Hostname: page-component-74656f9884-8r8nv Total loading time: 0 Render date: 2026-11-10T09:06:09.363Z Has data issue: false hasContentIssue false
  • English
  • Français

Pointwise-recurrent dendrite maps

Published online by Cambridge University Press:  09 July 2012

ISSAM NAGHMOUCHI
ISSAM NAGHMOUCHI*
Affiliation:
Department of Mathematics, Faculty of Sciences of Bizerte, University of Carthage, Jarzouna, 7021, Tunisia (email: issam.nagh@gmail.com)
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the 'Save PDF' action button.

Let $D$ be a dendrite and $f:D\rightarrow D$ a continuous map. Denote by $E(D)$ and $B(D)$ the sets of endpoints and branch points of $D$, respectively. We show that if $E(D)$ is countable, then $f$ is pointwise-recurrent if and only if $f$ is a pointwise-periodic homeomorphism; also, if $B(D)$ is discrete, then $f$ is pointwise-recurrent if and only if every point in $D\backslash E(D)$is periodic.

Information

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 33 , Issue 4 , August 2013 , pp. 1115 - 1123
Copyright
Copyright © 2012 Cambridge University Press 

References

[1]Acosta, G. and Eslami, P.. On open maps between dendrites. Houston J. Math. 33 (2007), 753–770.Google Scholar
[2]Arevalo, D., Charatonik, W. J., Covarrubias, P. P. and Simon, L.. Dendrites with a closed set of end points. Topology Appl. 115 (2001), 1–17.Google Scholar
[3]Balibrea, F., Hric, R. and Snoha, L.. Minimal sets on graphs and dendrites. Int. J. Bifurcation Chaos 13 (2003), 1721–1725.CrossRefGoogle Scholar
[4]Beardon, A. F.. Iteration of Rational Functions. Springer, New York, 1991.Google Scholar
[5]Blanchard, F., Glasner, E., Kolyada, S. and Maas, A.. On Li-Yorke pairs. J. Reine Angew. Math. 547 (2002), 51–68.Google Scholar
[6]Blokh, A.. On transitive maps of one-dimensional branched manifolds. Differential-Difference Equations and Problems of Mathematical Physics. Inst. Mat. Akad. Nauk Ukrain. SSR, Kiev, 1984, pp. 3–9 (in Russian).Google Scholar
[7]Blokh, A.. Dynamical systems on one-dimensional branched manifolds, 1. Theory of Functions, Functional Analysis and Applications (Kharkov) 46 (1986), 8–18 (in Russian). Translation in J. Soviet Math. 48(5) (1990), 500–508.Google Scholar
[8]Blokh, A.. Dynamical systems on one-dimensional branched manifolds, 2. Theory of Functions, Functional Analysis and Applications (Kharkov) 47 (1987), 67–77 (in Russian). Translation in J. Soviet Math. 48(6) (1990), 668–674.Google Scholar
[9]Blokh, A.. Dynamical systems on one-dimensional branched manifolds, 3. Theory of Functions, Functional Analysis and Applications (Kharkov) 48 (1987), 32–46 (in Russian). Translation in J. Soviet Math. 49(2) (1990), 875–883.Google Scholar
[10]Block, L. S. and Coppel, W. A.. Dynamics in One Dimension (Lecture Notes in Mathematics, 1513). Springer, Berlin, 1992.Google Scholar
[11]Efremova, L. and Makhrova, E. N.. The dynamics of monotone maps of dendrites. Sb. Math. 192 (2001), 807–821.Google Scholar
[12]Gordh, G. R. and Lum, L.. Monotone retracts and some characterizations of dendrites. Proc. Amer. Math. Soc. 59 (1976), 156–158.Google Scholar
[13]Gottschalk, W. H.. Powers of homeomorphisms with almost periodic properties. Bull. Amer. Math. Soc. 50 (1944), 222–227.Google Scholar
[14]Kolev, B. and Pérouème, M.-C.. Recurrent surface homeomorphisms. Math. Proc. Cambridge Philos. Soc. 124 (1998), 161–168.Google Scholar
[15]Kuratowski, K.. Topology, Vol. 2. Academic Press, New York, 1968.Google Scholar
[16]Mai, J. H.. Pointwise-recurrent graph maps. Ergod. Th. & Dynam. Sys. 25 (2005), 629–637.CrossRefGoogle Scholar
[17]Mai, J. H.. $\overline {R}=\overline {P}$ for maps of dendrites $X$ with $\mathrm {Card}(\mathrm {End}(X))\lt c$. Int. J. Bifurcation Chaos 19(4) (2009), 1391–1396.Google Scholar
[18]Montgomery, D.. Pointwise periodic homeomorphisms. Amer. J. Math. 59 (1937), 118–120.Google Scholar
[19]Montgomery, D. and Zippin, L.. Topological Transformation Groups. Interscience Publishers, New York, 1955.Google Scholar
[20]Nadler, S. B.. Continuum Theory: An Introduction (Monographs and Textbooks in Pure and Applied Mathematics, 158). Marcel Dekker, New York, 1992.Google Scholar
[21]Naghmouchi, I.. Dynamics of monotone graph, dendrite and dendroid maps. Int. J. Bifurcation Chaos 21(11) (2011), 1–11.Google Scholar
[22]Naghmouchi, I.. Dynamical properties of monotone dendrite maps. Topology Appl. 159 (2012), 144–149.Google Scholar
[23]Oversteegen, L. G. and Tymchatyn, E. D.. Recurrent homeomorphisms on $\mathbb {R}^{2}$ are periodic. Proc. Amer. Math. Soc. 110 (1990), 1083–1088.Google Scholar
[24]Weaver, N.. Pointwise periodic homeomorphisms of continua. Ann. of Math. (2) 95 (1972), 83–85.Google Scholar