mathematical Society, ProvidenceMATHGoogle Scholar
  30. 30.
    Denker M, Grillenberger C, Sigmund K (1978) Ergodic Theory on Compact Spaces. Lect. Notes in Math, vol 527. Springer, BerlinGoogle Scholar
  31. 31.
    Devaney RL (1989) An Introduction to Chaotic Dynamical Systems, 2nd edn. Addison‐Wesley Publishing Company, Redwood CityMATHGoogle Scholar
  32. 32.
    Douglas J (1939) Minimal surfaces of higher topological structure. Ann Math 40:205–298CrossRefGoogle Scholar
  33. 33.
    Ellis R (1969) Lectures on Topological Dynamics. WA Benjamin Inc, New YorkMATHGoogle Scholar
  34. 34.
    Fathi A, Herman M (1977) Existence de difféomorphismes minimaux. Soc Math de France, Astérique 49:37–59MathSciNetGoogle Scholar
  35. 35.
    Furstenberg H (1981) Recurrence in Ergodic Theory and Combinatorial Number Theory. Princeton Univ. Press, PrincetonMATHGoogle Scholar
  36. 36.
    Glasner E (2003) Ergodic Theory Via Joinings. American Mathematical Society, ProvidenceMATHGoogle Scholar
  37. 37.
    Glasner E, Weiss B (1993) Sensitive dependence on initial conditions. Nonlinearity 6:1067–1075MathSciNetMATHCrossRefGoogle Scholar
  38. 38.
    Gottschalk W, Hedlund G (1955) Topological Dynamics. American Mathematical Society, ProvidenceMATHGoogle Scholar
  39. 39.
    Hartman P (1964) Ordinary Differential Equations. John Wiley, New YorkMATHGoogle Scholar
  40. 40.
    Hirsch MW, Smale S, Devaney RL (2004) Differential Equations, Dynamical Systems and an Introduction to Chaos, 2nd edn. Elsevier Academic Press, AmsterdamMATHGoogle Scholar
  41. 41.
    Hofbauer J, Sigmund K (1988) The Theory of Evolution and Dynamical Systems. Cambridge Univ. Press, CambridgeMATHGoogle Scholar
  42. 42.
    Huang W, Ye X (2002) Devaney's chaos or 2‐scattering implies Li–Yorke's chaos. Topol Appl 117:259–272MathSciNetMATHCrossRefGoogle Scholar
  43. 43.
    Hunt B, Kennedy JA, Li T-Y, Nusse H (eds) (2004) The Theory of Chaotic Attractors. Springer, BerlinMATHGoogle Scholar
  44. 44.
    Iwanik A (1989) Independent sets of transitive points. In: Dynamical Systems and Ergodic Theory. Banach Cent Publ 23:277–282MathSciNetGoogle Scholar
  45. 45.
    Katok A, Hasselblatt B (1995) Introduction to the Modern Theory of Dynamical Systems. Cambridge Univ. Press, CambridgeMATHCrossRefGoogle Scholar
  46. 46.
    Kuratowski K (1973) Applications of the Baire‐category method to the problem of independent sets. Fundam Math 81:65–72MathSciNetGoogle Scholar
  47. 47.
    Li TY, Yorke J (1975) Period three implies chaos. Am Math Mon 82:985–992MathSciNetMATHCrossRefGoogle Scholar
  48. 48.
    Lorenz E (1993) The Essence of Chaos. University of Washington Press, SeattleMATHCrossRefGoogle Scholar
  49. 49.
    May RM (1973) Stability and Complexity in Model Ecosystems. Princeton Univ. Press, PrincetonGoogle Scholar
  50. 50.
    Murdock J (1991) Perturbations. John Wiley and Sons, New YorkMATHGoogle Scholar
  51. 51.
    Mycielski J (1964) Independent sets in topological algebras. Fundam Math 55:139–147MathSciNetMATHGoogle Scholar
  52. 52.
    Nemytskii V, Stepanov V (1960) Qualitative Theory of Differential Equations. Princeton U Press, PrincetonMATHGoogle Scholar
  53. 53.
    Oxtoby J (1980) Measure and Category, 2nd edn. Springer, BerlinMATHCrossRefGoogle Scholar
  54. 54.
    Peterson K (1983) Ergodic Theory. Cambridge Univ. Press, CambridgeCrossRefGoogle Scholar
  55. 55.
    Robinson C (1995) Dynamical Systems: Stability, Symbolic Dynamics and Chaos. CRC Press, Boca RatonMATHGoogle Scholar
  56. 56.
    Rudolph D (1990) Fundamentals of Measurable Dynamics: Ergodic Theory on Lebesgue Spaces. Oxford Univ. Press, OxfordMATHGoogle Scholar
  57. 57.
    Shub M (1987) Global Stability of Dynamical Systems. Springer, New YorkMATHGoogle Scholar
  58. 58.
    Shub M, Smale S (1972) Beyond hyperbolicity. Ann Math 96:587–591MathSciNetMATHCrossRefGoogle Scholar
  59. 59.
    Sigmund K (1993) Games of Life: Explorations in Ecology, Evolution and Behavior. Oxford Univ. Press, OxfordGoogle Scholar
  60. 60.
    Smale S (1967) Differentiable dynamical systems. Bull Am Math Soc 73:747–817MathSciNetMATHCrossRefGoogle Scholar
  61. 61.
    Smale S (1980) The Mathematics of Time. Springer, BerlinMATHCrossRefGoogle Scholar
  62. 62.
    Stewart I (1989) Does God Play Dice? Basil Blackwell, OxfordGoogle Scholar
  63. 63.
    Ura T (1953) Sur les courbes défines par les équations différentielles dans l'espace à m dimensions. Ann Sci Ecole Norm Sup 70(3):287–360MathSciNetMATHGoogle Scholar
  64. 64.
    Ura T (1959) Sur le courant extérieur à une région invariante. Funk Ehv 2:143–200MathSciNetMATHGoogle Scholar
  65. 65.
    de Vries J (1993) Elements of Topological Dynamics. Kluwer, DordechtMATHGoogle Scholar

Copyright information

© Springer-Verlag 2012

Authors and Affiliations

  • Ethan Akin
    • 1
  1. 1.Mathematics DepartmentThe City CollegeNew York CityUSA