PapersNo Access

SETS OF PERIODS FOR PIECEWISE MONOTONE TREE MAPS

Departament de Matemàtiques, Edifici Cc, Universitat Autònoma de Barcelona, 08913 Cerdanyola del Vallès, Barcelona, Spain

On leave at: Departament de Matemàtica Aplicada I, Universitat Politècnica de Catalunya, Diagonal 647, 08028 Barcelona, Spain.

Search for more papers by this author

D. JUHER

Departament d'Informàtica i Matemàtica Aplicada, Universitat de Girona, Lluís Santaló s/n, 17071 Girona, Spain

Search for more papers by this author

, and

P. MUMBRÚ

Departament de Matemàtica Aplicada i Anàlisi, Universitat de Barcelona, Gran Via 585, 08071 Barcelona, Spain

Search for more papers by this author

https://doi.org/10.1142/S021812740300656XCited by:4 (Source: Crossref)

Abstract

We study the set of periods of tree maps f : T → T which are monotone between any two consecutive points of a fixed periodic orbit P. This set is characterized in terms of some integers which depend only on the combinatorics of f|_P and the topological structure of T. In particular, a typep ≥ 1 of P is defined as a generalization of the notion introduced by Baldwin in his characterization of the set of periods of star maps. It follows that there exists a divisor k of the period of P such that if the set of periods of f is not finite then it contains either all the multiples of kp or an initial segment of the _kp≥ Baldwin's ordering, except for a finite set which is explicitly bounded. Conversely, examples are given where f has precisely these sets of periods.

Keywords:

References

Ll. Alsedà, J. Llibre and M. Misiurewicz, Trans. Amer. Math. Soc. 313, 475 (1989). Crossref, Web of Science, Google Scholar
Ll. Alsedà and X. Ye, Ergod. Th. Dyn. Syst. 15, 221 (1995). Crossref, Web of Science, Google Scholar
Ll. Alsedàet al., Topology 36, 1123 (1997). Crossref, Google Scholar
Ll. Alsedà , J. Llibre and M. Misiurewicz , Combinatorial Dynamics and Entropy in Dimension One , 2nd edn. , Advanced Series in Nonlinear Dynamics 5 ( World Scientific , Singapore , 2000 ) . Link, Google Scholar
Ll. Alsedà, D. Juher and P. Mumbrú, Proc. Amer. Math. Soc. 129(10), 2941 (2001). Crossref, Web of Science, Google Scholar
S. Baldwin, Discrete Math. 67, 111 (1987). Crossref, Web of Science, Google Scholar
S. Baldwin, Ergod. Th. Dyn. Syst. 11, 249 (1991). Crossref, Web of Science, Google Scholar
L. Blocket al., Global Theory of Dynamical Systems, SLNM 819 (Springer, Berlin, 1980) pp. 18–34. Crossref, Google Scholar
L. Block, Proc. Amer. Math. Soc. 82, 481 (1981). Crossref, Web of Science, Google Scholar
Blokh, A. M. [1991] "On some properties of graph maps: spectral decomposition, Misiurewicz conjecture and abstract sets of periods," preprint, Max Planck Institut für Mathematik, Bonn . Google Scholar
A. M. Blokh, Nonlinearity 5, 1375 (1992). Crossref, Web of Science, Google Scholar
M. Denker , C. Grillenberger and K. Sigmund , Ergodic Theory on Compact Spaces , SLNM 527 ( Springer , Berlin , 1976 ) . Crossref, Google Scholar
L. S. Efremova, Diff. Integr. Eqs. (Gor'kii) 2, 109 (1978). Web of Science, Google Scholar
W. Imrich and R. Kalinowski, Ann. Discr. Math. 27, 443 (1985). Web of Science, Google Scholar
W. Imrich and R. Kalinowski, Ann. Discr. Math. 27, 447 (1985). Google Scholar
T.-Y. Liet al., Trans. Amer. Math. Soc. 273, 191 (1982). Crossref, Web of Science, Google Scholar
J. Llibre and M. Misiurewicz, Topology 32, 649 (1993). Crossref, Web of Science, Google Scholar
M. Misiurewicz, Ergod. Th. Dyn. Syst. 2, 221 (1982). Crossref, Google Scholar
M. Misiurewicz and Z. Nitecki, Mem. Amer. Math. Soc. 94(456), (1991). Google Scholar
Sharkovskii, A. N. [1964] "Co-existence of the cycles of a continuous mapping of the line into itself," (in Russian), Ukrain. Math. Zh. 16(1), 61–71; English translation [1994] Proc. Conf., "Thirty years after Sharkovskii's theorem: New perspectives" (Murcia); [1995] Int. J. Bifurcation and Chaos 5, 1263–1273 . Google Scholar