| Title:
|
Distributional chaos on tree maps: the star case (English) |
| Author:
|
Cánovas, Jose S. |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
42 |
| Issue:
|
3 |
| Year:
|
2001 |
| Pages:
|
583-590 |
| . |
| Category:
|
math |
| . |
| Summary:
|
Let $\Bbb X =\{z\in \Bbb C:z^n\in [0,1]\}$, $n\in \Bbb N$, and let $f:\Bbb X \rightarrow \Bbb X$ be a continuous map having the branching point fixed. We prove that $f$ is distributionally chaotic iff the topological entropy of $f$ is positive. (English) |
| Keyword:
|
distributional chaos |
| Keyword:
|
topological entropy |
| Keyword:
|
star maps |
| MSC:
|
37B40 |
| MSC:
|
37D45 |
| MSC:
|
37E25 |
| MSC:
|
54H20 |
| idZBL:
|
Zbl 1052.37032 |
| idMR:
|
MR1860247 |
| . |
| Date available:
|
2009-01-08T19:16:25Z |
| Last updated:
|
2012-04-30 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/119273 |
| . |
| Reference:
|
[1] Adler R.L., Konheim A.G., McAndrew M.H.: Topological entropy.Trans. Amer. Math. Soc. 114 (1965), 309-319. Zbl 0127.13102, MR 0175106 |
| Reference:
|
[2] Alsedá L., Llibre J., Misiurewicz M.: Combinatorial Dynamics and Entropy in Dimension One.World Scientific Publishing, 1993. MR 1255515 |
| Reference:
|
[3] Alsedá L., Moreno J.M.: Linear orderings and the full periodicity kernel for the $n$-star.J. Math. Anal. Appl. 180 (1993), 599-616. MR 1251878 |
| Reference:
|
[4] Alsedá L., Ye X.: Division for star maps with the branching point fixed.Acta Math. Univ. Comenian. 62 (1993), 237-248. MR 1270511 |
| Reference:
|
[5] Babilonová M.: Distributional chaos for triangular maps.Ann. Math. Sil. 13 (1999), 33-38. MR 1735188 |
| Reference:
|
[6] Baldwin S.: An extension of Sarkovskii's Theorem to the n-od.Ergodic Theory Dynamical Systems 11 (1991), 249-271. MR 1116640 |
| Reference:
|
[7] Block L.S., Coppel W.A.: Dynamics in one dimension.Lecture Notes in Math. Springer-Verlag, 1992. Zbl 0746.58007, MR 1176513 |
| Reference:
|
[8] Blokh A.: The spectral decomposition for one-dimensional maps.Dynamics Reported (Jones et al, eds.) 4, Springer-Verlag, Berlin, 1995. Zbl 0828.58009, MR 1346496 |
| Reference:
|
[9] Cánovas J.S., Ruíz-Marín M., Soler-López G.: Distributional chaos in duopoly games.preprint, 2000. |
| Reference:
|
[10] Forti G.L., Paganoni L.: A distributionally chaotic triangular map with zero topological sequence entropy.Math. Pannon. 9 (1998), 147-152. MR 1620434 |
| Reference:
|
[11] Forti G.L., Paganoni L., Smítal J.: Dynamics of homeomorphisms on minimal sets generated by triangular mappings.Bull. Austral. Math. Soc. 59 (1999), 1-20. MR 1672771 |
| Reference:
|
[12] Hric R.: Topological sequence entropy for maps of the circle.Comment. Math. Univ. Carolinae 41 (2000), 53-59. Zbl 1039.37007, MR 1756926 |
| Reference:
|
[13] Li T.Y., Yorke J.A.: Period three implies chaos.Amer. Math. Monthly 82 (1975), 985-992. Zbl 0351.92021, MR 0385028 |
| Reference:
|
[14] Liao G., Fan Q.: Minimal subshifts which display Schweizer-Smítal chaos and have zero topological entropy.Science in China 41 (1998), 33-38. Zbl 0931.54034, MR 1612875 |
| Reference:
|
[15] Málek M.: Distributional chaos for continuous mappings of the circle.Ann. Math. Sil. 13 (1999), 205-210. MR 1735203 |
| Reference:
|
[16] Llibre J., Misiurewicz M.: Horseshoes, entropy and periods for graph maps.Topology 32 (1993), 649-664. Zbl 0787.54021, MR 1231969 |
| Reference:
|
[17] Schweizer B., Smítal J.: Measures of chaos and a spectral decomposition of dynamical systems on the interval.Trans. Amer. Math. Soc. 344 (1994), 737-754. MR 1227094 |
| Reference:
|
[18] Smítal J.: Chaotic functions with zero topological entropy.Trans. Amer. Math. Soc. 297 (1986), 269-282. MR 0849479 |
| . |
