# American Institute of Mathematical Sciences

• Previous Article
Long-term dynamics of semilinear wave equation with nonlinear localized interior damping and a source term of critical exponent
• DCDS Home
• This Issue
• Next Article
Multiscale asymptotic expansion for second order parabolic equations with rapidly oscillating coefficients
2008, 20(3): 511-541. doi: 10.3934/dcds.2008.20.511

## Minimal dynamics for tree maps

 1 Departament de Matemàtiques, Edifici Cc, Universitat Autònoma de Barcelona, 08913 Cerdanyola del Vallès, Barcelona, Spain 2 Departament d’Informàtica i Matemàtica Aplicada, Universitat de Girona, Lluís Santaló s/n, 17071 Girona, Spain 3 n/a, Spain

Received  December 2006 Revised  September 2007 Published  December 2007

We prove that, given a tree pattern $\mathcal{P}$, the set of periods of a minimal representative $f: T\rightarrow T$ of $\mathcal{P}$ is contained in the set of periods of any other representative. This statement is an immediate corollary of the following stronger result: there is a period-preserving injection from the set of periodic points of $f$ into that of any other representative of $\mathcal{P}$. We prove this result by extending the main theorem of [6] to negative cycles.
Citation: Lluís Alsedà, David Juher, Pere Mumbrú. Minimal dynamics for tree maps. Discrete & Continuous Dynamical Systems - A, 2008, 20 (3) : 511-541. doi: 10.3934/dcds.2008.20.511
 [1] Boju Jiang, Jaume Llibre. Minimal sets of periods for torus maps. Discrete & Continuous Dynamical Systems - A, 1998, 4 (2) : 301-320. doi: 10.3934/dcds.1998.4.301 [2] Daniel Coronel, Andrés Navas, Mario Ponce. On bounded cocycles of isometries over minimal dynamics. Journal of Modern Dynamics, 2013, 7 (1) : 45-74. doi: 10.3934/jmd.2013.7.45 [3] Cezar Joiţa, William O. Nowell, Pantelimon Stănică. Chaotic dynamics of some rational maps. Discrete & Continuous Dynamical Systems - A, 2005, 12 (2) : 363-375. doi: 10.3934/dcds.2005.12.363 [4] Paweł Góra, Abraham Boyarsky, Zhenyang LI, Harald Proppe. Statistical and deterministic dynamics of maps with memory. Discrete & Continuous Dynamical Systems - A, 2017, 37 (8) : 4347-4378. doi: 10.3934/dcds.2017186 [5] Begoña Alarcón, Sofia B. S. D. Castro, Isabel S. Labouriau. Global dynamics for symmetric planar maps. Discrete & Continuous Dynamical Systems - A, 2013, 33 (6) : 2241-2251. doi: 10.3934/dcds.2013.33.2241 [6] Xu Zhang, Guanrong Chen. Polynomial maps with hidden complex dynamics. Discrete & Continuous Dynamical Systems - B, 2019, 24 (6) : 2941-2954. doi: 10.3934/dcdsb.2018293 [7] Yajing Zhang, Xinfu Chen, Jianghao Hao, Xin Lai, Cong Qin. Dynamics of spike in a Keller-Segel's minimal chemotaxis model. Discrete & Continuous Dynamical Systems - A, 2017, 37 (2) : 1109-1127. doi: 10.3934/dcds.2017046 [8] Jeffrey Diller, Han Liu, Roland K. W. Roeder. Typical dynamics of plane rational maps with equal degrees. Journal of Modern Dynamics, 2016, 10: 353-377. doi: 10.3934/jmd.2016.10.353 [9] Claudio Bonanno, Carlo Carminati, Stefano Isola, Giulio Tiozzo. Dynamics of continued fractions and kneading sequences of unimodal maps. Discrete & Continuous Dynamical Systems - A, 2013, 33 (4) : 1313-1332. doi: 10.3934/dcds.2013.33.1313 [10] Simion Filip. Tropical dynamics of area-preserving maps. Journal of Modern Dynamics, 2019, 14: 179-226. doi: 10.3934/jmd.2019007 [11] Alexander Blokh, Michał Misiurewicz. Dense set of negative Schwarzian maps whose critical points have minimal limit sets. Discrete & Continuous Dynamical Systems - A, 1998, 4 (1) : 141-158. doi: 10.3934/dcds.1998.4.141 [12] Jun Hu, Oleg Muzician, Yingqing Xiao. Dynamics of regularly ramified rational maps: Ⅰ. Julia sets of maps in one-parameter families. Discrete & Continuous Dynamical Systems - A, 2018, 38 (7) : 3189-3221. doi: 10.3934/dcds.2018139 [13] Michael Gekhtman, Michael Shapiro, Serge Tabachnikov, Alek Vainshtein. Higher pentagram maps, weighted directed networks, and cluster dynamics. Electronic Research Announcements, 2012, 19: 1-17. doi: 10.3934/era.2012.19.1 [14] Lorenzo Sella, Pieter Collins. Computation of symbolic dynamics for two-dimensional piecewise-affine maps. Discrete & Continuous Dynamical Systems - B, 2011, 15 (3) : 739-767. doi: 10.3934/dcdsb.2011.15.739 [15] Alba Málaga Sabogal, Serge Troubetzkoy. Minimality of the Ehrenfest wind-tree model. Journal of Modern Dynamics, 2016, 10: 209-228. doi: 10.3934/jmd.2016.10.209 [16] Vincent Delecroix. Divergent trajectories in the periodic wind-tree model. Journal of Modern Dynamics, 2013, 7 (1) : 1-29. doi: 10.3934/jmd.2013.7.1 [17] Miaohua Jiang, Qiang Zhang. A coupled map lattice model of tree dispersion. Discrete & Continuous Dynamical Systems - B, 2008, 9 (1) : 83-101. doi: 10.3934/dcdsb.2008.9.83 [18] Frédéric Bernicot, Bertrand Maury, Delphine Salort. A 2-adic approach of the human respiratory tree. Networks & Heterogeneous Media, 2010, 5 (3) : 405-422. doi: 10.3934/nhm.2010.5.405 [19] Amadeu Delshams, Marina Gonchenko, Sergey V. Gonchenko, J. Tomás Lázaro. Mixed dynamics of 2-dimensional reversible maps with a symmetric couple of quadratic homoclinic tangencies. Discrete & Continuous Dynamical Systems - A, 2018, 38 (9) : 4483-4507. doi: 10.3934/dcds.2018196 [20] Antonio Pumariño, José Ángel Rodríguez, Joan Carles Tatjer, Enrique Vigil. Expanding Baker Maps as models for the dynamics emerging from 3D-homoclinic bifurcations. Discrete & Continuous Dynamical Systems - B, 2014, 19 (2) : 523-541. doi: 10.3934/dcdsb.2014.19.523

2017 Impact Factor: 1.179