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A note on the periodic orbits and topological entropy of graph maps


Authors: Ll. Alsedà, D. Juher and P. Mumbrú
Journal: Proc. Amer. Math. Soc. 129 (2001), 2941-2946
MSC (2000): Primary 37E25, 37B40; Secondary 54H20, 54C70
DOI: https://doi.org/10.1090/S0002-9939-01-06134-2
Published electronically: April 17, 2001
MathSciNet review: 1840097
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Abstract:

This paper deals with the relationship between the periodic orbits of continuous maps on graphs and the topological entropy of the map. We show that the topological entropy of a graph map can be approximated by the entropy of its periodic orbits.


References [Enhancements On Off] (What's this?)

  • 1. R. Adler, A. Konheim and M. McAndrew, Topological entropy, Trans. Am. Math. Soc. 114 (1965), 309-319. MR 30:5291
  • 2. Ll. Alsedà, J. Llibre and M. Misiurewicz, Combinatorial Dynamics and entropy in dimension one, Advanced Series in Nonlinear Dynamics 5, World Scientific, Singapore, 1993. MR 95j:58042
  • 3. Ll. Alsedà, F. Mañosas and P. Mumbrú, Minimizing topological entropy for continuous maps on graphs, Ergod. Th. & Dynam. Sys. 20 (2000), 1559-1576. CMP 2001:06
  • 4. S. Baldwin and E. Slaminka, Calculating topological entropy, J. Stat. Phys. 89 (1997), 1017-1033. MR 99a:58100
  • 5. L. Block and E. Coven, Approximating entropy of maps of the interval, Proceedings of the semester on Ergodic Theory and Dynamical Systems, 237-241, Banach Center Pub. 23, PWN, Warsaw, 1989. MR 92c:58106
  • 6. R. Bowen, Entropy for group endomorphisms and homogeneous spaces, Trans. Am. Math. Soc. 153 (1971), 401-414 MR 43:469; erratum: Trans. Am. Math. Soc. 181 (1973), 509-510. MR 47:8810
  • 7. P. Góra and A. Boyarsky, Computing the topological entropy of general one-dimensional maps, Trans. Am. Math. Soc. 323 (1991), 39-49. MR 92a:58083
  • 8. J. Llibre and M. Misiurewicz, Horseshoes, entropy and periods for graph maps, Topology 32 (1993), 649-664. MR 94k:58113
  • 9. M. Misiurewicz and Z. Nitecki, Combinatorial patterns for maps of the interval, Mem. Amer. Math. Soc. 94, no. 456 (1991). MR 92h:58105
  • 10. M. Misiurewicz and W. Szlenk, Entropy of piecewise monotone mappings, Studia Math. 67 (1980), 45-63. MR 82a:58030
  • 11. S. Newhouse and T. Pignataro, On the estimation of topological entropy, J. Stat. Phys. 72 (1993), 1331-1351. MR 94i:58116
  • 12. Y. Takahashi, A formula for topological entropy of one-dimensional dynamics, Sci. Papers College Gen. Ed. Univ. Tokyo 30 (1980), 11-22. MR 82i:58057

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Additional Information

Ll. Alsedà
Affiliation: Departament de Matemàtiques, Edifici Cc, Universitat Autònoma de Barcelona, 08913 Cerdanyola del Vallès, Barcelona, Spain
Email: alseda@mat.uab.es

D. Juher
Affiliation: Departament d’Informàtica i Matemàtica Aplicada, Universitat de Girona, Lluís Santaló s/n, 17071 Girona, Spain
Email: juher@ima.udg.es

P. Mumbrú
Affiliation: Departament de Matemàtica Aplicada i Anàlisi, Universitat de Barcelona, Gran Via 585, 08071 Barcelona, Spain
Email: mumbru@mat.ub.es

DOI: https://doi.org/10.1090/S0002-9939-01-06134-2
Keywords: Graph maps, periodic orbits, topological entropy
Received by editor(s): February 10, 2000
Published electronically: April 17, 2001
Additional Notes: The authors have been partially supported by the DGES grant number PB96-1153 and the INTAS OPEN 97 grant number 97-1843.
Communicated by: Michael Handel
Article copyright: © Copyright 2001 American Mathematical Society

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