Abstract
We describe a spectral decomposition of the set \(\omega (f) = \bigcup\limits_{x \in X} {\omega (x)}\)for a continuous map f : X → X 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 : X → X 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 : X → X be a continuous graph map. Then the following statements are equivalent.
-
1)
The map f has positive entropy.
-
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
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
Blokh, A.M.: On the limit behaviour of one-dimensional dynamical systems. 1,2. (in russ.) Preprints NN 1156-82,2704-82, Moscow, 1982.
Blokh, A.M.: Decomposition of dynamical systems on an interval. Russ. Math. Surv., vol. 38, no. 5, (1983)133–134.
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.
Blokh, A.M.: On Misiurewicz conjecture for tree maps. (1990) to appear.
Denker, M., Grillenberger, C., Sigmund, K.: Ergodic theory on compact spaces. Lecture Notes in Math., vol. 527, Berlin, 1976.
Hofbauer, F.: The structure of piecewise-monotonic transformations. Erg. Theory & Dyn. Syst. 1(1981)159–178.
Jonker, L., Rand, D.: Bifurcations in one dimension. 1: The non-wandering set. Inv. Math. 62(1981)347–365.
Nitecki, Z.: Topological dynamics on the interval, Erg. Theory and Dyn. Syst. 2, Progress in Math. vol. 21, Boston, (1982)1–73.
Sharkovsky, A.N.: Partially ordered systems of attracting sets. (in russ.), DAN SSSR vol.170, no. 6, (1966)1276–1278.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights 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
