Hostname: page-component-8448b6f56d-42gr6 Total loading time: 0 Render date: 2024-04-19T22:18:50.955Z Has data issue: false hasContentIssue false
  • English
  • Français

No division and the set of periods for tree maps

Published online by Cambridge University Press:  19 September 2008

Lluís Alsedà  and
Xiangdong Ye
Lluís Alsedà
Affiliation:
Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193-Bellaterra, Barcelona, Spain
Xiangdong Ye
Affiliation:
Section of Mathematics, International Center for Theoretic Physics, PO Box 586, 34100 Trieste, Italy
Get access Rights & Permissions [Opens in a new window]

Abstract

We extend the notion of no division for star maps to tree maps and prove that the set of periods of a tree map is cofinite if there exists some periodic orbit of the given map with period larger than one having no division. Using this result we obtain somesimple proofs of known results about the set of periods of a tree map. Also we show that this set is a union of initial segments of a finite number of linear orderings which depend only on the given tree minus a finite subset of ℕ.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 15 , Issue 2 , April 1995 , pp. 221 - 237
Copyright
Copyright © Cambridge University Press 1995

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]Alsedà, Ll., Llibre, J. and Misiurewicz, M.. Periodic orbits of maps of Y. Trans. Amer. Math. Soc. 313 (1989), 475–538.Google Scholar
[2]Alsedà, Ll., Llibre, J. and Misiurewicz, M.. Combinatorical dynamics and entropy in dimension one. Advanced Series in Nonlinear Dynamics. Vol. 5. World Scientific: Singapore, 1993.Google Scholar
[3]Alsedà, Ll., Moreno, J. M.. Linear orderings and the full periodicity kernel for the n—star. J. Math. Anal. Appl. 180 (1993), 599–616.CrossRefGoogle Scholar
[4]Alsedà, Ll. and Ye, X. D.. Minimal sets of maps of Y. J. Math. Anal. Appl. 187 (1994), 324–338.CrossRefGoogle Scholar
[5]Alsedà, Ll. and Ye, X. D.. Division for star maps with the central point fixed. Acta Math. Univ. Comenianae LXII (1993), 237–248.Google Scholar
[6]Baldwin, S.. An extension of Sarkovskii's theorem to n-od. Ergod. Th. & Dynam. Sys. 11 (1991), 249–271.CrossRefGoogle Scholar
[7]Baldwin, S.. On the possible sets of periods for functions on trees. Preliminary report. Auburn University, 1992.Google Scholar
[8]Blokh, A. M.. On some properties of graph maps: spectral decomposition, Misiurewicz conjecture and abstract sets of periods. Preprint. 1991.Google Scholar
[9]Blokh, A. M.. Periods implying almost all periods, trees with snowflakes, and zero entropy maps. Preprint. SUNY Stony Brook, # 1991/13.Google Scholar
[10]Blokh, A. M.. Trees with snowflakes and zero topological entropy maps. Preprint. 1992.Google Scholar
[11]Denker, M., Grillenberger, C. and Sigmund, K.. Ergodic theory on compact spaces. Springer Lecture Notes in Mathematics 527. Springer: Berlin, 1976.Google Scholar
[12]Imrich, W. and Kalinowski, R.. Periodic points of continuous mappings of trees. Ann. Discrete Math. 27 (1985), 443–446.Google Scholar
[13]Llibre, J. and Misiurewicz, M.. Horseshoes, entropy and periods for graph maps. Topology 32 (1993), 649–664.CrossRefGoogle Scholar
[14]Li, T.-Y., Misiurewicz, M., Pianigiani, G. and Yorke, J.. No division implies chaos. Trans. Amer. Math. Soc. 273 (1982), 191–199.CrossRefGoogle Scholar
[15]Misiurewicz, M.. Horseshoes for mappings of an interval. Bull. Acad. Pol. Sci., Sér. Sci. Math. 27 (1979), 167–169.Google Scholar
[16]Misiurewicz, M.. Minor cycles for interval maps. Preprint. 1992.Google Scholar
[17]Sharkovskii, A. N.. Coexistence of cycles of a continuous map of a line into itself. Ukrain. Math. J. 16 (1964), 61–71.Google Scholar
[18]Ye, X. D.. The center and the depth of the center of tree maps. Bull. Austral. Math. Soc. 48 (1993), 347–350.CrossRefGoogle Scholar