July  2006, 14(3): 399-408. doi: 10.3934/dcds.2006.14.399

Vertex maps for trees: Algebra and periods of periodic orbits

1. 

Department of Mathematics and Computer Science, Fairfield University, Fairfield, CT 06824, United States

Received  December 2004 Revised  July 2005 Published  December 2005

Let $T$ be a tree with $n$ vertices. Let $f: T \rightarrow T$ be continuous and suppose that the $n$ vertices form a periodic orbit under $f$. The combinatorial information that comes from possible permutations of the vertices gives rise to an irreducible representation of $S_n$. Using the algebraic information it is shown that $f$ must have periodic orbits of certain periods. Finally, a family of maps is defined which shows that the result about periods is best possible if $n=2^k+2^l$ for $k, l \geq 0$.
Citation: Chris Bernhardt. Vertex maps for trees: Algebra and periods of periodic orbits. Discrete & Continuous Dynamical Systems - A, 2006, 14 (3) : 399-408. doi: 10.3934/dcds.2006.14.399
[1]

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

[2]

G. Conner, Christopher P. Grant, Mark H. Meilstrup. A Sharkovsky theorem for non-locally connected spaces. Discrete & Continuous Dynamical Systems - A, 2012, 32 (10) : 3485-3499. doi: 10.3934/dcds.2012.32.3485

[3]

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

[4]

Lluís Alsedà, Sylvie Ruette. On the set of periods of sigma maps of degree 1. Discrete & Continuous Dynamical Systems - A, 2015, 35 (10) : 4683-4734. doi: 10.3934/dcds.2015.35.4683

[5]

Zhibin Liang, Xuezhi Zhao. Self-maps on flat manifolds with infinitely many periods. Discrete & Continuous Dynamical Systems - A, 2012, 32 (6) : 2223-2232. doi: 10.3934/dcds.2012.32.2223

[6]

Rabah Amir, Igor V. Evstigneev. On Zermelo's theorem. Journal of Dynamics & Games, 2017, 4 (3) : 191-194. doi: 10.3934/jdg.2017011

[7]

John Hubbard, Yulij Ilyashenko. A proof of Kolmogorov's theorem. Discrete & Continuous Dynamical Systems - A, 2004, 10 (1&2) : 367-385. doi: 10.3934/dcds.2004.10.367

[8]

Roberta Fabbri, Carmen Núñez, Ana M. Sanz. A perturbation theorem for linear Hamiltonian systems with bounded orbits. Discrete & Continuous Dynamical Systems - A, 2005, 13 (3) : 623-635. doi: 10.3934/dcds.2005.13.623

[9]

Hahng-Yun Chu, Se-Hyun Ku, Jong-Suh Park. Conley's theorem for dispersive systems. Discrete & Continuous Dynamical Systems - S, 2015, 8 (2) : 313-321. doi: 10.3934/dcdss.2015.8.313

[10]

Sergei Ivanov. On Helly's theorem in geodesic spaces. Electronic Research Announcements, 2014, 21: 109-112. doi: 10.3934/era.2014.21.109

[11]

Miguel Mendes. A note on the coding of orbits in certain discontinuous maps. Discrete & Continuous Dynamical Systems - A, 2010, 27 (1) : 369-382. doi: 10.3934/dcds.2010.27.369

[12]

Héctor E. Lomelí. Heteroclinic orbits and rotation sets for twist maps. Discrete & Continuous Dynamical Systems - A, 2006, 14 (2) : 343-354. doi: 10.3934/dcds.2006.14.343

[13]

Betseygail Rand, Lorenzo Sadun. An approximation theorem for maps between tiling spaces. Discrete & Continuous Dynamical Systems - A, 2011, 29 (1) : 323-326. doi: 10.3934/dcds.2011.29.323

[14]

Robert W. Ghrist. Flows on $S^3$ supporting all links as orbits. Electronic Research Announcements, 1995, 1: 91-97.

[15]

V. Niţicâ. Journé's theorem for $C^{n,\omega}$ regularity. Discrete & Continuous Dynamical Systems - A, 2008, 22 (1&2) : 413-425. doi: 10.3934/dcds.2008.22.413

[16]

Jacques Féjoz. On "Arnold's theorem" on the stability of the solar system. Discrete & Continuous Dynamical Systems - A, 2013, 33 (8) : 3555-3565. doi: 10.3934/dcds.2013.33.3555

[17]

Dmitry Kleinbock, Barak Weiss. Dirichlet's theorem on diophantine approximation and homogeneous flows. Journal of Modern Dynamics, 2008, 2 (1) : 43-62. doi: 10.3934/jmd.2008.2.43

[18]

Lena Noethen, Sebastian Walcher. Tikhonov's theorem and quasi-steady state. Discrete & Continuous Dynamical Systems - B, 2011, 16 (3) : 945-961. doi: 10.3934/dcdsb.2011.16.945

[19]

Fatiha Alabau-Boussouira, Piermarco Cannarsa. A constructive proof of Gibson's stability theorem. Discrete & Continuous Dynamical Systems - S, 2013, 6 (3) : 611-617. doi: 10.3934/dcdss.2013.6.611

[20]

Koray Karabina, Edward Knapp, Alfred Menezes. Generalizations of Verheul's theorem to asymmetric pairings. Advances in Mathematics of Communications, 2013, 7 (1) : 103-111. doi: 10.3934/amc.2013.7.103

2018 Impact Factor: 1.143

Metrics

  • PDF downloads (11)
  • HTML views (0)
  • Cited by (5)

Other articles
by authors

[Back to Top]