Probability Theory and Related Fields

, Volume 72, Issue 3, pp 359–386

Piecewise invertible dynamical systems

  • Franz Hofbauer
Article

Summary

The aim of the paper is the investigation of piecewise monotonic maps T of an interval X. The main tool is an isomorphism of (X, T) with a topological Markov chain with countable state space which is described by a 0–1-transition matrix M. The behavior of the orbits of points in X under T is very similar to the behavior of the paths of the Markov chain. Every irreducible submatrix of M gives rise to a T-invariant subset L of X such that L is the set ω(x) of all limit points of the orbit of an xX. The topological entropy of L is the logarithm of the spectral radius of the irreducible submatrix, which is a l1-operator. Besides these sets L there are two T-invariant sets Y and P, such that for all xX the set ω(x) is either contained in one of the sets L or in Y or in P. The set P is a union of periodic orbits and Y is contained in a finite union of sets ω(y) with yX and has topological entropy zero. This isomorphism of (X, T) with a topological Markov chain is also an important tool for the investigation of T-invariant measures on X. Results in this direction, which are published elsewhere, are described at the end of the paper. Furthermore, a part of the proofs in the paper is purely topological without using the order relation of the interval X, so that some results hold for more general dynamical systems (X, T).

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Feller, W.: Introduction to probability theory and its applications. New York: J. Wiley & Sons Inc. 1950Google Scholar
  2. 2.
    Hamedinger, W.: Die topologische Struktur stückweise monotoner Transformationen. Diplomarbeit. Wien 1984Google Scholar
  3. 3.
    Hofbauer, F.: On intrinsic ergodicity of piecewise monotonic transformations with positive entropy. Isr. J. Math. 34, 213–237 (1979). Part II: Isr. J. Math. 38, 107–115 (1981)Google Scholar
  4. 4.
    Hofbauer, F.: The structure of piecewise monotonic transformations. Ergodic Theory Dyn. Syst. 1, 159–178 (1981)Google Scholar
  5. 5.
    Hofbauer, F.: Maximal measures for simple piecewise monotonic transformations. Z. Wahrscheinlichkeitstheor. Verw. Geb. 52, 289–300 (1980)Google Scholar
  6. 6.
    Hofbauer, F.: The maximal measure for linear mod one transformations. J.L.M.S. 23, 92–112 (1981)Google Scholar
  7. 7.
    Hofbauer, F.: Monotonic mod one transformations. Studia Math. 80, 17–40 (1984)Google Scholar
  8. 8.
    Hofbauer, F.: Periodic points for piecewise monotonic transformations. Ergodic Theory Dyn. Syst. 5, 237–256 (1985)Google Scholar
  9. 9.
    Hofbauer, F., Keller, G.: Ergodic properties of invariant measures for piecewise monotonic transformations. Math. Z. 180, 119–140 (1982)Google Scholar
  10. 10.
    Hofbauer, F., Keller, G.: Equilibrium states for piecewise monotonic transformations. Ergodic Theory Dyn. Syst. 2, 23–43 (1982)Google Scholar
  11. 11.
    Hofbauer, F., keller, G.: Zeta-functions and transfer operators for piecewise linear transformations. J. Reine Angew. Math. 352, 100–113 (1984)Google Scholar
  12. 12.
    Jonker, L., Rand, D.: Bifurcations in one dimension. I: The nonwandering set. Invent. Math. 62, 347–365 (1981)Google Scholar
  13. 13.
    Nitecki, Z.: Topological dynamics on the interval. Ergodic Theory and Dynamical Systems, vol. II. Progress in Math. Boston: Birkhäuser 1981Google Scholar
  14. 14.
    Preston, C.: Iterates of piecewise monotone maps on an interval. Preprint. Bielefeld 1984Google Scholar
  15. 15.
    Rychlik, M.: Bounded variation and invariant measures. Studia Math. 76, 69–80 (1983)Google Scholar
  16. 16.
    Takahashi, Y.: Isomorphism of β-automorphisms to Markov automorphisms. Osaka J. Math. 10, 175–184 (1973)Google Scholar
  17. 17.
    Walters, P.: An introduction to ergodic theory. Berlin-Heidelberg-New York: Springer 1982Google Scholar

Copyright information

© Springer-Verlag 1986

Authors and Affiliations

  • Franz Hofbauer
    • 1
  1. 1.Institut für MathematikUniversität WienWienAustria

Personalised recommendations