Skip to main content

Spectral decomposition, periods of cycles and a conjecture of M. Misiurewicz for graph maps

Part of the Lecture Notes in Mathematics book series (LNM,volume 1514)

Abstract

We describe a spectral decomposition of the set \(\omega (f) = \bigcup\limits_{x \in X} {\omega (x)}\)for a continuous map f : XX of a one-dimensional branched manifold (“graph”) into itself similar to that of Jonker-Rand [JR], Hofbauer [H] and Nitecki [N] (see also [B1–B3]; the analogous decomposition holds for the sets Θ(f), \(\overline {Per(f)}\). Denoting by P(f) the set of all periods of cycles of a map f we then verify the following Misiurewicz conjecture: for a graph X there exists an interger L=L(X) such that for a continuous map f : XX the inclusion P(f) ⊃ {1,...,L} implies that P(f)=N (we prove also that such a map f has a positive entropy). It allows us to prove the following Theorem. Let f : XX be a continuous graph map. Then the following statements are equivalent.

  1. 1)

    The map f has positive entropy.

  2. 2)

    There exists such n that P(f) ⊃ nN={i · n|i ∈ N}.

Keywords

  • Invariant Measure
  • Periodic Point
  • Spectral Decomposition
  • Minimal Period
  • Positive Entropy

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

This is a preview of subscription content, access via your institution.

Buying options

Chapter
USD   29.95
Price excludes VAT (Canada)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   54.99
Price excludes VAT (Canada)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   69.99
Price excludes VAT (Canada)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Blokh, A.M.: On the limit behaviour of one-dimensional dynamical systems. 1,2. (in russ.) Preprints NN 1156-82,2704-82, Moscow, 1982.

    Google Scholar 

  2. Blokh, A.M.: Decomposition of dynamical systems on an interval. Russ. Math. Surv., vol. 38, no. 5, (1983)133–134.

    CrossRef  MathSciNet  MATH  Google Scholar 

  3. Blokh, A.M.: On dynamical systems on one-dimensional branched manifolds, 1,2,3. (in russ.) Theory of functions, functional analysis and appl. 46(1986)8–18, 47(1987)67–77, 48(1987)32–46.

    Google Scholar 

  4. Blokh, A.M.: On Misiurewicz conjecture for tree maps. (1990) to appear.

    Google Scholar 

  5. Denker, M., Grillenberger, C., Sigmund, K.: Ergodic theory on compact spaces. Lecture Notes in Math., vol. 527, Berlin, 1976.

    Google Scholar 

  6. Hofbauer, F.: The structure of piecewise-monotonic transformations. Erg. Theory & Dyn. Syst. 1(1981)159–178.

    CrossRef  MathSciNet  MATH  Google Scholar 

  7. Jonker, L., Rand, D.: Bifurcations in one dimension. 1: The non-wandering set. Inv. Math. 62(1981)347–365.

    CrossRef  MathSciNet  MATH  Google Scholar 

  8. Nitecki, Z.: Topological dynamics on the interval, Erg. Theory and Dyn. Syst. 2, Progress in Math. vol. 21, Boston, (1982)1–73.

    MathSciNet  Google Scholar 

  9. Sharkovsky, A.N.: Partially ordered systems of attracting sets. (in russ.), DAN SSSR vol.170, no. 6, (1966)1276–1278.

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and Permissions

Copyright information

© 1992 Springer-Verlag

About this paper

Cite this paper

Blokh, A.M. (1992). Spectral decomposition, periods of cycles and a conjecture of M. Misiurewicz for graph maps. In: Krengel, U., Richter, K., Warstat, V. (eds) Ergodic Theory and Related Topics III. Lecture Notes in Mathematics, vol 1514. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0097525

Download citation

  • DOI: https://doi.org/10.1007/BFb0097525

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-55444-8

  • Online ISBN: 978-3-540-47076-2

  • eBook Packages: Springer Book Archive