Distributional chaos on tree maps: the star case

Jose S. Cánovas

Commentationes Mathematicae Universitatis Carolinae (2001)

  • Volume: 42, Issue: 3, page 583-590
  • ISSN: 0010-2628

Abstract

top
Let 𝕏 = { z ∈ ℂ : z n ∈ [ 0 , 1 ] } , n ∈ ℕ , and let f : 𝕏 → 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.

How to cite

top

Cánovas, Jose S.. "Distributional chaos on tree maps: the star case." Commentationes Mathematicae Universitatis Carolinae 42.3 (2001): 583-590. <http://eudml.org/doc/248770>.

@article{Cánovas2001,
abstract = {Let $\mathbb \{X\} =\lbrace z\in \mathbb \{C\}:z^n\in [0,1]\rbrace $, $n\in \mathbb \{N\}$, and let $f:\mathbb \{X\} \rightarrow \mathbb \{X\}$ be a continuous map having the branching point fixed. We prove that $f$ is distributionally chaotic iff the topological entropy of $f$ is positive.},
author = {Cánovas, Jose S.},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {distributional chaos; topological entropy; star maps; distributional chaos; topological entropy; star maps},
language = {eng},
number = {3},
pages = {583-590},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Distributional chaos on tree maps: the star case},
url = {http://eudml.org/doc/248770},
volume = {42},
year = {2001},
}

TY - JOUR
AU - Cánovas, Jose S.
TI - Distributional chaos on tree maps: the star case
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2001
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 42
IS - 3
SP - 583
EP - 590
AB - Let $\mathbb {X} =\lbrace z\in \mathbb {C}:z^n\in [0,1]\rbrace $, $n\in \mathbb {N}$, and let $f:\mathbb {X} \rightarrow \mathbb {X}$ be a continuous map having the branching point fixed. We prove that $f$ is distributionally chaotic iff the topological entropy of $f$ is positive.
LA - eng
KW - distributional chaos; topological entropy; star maps; distributional chaos; topological entropy; star maps
UR - http://eudml.org/doc/248770
ER -

References

top
  1. Adler R.L., Konheim A.G., McAndrew M.H., Topological entropy, Trans. Amer. Math. Soc. 114 (1965), 309-319. (1965) Zbl0127.13102MR0175106
  2. Alsedá L., Llibre J., Misiurewicz M., Combinatorial Dynamics and Entropy in Dimension One, World Scientific Publishing, 1993. MR1255515
  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. (1993) MR1251878
  4. Alsedá L., Ye X., Division for star maps with the branching point fixed, Acta Math. Univ. Comenian. 62 (1993), 237-248. (1993) MR1270511
  5. Babilonová M., Distributional chaos for triangular maps, Ann. Math. Sil. 13 (1999), 33-38. (1999) MR1735188
  6. Baldwin S., An extension of Sarkovskii's Theorem to the n-od, Ergodic Theory Dynamical Systems 11 (1991), 249-271. (1991) MR1116640
  7. Block L.S., Coppel W.A., Dynamics in one dimension, Lecture Notes in Math. Springer-Verlag, 1992. Zbl0746.58007MR1176513
  8. Blokh A., The spectral decomposition for one-dimensional maps, Dynamics Reported (Jones et al, eds.) 4, Springer-Verlag, Berlin, 1995. Zbl0828.58009MR1346496
  9. Cánovas J.S., Ruíz-Marín M., Soler-López G., Distributional chaos in duopoly games, preprint, 2000. 
  10. Forti G.L., Paganoni L., A distributionally chaotic triangular map with zero topological sequence entropy, Math. Pannon. 9 (1998), 147-152. (1998) MR1620434
  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. (1999) MR1672771
  12. Hric R., Topological sequence entropy for maps of the circle, Comment. Math. Univ. Carolinae 41 (2000), 53-59. (2000) Zbl1039.37007MR1756926
  13. Li T.Y., Yorke J.A., Period three implies chaos, Amer. Math. Monthly 82 (1975), 985-992. (1975) Zbl0351.92021MR0385028
  14. Liao G., Fan Q., Minimal subshifts which display Schweizer-Smítal chaos and have zero topological entropy, Science in China 41 (1998), 33-38. (1998) Zbl0931.54034MR1612875
  15. Málek M., Distributional chaos for continuous mappings of the circle, Ann. Math. Sil. 13 (1999), 205-210. (1999) MR1735203
  16. Llibre J., Misiurewicz M., Horseshoes, entropy and periods for graph maps, Topology 32 (1993), 649-664. (1993) Zbl0787.54021MR1231969
  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. (1994) MR1227094
  18. Smítal J., Chaotic functions with zero topological entropy, Trans. Amer. Math. Soc. 297 (1986), 269-282. (1986) MR0849479

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.